In an ideal world, everyone would be equivalently, severely punished.
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Artists are hard-pressed to find equivalently generous funding in America.
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Agent Provocateur: Incredibly luxe — and equivalently spendy — fantasy gear at your disposal.
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The Plan poses an equivalently disproportionate threat to minority communities and their families.
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Still, noncriminal arrests did not soar equivalently in other sanctuary cities last year.
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THE term "wirkungsgleich" is hard to translate into English but means something like "equivalently effective".
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Black and white women born into equivalently wealthy families enjoy basically the same economic outcomes.
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Energy sanctions or something equivalently significant would be appropriate and would be a strong signal.
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Although definitions of what is unlawful speech and equivalently unlawful will ultimately rest with courts.
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Equivalently, employees often gain from interacting with each other in meetings or through random exchanges.
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Unsurprisingly, such a dramatic and iconoclastic painting was made during an equivalently turbulent period in Florence.
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But if someone had yelled an equivalently demeaning remark at Obama -- like, say, 'Shine my shoes!
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But it wasn't fast, certainly not as fast as an equivalently priced Intel device would be.
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This depreciation of the CNY (equivalently, appreciation of the dollar) would amplify the appreciation of the dollar basket.
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One thing they all have in common is a large amount of money in cash or equivalently liquid securities.
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It also did something called "cash benchmarking": seeing how well a program works compared to an equivalently priced cash transfer.
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Jealousy among equivalently poor households in the same village was less likely this time around, because all eligible households got paid.
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This is why travel psychology envisions man in equivalently weighted situations, without trying to lend his life any — even approximate — continuity.
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I don't think that all religions are the same, but I do believe that they're equivalently untrue in the conventional sense of that term.
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But all other EU countries have equivalently strong legal commitments to guaranteeing asylum to the persecuted, including via international conventions that have constitutional force.
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The One's sound blows equivalently priced rivals out of the water, and it can wirelessly hook up to other Sonos devices for multi-room sound.
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Take your annual salary and divide by those 2,000 hours (equivalently, drop the last three zeros and divide by two), and voilà, you get your hourly rate.
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Often in Pousette-Dart's work, gravity and levity, bulkiness and softness are so equivalently palpable that the viewer feels both drawn into and levitated by their content.
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Do you expect something similar with Mr. Buttigieg or are there not enough people who will think that the first gay president is an equivalently significant milestone?
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It's spoken of as "Obamacare repeal," but it's equivalently a large tax cut for the rich paid for by taking insurance away from the working class and poor.
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User choice in this scenario is radically denuded, and competition increasingly gets pushed back and even locked out, without such easy access to equivalently massive pools of data.
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While the A/R facility is not rated, Fitch assumes it would be replaced by an equivalently sized super-senior facility in bankruptcy, and includes this in the waterfall.
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He researched the butterfly and learned that when in caterpillar form, it only feeds on one plant: the California pipevine (Aristolochia californica), an equivalently rare flora in the city.
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It asks students to gauge the impact of a corporate rate reduction on wages in a so called "Ramsey" model or equivalently in a small fully open economy, with perfect capital mobility.
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If South Dakota prevails, these small e-commerce businesses will be subject to multi-state tax burdens that are significantly greater than those imposed on their equivalently sized brick-and-mortar counterparts.
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Not only are these profits enormous, but the financial fallout is equivalently large: the total cost in terms of the damage done to ecosystems from environmental crime stands at billions of dollars annually.
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And it's an idea that's extremely difficult to reconcile with this study's finding that black women born into equivalently affluent families earn the same amount as white women, while black men do not.
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At the time, our Ara poll showed that 75 percent of those who responded would only buy Ara if it was priced the same or less than an equivalently specced iPhone or Samsung Galaxy flagship.
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And though one can hardly call Seuss' work equivalently lurid, it is vulgar in the best, positive sense: bracingly direct and unafraid of silliness, obviously easy to enjoy and always unabashedly fun in the first instance.
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Their objective: to limit global warming to 7003 to 2 degrees Celsius (2.7 to 3.6 degrees Fahrenheit) above preindustrial temperatures, or equivalently about 0.5 to 1.0°C (0.9 to 1.8°F) above the current global average temperature.
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First of all, she used seemingly random emoji sometimes, so I decided to send the same or equivalently off-topic emoji back in response—the occasional dolphin or pair of eyes after a question, because why not.
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These lower prices do not seem to be available for equivalently sized retail funds Fund managers benefit from economies of scale; it does not cost ten times as much to manage £100m as it does to manage £10m.
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"We think that this should be treated equivalently to alcohol in many respects and allowing for safe and regulated public consumption is just one aspect of that," said Morgan Fox, a spokesman for the National Cannabis Industry Association.
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So to just put it out there that one side says this and the other side says that, as if they are equivalently close to reality, does not serve the reader and the purpose of journalism, I think.
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If all schools in the district had enrollments that were 90% black, we would have low unevenness but high black isolation (or, equivalently, low black-white exposure) because the average black student would attend a predominantly black school.
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It wasn't that America was equivalent to Germany or that American propaganda was equivalently awful; rather, it was that America's culture industry smuggled its consumerist ethos into its art with a similar goal of producing conformity of thought and behavior.
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They argued that it is currently possible to possess a lot of fentanyl — tens of grams — with an intent to distribute without triggering a mandatory minimum that would kick in for much lower amounts of equivalently deadly doses of other illicit substances.
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"I would assume that if China shuts down trading on continuous order books of the large exchanges, the price would drop below $4,000, or the price of the U.S. dollar price of bitcoin would catch up to where it's trading equivalently in China," he said.
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"This type of intensive intervention package — or something equivalently effective at reducing transmission — will need to be maintained until a vaccine becomes available (potentially 18 months or more) — given that we predict that transmission will quickly rebound if interventions are relaxed," the authors write.
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The $1.5 billion short bet against cannabis stocks is costing $2.4 million a day, or 200 times more than an equivalently sized bet against a basket of stocks including Apple, Amazon, IBM and Goldman Sachs, said Ihor Dusaniwsky, head of predictive analytics at S3 Partners.
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An earlier version of this article misstated the relative size of the fees that short sellers betting against cannabis stocks are paying compared with an equivalently sized bet against a basket of stocks including Apple, Amazon, IBM and Goldman Sachs, according to an analyst.
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The $1.5 billion short bet against cannabis stocks is costing $2.4 million a day, or 200 times more than an equivalently sized bet against a basket of stocks including Apple, Amazon, IBM and Goldman Sachs, said Ihor Dusaniwsky, head of predictive analytics at S3 Partners.
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My preferred method would be to amend the Constitution to line up House, Senate, and presidential terms, so that each had an equivalently long term (four or six years) and every House member, senator, and president is up for reelection at the exact same time.
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"The Ground Floor where everyone can come is pretty spectacular," says Soho House founder Nick Jones, citing it as his favorite space in the new venture and observing that no-one would design an equivalently grand space now as it would never get through a corporate boardroom.
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Literature generally suggests that between 3.5 and 5 nanograms/ML of THL in a person's blood is the point in which someone is equivalently impaired to that of being too drunk to drive—Colorado and Washington both have set 5 ng/ml as the legal limit.
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To make that assessment, we need to look at net sales of assets to foreigners by the U.S. as a whole - or, equivalently, at our net sales of goods and services, because the balance of payments always balances: the trade balance is by definition the inverse of the net capital inflow.
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Among men, however, there is both a large gap between men who grew up in rich or poor families and a huge related gap between black and white men, even those who grew up in equivalently wealthy families: Children born to poorer parents are far, far likelier to find themselves in prison than children born to richer parents.
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The CJEU goes into some detail on "specific elements" it says are needed for non-identical illegal speech to be judged equivalently unlawful, and also on the limits of the burden that should be placed on platforms so they are not under a general obligation to monitor content — ultimately implying that technology filters, not human assessments, should be used to identify equivalent speech.
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" Declaring victory over the Disrupters is "the Resistance," equivalently endowed with a capital letter but otherwise overmatched by their opponents: "The Disrupters are led by a cabal of self-centered billionaires who love to exercise their power over others," Ravitch helpfully explains, "and the Resistance is led by tireless parents, students, grandparents, public school graduates, teachers, retired teachers, community leaders and religious leaders who are devoted to the principle of separation of church and state.
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A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.
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Equivalently, if every closed bounded subset is compact. Every proper metric space is complete.
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Since behavioural traits are highly variable, mutation accumulation does not negatively affect all traits equivalently.
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Equivalently, one must specify whether the movement from point to point is clockwise or counterclockwise.
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Equivalently, a loop in X is a continuous map from the unit circle S1 into X.
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In mathematics, more specifically in the area of abstract algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that G(1) = G (the commutator subgroup equals the group), or equivalently one such that Gab = {1} (its abelianization is trivial).
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According to a GAO assessment, the value of lands were not appraised equivalently and probably not fairly.
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The concept is addressed in technical systems and in information technology equivalently and generally addressed as redundancy.
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However most have been amalgamated in a single Junior Mixed Infant (JMI) school or (equivalently) primary school.
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It can equivalently be regarded as the expected information gain from learning the outcome of the random variable.
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Equation () can be stated equivalently as where is the by identity matrix and 0 is the zero vector.
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A morphism or homomorphism of posemigroups is a semigroup homomorphism that preserves the order (equivalently, that is monotonically increasing).
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They are equivalently symbolism for Shiva, Vishnu, and Brahma, they being equivalent of the three aspects found in Shaivism.
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A Farewell to Entropy: Statistical Thermodynamics Based on Information, World Scientific, New Jersey, . Equivalently, in a thermodynamic process, energy spreads.
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More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles.
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Equivalently, a diffeomorphism of M is isotopic to the identity if and only if it is pseudo-isotopic to the identity.
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However, according to the solution of the carpenter's rule problem, every folded state can be constructed (or equivalently, can be unfolded).
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A preference relation that as above satisfies completeness, transitivity, and, in addition, continuity, can be equivalently represented by a utility function.
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When compared to the native GPD promoter, the presence of UAS caused the transcriptional activity to remain equivalently enhanced under induced conditions.
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A Daewi in the North Korean Navy would correspond, equivalently, to a senior lieutenant, a rank between U.S. Navy lieutenant and lieutenant commander.
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An arborescence can equivalently be defined as a rooted digraph in which the path from the root to any other vertex is unique.
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In mathematics, and more specifically in algebra, a domain is a nonzero ring in which implies or .Lam (2001), p. 3 (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.
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In mathematics, a Hopfian group is a group G for which every epimorphism :G -> G is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group G is co-Hopfian if every monomorphism :G -> G is an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups.
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In mathematics, a bundle gerbe is a geometrical model of certain 1-gerbes with connection, or equivalently of a 2-class in Deligne cohomology.
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Equivalently, P(x,t\mid x',t') is the solution of the Fokker-Planck equation with initial condition P(x,t') = \delta(x - x').
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Equivalently, the injective dimension of M is the minimal integer (if there is such, otherwise ∞) n such that Ext(–,M) = 0 for all N > n.
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Equivalently, it is impossible for a square (the geometric shape) and a right triangle to have both equal areas and all sides commensurate with each other.
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However, the hairy ball theorem says there exists no continuous function that can do this for every point on the sphere (equivalently, for every given vector).
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Constraint languages for which no squashing function exist are called reduced languages; equivalently, these are languages on which all reductions via squashing functions have been applied.
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But since x is join-irreducible, at least one of the two terms in this join must be x itself, showing that either x = x ∧ y (equivalently x ≤ y) or x = x ∧ z (equivalently x ≤ z). The lattice ordering on the subset of join-irreducible elements forms a partial order; Birkhoff's theorem states that the lattice itself can be recovered from the lower sets of this partial order.
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A group G is an abelian group if and only if the derived group is trivial: [G,G] = {e}. Equivalently, if and only if the group equals its abelianization. See above for the definition of a group's abelianization. A group G is a perfect group if and only if the derived group equals the group itself: [G,G] = G. Equivalently, if and only if the abelianization of the group is trivial.
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DES also has four so-called weak keys. Encryption (E) and decryption (D) under a weak key have the same effect (see involution): :E_K(E_K(P)) = P or equivalently, E_K = D_K. There are also six pairs of semi-weak keys. Encryption with one of the pair of semiweak keys, K_1, operates identically to decryption with the other, K_2: :E_{K_1}(E_{K_2}(P)) = P or equivalently, E_{K_2} = D_{K_1}.
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This is the definition used, e.g., by . Equivalently, it is an undirected graph in which each connected component has no more edges than vertices.This is the definition in .
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Multidimensional variants of the various DCT types follow straightforwardly from the one-dimensional definitions: they are simply a separable product (equivalently, a composition) of DCTs along each dimension.
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To calculate the probability under the real ("physical") probability measure, additional information is required—the drift term in the physical measure, or equivalently, the market price of risk.
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Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.Giancoli, Douglas C. Physics: Principles with Applications. 6th ed., Pearson/Prentice Hall, 2005.
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In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.
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Suppose that is a topological vector space and is a convex balanced and radial set. Then } is a neighborhood basis at the origin for some locally convex topology on . This TVS topology is given by the Minkowski functional formed by , , which is a seminorm on defined by . The topology is Hausdorff if and only if is a norm, or equivalently, if and only if } or equivalently, for which it suffices that be bounded in .
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In very abstract terms, general position is a discussion of generic properties of a configuration space; in this context one means properties that hold on the generic point of a configuration space, or equivalently on a Zariski-open set. This notion coincides with the measure theoretic notion of generic, meaning almost everywhere on the configuration space, or equivalently that points chosen at random will almost surely (with probability 1) be in general position.
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Equivalently, a and b are associates if a = ub for some unit u. An irreducible element is a nonzero non-unit that cannot be written as a product of two non-units. A nonzero non-unit p is a prime element if, whenever p divides a product ab, then p divides a or p divides b. Equivalently, an element p is prime if and only if the principal ideal (p) is a nonzero prime ideal.
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Thus instead of five points being needed, only 5 – 2 = 3 are needed, coinciding with the 3 parameters a, b, k (equivalently a, b, r) that need to be identified.
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A tree T with 2n vertices, is bivariegated if and only if the independence number of T is n, or, equivalently, if and only if it has a perfect matching.
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This orbital angular momentum corresponds to helical wavefronts, or, equivalently, a phase proportional to the azimuthal angle. Electron beams with quantized orbital angular momentum are also called electron vortex beams.
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Minimizing (or, equivalently, maximizing) a pseudo-Boolean function is NP-hard. This can easily be seen by formulating, for example, the maximum cut problem as maximizing a pseudo- Boolean function.
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Equivalently, T is the conjugation of the pointwise multiplication operator by the Fourier transform. Thus one can think of multiplier operators as operators which are diagonalized by the Fourier transform.
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In polar coordinates, the Kampyle has the equation :r = a\sec^2\theta. Equivalently, it has a parametric representation as :x=a\sec(t), \quad y=a\tan(t)\sec(t).
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A googolplex is the number 10, or equivalently, 10. Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes, that is, a 1 followed by a googol zeroes.
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A separating equilibrium is an equilibrium where the first player's action fully reveals her private information. Equivalently, in such an equilibrium, the first player's action is different for different private information.
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One equivalently says that ω is a modulus of continuity (resp., at x) for f, or shortly, f is ω-continuous (resp., at x). Here, we mainly treat the global notion.
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In the Bruhat decomposition of the flag variety of a Lie group, the fundamental class corresponds to the top- dimension Schubert cell, or equivalently the longest element of a Coxeter group.
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The canonical Gaussian cylinder set measure on an infinite-dimensional Hilbert space can never be a bona fide measure; equivalently, the identity function on such a space cannot be γ-radonifying.
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A forest is an undirected graph in which any two vertices are connected by at most one path. Equivalently, a forest is an undirected acyclic graph. Equivalently, a forest is an undirected graph, all of whose connected components are trees; in other words, the graph consists of a disjoint union of trees. As special cases, the order-zero graph (a forest consisting of zero trees), a single tree, and an edgeless graph, are examples of forests.
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In an interview with MistaJam on BBC Radio 1Xtra, Plan B explained that the use of the word chav was equivalently offensive to "terms used to be derogatory towards race and sex".
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200px The pentakis dodecahedron in a model of buckminsterfullerene: each surface segment represents a carbon atom. Equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom.
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Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph G is called the period of G.
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The size of an interval (also known as its width or height) can be represented using two alternative and equivalently valid methods, each appropriate to a different context: frequency ratios or cents.
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A geometric morphism (u∗,u∗) is essential if u∗ has a further left adjoint u!, or equivalently (by the adjoint functor theorem) if u∗ preserves not only finite but all small limits.
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To delete an element from the heap, decrease its key to negative infinity (or equivalently, to some value lower than any element in the heap) and then delete the minimum in the heap.
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Suppose is a vector space over , a subfield of the complex numbers (normally itself or ). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.
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A subring of a ring is a subset S of R that preserves the structure of the ring, i.e. a ring with . Equivalently, it is both a subgroup of and a submonoid of .
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Equivalently the bridge number is the minimal number of local maxima of the projection of the knot onto a vector, where we minimize over all projections and over all conformations of the knot.
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Consequently, the orbits are perpendicular to all gradients of all these independent isosurfaces, five in this specific problem, and hence are determined by the generalized cross products of all of these gradients. As a result, all superintegrable systems are automatically describable by Nambu mechanics, alternatively, and equivalently, to Hamiltonian mechanics. Maximally superintegrable systems can be quantized using commutation relations, as illustrated below. Nevertheless, equivalently, they are also quantized in the Nambu framework, such as this classical Kepler problem into the quantum hydrogen atom.
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If the number of such constraints is , then :C = M - Z + R'. Equivalently, if is the number of independent reactions that can take place, then :C = N - Z - R. The constants are related by .
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It is easy to see that (A, A∗) is a cut if and only if the sets A and A∗ are almost invariant (equivalently, if and only if the set A is almost invariant).
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If we accept the hypothesis that every calculation/computation can be done by either method equivalently we have accepted both Kleene's Theorem XXX (the equivalence) and the Church–Turing Thesis (the hypothesis of "every").
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In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function.
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The discrete spectrum is defined as the set of normal eigenvalues. Equivalently, it can be characterized as the set of isolated points of the spectrum such that the corresponding Riesz projector is of finite rank.
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In broadcasting, a channel or frequency channel is a designated radio frequency (or, equivalently, wavelength), assigned by a competent frequency assignment authority for the operation of a particular radio station, television station or television channel.
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Whitehead, section 6.1; page 257. One can define homology with coefficients in a bundle of abelian groups.Whitehead, section 6.2. When satisfies certain conditions, a local system can be equivalently described as a locally constant sheaf.
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Equivalently, the vertices of every 2-vertex-connected graph G may be arranged into a cyclic order such that adjacent vertices in this order are at distance at most two from each other in G.
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Equivalently, a strong reducibility relation is one whose degrees form a finer equivalence relation than the Turing degrees, while a weak reducibility relation is one whose degrees form a coarser equivalence relation than Turing equivalence.
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The perfect graph theorem states: :The complement of a perfect graph is perfect. Equivalently, in a perfect graph, the size of the maximum independent set equals the minimum number of cliques in a clique cover.
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A Noetherian domain is a G-domain iff its rank is at most one, and has only finitely many maximal ideals (or equivalently, prime ideals).Kaplansky, Irving. Commutative Algebra. Polygonal Publishing House, 1974, p. 19.
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The Latin binomial Crataegus × grignonensis (or equivalently Crataegus grignonensis) was published in 1890 on the basis of a description written by Pierre Mouillefert, who was the professor of sylviculture at l'École nationale d'Agriculture de Grignon.
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In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).
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Equivalently M is spin if the SO(n) principal bundle of orthonormal bases of the tangent fibers of M is a Z2 quotient of a principal spin bundle. If the manifold has a cell decomposition or a triangulation, a spin structure can equivalently be thought of as a homotopy-class of trivialization of the tangent bundle over the 1-skeleton that extends over the 2-skeleton. If the dimension is lower than 3, one first takes a Whitney sum with a trivial line bundle.
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Prüfer domains can be characterized as integral domains whose localizations at all prime (equivalently, at all maximal) ideals are valuation domains. So the localization of a Bézout domain at a prime ideal is a valuation domain. Since an invertible ideal in a local ring is principal, a local ring is a Bézout domain iff it is a valuation domain. Moreover, a valuation domain with noncyclic (equivalently non-discrete) value group is not Noetherian, and every totally ordered abelian group is the value group of some valuation domain.
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Therefore, an arbitrary unrestricted grammar can always be equivalently converted to obey the latter form, by converting it to a Turing machine and back again. Some authors use the latter form as definition of unrestricted grammar.
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Alternatively and equivalently, in this case every equilibrium load on the framework may be resolved by a stress that generates an equal and opposite set of forces, and the framework is said to be statically rigid.
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In mathematics, a unimodular polynomial matrix is a square polynomial matrix whose inverse exists and is itself a polynomial matrix. Equivalently, a polynomial matrix A is unimodular if its determinant det(A) is a nonzero constant.
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The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is no cut-free proof of falsity, there is no contradiction in general.
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We assume that each agent has a preference-relation on the set of all bundles of commodities. The preferences are complete, transitive, and closed. Equivalently, each preference relation can be represented by a continuous utility function.
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VII §1 no. 3 th. 2 A polynomial ring over a Mori domain need not be a Mori domain. Also, the complete integral closure of a Mori domain need not be a Mori (or, equivalently, Krull) domain.
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A ternary equivalence relation on a set is a relation , written , that satisfies the following axioms: #Symmetry: If then and . (Therefore also , , and .) #Reflexivity: . Equivalently, if , , and are not all distinct, then . #Transitivity: If and and then .
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The game developed differently at different locations. Archaeologists have found editions in places such as Ireland and Ukraine. Hnefatafl literally translates to "fist table," from the Old Norse (equivalently in modern Icelandic) hnef, 'fist', and tafl, 'table'.
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The elementary diagram of A is the set eldiagA of all first-order σ'-sentences that are satisfied by A or, equivalently, the complete (first-order) theory of the natural expansion of A to the signature σ'.
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In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category.
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Using linear algebra, a projective space of dimension is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space of dimension . Equivalently, it is the quotient set of by the equivalence relation "being on the same vector line". As a vector line intersects the unit sphere of in two antipodal points, projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane.
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An abelian group is p-divisible for a prime p if for every g \in G, there exists y \in G such that py=g. Equivalently, an abelian group is p-divisible if and only if pG=G.
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Some of their well-known models having expensive options, e.g., one review lists the Don Edwards signature model "Cowboy Singer" at (approx. as of March 2012). Other models are priced similar to equivalently featured instruments from other makers.
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Venn diagram of P leftarrow Q (the red area is true) In logic, converse nonimplicationLehtonen, Eero, and Poikonen, J.H. is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication).
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In physics, an ultraviolet divergence or UV divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with unbounded energy, or, equivalently, because of physical phenomena at infinitesimal distances.
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Inside money is thus a liability (equivalently a negative asset) to the issuer, so the net amount of assets associated with inside money in an economy is zero. Most money circulating in a modern economy is inside money.
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A connected 3-manifold M is prime if it cannot be obtained as a connected sum N_1\\# N_2 of two manifolds neither of which is the 3-sphere S^3 (or, equivalently, neither of which is homeomorphic to M).
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In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first- order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson.
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Garcia T, Miller B. Arrhythmia Recognition: The Art of Interpretation. Jones and Bartlett, Sudbury MA: 2004. Clinically, an agonal rhythm is regarded as asystole and should be treated equivalently, with cardiopulmonary resuscitation and administration of intravenous adrenaline.UK Resuscitation Council.
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Equivalently, the test demonstrated the general relativity prediction that clocks should run at different rates in different places of a gravitational field. It is considered to be the experiment that ushered in an era of precision tests of general relativity.
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Rationalisation can be extended to all algebraic numbers and algebraic functions (as an application of norm forms). For example, to rationalise a cube root, two linear factors involving cube roots of unity should be used, or equivalently a quadratic factor.
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In formal language theory, a context-sensitive language is a language that can be defined by a context-sensitive grammar (and equivalently by a noncontracting grammar). Context-sensitive is one of the four types of grammars in the Chomsky hierarchy.
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The board is an 8 x 8 square board with pieces played on the intersection points. Equivalently, the board is a 9 x 9 square grid. The board is further designed with marked lines of two colors (e.g. blue and red).
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Anaximandre de Millet, ou la naissance de la science, pg. 75. The foundation of science, therefore, is not certainty but the very opposite, a radical uncertainty about our own knowledge, or equivalently, an acute awareness of the extent of our ignorance.
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The Abel equation, named after Niels Henrik Abel, is a type of functional equation which can be written in the form :f(h(x)) = h(x + 1) or, equivalently, :\alpha(f(x))=\alpha(x)+1 and controls the iteration of .
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Equivalently, a regular n-gon is constructible if and only if the cosine of its common angle is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.
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A toroidal polyhedron is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose genus is 1 or greater. Topologically, the surfaces of such polyhedra are torus surfaces having one or more holes through the middle.
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This interpretation can equivalently be restated in the language of differential forms. The pair u,v satisfy the Cauchy–Riemann equations if and only if the one-form v\,dx + u\, dy is both closed and coclosed (a harmonic differential form).
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Recall that N(v) is the set of neighbors of v. Equivalently, \sigma is a DFS ordering if, for all 1\le i with v_i\in N(v_k)\setminus N(v_j), there exists a neighbor v_m of v_j such that i.
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A linear algebraic group over a field k is defined as a smooth closed subgroup scheme of GL(n) over k, for some positive integer n. Equivalently, a linear algebraic group over k is a smooth affine group scheme over k.
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A topological space X is called a Gδ spaceSteen & Seebach, p. 162 if every closed subset of X is a Gδ set. Dually and equivalently, a Gδ space is a space in which every open set is an Fσ set.
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In mathematics, a Loewy ring or semi-Artinian ring is a ring in which every non-zero module has a non-zero socle, or equivalently if the Loewy length of every module is defined. The concepts are named after Alfred Loewy.
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When , this is called the -double coset of . Equivalently, is the equivalence class of under the equivalence relation : if and only if there exist in and in such that . The set of all double cosets is denoted :H \backslash G / K.
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Method equivalence summarizes three further levels of equivalence: sample equivalence, instrument equivalence, and administration equivalence.cf. Wirth & Kolb (2004), p. 103. If the units of analysis or interview partners, e.g. campaign consultants, are equivalently selected in every considered country, sample equivalence exists.
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However, this article will reserve use of for the canonical evaluation map (defined below) so as to avoid confusion for readers not familiar with this subject. A pairing is called a dual system, a dual pair, or a duality over if it satisfies the following two separation axioms: # separates/distinguishes points of : if is such that then ; or equivalently, for all non-zero , the map is not identically (i.e. there exists a such that ); # separates/distinguishes points of : if is such that then ; or equivalently, for all non-zero , the map is not identically (i.e. there exists an such that ).
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Such posets may be equivalently characterized as those with no induced subposet isomorphic to the pair of two-element chains, in other words as the (2+2)-free posets . The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form (\ell_i, \ell_i + 1), is precisely the semiorders. The complement of the comparability graph of an interval order (X, ≤) is the interval graph (X, \cap). Interval orders should not be confused with the interval-containment orders, which are the inclusion orders on intervals on the real line (equivalently, the orders of dimension ≤ 2).
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Cauchy continuity makes sense in situations more general than metric spaces, but then one must move from sequences to nets (or equivalently filters). The definition above applies, as long as the Cauchy sequence (x1, x2, …) is replaced with an arbitrary Cauchy net. Equivalently, a function f is Cauchy-continuous if and only if, given any Cauchy filter F on X, then f(F) is a Cauchy filter base on Y. This definition agrees with the above on metric spaces, but it also works for uniform spaces and, most generally, for Cauchy spaces. Any directed set A may be made into a Cauchy space.
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As in the computation of, for example, standard deviation, the estimation of a quantile depends upon whether one is operating with a statistical population or with a sample drawn from it. For a population, of discrete values or for a continuous population density, the -th -quantile is the data value where the cumulative distribution function crosses . That is, is a -th -quantile for a variable if : or, equivalently, and : and . For a finite population of equally probable values indexed from lowest to highest, the -th -quantile of this population can equivalently be computed via the value of .
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An element X of a semisimple Lie algebra g is called nilpotent if its adjoint endomorphism : ad X: g -> g, ad X(Y) = [X,Y] is nilpotent, that is, (ad X)n = 0 for large enough n. Equivalently, X is nilpotent if its characteristic polynomial pad X(t) is equal to tdim g. A semisimple Lie group or algebraic group G acts on its Lie algebra via the adjoint representation, and the property of being nilpotent is invariant under this action. A nilpotent orbit is an orbit of the adjoint action such that any (equivalently, all) of its elements is (are) nilpotent.
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With the exception of 7, a safe prime q is of the form 6k − 1 or, equivalently, q ≡ 5 (mod 6) — as is p > 3. Similarly, with the exception of 5, a safe prime q is of the form 4k − 1 or, equivalently, q ≡ 3 (mod 4) — trivially true since (q − 1) / 2 must evaluate to an odd natural number. Combining both forms using lcm(6,4) we determine that a safe prime q > 7 also must be of the form 12k−1 or, equivalently, q ≡ 11 (mod 12). It follows that 3 (also 12) is a quadratic residue mod q for any safe prime q > 7. (Thus, 12 is not a primitive root of any safe prime q > 7, and the only safe primes that are also full reptend primes in base 12 are 5 and 7) If p is a Sophie Germain prime greater than 3, then p must be congruent to 2 mod 3\.
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An embedded Lie subgroup is closed Corollary 15.30. so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently, is an embedded Lie subgroup if and only if its group topology equals its relative topology. Problem 2.
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Equivalently, the probabilities of an event and its complement must always total to 1. This does not, however, mean that any two events whose probabilities total to 1 are each other's complements; complementary events must also fulfill the condition of mutual exclusivity.
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In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.
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Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere (or equivalently if every Jordan curve separates it), then it is conformally equivalent to an open subset of the complex sphere.
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Mar de Grises was a Chilean death/doom metal band created in 2000. A good translation of "Mar de Grises" would be "Sea of Grief," or equivalently, as grey is the Spanish color of sorrows as blue is in English, "Sea of Greys".
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O'Neil, M.J. (2001): The Merck Index, 13th Edition, page 803. Pure glyoxal is not commonly encountered because it forms hydrates, which oligomerize. For many purposes, these hydrated oligomers behave equivalently to glyoxal. It is produced industrially as a precursor to many products.
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In the 1965 paper by Krohn and Rhodes, the proof of the theorem on the decomposition of finite automata (or, equivalently sequential machines) made extensive use of the algebraic semigroup structure. Later proofs contained major simplifications using finite wreath products of finite transformation semigroups.
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Likewise the signature is equal for two congruent matrices and classifies a matrix up to congruency. Equivalently, the signature is constant on the orbits of the general linear group GL(V) on the space of symmetric rank 2 contravariant tensors S2V∗ and classifies each orbit.
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Though the universe might in theory have a longer history, the International Astronomical Union presently uses the term "age of the universe" to mean the duration of the Lambda-CDM expansion, or equivalently the elapsed time since the Big Bang in the current observable universe.
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In philosophy, mereological nihilism (also called compositional nihilism, or rarely simply nihilism) is the metaphysical thesis that there are no objects with proper parts. Equivalently, mereological nihilism says that mereological simples, or objects without any proper parts, are the only material objects that exist .
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Using the interval notation, the property "a is covered by b" can be rephrased equivalently as [a, b] = {a, b}. This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval orders.
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Octagonal zonogon Tessellation by irregular hexagonal zonogons Regular octagon tiled by squares and rhombi In geometry, a zonogon is a centrally symmetric convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations.
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A nondegenerate or nonsingular form is one that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if :f(x,y)=0\, for all y \in V implies that x = 0.
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A volcanic group (or, equivalently, a volcanic complex) is a collection of related volcanoes or volcanic landforms. The term is also used in a different sense when it denotes a suite of associated rock strata largely of volcanic origin; see group (stratigraphy) for details.
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In physics, an infrared divergence (also IR divergence or infrared catastrophe) is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very small energy approaching zero, or, equivalently, because of physical phenomena at very long distances.
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Of the three stereoisomers, two are enantiomers (levo- and dextro-2,3-butanediol) and one is a meso compound. The enantiomeric pair have (2R, 3R) and (2S, 3S) configurations at carbons 2 and 3, while the meso compound has configuration (2R, 3S) or, equivalently, (2S, 3R).
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Equivalently, a weak truth-table reduction is a Turing reduction for which the use of the reduction is bounded by a computable function. For this reason, they are sometimes referred to as bounded Turing (bT) reductions rather than as weak truth-table (wtt) reductions.
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To avoid item bias questionnaires need to be pretested, e.g. by using a translation procedure to deal with linguistic diversity. Construct equivalence does exist, if the object of investigation is equivalently embedded into the higher system level in all countries.cf. Wirth & Kolb (2004), p. 88.
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The layered permutations can also be equivalently described as the permutations that do not contain the permutation patterns 231 or 312. That is, no three elements in the permutation (regardless of whether they are consecutive) have the same ordering as either of these forbidden triples.
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When the values are quadratic in the amplitude (e.g. power), they are first linearised by taking the square root before the logarithm is taken, or equivalently the result is halved. In the International System of Quantities, the neper is defined as .Thor, A. J. (1994).
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A chief series is a maximal normal series for a group. Equivalently, a chief series is a composition series of the group G under the action of inner automorphisms. In detail, if G is a group, then a chief series of G is a finite collection of normal subgroups Ni ⊆ G, :1=N_0\subseteq N_1\subseteq N_2\subseteq\cdots\subseteq N_n=G, such that each quotient group Ni+1/Ni, for i = 1, 2,..., n − 1, is a minimal normal subgroup of G/Ni. Equivalently, there does not exist any subgroup A normal in G such that Ni < A < Ni+1 for any i.
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A situation is said to be Kaldor–Hicks efficient, or equivalently is said to satisfy the Kaldor–Hicks criterion, if no potential Kaldor–Hicks improvement from that situation exists. If an outcome is the highest it could possibly be, it is called a Hicks-optimal outcome.
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Berge conjectured that every Berge graph is perfect, or equivalently that the perfect graphs and the Berge graphs define the same class of graphs. This became known as the strong perfect graph conjecture, until its proof in 2002, when it was renamed the strong perfect graph theorem.
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The boolean hierarchy is the hierarchy of boolean combinations (intersection, union and complementation) of NP sets. Equivalently, the boolean hierarchy can be described as the class of boolean circuits over NP predicates. A collapse of the boolean hierarchy would imply a collapse of the polynomial hierarchy.
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If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring that is a topological group (for +) in which multiplication is continuous, too.
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Equivalently, in every finite partially ordered set that is not totally ordered, there exists a pair of elements x and y with the property that at least 1/3 and at most 2/3 of the linear extensions of the partial order place x earlier than y.
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Since the end of the 1990s, however, astronomical evidence indicating an accelerating expansion consistent with a cosmological constant – or, equivalently, with a particular and ubiquitous kind of dark energy – has steadily been accumulating.Einstein's original paper is ; good descriptions of more modern developments can be found in and .
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The orientation of Listing's plane (equivalently, the location of the primary position) of an individual can be measured using scleral coils. It can also be measured using a synoptometer. Alternatively, it can be measured using eye tracking (see also Eye tracking on the ISS for an example).
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In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).
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Tube amplifiers require more maintenance than solid state transistor amplifiers, such as replacing vacuum tubes or rectifying the tubes. Tube amplifiers are usually heavier than an equivalently powered transistor amplifier. As tubes are made of glass, tube amplifiers are more fragile than a solid state amp.
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In physics, the meter water equivalent (often m.w.e. or mwe) is a standard measure of cosmic ray attenuation in underground laboratories. A laboratory at a depth of 1000 m.w.e is shielded from cosmic rays equivalently to a lab below the surface of a body of water.
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Equivalently, a graph is line perfect if and only if each of its biconnected components is either bipartite or of the form K4 (the tetrahedron) or K1,1,n (a book of one or more triangles all sharing a common edge). Every line perfect graph is itself perfect.
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For higher configurations, more licenses must be purchased equivalently. A computer with 4 CPU cores, 16 GB RAM and 386 CUDA cores is recommended. Meshing, coloring and texturing are completely out-of-core in RC, which is intended to avoid RAM performance loss during these processes.
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In the mathematical field of graph theory, the intersection number of a graph is the smallest number of elements in a representation of as an intersection graph of finite sets. Equivalently, it is the smallest number of cliques needed to cover all of the edges of ...
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Every sufficiently small local patch of an algebraic manifold is isomorphic to km where k is the ground field. Equivalently the variety is smooth (free from singular points). The Riemann sphere is one example of a complex algebraic manifold, since it is the complex projective line.
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As show, a minimum connected dominating set (or equivalently a spanning tree with the maximum possible number of leaves) can be found in polynomial time on a distance-hereditary graph. A Hamiltonian cycle or Hamiltonian path of any distance-hereditary graph can also be found in polynomial time.; .
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The gamma distribution is a two-parameter exponential family with natural parameters k − 1 and −1/θ (equivalently, α − 1 and −β), and natural statistics X and ln(X). If the shape parameter k is held fixed, the resulting one-parameter family of distributions is a natural exponential family.
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A variety X is said to be rational if it is birational to affine space (or equivalently, to projective space) of some dimension. Rationality is a very natural property: it means that X minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset.
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Several fragments of TGDs have been defined. For instance, full TGDs are TGDs which do not use the existential quantifier. Full TGDs can equivalently be seen as programs in the Datalog query language. There are also some fragments of TGDs that can be expressed in guarded logic, e.g.
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In particular, he predicted a number of hydrogen atoms in the Universe 136 × 2256, or equivalently the half of the total number of particles protons + electrons. He did not complete this line of research before his death in 1944; his book Fundamental Theory was published posthumously in 1948.
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23–24 The ratio of these two values (Mw / Mn) is the dispersity (Đ), which is commonly used to express the width of the molecular weight distribution.Painter, p. 22 The physical properties of polymer strongly depend on the length (or equivalently, the molecular weight) of the polymer chain.Rubinstein, p.
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The name Thomas (Koine Greek: Θωμᾶς) given for the apostle in the New Testament is derived from the Aramaic תְּאוֹמָא or Tāʾwma/Tʾōmā, equivalently from Hebrew תְּאוֹם tʾóm, meaning "twin". The equivalent term for twin in Greek, which is also used in the New Testament, is Δίδυμος Didymos.
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Several branches of programming language semantics, such as domain theory, are formalized using topology. In this context, Steve Vickers, building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties.
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The Nagel point is the isotomic conjugate of the Gergonne point. The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line. The incenter is the Nagel point of the medial triangle; equivalently, the Nagel point is the incenter of the anticomplementary triangle.
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In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field F with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of F with coefficients in Z/2Z. It was proved by .
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A Deputy Conservator of Forests (Hindi: उप वन संरक्षक) or, equivalently a Divisional Forest Officer (Hindi: वन मण्डल अधिकारी), is an officer belonging to the Indian Forest Service (IFS). The Deputy Conservator of Forests is responsible for managing the forests, environment and wildlife related issues of a Forest Division of a state or a union territory of India. He/She is assisted by the officers of the State Forest Service and other state forest and wildlife officials, and reports to the Conservator of Forests (CF) of the Forest Department. ' _Assistant Conservator Forest (ACF)_ (Hindi: सहायक वन संरक्षक) Or equivalently a ' Sub Divisional Officer ' (Hindi: उप वन मंडलाधिकारी ) ACF is an important officer in the administration of forest divisions .
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A finite-dimensional, unital, associative algebra A defined over a field k is said to be a Frobenius algebra if A is equipped with a nondegenerate bilinear form σ:A × A → k that satisfies the following equation: σ(a·b,c)=σ(a,b·c). This bilinear form is called the Frobenius form of the algebra. Equivalently, one may equip A with a linear functional λ : A → k such that the kernel of λ contains no nonzero left ideal of A. A Frobenius algebra is called symmetric if σ is symmetric, or equivalently λ satisfies λ(a·b) = λ(b·a). There is also a different, mostly unrelated notion of the symmetric algebra of a vector space.
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In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type. They are the canonical singularities (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by and Felix Klein. The Du Val singularities also appear as quotients of C2 by a finite subgroup of SL2(C); equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups.
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Since a standard torus is the orbit of a point under a two dimensional abelian subgroup of the Möbius group, it follows that the cyclides also are, and this provides a second way to define them. A third property which characterizes Dupin cyclides is that their curvature lines are all circles (possibly through the point at infinity). Equivalently, the curvature spheres, which are the spheres tangent to the surface with radii equal to the reciprocals of the principal curvatures at the point of tangency, are constant along the corresponding curvature lines: they are the tangent spheres containing the corresponding curvature lines as great circles. Equivalently again, both sheets of the focal surface degenerate to conics.
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The elliptic functions thus have a double periodicity. Equivalently one can say that they have two complex periods . Their zeros and poles will thus form a regular, two-dimensional lattice. Corresponding properties of the lemniscatic elliptic functions had also been established by Gauss, but not published before after his death.
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In mathematics, an element p of a partial order (P, ≤) is a meet prime element when p is the principal element of a principal prime ideal. Equivalently, if P is a lattice, p ≠ top, and for all a, b in P, :a∧b ≤ p implies a ≤ p or b ≤ p.
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Equivalently, it is a surjective TVS embedding. Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms. ;A necessary condition for a vector topology All of the above conditions are consequently a necessity for a topology to form a vector topology.
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The expansion factor of a sawtooth is the limit of the ratio of successive heights (or equivalently, widths) of the "teeth" in plots of population versus generation number. Some sawtooths do not have an expansion factor under its standard definition because some sawtooths have growth that is not exponentially-spaced.
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In graph theory, a graph or digraph whose adjacency matrix is circulant is called a circulant graph (or digraph). Equivalently, a graph is circulant if its automorphism group contains a full-length cycle. The Möbius ladders are examples of circulant graphs, as are the Paley graphs for fields of prime order.
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A semicircle of radius r. In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line of symmetry (reflection symmetry).
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Optical configuration for imaging ptychography. A lens is used to make a conventional image. An aperture in the image plane acts equivalently to the illumination in conventional ptychography, while the image corresponds to the specimen. The detector lies in the Fraunhofer or Fresnel diffraction plane downstream of the image and aperture.
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East Lake High School which has managed to rate equivalently with Palm Harbor's University High School (both A+ rated), despite not having the attraction of an International Baccalaureate program like Palm Harbor University High. Also of benefit are the top-flight youth recreational programs offered by East Lake/City of Oldsmar.
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The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope of the normals to a curve. The evolute of a curve, a surface, or more generally a submanifold, is the caustic of the normal map. Let be a smooth, regular submanifold in .
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In mathematics, a protorus is a compact connected topological abelian group. Equivalently, it is a projective limit of tori (products of a finite number of copies of the circle group), or the Pontryagin dual of a discrete torsion-free abelian group. Some examples of protori are given by solenoid groups.
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An algebraic number is a number that is a solution of a non-zero polynomial equation in one variable with rational coefficients (or equivalently — by clearing denominators — with integer coefficients). Numbers such as that are not algebraic are said to be transcendental. Almost all real and complex numbers are transcendental.
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Equivalently, combine each input block with a key bit, and map the result through a 5→4 bit S-box. ## Mix adjacent 4-bit blocks using a maximum distance separable code over GF(24). ## Permute 4-bit blocks so that they will be adjacent to different blocks in following rounds.
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Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint. The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle subtending the same arc. The inscribed angle theorem appears as Proposition 20 on Book 3 of Euclid’s "Elements".
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For the purposes of this article, it will be taken as US Highway 2, running over Stevens Pass, or equivalently, the Skykomish River, Nason Creek, and the lower Wenatchee River. This roughly follows Beckey's division. Sometimes the southern boundary is defined by Snoqualmie Pass and the approximate route of Interstate 90.
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A distortionmeter is actually a levelmeter with two switchable parallel circuits at the input. The first circuit measures the total signal at the output of a system. (For low distortion levels this will be almost equal to fundamental). That value is adjusted to read 100% or, equivalently, to 0 dB.
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A point at infinity of the curve is a zero of p of the form (a, b, 0). Equivalently, (a, b) is a zero of pd. The fundamental theorem of algebra implies that, over an algebraically closed field (typically, the field of complex numbers), pd factors into a product of linear factors.
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In game theory, an outcome is a situation which results from a combination of player's strategies. Formally, a path through the game tree, or equivalently a terminal node of the game tree. A primary purpose of game theory is to determine the outcomes of games according to a solution concept (e.g. Nash equilibrium).
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A disulfite, commonly known as metabisulfite or pyrosulfite, is a chemical compound containing the ion . It is a colorless dianion that is primarily marketed in the form of sodium metabisulfite or potassium metabisulfite. When dissolved in water, these salts release the bisulfite ion. These salts act equivalently to sodium bisulfite or potassium bisulfite.
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Equivalently, taxing saving may reduce the implicit subsidy on saving created by the borrowing constraints and thus restore efficiency in saving. Furthermore, Aiyagari (1995) and Chamley (2001) show that capital income taxation is desirable when consumption is positively correlated with savings in a model featuring borrowing-constrained agents with infinite lives and uncertainty.
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Because the two opposite directions along an easy axis are usually equivalently easy to magnetize along, and the actual direction of magnetization can just as easily settle into either direction, which is an example of spontaneous symmetry breaking. Magnetic anisotropy is a prerequisite for hysteresis in ferromagnets: without it, a ferromagnet is superparamagnetic.
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This can be equivalently conceived of as the system lowering its threshold, and this is accomplished such that the ability to detect suprathreshold signals is not degraded. Stochastic resonance in large- scale physiological systems of neurons (above the single-neuron level but below the behavioral level) has not yet been investigated experimentally.
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Between Trollhättan and Surte there is a 52 km long motorway, finished in 2012. Between Surte and Gothenburg there is a 17 km road designed equivalently to a motorway. The exception is that there are two gaps in the Trollhättan–Surte motorway and there are two traffic lights along the Surte–Gothenburg road.
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The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned. Equivalently it is the minimum number of spanning forests needed to cover all the edges of the graph. The Nash-Williams theorem provides necessary and sufficient conditions for when a graph is k-arboric.
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Equivalently, if the risk of an outcome is decreased by the exposure, the term absolute risk reduction (ARR) is used, and computed as I_u - I_e. The inverse of the absolute risk reduction is the number needed to treat, and the inverse of the absolute risk increase is the number needed to harm.
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In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) k-coloring up to permutation of the colors. Equivalently, there is only one way to partition its vertices into k independent sets and there is no way to partition them into k−1 independent sets.
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The detection threshold for the (+)-(R) enantiomer of grapefruit mercaptan is 2×10−5 ppb, or equivalently a concentration of 2×10−14. This corresponds to being able to detect 2×10−5 mg in one metric ton of water - one of the lowest detection thresholds ever recorded for a naturally occurring compound.
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A 2015 study on racial bias in teacher evaluations in Brazil found that Brazilian math teachers gave better grading assessments of white students than equally proficient and equivalently well-behaved black students. A 2018 paper found that discriminatory hiring and retention policies accounted for 6-8% of the overall racial wage gap.
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In linear algebra, two matrices A and B are said to commute if AB=BA and equivalently, their commutator [A,B]= AB-BA is zero. A set of matrices A_1,\ldots,A_k is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other.
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The intersection of two sets A and B, represented by circles. A \cap B is in red. In mathematics, the intersection of two sets A and B, denoted by A ∩ B, is the set containing all elements of A that also belong to B (or equivalently, all elements of B that also belong to A).
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Let C be an additive category, or more generally an additive -linear category for a commutative ring . We call C a Krull–Schmidt category provided that every object decomposes into a finite direct sum of objects having local endomorphism rings. Equivalently, C has split idempotents and the endomorphism ring of every object is semiperfect.
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For every subset Q of a real vector space, its is the minimal convex set that contains Q. Thus Conv(Q) is the intersection of all the convex sets that cover Q. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in Q.
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Work was carried out in two shifts. Thirty farms were partially expropriated. Half a million cubic meters (18 million cu ft) of rock were blasted and a similar amount of earthwork moved in the construction process. The work consisted of a runway and an equivalently long taxiway, although it was only half the width.
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The work consisted of a runway and an equivalently long taxiway, although it was only half the width. The terminal building cost NOK 200,000 and was located next to a parking lot with place for seventy cars. A long barbed wire fence circumferenced the airport.Østerbø: 56 The airport had instrument landing system from the start.
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Equivalently, it concerns the density of a different class of graphs, the locally linear graphs in which the neighborhood of every vertex is an induced matching. Neither of these types of graph can have a quadratic number of edges, but constructions are known for graphs of this type with nearly-quadratic numbers of edges.
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A direct computation shows that ∂2 = 0. In geometric terms, this says that the boundary of anything has no boundary. Equivalently, the abelian groups :(C_k, \partial_k) form a chain complex. Another equivalent statement is that Bk is contained in Zk. As an example, consider a tetrahedron with vertices oriented as w,x,y,z.
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See the footnote at the end of Soare: 1996. showing that a general solution to the ' is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis.
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Once the data are sorted into this ordering, any one-dimensional data structure can be used such as binary search trees, B-trees, skip lists or (with low significant bits truncated) hash tables. The resulting ordering can equivalently be described as the order one would get from a depth-first traversal of a quadtree.
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In electronics, when describing a voltage or current step function, rise time is the time taken by a signal to change from a specified low value to a specified high value. These values may be expressed as ratiosSee for example , and . or, equivalently, as percentagesSee for example , and . with respect to a given reference value.
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The box has a depth of d, a height of r2-r1cos(α) and a width of r1sin(a). Equivalently, the law of cosines may be used to first compute the length of the spoke as projected on the wheel's plane (as illustrated in the diagram), followed by an application of the Pythagorean theorem.
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A spring system can be thought of as the simplest case of the finite element method for solving problems in statics. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem.
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A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X., p. 28. Every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. Disjoint-set data structures. and partition refinement.
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Equivalently is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In more general terms, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space.
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A perfect conductor or perfect electric conductor (PEC) is an idealized material exhibiting infinite electrical conductivity or, equivalently, zero resistivity (cf. perfect dielectric). While perfect electrical conductors do not exist in nature, the concept is a useful model when electrical resistance is negligible compared to other effects. One example is ideal magnetohydrodynamics, the study of perfectly conductive fluids.
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A space with quadratic form is split (or metabolic) if there is a subspace which is equal to its own orthogonal complement; equivalently, the index of isotropy is equal to half the dimension. The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.
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The fine topology on the Euclidean space \R^n is defined to be the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous. Concepts in the fine topology are normally prefixed with the word 'fine' to distinguish them from the corresponding concepts in the usual topology, as for example 'fine neighbourhood' or 'fine continuous'.
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Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.See and .
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A topological space X is called countably generated if V is closed in X whenever for each countable subspace U of X the set V \cap U is closed in U. Equivalently, X is countably generated if and only if the closure of any A \subset X equals the union of closures of all countable subsets of A.
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Besharmi Morcha () is most often translated into English as "shameless protest" or "shameless front". It has also been translated as “March of the Shameless” and "Shameless Campaign". Besharmi Morcha was chosen as the name in India to address its diverse cultural aspects and languages. The word "slut" does not translate directly or equivalently into Indian languages such as Hindi.
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The notion of boundary layers—introduced by Prandtl in 1904, founded on both theory and experiments—explained the causes of drag at high Reynolds numbers. The boundary layer is the thin layer of fluid close to the object's boundary, where viscous effects remain important even when the viscosity is very small (or equivalently the Reynolds number is very large).
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Restoration of P. engelhardti and contemporary animals from the Trossingen Formation Plateosaurus gracilis, the older species, is found in the Löwenstein Formation (Lower Norian). P. engelhardti stems from the upper Löwenstein Formation (Upper Norian), the Trossingen Formation (Upper Norian), and equivalently aged rock units. Plateosaurus thus lived probably from approximately 214 to 204 million years ago.
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N-th harmonic generation Harmonic generation (HG, also called multiple harmonic generation) is a nonlinear optical process in which n photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with n times the energy of the initial photons (equivalently, n times the frequency and the wavelength divided by n).
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A serial relation can be equivalently characterized as every element having a non-empty successor neighborhood. Similarly an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood". More commonly, an inverse serial relation is called a surjective relation, and is specified by a serial converse relation. Definition 5.8, page 57.
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29, No. 5 (Nov., 1998), pp. 354-370. (abstract) (JSTOR) One can represent the twelve equal-tempered pitch classes by the cyclic group of order twelve, or equivalently, the residue classes modulo twelve, Z/12Z. The group Z_{12} has four generators, which can be identified with the ascending and descending semitones and the ascending and descending perfect fifths.
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A strictly convex curve is a convex curve that does not contain any line segments. Equivalently, a strictly convex curve is a curve that intersects any line in at most two points,.. or a simple curve in convex position, meaning that none of its points is a convex combination of any other subset of its points.
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A default is categorical or prerequisite-free if it has no prerequisite (or, equivalently, its prerequisite is tautological). A default is normal if it has a single justification that is equivalent to its conclusion. A default is supernormal if it is both categorical and normal. A default is seminormal if all its justifications entail its conclusion.
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In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset.Steen & Seebach, p. 33Engelking, p. 59 Equivalently, every nonempty subset A of X contains a point isolated in A. A subset of a topological space is called a scattered set if it is a scattered space with the subspace topology.
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In mathematics, a uniform tree is a locally finite tree which is the universal cover of a finite graph. Equivalently, the full automorphism group G=Aut(X) of the tree, which is a locally compact topological group, is unimodular and G\X is finite. Also equivalent is the existence of a uniform X-lattice in G.
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MakeSet creates 8 singletons. After some operations of Union, some sets are grouped together. In computer science, a disjoint-set data structure, also called a union–find data structure or merge–find set, is a data structure that stores a collection of disjoint (non-overlapping) sets. Equivalently, it stores a partition of a set into disjoint subsets.
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By definition, :1 reyn = 1 lbf s in−2. It follows that the relation between the reyn and the poise is approximately :1 reyn = 6.89476 × 10 6 cP. In SI units, viscosity is expressed in newton-seconds per square meter, or equivalently in pascal-seconds. The conversion factor between the two is approximately :1 reyn = 6890 Pa s.
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Any computation can be expressed declaratively as a sequence of state transitions. As an example, an optimizing compiler with three optimization passes could be implemented as a relation between an initial program and its optimized form: program_optimized(Prog0, Prog) :- optimization_pass_1(Prog0, Prog1), optimization_pass_2(Prog1, Prog2), optimization_pass_3(Prog2, Prog). or equivalently using DCG notation: program_optimized --> optimization_pass_1, optimization_pass_2, optimization_pass_3.
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In mathematics, a topological space X is said to be ultraconnected if no pair of nonempty closed sets of X is disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no T_1 space with more than 1 point is ultraconnected.Steen and Seeback, Sect.
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The chemical structure of gallocyanin The gallocyanin stain, also known as the gallocyanin-chromalum stain, is a stain of the oxazine group for total nucleic acids. It is prepared from gallocyanin and is an ideal method for numerous slides that need to be stained serially, equivalently, and reproducible. Structures containing basophilic compounds take on a bluish color.
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Consider an experiment that can produce a number of outcomes. The set of all outcomes is called the sample space of the experiment. The power set of the sample space (or equivalently, the event space) is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results.
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An operator A on an (infinite dimensional) Banach space or Hilbert space H has a cyclic vector f if the vectors f, Af, A2f,... span H. Equivalently, f is a cyclic vector for A in case the set of all vectors of the form p(A)f, where p varies over all polynomials, is dense in H.
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In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three coordinate axes. Equivalently, they are the contributions of each component of the basis to a unit vector in that direction. Direction cosines are an analogous extension of the usual notion of slope to higher dimensions.
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Most other languages lack a prop-word used equivalently to English one. Instead, they may make more use of demonstrative pronouns (similar to this or that when used without a following noun or prop-word). Phrases like "the red one" may be translated simply as "the red", i.e. using a nominalized adjective, as in the German der Rote.
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A reflexive relation on a nonempty set X can neither be irreflexive, nor asymmetric, nor antitransitive. The reflexive closure ≃ of a binary relation ~ on a set X is the smallest reflexive relation on X that is a superset of ~. Equivalently, it is the union of ~ and the identity relation on X, formally: (≃) = (~) ∪ (=). For example, the reflexive closure of (<) is (≤).
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Crossed ladders of lengths a and b. h is half the harmonic mean of A and B; equivalently, the reciprocals of A and B sum to the reciprocal of h (the optic equation). Given a, b, and h, find w. Two ladders of lengths a and b lie oppositely across an alley, as shown in the figure.
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The notation A ≡T B indicates A and B are Turing equivalent. The relation ≡T is an equivalence relation known as Turing equivalence. A Turing degree is a collection of sets of natural numbers, such that any two sets in the collection are Turing equivalent. Equivalently, a Turing degree is an equivalence class of the relation ≡T.
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For a distributive lattice, the complement of x, when it exists, is unique. In the case the complement is unique, we write and equivalently, . The corresponding unary operation over L, called complementation, introduces an analogue of logical negation into lattice theory. Heyting algebras are an example of distributive lattices where some members might be lacking complements.
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Alternatively, a Reuleaux triangle may be constructed from an equilateral triangle T by drawing three arcs of circles, each centered at one vertex of T and connecting the other two vertices. Or, equivalently, it may be constructed as the intersection of three disks centered at the vertices of T, with radius equal to the side length of T.
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Riemann further proved the Riemann singularity theorem, identifying the multiplicity of a point p = class(D) on Wg − 1 as the number of linearly independent meromorphic functions with pole divisor dominated by D, or equivalently as h0(O(D)), the number of linearly independent global sections of the holomorphic line bundle associated to D as Cartier divisor on C.
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The geometric median is the point to which the population has the smallest possible sum of distances (or equivalently, the smallest average distance). Because of this property, it is also known as the point of minimum aggregate travel. Unfortunately, there is no direct closed-form expression for the geometric median; it is typically computed using iterative methods.
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Let N be a (finite or infinite) nonempty set of individuals. The subsets of N are called coalitions. A simple game (voting game) is a collection W of coalitions. (Equivalently, it is a coalitional game that assigns either 1 or 0 to each coalition.) We assume that W is nonempty and does not contain an empty set.
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In this point of view the quasistatic approximation is obtained by using time instead of retarded time or equivalently to assume that the speed of light is infinite. To first order, the mistake of using only Biot–Savart's law rather than both terms of Jefimenko's magnetic field equation fortuitously cancel. Griffiths, David J., Introduction to Electrodynamics -3rd Ed., 1999.
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It is obtained by taking intersection of all \sigma-algebras on X containing \tau . ;Borel set: A Borel set is an element of a Borel algebra. ;Boundary: The boundary (or frontier) of a set is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement.
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In Pakistan, the twelfth grade is the second year of college (the first being the eleventh grade) and is equivalently called the "second year." Students who graduate from the second year are normally between seventeen and eighteen years old. For entrance into universities, students undergo standardized exams like ECAT, MDCAT and exams conducted by the NTS.
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Note that transitivity of integrality above implies that if B is integral over A, then B is a union (equivalently an inductive limit) of subrings that are finitely generated A-modules. If A is noetherian, transitivity of integrality can be weakened to the statement: :There exists a finitely generated A-submodule of B that contains A[b].
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For example, the compound subject you and I is treated equivalently to we, taking appropriate pronominal agreement ("our car", not "your car", "their car", etc.). In languages with more extensive subject-verb agreement (e.g. Spanish or Arabic), the verb agreement is clearly revealed as also being first-person plural. Compound subjects joined by “and” generally take a plural verb.
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In theoretical physics, conformal supergravity is the study of the supersymmetrized version of conformal gravity with Weyl transformations. Equivalently, it is the extension of ordinary supergravity to include Weyl transformations. Often, nonconformal gravity is described by conformal gravity with a conformal compensator. For a review of conformal supergravity see E.S. Fradkin and A.A. Tseytlin, "Conformal Supergravity", Phys. Rep.
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The decomposition of ammonium nitrate is also a common laboratory method for preparing the gas. Equivalently, it can be obtained by heating a mixture of sodium nitrate and ammonium sulfate:"Nitrogen Family". chemistry.tutorvista.com :2 + () → + 2 + 4 . Another method involves the reaction of urea, nitric acid and sulfuric acid:"Preparation of Nitrous Oxide from Urea, Nitric Acid and Sulfuric Acid".
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In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation.
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A subset of the Euclidean -space is open if, for every point in , there exists a positive real number (depending on ) such that a point in belongs to as soon as its Euclidean distance from is smaller than . Equivalently, a subset of is open if every point in is the center of an open ball contained in .
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In the mathematics of permutations, a layered permutation is a permutation that reverses contiguous blocks of elements. Equivalently, it is the direct sum of decreasing permutations. One of the earlier works establishing the significance of layered permutations was , which established the Stanley–Wilf conjecture for classes of permutations forbidding a layered permutation, before the conjecture was proven more generally.
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The elements x and y generate a normal subgroup isomorphic to the quaternion group of order 8. The center is cyclic of order 2m. It is generated by the elements z3 and x2 = y2, and the quotient by the center is the tetrahedral group, equivalently, the alternating group A4. When m = 1 this group is the binary tetrahedral group.
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A partial word is a string whose characters may either belong to a given alphabet or be a wildcard character. Such a word can represent a set of strings over the alphabet without wildcards, by allowing each wildcard character to be replaced by any single character of the alphabet, independently of the replacements of the other wildcard characters. Two partial words are compatible when they agree on their non-wildcard characters, or equivalently when there is a string that they both match; one partial word x contains another partial word y if they are compatible and the non-wildcard positions of x contain those of y; equivalently, the strings matched by x are a subset of those matched by y. The book has 12 chapters, which can be grouped into five larger parts.
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Apollonian networks are examples of maximal planar graphs, graphs to which no additional edges can be added without destroying planarity, or equivalently graphs that can be drawn in the plane so that every face (including the outer face) is a triangle. They are also chordal graphs, graphs in which every cycle of four or more vertices has a diagonal edge connecting two non-consecutive cycle vertices, and the order in which vertices are added in the subdivision process that forms an Apollonian network is an elimination ordering as a chordal graph. This forms an alternative characterization of the Apollonian networks: they are exactly the chordal maximal planar graphs or equivalently the chordal polyhedral graphs.The equivalence of planar 3-trees and chordal maximal planar graphs was stated without proof by .
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In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a topological group, every open neighborhood of the identity contains a normal subgroup whose corresponding quotient group is a solvable group.
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An inverse monoid is a monoid where for every a in M, there exists a unique a−1 in M such that and . If an inverse monoid is cancellative, then it is a group. In the opposite direction, a zerosumfree monoid is an additively written monoid in which implies that and : equivalently, that no element other than zero has an additive inverse.
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But in signal processing, decimation by a factor of 10 actually means keeping only every tenth sample. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, if compact disc audio at 44,100 samples/second is decimated by a factor of 5/4, the resulting sample rate is 35,280. A system component that performs decimation is called a decimator.
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In the catalytic cycle, W(VI) (tungsten "six") converts to W(IV) concomitant with oxidation of the aldehyde to a carboxylic acid (equivalently, a carboxylate). A W(V) intermediate can be detected by EPR spectroscopy. AOR mechanism at the active site. General Reaction Mechanism of AOR: :RCHO + H2O → RCO2H + 2H+ \+ 2 e− The redox equivalents are provided by the 4Fe-4S cluster.
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Detail of Cambridge North railway station cladding The Cambridge North railway station is decorated with architectural panels displaying the evolution of Rule 30 (or equivalently under black-white reversal, Rule 135). The design was described by its architect as inspired by Conway's Game of Life, a different cellular automaton studied by Cambridge mathematician John Horton Conway, but is not actually based on Life.
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Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction. A spread of is a set of lines, with no two intersecting. Equivalently, it is a partition of the points of into lines. Given a spread of , the André/Bruck- Bose construction produces a translation plane of order as follows: Embed as a hyperplane of .
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There are several equivalent statements of the Smale conjecture. One is that the component of the unknot in the space of smooth embeddings of the circle in 3-space has the homotopy-type of the round circles, equivalently, O(3). Another equivalent statement is that the group of diffeomorphisms of the 3-ball which restrict to the identity on the boundary is contractible.
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"Sekele" itself derives from Vasekele, the Angolan Bantu name. It is also known as Northern ǃKung (or equivalently "Northern ǃXuun", "Northern Ju" and several other variants). Two of the Angolan varieties have gone by the outdated term ǃʼOǃKung (or ǃʼO ǃuŋ "Forest ǃKung") and Maligo (short for "Sekele Maligo"). There are several Namibian dialects, of which the best known is Ekoka.
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Inflation is defined either as the devaluation of a currency or equivalently the rise of prices relative to a currency. Since inflation lowers real wages, Keynesians view inflation as the solution to involuntary unemployment. However, "unanticipated" inflation leads to lender losses as the real interest rate will be lower than expected. Thus, Keynesian monetary policy aims for a steady rate of inflation.
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Therefore, any Moore graph has the maximum number of vertices possible among all graphs with maximum degree d and diameter k. Later, showed that Moore graphs can equivalently be defined as having diameter k and girth 2k+1; these two requirements combine to force the graph to be d-regular for some d and to satisfy the vertex-counting formula.
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The elementary charge, usually denoted by or sometimes e, is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. Somewhat confusingly, in atomic physics, e sometimes denotes the electron charge, i.e. the negative of the elementary charge.
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The rotation group is a group under function composition (or equivalently the product of linear transformations). It is a subgroup of the general linear group consisting of all invertible linear transformations of the real 3-space \R^3.n × n real matrices are identical to linear transformations of \R^n expressed in its standard basis. Furthermore, the rotation group is nonabelian.
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Thus, no current will flow into the buffer's input (or equivalently no voltage will be present at the buffer's input). In practice, due to device mismatches the collector currents are unequal and this results in the difference flowing into the buffer's input resulting in an offset at its output. This is corrected by adjusting the input bias or adding offset nulling circuitry.
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TC0 is a complexity class used in circuit complexity. It is the first class in the hierarchy of TC classes. TC0 contains all languages which are decided by Boolean circuits with constant depth and polynomial size, containing only unbounded fan-in AND gates, OR gates, NOT gates, and majority gates. Equivalently, threshold gates can be used instead of majority gates.
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Another equivalently famous apparel is the Xhubleta- clothing. A xhubleta is a bell wavy skirt which is held by two straps on the shoulders, worn on top of a long sleeved white linen shirt. Its texture consists of long suspended long black straps etched in the material, which was usually chestnut velvet. The socks and shoes were the same as the pështjellak clothing.
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Owing to their superior hunting ability, leaders eat more meat and fat and fewer carbohydrates than other men. Men and women basically have equal power. Issues are discussed and decisions are made by consensus at fire camps; men and women engage in the conversations equivalently. If there is a disagreement, misdemeanor, or offense, then the offender may be banished, beaten, or scorned.
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In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, or, equivalently, the angle between its equatorial plane and orbital plane. It differs from orbital inclination. At an obliquity of 0 degrees, the two axes point in the same direction; i.e., the rotational axis is perpendicular to the orbital plane.
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Shooting down a satellite approximately present in a 300-km LEO is more challenging than shooting down a satellite present in a higher orbit. The transverse orbital speed of a satellite is inversely proportional to its distance from the centre of Earth, which is a direct consequence of the law of conservation of angular momentum, or equivalently, Kepler's second law.
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For a polygon P with n sides, there is one face (the entire polygon), which has internal angle 2\pi, and n edges, each of which has internal angle \pi. Let A be the sum of the internal angles of the corners. The Gram-Euler theorem then tells us that A - \pi n + 2\pi = 0, or equivalently, A = \pi (n - 2).
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In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set S is a bijection between two specified subsets of S. That is, it is defined by two subsets U and V of equal size, and a one-to- one mapping from U to V. Equivalently, it is a partial function on S that can be extended to a permutation...
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For a continuous random variable \theta_i distributed about the unit circle, the Von Mises distribution maximizes the entropy when the real and imaginary parts of the first circular moment are specified or, equivalently, the circular mean and circular variance are specified. When the mean and variance of the angles \theta_i modulo 2\pi are specified, the wrapped normal distribution maximizes the entropy.
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A matrix or, equivalently, a linear operator T on a finite- dimensional vector space V is called semi-simple if every T-invariant subspace has a complementary T-invariant subspace.Lam (2001), [ p. 39] This is equivalent to the minimal polynomial of T being square-free. For vector spaces over an algebraically closed field F, semi-simplicity of a matrix is equivalent to diagonalizability.
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Küssner derived an approximate model for an airfoil encountering a sudden step-like change in the transverse gust velocity—or, equivalently, as seen from a frame of reference moving with the airfoil: a sudden change in the angle of attack. The airfoil is modelled as a flat plate in a potential flow, moving with constant horizontal velocity.Cebeci (2005) pp. 15–16.
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In terms of a security, such as a stock or a bond, or equivalently to be long in a security, means the holder of the position owns the security, on the expectation that the security will increase in value, and will profit if the price of the security goes up. Going long a security is the more conventional practice of investing.
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The situation is analogous to that of weak acids and strong bases. :H3O+ \+ B H2O + BH+ The pH of the neutralized solution depends on the acid dissociation constant of the base, pKa, or, equivalently, on the base association constant, pKb. The most suitable indicator to use for this type of titration is one, like Methyl orange, that changes color at low pH.
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A totally unimodular matrix The term was coined by Claude Berge, see (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. Equivalently, every square submatrix has determinant 0, +1 or −1\. A totally unimodular matrix need not be square itself. From the definition it follows that any submatrix of a totally unimodular matrix is itself totally unimodular (TU).
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A topological space X is said to be locally regular if and only if each point, x, of X has a neighbourhood that is regular under the subspace topology. Equivalently, a space X is locally regular if and only if the collection of all open sets that are regular under the subspace topology forms a base for the topology on X.
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On the other hand, if it is symmetric, that is, if implies , then it is an equivalence relation. A preorder is total if or for all a, b. Equivalently, the notion of a preordered set P can be formulated in a categorical framework as a thin category; i.e., as a category with at most one morphism from an object to another.
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If a maximal planar graph has v vertices with v > 2, then it has precisely 3v − 6 edges and 2v − 4 faces. Apollonian networks are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles. Equivalently, they are the planar 3-trees. Strangulated graphs are the graphs in which every peripheral cycle is a triangle.
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A line m and a plane q in three-dimensional space, the line not lying in that plane, are parallel if and only if they do not intersect. Equivalently, they are parallel if and only if the distance from a point P on line m to the nearest point in plane q is independent of the location of P on line m.
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Photon transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation (RTE). However, the RTE is difficult to solve without introducing approximations. A common approximation summarized here is the diffusion approximation. Overall, solutions to the diffusion equation for photon transport are more computationally efficient, but less accurate than Monte Carlo simulations.
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As we have thus seen, any correlations seen in the wavefunction of each observer's minds are only concrete after interaction between the different polarizers. The correlations on the level of individual minds correspond to the appearance of quantum non- locality (or equivalently, violation of Bell's inequality). So the many world is non-local, or it cannot explain EPR-GHZ correlations.
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These zeros play an important role in numerical integration based on Gaussian quadrature. The specific quadrature based on the P_n's is known as Gauss-Legendre quadrature. From this property and the facts that P_n(\pm 1) e 0 , it follows that P_n(x) has n-1 local minima and maxima in (-1,1) . Equivalently, dP_n(x)/dx has n -1 zeros in (-1,1) .
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If every Cauchy net (or equivalently every Cauchy filter) has a limit in X, then X is called complete. One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces. The most general situation in which Cauchy nets apply is Cauchy spaces; these too have a notion of completeness and completion just like uniform spaces.
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Because the costs are additive, they behave like the logarithm of the probability (since log-likelihoods are additive), or equivalently, somewhat like the entropy (since entropies are additive). This makes Link Grammar compatible with machine learning techniques such as hidden Markov models and the Viterbi algorithm, because the link costs correspond to the link weights in Markov networks or Bayesian networks.
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Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the primes are evenly distributed (asymptotically) among the congruence classes modulo d containing a's coprime to d.
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Equivalently, the left wing's effective angle of attack is decreased due to the positive roll. By definition, lift is perpendicular to the oncoming flow. The upward deflection of oncoming air causes the lift vector to be deflected backward. Conversely, as the right wing descends, its vector pointing towards the oncoming air is deflected downward and its lift vector is deflected forward.
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The NAND Boolean function has the property of functional completeness. This means, any Boolean expression can be re-expressed by an equivalent expression utilizing _only_ NAND operations. For example, the function NOT(x) may be equivalently expressed as NAND(x,x). In the field of digital electronic circuits, this implies that we can implement any Boolean function using just NAND gates.
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The remaining 1-1/n of the time, the value of E[(x_u-x_v)^2] is the expected squared difference between two independent observations from the original distribution. Therefore, dividing the sample expected squared difference by (1-1/n), or equivalently multiplying by 1/(1-1/n) = n/(n-1), gives an unbiased estimate of the original expected squared difference.
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The semantics are rather more complicated than for the classical case. A model theory can be given by Heyting algebras or, equivalently, by Kripke semantics. Recently, a Tarski- like model theory was proved complete by Bob Constable, but with a different notion of completeness than classically. Unproved statements in intuitionistic logic are not given an intermediate truth value (as is sometimes mistakenly asserted).
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Boundary of a set A is denoted by \partial A or bd A. ;Bounded: A set in a metric space is bounded if it has finite diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A function taking values in a metric space is bounded if its image is a bounded set.
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The first-order functions that are provably total in second-order arithmetic are precisely the same as those representable in system F (Girard and Taylor 1987, pp. 122-123). Almost equivalently, system F is the theory of functionals corresponding to second-order arithmetic in a manner parallel to how Gödel's system T corresponds to first-order arithmetic in the Dialectica interpretation.
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Void safety (also known as null safety) is a guarantee within an object- oriented programming language that no object references will have null or void values. In object-oriented languages, access to objects is achieved through references (or, equivalently, pointers). A typical call is of the form: x.f(a, ...) where f denotes an operation and x denotes a reference to some object.
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But the vine may also be stretched out to maximize its extent (or length). In this case, the torsion of the vine is related to the torsion of the pair of filaments (or equivalently the surface torsion of the ribbon connecting the filaments), and it reflects the difference between the length-maximizing (geodesic) configuration of the vine and its energy-minimizing configuration.
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This concept applies equivalently close to the focus. According to the user manual for the Zemax optical design software, the correct approximation for propagation in the near field follows the angular spectrum method. This approximation works well when at the observation point the distance to the aperture is of the same order as the aperture size. This propagation regime satisfies \ F \gg 1.
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An infrared cutoff (long-distance cutoff) is the minimal value of energy – or, equivalently, the maximal wavelength (usually a very large distance) – that will be taken into account in a calculation, typically an integral. At the opposite end of the energy scale, an ultraviolet cutoff is the maximal allowed energy or the shortest allowed distance (usually a very short length scale).
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An algebraic manifold is an algebraic variety that is also an m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to km. Equivalently, the variety is smooth (free from singular points). When is the real numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions.
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Two skew semistandard tableaux T and S are said to be jeu-de-taquin equivalent if one can transform one of them into the other using a sequence (possibly empty) of slides (both inward and outward slides being allowed). Equivalently, two skew semistandard tableaux T and S are jeu-de-taquin equivalent if and only if they have the same rectification.
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In mathematics, an extensive category is a category C with finite coproducts that are disjoint and well-behaved with respect to pullbacks. Equivalently, C is extensive if the coproduct functor from the product of the slice categories C/X × C/Y to the slice category C/(X + Y) is an equivalence of categories for all objects X and Y of C.
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This formalises the classical theory of the "moving frame", favoured by French authors. Lifts of loops about a point give rise to the holonomy group at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly small loops at the point. Equivalently curvature can be calculated directly infinitesimally in terms of Lie brackets of lifted vector fields.
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In the framework of that formalism, there is also a straightforward way to define globally quantities like energy or, equivalently, mass (so-called ADM mass/energy) which, in general relativity, is not trivial at all. With L. Abbott, Deser extended the notion of energy for gravity with a cosmological constant. And with Claudio Teitelboim he showed that supergravity has positive energy.
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In the mathematical field of order theory, an element a of a partially ordered set with least element 0 is an atom if 0 < a and there is no x such that 0 < x < a. Equivalently, one may define an atom to be an element that is minimal among the non-zero elements, or alternatively an element that covers the least element 0.
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In mathematics, the Duistermaat–Heckman formula, due to , states that the pushforward of the canonical (Liouville) measure on a symplectic manifold under the moment map is a piecewise polynomial measure. Equivalently, the Fourier transform of the canonical measure is given exactly by the stationary phase approximation. and, independently, showed how to deduce the Duistermaat–Heckman formula from a localization theorem for equivariant cohomology.
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The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points. If S and S' are the isodynamic points of a triangle ABC, then the three products of distances AS\cdot BC=BS\cdot AC=CS\cdot AB are equal. The analogous equalities also hold for S'. states that this property is the reason for calling these points "isodynamic". Equivalently to the product formula, the distances AS, BS, and CS are inversely proportional to the corresponding triangle side lengths BC, AC, and AB. S and S' are the common intersection points of the three circles of Apollonius associated with triangle of a triangle ABC, the three circles that each pass through one vertex of the triangle and maintain a constant ratio of distances to the other two vertices.
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Glulam beams were used to support the second floor and the roof. This allowed the appearance of wooden columns, while providing the necessary strength. The method for calculating the size of a flitch beam to be used in construction is straightforward, using the transformed-section method. The steel piece of the beams is imagined to be an equivalently flexible (although much thicker) piece of wood.
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Avoiding existential quantifiers is important in constructive mathematics and computing.. See also Heyting field. One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two constants and , since and .The a priori twofold use of the symbol "−" for denoting one part of a constant and for the additive inverses is justified by this latter condition.
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Equivalently, a field is an algebraic structure of type , such that is not defined, and are abelian groups, and · is distributive over +. Some elementary statements about fields can therefore be obtained by applying general facts of groups. For example, the additive and multiplicative inverses and are uniquely determined by . The requirement follows, because 1 is the identity element of a group that does not contain 0.
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A field is algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any polynomial equation :, with coefficients , has a solution . By the fundamental theorem of algebra, is algebraically closed, i.e., any polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are not algebraically closed since the equation : does not have any rational or real solution.
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These concepts generalize respectively those of preordered set, partially ordered set and totally ordered set. However, it is difficult to work with them as in the small case because many constructions common in a set theory are no longer possible in this framework. Equivalently, a preordered class is a thin category, that is, a category with at most one morphism from an object to another.
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In compilers, live variable analysis (or simply liveness analysis) is a classic data-flow analysis to calculate the variables that are live at each point in the program. A variable is live at some point if it holds a value that may be needed in the future, or equivalently if its value may be read before the next time the variable is written to.
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A non-vertical line can be defined by its slope , and its -intercept (the coordinate of its intersection with the -axis). In this case its linear equation can be written :y=mx+y_0. If, moreover, the line is not horizontal, it can be defined by its slope and its -intercept . In this case, its equation can be written :y=m(x-x_0), or, equivalently, :y=mx-mx_0.
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Petkovšek's algorithm (also Hyper) is a computer algebra algorithm that computes a basis of hypergeometric terms solution of its input linear recurrence equation with polynomial coefficients. Equivalently, it computes a first order right factor of linear difference operators with polynomial coefficients. This algorithm was developed by Marko Petkovšek in his PhD- thesis 1992. The algorithm is implemented in all the major computer algebra systems.
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Thiosulfoxides are orthogonally isomeric with disulfides, having the second sulfur branching from the first and not partaking in a continuous chain, i.e. >S=S rather than −S−S−. Disulfide bonds are analogous but more common than related peroxide, thioselenide, and diselenide bonds. Intermediate compounds of these also exist, for example thioperoxides (also known as oxasulfides) such as hydrogen thioperoxide, have the formula R1OSR2 (equivalently R2SOR1).
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Affinity describes how well a drug can bind to a receptor. Faster or stronger binding is represented by a higher affinity, or equivalently a lower dissociation constant. The EC50 should not be confused with the affinity constant, Kd. While the former reflects the drug concentration needed for a level of tissue response, the latter reflects the drug concentration needed for an amount of receptor binding.
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In the kinetic proofreading schema, a time delay (equivalently, an irreversible intermediate stage) is introduced during the formation of the correct or incorrect complexes. This time delay reduces the production rates of both complexes but enhances the fidelity beyond the equilibrium limit. The irreversibility of the scheme requires an energy source. The time delay in kinetic proofreading is analogous to the spatial difference in conformational proofreading.
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The density fluctuations associated with sound waves can be experimentally observed by Brillouin scattering. On supercooling a liquid towards the glass transition, the crossover from liquid-like to solid-like response moves from GHz to MHz, kHz, Hz, ...; equivalently, the characteristic time of structural relaxation increases from ns to μs, ms, s, ... This is the microscopic explanation for the above-mentioned viscoelastic behaviour of glass-forming liquids.
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Dual numbers find applications in physics, where they constitute one of the simplest non-trivial examples of a superspace. Equivalently, they are supernumbers with just one generator; supernumbers generalize the concept to distinct generators , each anti- commuting, possibly taking to infinity. Superspace generalizes supernumbers slightly, by allowing multiple commuting dimensions. The motivation for introducing dual numbers into physics follows from the Pauli exclusion principle for fermions.
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Under Rayleigh fading and the power-law path-loss function, outage (or equivalently, coverage) probability expressions were derived by treating the interference term as a shot noise and using Laplace transforms models,J.-P. Linnartz. Exact analysis of the outage probability in multiple-user mobile radio. Communications, IEEE Transactions on, 40(1):20–23, 1992. which was later extended to a general path-loss function,F.
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Probabilistic computing engines, e.g. use of probabilistic graphical model such as Bayesian network. Such computational techniques are referred to as randomization, yielding probabilistic algorithms. When interpreted as a physical phenomenon through classical statistical thermodynamics, such techniques lead to energy savings that are proportional to the probability p with which each primitive computational step is guaranteed to be correct (or equivalently to the probability of error, (1–p).
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The index contains all the companies that are in the Swiss Performance Index (SPI), but not among the 20 blue chips that are in the Swiss Market Index (SMI), or equivalently the SPI 20. While the SPI Extra overlaps with the SPI Mid+Small, it does not fully coincide because the SMI criteria are not only based on market capitalisation, but also on liquidity.
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Consequently, the Duron inherited the Thunderbird's reduction in sensitivity to L2 cache size, allowing AMD to make their L2 cache higher latency and lower bandwidth to lessen processor complexity and allow better manufacturing yields without incurring a significant performance loss. The net result was that the budget Duron "Spitfire" CPU was roughly only 10% slower than an equivalently clocked (and significantly more expensive) Athlon "Thunderbird".
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In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding orbit. Heuristically, Krener's theorem prohibits attainable sets from being hairy.
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A cactus graph In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cactus) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle.
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To specify that all customers insert the same card — however unrealistic that situation seems — we can write: :A card is inserted by every customer. or, equivalently: :There is a card that every customer inserts. To state that every card is inserted by a customer we write: :Every card is inserted by a customer. or, somewhat indirectly: :For every card there is a customer who inserts it.
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Circular knitting is usually carried out on a single circular needle. In such cases, the knitter can resort to a variety of alternative techniques, such as double-pointed needles, knitting on two circular needles, a Möbius strip-like "magic needle" approach (commonly known as "Magic Loop"), or careful use of slip-stitch knitting or equivalently double knitting to knit the back and front of the tube.
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In a topological space, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. This is not to be confused with a closed manifold.
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Codimension 0 immersions are equivalently relative dimension 0 submersions, and are better thought of as submersions. A codimension 0 immersion of a closed manifold is precisely a covering map, i.e., a fiber bundle with 0-dimensional (discrete) fiber. By Ehresmann's theorem and Phillips' theorem on submersions, a proper submersion of manifolds is a fiber bundle, hence codimension/relative dimension 0 immersions/submersions behave like submersions.
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A Kähler manifold is called Kähler–Einstein if it has constant Ricci curvature. Equivalently, the Ricci curvature tensor is equal to a constant λ times the metric tensor, Ric = λg. The reference to Einstein comes from general relativity, which asserts in the absence of mass that spacetime is a 4-dimensional Lorentzian manifold with zero Ricci curvature. See the article on Einstein manifolds for more details.
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A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. In graph theory, a line perfect graph is a graph whose line graph is a perfect graph. Equivalently, these are the graphs in which every odd-length simple cycle is a triangle.
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Intense localized heat can rapidly expand a boulder. The thermal heat from wildfire can cause significant weathering of rocks and boulders, heat can rapidly expand a boulder and thermal shock can occur. The differential expansion of a thermal gradient can be understood in terms of stress or of strain, equivalently. At some point, this stress can exceed the strength of the material, causing a crack to form.
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In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class w_2(M) vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H^2(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952.
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In mathematics, an algebraic function field (often abbreviated as function field) of n variables over the field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K = k(x1,...,xn) of rational functions in n variables over k.
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Networks of various sorts play an integral part in many theories of semantic memory. Generally speaking, a network is composed of a set of nodes connected by links. The nodes may represent concepts, words, perceptual features, or nothing at all. The links may be weighted such that some are stronger than others or, equivalently, have a length such that some links take longer to traverse than others.
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Pullum, Geoffrey K. Language Log: An ivory-billed relative clause, Language Log. 1 December 2005. Equivalently, the two cases would be applied where the statements are logically: :(1) "which", non-restrictive: (The building company erects very fine houses) AND (The building company will make a large profit). :(2) "that", restrictive: (The building company erects very fine houses) IMPLIES (The building company will make a large profit).
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In this case there is no primitive solution. To find non-primitive solutions where , note that the existence of such a solution implies that divides (and equivalently, that if is square-free, then all solutions are primitive). Thus the above algorithm can be used to search for a primitive solution to . If such a solution is found, then will be a solution to the original equation.
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High-density liquids such as diiodomethane can be used for this purpose, but these liquids are all highly toxic and therefore are usually avoided. A more practical method is to compare the expected size and weight of a suspect diamond to its measured parameters: for example, a cubic zirconia (SG 5.6–6) will be 1.7 times the expected weight of an equivalently sized diamond.
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In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far. The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).
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It has been represented by the Greek letter "" since the mid-18th century, and is spelled out as "'pi". It is also referred to as Archimedes' constant. Being an irrational number, cannot be expressed as a common fraction, although fractions such as 22/7 are commonly used to approximate it. Equivalently, its decimal representation never ends and never settles into a permanently repeating pattern.
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All data should be on the same scale for CA to be applicable, keeping in mind that the method treats rows and columns equivalently. It is traditionally applied to contingency tables — CA decomposes the chi-squared statistic associated with this table into orthogonal factors. Because CA is a descriptive technique, it can be applied to tables whether or not the \chi^2 statistic is appropriate.
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Equivalently, every edge uv of such a graph belongs to a unique triangle uvw. Locally linear graphs have also been called locally matched graphs. Examples of locally linear graphs include the triangular cactus graphs, the line graphs of 3-regular triangle-free graphs, and the Cartesian products of smaller locally linear graphs. Certain Kneser graphs, and certain strongly regular graphs, are also locally linear.
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Over a field, every nonzero polynomial is associated to a unique monic polynomial. Given two polynomials, and , one says that divides , is a divisor of , or is a multiple of , if there is a polynomial such that . A polynomial is irreducible if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree.
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That purely mathematical abstraction is sometimes referred to as impulse sampling. Most sampled signals are not simply stored and reconstructed. But the fidelity of a theoretical reconstruction is a customary measure of the effectiveness of sampling. That fidelity is reduced when s(t) contains frequency components whose periodicity is smaller than two samples; or equivalently the ratio of cycles to samples exceeds ½ (see Aliasing).
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The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two. The rank of a matrix is also equal to the dimension of the column space.
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The number of decibels is ten times the number of bels (equivalently, a decibel is one-tenth of a bel). P and P0 must measure the same type of quantity, and have the same units before calculating the ratio. If in the above equation, then LP = 0. If P is greater than P0 then LP is positive; if P is less than P0 then LP is negative.
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Schwarz's theorem states that if the second derivatives are continuous the expression for the cross partial derivative is unaffected by which variable the partial derivative is taken with respect to first and which is taken second. That is, :\frac = \frac or equivalently f_=f_. Own and cross partial derivatives appear in the Hessian matrix which is used in the second order conditions in optimization problems.
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By the three > subgroups lemma (or equivalently, by the Hall-Witt identity), it follows > that [G, Z2] = G, G], Z2] = [G, G, Z2] = {1}. Therefore, Z2 ⊆ Z1 = Z(G), and > the center of the quotient group G ⁄ Z(G) is the trivial group. As a consequence, all higher centers (that is, higher terms in the upper central series) of a perfect group equal the center.
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Although one methyl group is only attached to the thorium atom (Th–C distance 257.1 pm) and the other six connect the lithium and thorium atoms (Th–C distances 265.5–276.5 pm) they behave equivalently in solution. Tetramethylthorium, , is not known, but its adducts are stabilised by phosphine ligands. Some coordination complexes with carboxylates and acetylacetonates are also known, although these are not organothorium compounds.
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For practical purposes there is no distinction between expressions and statements. Line breaks are significant and taken as the end of a statement; a semicolon may be equivalently used. Unlike Python, indentation is not significant. One of the differences from Python and Perl is that Ruby keeps all of its instance variables completely private to the class and only exposes them through accessor methods (`attr_writer`, `attr_reader`, etc.).
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Twelfth grade, senior year, or grade 12 is the final year of secondary school in most of North America. In other regions it is also equivalently referred to as class 12 or Year 13. In most countries, students then often graduate at ages between 17 and 18. Some countries have a thirteenth grade, while other countries do not have a 12th grade/year at all.
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Thus, the problem may be solved in polynomial time whenever is a fixed constant. However, when does not have a fixed value, but instead may vary as part of the input to the problem, the time is exponential.E.g., see . The simplest nontrivial case of the clique-finding problem is finding a triangle in a graph, or equivalently determining whether the graph is triangle-free.
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Cross vanna measures the rate of change of vega in one underlying due to a change in the level of another underlying. Equivalently, it measures the rate of change of delta in the second underlying due to a change in the volatility of the first underlying. Cross volga measures the rate of change of vega in one underlying to a change in the volatility of another underlying.
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In fact, Beer's law, i.e. the concentration dependence of absorbance, can be derived directly from the Lorentz-Lorenz relation (or, equivalently, the Clausius-Mossotti relation). Correspondingly, it can be demonstrated that there is a twin law according to which the change of the refractive index is approximately linear to the molar concentration for diluted solutions. This twin law can also be derived from the Lorentz-Lorenz relation.
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However, differences that arise directly from gene activity have also been observed. Language areas of the brain: It has also been demonstrated that brain processing responds to the external environment. Learning, both of ideas and behaviors, appears to be coded in brain processes. It also appears that in several simplified cases this coding operates differently, but in some ways equivalently, in the brains of men and women.
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If there is reason to believe that none of the x_js is caused by y, then estimates of the coefficients a_j are obtained. If the null hypothesis that a_j=0 is rejected, then the alternative hypothesis that a_{j} e 0 and equivalently that x_j causes y cannot be rejected. On the other hand, if the null hypothesis that a_j=0 cannot be rejected, then equivalently the hypothesis of no causal effect of x_j on y cannot be rejected. Here the notion of causality is one of contributory causality: If the true value a_j e 0, then a change in x_j will result in a change in y unless some other causative variable(s), either included in the regression or implicit in the error term, change in such a way as to exactly offset its effect; thus a change in x_j is not sufficient to change y.
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Of the eight solar planets, all but Venus and Uranus have prograde rotation—that is, they rotate more than once per year in the same direction as they orbit the Sun, so the Sun rises in the east. Venus and Uranus, however, have retrograde rotation. For prograde rotation, the formula relating the lengths of the sidereal and solar days is: or, equivalently: On the other hand, the formula in the case of retrograde rotation is: or, equivalently: All the solar planets more distant from the Sun than Earth are similar to Earth in that, since they experience many rotations per revolution around the Sun, there is only a small difference between the length of the sidereal day and that of the solar day – the ratio of the former to the latter never being less than Earth's ratio of 0.997. But the situation is quite different for Mercury and Venus.
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Victorian Literature Group. Page 603. Black dogs were also at times called "stampes" or "stampees," as they were typically the coins of other colonial powers—French coins worth 2 sous or, equivalently, 24 diniers—stamped to make them British currency. A dog and a stampee were not necessarily of equal value. For example, the Spanish dollar was subdivided into bits, each worth 9 pence, 6 black dogs or 4 stampees.
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The ANOVA model could be equivalently written as each group parameter \tau_i being an offset from some overall reference. Typically this reference point is taken to be one of the groups under consideration. This makes sense in the context of comparing multiple treatment groups to a control group and the control group is considered the "reference". In this example, group 1 was chosen to be the reference group.
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Working in the opposite direction again, the second expression asserts that at least one of "not A" and "not B" must be true, or equivalently that at least one of A and B must be false. Since at least one of them must be false, then their conjunction would likewise be false. Negating said conjunction thus results in a true expression, and this expression is identical to the first claim.
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Paramagnetism is a form of magnetism which occurs only in the presence of an externally applied magnetic field. Paramagnetic materials are attracted to magnetic fields, hence have a relative magnetic permeability greater than one (or, equivalently, a positive magnetic susceptibility). The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect.
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A square pyramid of cannonballs in a square frame In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1.
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"Sunset", also known as the "Retreat Call", is a bugle call played in United Kingdom and British Commonwealth countries to signal the end of the official military day.Royal Marines Band - "beating retreat" royalmarinesbands.co.uk, Retrieved 18 July 2010 In common with all bugle calls, it consists only of notes from a single overtone series. This allows it to be playable on a bugle or equivalently on a trumpet without moving the valves.
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In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence :1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\, where H and K are cyclic. Equivalently, a metacyclic group is a group G having a cyclic normal subgroup N, such that the quotient G/N is also cyclic.
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It implies that there will be considerable variability in the density along different rays. In particular, the density is highly sensitive to the discriminant of the polynomial, b2 − 16c. The primes of the form 4x2 − 2x + 41 with x = 0, 1, 2, ... have been highlighted in purple. The prominent parallel line in the lower half of the figure corresponds to 4x2 \+ 2x + 41 or, equivalently, to negative values of x.
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A suffix code is a set of words none of which is a suffix of any other; equivalently, a set of words which are the reverse of a prefix code. As with a prefix code, the representation of a string as a concatenation of such words is unique. A bifix code is a set of words which is both a prefix and a suffix code.Berstel et al (2010) p.
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If E eq 0 is a module that satisfies the ACC and DCC on submodules (that is, it is both Noetherian and Artinian or – equivalently – of finite length), then E is a direct sum of indecomposable modules. Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism. In general, the theorem fails if one only assumes that the module is Noetherian or Artinian.
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In addition, one then also has : \varphi(u + \omega \pm i\omega') = \varphi(u) so that the function can equivalently be said to have the two complex periods . Since φ(0) = 0, the function will also be zero in all points where m and n are integers. These zeros thus form a regular lattice in the complex plane as the poles also will. For the two other functions Abel found and .
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This is mainly based on the observed phenomena that in smooth gravity- or pressure-driven geophysical and industrial mass flows, the momentum transfer in the direction normal to the main flow direction is negligible. Or in other words, the aspect ratio (typical flow depth divided by typical flow extent) is very small, or to say equivalently, the flow depth is much smaller than a typical wave length (long wave approximation).
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US real average hourly wage (red, in constant 2017 dollars, for private sector production/nonsupervisory workers) trended up to a peak in 1973, trended down to a low in 1995, then trended up through 2018. Nominal hourly wage shown in grey. US net productivity compared to real wages. Real wages are wages adjusted for inflation, or, equivalently, wages in terms of the amount of goods and services that can be bought.
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As the Aṣṭasāhasrikā Prajñāpāramitā Sūtra states, these terms are generally used equivalently: "As the suchness (tathatā) of dharmas is immovable (acalitā), and the suchness (tathatā) of dharmas is the Tathāgata.".Orsborn, Matthew Bryan. “Chiasmus in the Early Prajñāpāramitā: Literary Parallelism Connecting Criticism & Hermeneutics in an Early Mahāyāna Sūtra”, University of Hong Kong , 2012, page 233. The Tathāgata is said in the Aṣṭasāhasrikā Prajñāpāramitā Sūtra to "neither come nor go".
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Specifically, the point of tangency between any isoquant and an isocost line gives the lowest-cost combination of inputs that can produce the level of output associated with that isoquant. Equivalently, it gives the maximum level of output that can be produced for a given total cost of inputs. A line joining tangency points of isoquants and isocosts (with input prices held constant) is called the expansion path.Salvatore, Dominick (1989).
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Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element x, the operations of multiplying by powers x^n all commute. Jordan algebras were first introduced by to formalize the notion of an algebra of observables in quantum mechanics. They were originally called "r-number systems", but were renamed "Jordan algebras" by , who began the systematic study of general Jordan algebras.
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English units of length The English palm, handbreadth, or handsbreadth is three inches (7.62cm) or, equivalently, four digits. The measurement was, however, not always well distinguished from the hand or handful, which became equal to four inches by a 1541 statute of Henry VIII. The palm was excluded from the British Weights and Measures Act of 1824 that established the imperial system and is not a standard US customary unit.
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Late selection models argue that information is selected after processing for meaning, as opposed to during the earlier stages of processing. According to these models, all information is attended to, whether intentionally or unintentionally. Information inputs are processed equivalently, until semantic encoding and analysis can be performed. The filter merely acts as an information attenuator; it intensifies the pertinent information and attenuates the intensity of the stimuli deemed to be unimportant.
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The simulants imitate the look and color of the real stone but possess neither their chemical nor physical characteristics. In general, all are less hard than diamond. Moissanite actually has a higher refractive index than diamond, and when presented beside an equivalently sized and cut diamond will show more "fire". Cultured, synthetic, or "lab-created" gemstones are not imitations: The bulk mineral and trace coloring elements are the same in both.
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Equivalently, every -critical graph (a graph that requires colors but for which every proper subgraph requires fewer colors) is -constructible.A proof can also be found in . Alternatively, every graph that requires colors may be formed by combining the Hajós construction, the operation of identifying any two nonadjacent vertices, and the operations of adding a vertex or edge to the given graph, starting from the complete graph ., p. 184.
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As cellular decompositions of the projective plane, they have Euler characteristic 1, while spherical polyhedra have Euler characteristic 2. The qualifier "globally" is to contrast with locally projective polyhedra, which are defined in the theory of abstract polyhedra. Non-overlapping projective polyhedra (density 1) correspond to spherical polyhedra (equivalently, convex polyhedra) with central symmetry. This is elaborated and extended below in relation with spherical polyhedra and relation with traditional polyhedra.
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Letting S denote an arbitrary subset of a topological space, write kS for the closure of S, and cS for the complement of S. The following three identities imply that no more than 14 distinct sets are obtainable: (1) kkS = kS. (The closure operation is idempotent.) (2) ccS = S. (The complement operation is an involution.) (3) kckckckcS = kckcS. (Or equivalently kckckckS = kckckckccS = kckS. Using identity (2).) The first two are trivial.
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The discriminant of the conic section's quadratic equation (or equivalently the determinant of the 2×2 matrix) and the quantity (the trace of the 2×2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes,Pettofrezzo, Anthony, Matrices and Transformations, Dover Publ., 1966, p. 110. as is the determinant of the 3×3 matrix above. The constant term and the sum are invariant under rotation only.
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The Local Education Authority is Hertfordshire County Council.Herts CC - Schools in the Hertsmere area There are still some linked pairs of infant schools and junior schools, with the infant school covering Reception and Key Stage 1 (Years 1 and 2) and the junior school covering Key Stage 2 (Years 3 to 6). However most have been amalgamated in a single Junior Mixed Infant (JMI) school or (equivalently) primary school.
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Bent's rule can be generalized to d-block elements as well. The hybridisation of a metal center is arranged so that orbitals with more s character are directed towards ligands that form bonds with more covalent character. Equivalently, orbitals with more d character are directed towards groups that form bonds of greater ionic character. The validity of Bent's rule for 75 bond types between the main group elements was examined recently.
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Silsesquioxanes are known in molecular form with 6, 8, 10, and 12 Si vertices, as well as polymers. The cages are sometimes labeled T6 T8, T10, and T12, respectively (T = tetrahedral vertex). The T8 cages, the most widely studied members, have the formula [RSiO3/2]8, or equivalently R8Si8O12. In all cases each Si center is bonded to three oxo groups, which in turn connect to other Si centers.
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To construct one, take a complex vector space minus the origin and consider the action of the group of integers on this space by multiplication by exp(n). The quotient is a complex manifold whose first Betti number is one, so by the Hodge theory, it cannot be Kähler. A Calabi–Yau manifold can be defined as a compact Ricci-flat Kähler manifold or equivalently one whose first Chern class vanishes.
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In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name nonagon is a prefix hybrid formation, from Latin (nonus, "ninth" + gonon), used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century. The name enneagon comes from Greek enneagonon (εννεα, "nine" + γωνον (from γωνία = "corner")), and is arguably more correct, though less common than "nonagon".
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A linear operator is closed if for every sequence in converging to in such that as one has and . Equivalently, is closed if its graph is closed in the direct sum . Given a linear operator , not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of , and we say that is closable. Denote the closure of by .
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In abstract algebra, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. One should notice that several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple.
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Any secondary Hopf surface has a finite unramified cover that is a primary Hopf surface. Equivalently, its fundamental group has a subgroup of finite index in its center that is isomorphic to the integers. classified them by finding the finite groups acting without fixed points on primary Hopf surfaces. Many examples of secondary Hopf surfaces can be constructed with underlying space a product of a spherical space forms and a circle.
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Open interest (futures) is the number of "open" contracts or open interest of derivatives in the futures market. Open interest in a derivative is the sum of all contracts that have not expired, been exercised or physically delivered. Moreover, the open interest is the number of long positions or, equivalently, the number of short positions. Open interest is used as a technical indicator as it is a measure of market activity.
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A dominating set may or may not be an independent set. For example, figures (a) and (b) above show independent dominating sets, while figure (c) illustrates a dominating set that is not an independent set. The independent domination number i(G) of a graph G is the size of the smallest dominating set that is an independent set. Equivalently, it is the size of the smallest maximal independent set.
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But N has been defined by a finite number of words in this paragraph. It should therefore be in the set E. That is a contradiction. As with König's paradox, this paradox cannot be formalized in axiomatic set theory because it requires the ability to tell whether a description applies to a particular set (or, equivalently, to tell whether a formula is actually the definition of a single set).
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However, the prediction interval only bounds a single future sample, whereas a tolerance interval bounds the entire population (equivalently, an arbitrary sequence of future samples). In other words, a prediction interval covers a specified proportion of a population on average, whereas a tolerance interval covers it with a certain confidence level, making the tolerance interval more appropriate if a single interval is intended to bound multiple future samples.
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Equivalently, expressed as strings of binary digits, the codewords are: :00000,\quad 00111,\quad 01110,\quad 01001, :11100,\quad 11011,\quad 10010,\quad 10101. This, as every polynomial code, is indeed a linear code, i.e., linear combinations of code words are again code words. In a case like this where the field is GF(2), linear combinations are found by taking the XOR of the codewords expressed in binary form (e.g.
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Apothem of a hexagon The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems.
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Since the complex numbers contain the integers, their characteristic is 0. A Z/nZ-algebra is equivalently a ring whose characteristic divides n. This is because for every ring R there is a ring homomorphism Z → R, and this map factors through Z/nZ if and only if the characteristic of R divides n. In this case for any r in the ring, then adding r to itself n times gives .
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A computer model is the algorithms and equations used to capture the behavior of the system being modeled. By contrast, computer simulation is the actual running of the program that contains these equations or algorithms. Simulation, therefore, is the process of running a model. Thus one would not "build a simulation"; instead, one would "build a model", and then either "run the model" or equivalently "run a simulation".
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Given two closed model categories C and D, a Quillen adjunction is a pair :(F, G): C \leftrightarrows D of adjoint functors with F left adjoint to G such that F preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that G preserves fibrations and trivial fibrations. In such an adjunction F is called the left Quillen functor and G is called the right Quillen functor.
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The roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution. But the Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just for some of them) if n \ge 5\.
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Two affine varieties are birationally equivalent if there are two rational functions between them which are inverse one to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic. An affine variety is a rational variety if it is birationally equivalent to an affine space. This means that the variety admits a rational parameterization, that is a parametrization with rational functions.
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Glacier Peak is the dominant feature in the southern portion of the North Cascades. The southern boundary of the North Cascades is less definite. For the purposes of this article, it will be taken as U.S. Highway 2, running over Stevens Pass, or equivalently, the Skykomish River, Nason Creek, and the lower Wenatchee River. This roughly follows Beckey's geologic division in Cascade Alpine Guide and the definition used by Peakbagger.com.
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.
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A series with the additional property that Ai ≠ Ai+1 for all i is called a series without repetition; equivalently, each Ai is a proper subgroup of Ai+1. The length of a series is the number of strict inclusions Ai < Ai+1. If the series has no repetition then the length is n. For a subnormal series, the length is the number of non-trivial factor groups.
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He used this approach not only for 3-coloring but as part of a more general graph coloring algorithm, and similar approaches to graph coloring have been refined by other authors since.; . Other more complex problems can also be modeled as finding a clique or independent set of a specific type. This motivates the algorithmic problem of listing all maximal independent sets (or equivalently, all maximal cliques) efficiently.
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Q-analysis is a mathematical framework to describe and analyze set systems, or equivalently simplicial complexes. This idea was first introduced by Ronald Atkin in the early 1970s. Atkin was a British mathematician teaching at the University of Essex. Crediting the inspiration of his idea to Clifford Dowker’s paper (Homology Groups of Relations, Annals of Mathematics, 1952), he became interested in the algebra of relations in social structures.
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These considerations result in a count: 4 N \log_2 N - 6N + 8 real additions and multiplications, for N>1 a power of two. This count assumes that, for odd powers of 2, the leftover factor of 2 (after all the split-radix steps, which divide N by 4) is handled directly by the DFT definition (4 real additions and multiplications), or equivalently by a radix-2 Cooley–Tukey FFT step.
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A function between topological spaces is called continuous if for every x in X and every neighbourhood N of f(x) there is a neighbourhood M of x such that . This relates easily to the usual definition in analysis. Equivalently, f is continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function.
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If a_0 and a_n may be zero, the polynomials and may have a degree smaller than . In this case, above formulas and definition remain valid, if the discriminants are computed as if all polynomials would have the degree . This means that the discriminants must be computed with a_0 and a_n indeterminate, the substitution for them of their actual values being done after this computation. Equivalently, the formulas of must be used.
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In differential geometry the defining characteristic of a tangent space is that it approximates the smooth manifold to first order near the point of tangency. Equivalently, if we zoom in more and more at the point of tangency the manifold appears to become more and more straight, asymptotically tending to approach the tangent space. This turns out to be the correct point of view in geometric measure theory.
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In a maximal planar graph, or more generally in every polyhedral graph, the peripheral cycles are exactly the faces of a planar embedding of the graph, so a polyhedral graph is strangulated if and only if all the faces are triangles, or equivalently it is maximal planar. Every chordal graph is strangulated, because the only induced cycles in chordal graphs are triangles, so there are no longer cycles to delete.
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Alternate definitions include "the entropy of all systems and of all states of a system is smallest at absolute zero," or equivalently "it is impossible to reach the absolute zero of temperature by any finite number of processes". Absolute zero, at which all activity would stop if it were possible to achieve, is −273.15 °C (degrees Celsius), or −459.67 °F (degrees Fahrenheit), or 0 K (kelvin), or 0° R (degrees Rankine).
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Since the area of the diagonals of the square is 0, the probability that the dart will land exactly on a diagonal is 0. That is, the dart will almost never land on a diagonal (equivalently, it will almost surely not land on a diagonal), even though the set of points on the diagonals is not empty, and a point on a diagonal is no less possible than any other point.
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Similar to continuous functions between topological spaces, which preserve topological properties, are the uniformly continuous functions between uniform spaces, which preserve uniform properties. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a uniform isomorphism. A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers.
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An inner product space is a real or complex linear space, endowed with a bilinear or respectively sesquilinear form, satisfying some conditions and called an inner product. Every inner product space is also a normed space. A normed space underlies an inner product space if and only if it satisfies the parallelogram law, or equivalently, if its unit ball is an ellipsoid. Angles between vectors are defined in inner product spaces.
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In both cases, we assume that the charge distributions are localized, so that the potentials can be chosen to go to zero at infinity. Then, Green's reciprocity theorem states that, for integrals over all space: :\int \rho_1 \phi_2 dV = \int \rho_2 \phi_1 dV. This theorem is easily proven from Green's second identity. Equivalently, it is the statement that \int \phi_2 ( abla^2 \phi_1 ) dV = \int \phi_1 ( abla^2 \phi_2 ) dV, i.e.
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The Local Education Authority is Hertfordshire County Council.Herts CC - Schools in the Three Rivers area There are still some linked pairs of infant schools and junior schools, with the infant school covering Reception and Key Stage 1 (Years 1 and 2) and the junior school covering Key Stage 2 (Years 3 to 6). However most have been amalgamated in a single Junior Mixed Infant (JMI) school or (equivalently) primary school.
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In the science of fluid flow, Stokes' paradox is the phenomenon that there can be no creeping flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial steady-state solution for the Stokes equations around an infinitely long cylinder. This is opposed to the 3-dimensional case, where Stokes' method provides a solution to the problem of flow around a sphere.
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A subset U of a metric space is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with y also belongs to U. Equivalently, U is open if every point in U has a neighborhood contained in U. This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.
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Equivalently, P(τ = σ < ∞) = 0 for every predictable time σ. Examples of totally inaccessible stopping times include the jump times of Poisson processes. Every stopping time τ can be uniquely decomposed into an accessible and totally inaccessible time. That is, there exists a unique accessible stopping time σ and totally inaccessible time υ such that τ = σ whenever σ < ∞, τ = υ whenever υ < ∞, and τ = ∞ whenever σ = υ = ∞.
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To illustrate the significance of these formulae, the annihilation of a particle with its antiparticle in free space must result in the creation of at least two photons for the following reason. In the center of momentum frame, the colliding antiparticles have no net momentum, whereas a single photon always has momentum (since, as we have seen, it is determined by the photon's frequency or wavelength, which cannot be zero). Hence, conservation of momentum (or equivalently, translational invariance) requires that at least two photons are created, with zero net momentum. (However, it is possible if the system interacts with another particle or field for the annihilation to produce one photon, as when a positron annihilates with a bound atomic electron, it is possible for only one photon to be emitted, as the nuclear Coulomb field breaks translational symmetry.) The energy of the two photons, or, equivalently, their frequency, may be determined from conservation of four-momentum.
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If the null hypothesis that a_j=0 is rejected, then the alternative hypothesis that a_{j} e 0 and equivalently that x_j causes y cannot be rejected. On the other hand, if the null hypothesis that a_j=0 cannot be rejected, then equivalently the hypothesis of no causal effect of x_j on y cannot be rejected. Here the notion of causality is one of contributory causality as discussed above: If the true value a_j e 0, then a change in x_j will result in a change in y unless some other causative variable(s), either included in the regression or implicit in the error term, change in such a way as to exactly offset its effect; thus a change in x_j is not sufficient to change y. Likewise, a change in x_j is not necessary to change y, because a change in y could be caused by something implicit in the error term (or by some other causative explanatory variable included in the model).
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1-junctions behave opposite of 0-junctions. 1-junctions behave such that all flow values are equal across the bonds, but the sum of the effort values in equals the sum the effort values out, or equivalently, all efforts sum to zero. In an electrical circuit, the 1 junction represents a series connection among components. In a mechanical circuit, the 1-junction represents a velocity shared by all components connected to it.
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In geometry, the Beckman-Quarles theorem, named after F. S. Beckman and D. A. Quarles, Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all distances. Equivalently, every automorphism of the unit distance graph of the plane must be an isometry of the plane. Beckman and Quarles published this result in 1953;. it was later rediscovered by other authors...
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In theoretical physics, a logarithmic conformal field theory is a conformal field theory in which the correlators of the basic fields are allowed to be logarithmic at short distance, instead of being powers of the fields' distance. Equivalently, the dilation operator is not diagonalizable. Just like conformal field theory in general, logarithmic conformal field theory has been particularly well-studied in two dimensions. Examples of logarithmic conformal field theories include critical percolation.
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Moreover, the probability associated with a unit vector is a function of the density operator and the unit vector, and not of additional information like a choice of basis for that vector to be embedded in. Gleason's theorem establishes the converse: all assignments of probabilities to unit vectors (or, equivalently, to the operators that project onto them) that satisfy these conditions take the form of applying the Born rule to some density operator.
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This process results in cooling. Unlike entropy transfer between two "regular" objects which preserves the entropy of the system, entropy transfer to a heat bath is normally regarded as non-preserving. This is because the bath is normally not considered as a part of the relevant system, due to its size. Therefore, when transferring entropy to a heat bath, one can essentially lower the entropy of their system, or equivalently, cool it.
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Areas like volcanic islands are experiencing relative sea level rise as a result of isostatic cooling of the rock which causes the land to sink. On other planets that lack a liquid ocean, planetologists can calculate a "mean altitude" by averaging the heights of all points on the surface. This altitude, sometimes referred to as a "sea level" or zero-level elevation, serves equivalently as a reference for the height of planetary features.
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Cheeger and Gromoll's soul conjecture states: :Suppose is complete, connected and non-compact with sectional curvature , and there exists a point in where the sectional curvature (in all sectional directions) is strictly positive. Then the soul of is a point; equivalently is diffeomorphic to . Grigori Perelman proved this statement by establishing that in the general case , Sharafutdinov's retraction is a submersion. Cao and Shaw later provided a different proof that avoids Perelman's flat strip theorem.
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If two equivalently charged fields come in contact and are confined to a particular region, excessive force is needed to overcome the repulsive forces needed to elicit dislocation movement past one another. If two oppositely charged fields come into contact with one another they will merge with one another to form a jog. A jog can be modelled as a potential well that traps dislocations. Thus, excessive force is needed to force the dislocations apart.
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The set of complex numbers is defined as the quotient ring . This extension field contains two square roots of , namely (the cosets of) and , respectively. (The cosets of) and form a basis of as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs of real numbers.
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The star chromatic number of Dyck graph is 4, while the chromatic number is 2. In graph-theoretic mathematics, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by .
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One reasonable choice as the best hyperplane is the one that represents the largest separation, or margin, between the two classes. So we choose the hyperplane so that the distance from it to the nearest data point on each side is maximized. If such a hyperplane exists, it is known as the maximum-margin hyperplane and the linear classifier it defines is known as a maximum-margin classifier; or equivalently, the perceptron of optimal stability.
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An R-algebra S is called finitely generated (as an algebra) if there are finitely many elements s1, ..., sn such that any element of s is expressible as a polynomial in the si. Equivalently, S is isomorphic to :R[T1, ..., Tn] / I. A much stronger condition is that S is finitely generated as an R-module, which means that any s can be expressed as a R-linear combination of some finite set s1, ..., sn.
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The procedures of Bonferroni and Holm control the FWER under any dependence structure of the p-values (or equivalently the individual test statistics). Essentially, this is achieved by accommodating a `worst-case' dependence structure (which is close to independence for most practical purposes). But such an approach is conservative if dependence is actually positive. To give an extreme example, under perfect positive dependence, there is effectively only one test and thus, the FWER is uninflated.
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Since this is the only axiom of his 1925 axiom system that S + Regularity does not have, S + Regularity implies all the axioms of his 1925 system. These results imply: If S is consistent, then von Neumann's 1925 axiom system is consistent. Proof: If S is consistent, then S + Regularity is consistent (result 1). Using proof by contradiction, assume that the 1925 axiom system is inconsistent, or equivalently: the 1925 axiom system implies a contradiction.
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Correspondence analysis (CA) was developed by Jean-Paul Benzécri and is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently. It is traditionally applied to contingency tables. CA decomposes the chi-squared statistic associated to this table into orthogonal factors. Because CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not.
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In the same lattice, 4 is join-prime: whenever lcm(y,z) is divisible by 4, at least one of y and z must itself be divisible by 4. In any lattice, a join-prime element must be join-irreducible. Equivalently, an element that is not join-irreducible is not join-prime. For, if an element x is not join-irreducible, there exist smaller y and z such that x = y ∨ z.
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A Kähler manifold is a Riemannian manifold X of even dimension 2n whose holonomy group is contained in the unitary group U(n).Kobayashi & Nomizu (1996), v. 2, p. 149. Equivalently, there is a complex structure J on the tangent space of X at each point (that is, a real linear map from TX to itself with ) such that J preserves the metric g (meaning that ) and J is preserved by parallel transport.
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By the rank–nullity theorem, this is equivalent to the condition that the left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the definition of nondegeneracy: : Definition: B is nondegenerate if for all w implies . Given any linear map one can obtain a bilinear form B on V via : B(v, w) = A(v)(w). This form will be nondegenerate if and only if A is an isomorphism.
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The second equation follows since θ is measurable with respect to the conditional distribution P(x\mid\theta). An estimator is said to be unbiased if its bias is equal to zero for all values of parameter θ, or equivalently, if the expected value of the estimator matches that of the parameter. In a simulation experiment concerning the properties of an estimator, the bias of the estimator may be assessed using the mean signed difference.
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Since air is composed of approximately 21% oxygen and 79% nitrogen, the narcotic gases make up 100% of the mix, or equivalently the fraction of the total gases which are narcotic is 1.0. Oxygen is assumed equivalent in narcotic effect to nitrogen for this purpose. In contrast, the oxygen and nitrogen component in a trimix containing, for example, 40% helium accounts for only 60% of the mix, i.e. a fraction of 0.6.
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Fermions are classified according to how they interact (or equivalently, by what charges they carry). There are six quarks (up, down, charm, strange, top, bottom), and six leptons (electron, electron neutrino, muon, muon neutrino, tau, tau neutrino). Each class is divided into pairs of particles that exhibit a similar physical behavior called a generation (see the table). The defining property of quarks is that they carry color charge, and hence interact via the strong interaction.
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Since the expected value of an unbiased estimator is equal to the parameter value, E[T]=\theta. Therefore, MSE(T)=Var(T) as the (E[T]-\theta)^2 term drops out from being equal to 0. If an unbiased estimator of a parameter θ attains e(T) = 1 for all values of the parameter, then the estimator is called efficient. Equivalently, the estimator achieves equality in the Cramér–Rao inequality for all θ.
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Flux is analogous to the flow of a fluid through a surface since the angle of the surface to the direction of flow determines how much fluid can flow through the surface. Gauss' law states that the flux through a closed surface is proportional to the amount of charge enclosed. The SI units of flux are newton meters-squared per coulomb (N m^2 C^{-1}), or equivalently, volt-meters (V m).
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A solution to Laplace's equation defined on an annulus. The Laplace operator is the most famous example of an elliptic operator. In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions.
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Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determinant problem. Certain Hadamard matrices can almost directly be used as an error-correcting code using a Hadamard code (generalized in Reed–Muller codes), and are also used in balanced repeated replication (BRR), used by statisticians to estimate the variance of a parameter estimator.
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This spring change takes place at the end of the day of Nowruz, which is the Iranian New Year's Day and the most important festival in Iranian culture. DST likewise ends in Iran at 24:00 on 30 Shahrivar, which corresponds to either 20 or 21 September. (Equivalently, at 00:00 on 31 Shahrivar, either 21 or 22 September). Clocks move backward to 23:00 on 30 Shahrivar (20 or 21 September).
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As authorised by the Act, Elizabeth proclaimed that her title in the United Kingdom would be, equivalently in English and (or the first time) in Latin: "Elizabeth II, by the Grace of God of the United Kingdom of Great Britain and Northern Ireland and of Her other Realms and Territories Queen, Head of the Commonwealth, Defender of the Faith" and Elizabeth II, Dei Gratia Britanniarum Regnorumque Suorum Ceterorum Regina, Consortionis Populorum Princeps, Fidei Defensor.
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In grades 3, 6 and 9 national tests are administered to all students in the Swedish grundskola. The primary aim of these tests are to ensure all students are assessed equivalently. The test results should be given special consideration by teachers when setting the students final grade. In grade 3 the tests are framed around a story involving two children encountering Math and Swedish in everyday situations to make the test situation less intimidating.
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Idempotent semirings are special to semiring theory as any ring which is idempotent under addition is trivial.i.e. is a ring consisting of just one element, because rings have additive inverses, unlike semirings. One can define a partial order ≤ on an idempotent semiring by setting whenever (or, equivalently, if there exists an x such that ). It is easy to see that 0 is the least element with respect to this order: for all a.
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Quantum field theory, the framework in which all of modern particle physics is described, is usually described as a theory of quantized fields. However, although not widely appreciated, it has been known since Feynman that many quantum field theories may equivalently be described in terms of world lines. The world line formulation of quantum field theory has proved particularly fruitful for various calculations in gauge theories and in describing nonlinear effects of electromagnetic fields.
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The barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions. Barycentric coordinates for the incenter are given by :\ a : b : c where a, b, and c are the lengths of the sides of the triangle, or equivalently (using the law of sines) by :\sin(A):\sin(B):\sin(C) where A, B, and C are the angles at the three vertices.
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Carbonic acid is a chemical compound with the chemical formula (equivalently: OC(OH)2). It is also a name sometimes given to solutions of carbon dioxide in water (carbonated water), because such solutions contain small amounts of H2CO3. In physiology, carbonic acid is described as volatile acid or respiratory acid because it is the only acid excreted as a gas by the lungs.Acid-Base Physiology 2.1 – Acid-Base Balance by Kerry Brandis.
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One photon transitions are only allowed between states that differ in the inversion symmetry, i.e. g <-> u, while two photon transitions are only allowed between states that have the same inversion symmetry, i.e. g <->g and u <-> u. The relation between the number of photons - or, equivalently, order of the electronic transitions - involved in a TPA process (two) and the order of the corresponding nonlinear susceptibility (third) may be understood using the optical theorem.
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A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if no (infinite) strictly ascending sequence :a_1 < a_2 < a_3 < \cdots of elements of P exists. Equivalently,Proof: first, a strictly increasing sequence cannot stabilize, obviously. Conversely, suppose there is an ascending sequence that does not stabilize; then clearly it contains a strictly increasing (necessarily infinite) subsequence. Notice the proof does not use the full force of the axiom of choice.
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The principle of ellipsographs were known to Greek mathematicians such as Archimedes and Proklos. If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices. For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis). If this presumption is not fulfilled one has to know at least two conjugate diameters.
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This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates (, ) it can be described by the equation \, r=a+b\theta with real numbers and .
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In mathematics, a group, , is said to be complete if every automorphism of is inner, and it is centerless; that is, it has a trivial outer automorphism group and trivial center. Equivalently, a group is complete if the conjugation map, (sending an element to conjugation by ), is an isomorphism: injectivity implies that only conjugation by the identity element is the identity automorphism, meaning the group is centerless, while surjectivity implies it has no outer automorphisms.
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Theoretical isochrones for near-solar metallicity and a range of ages In stellar evolution, an isochrone is a curve on the Hertzsprung-Russell diagram, representing a population of stars of the same age. The Hertzsprung-Russell diagram plots a star's luminosity against its temperature, or equivalently, its color. Stars change their positions on the HR diagram throughout their life. Newborn stars of low or intermediate mass are born cold but extremely luminous.
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In fact, this is necessary and sufficient, because if is any nilpotent, then one of its powers will be nilpotent of order at most . In particular, if is a field then the Frobenius endomorphism is injective. The Frobenius morphism is not necessarily surjective, even when is a field. For example, let be the finite field of elements together with a single transcendental element; equivalently, is the field of rational functions with coefficients in .
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The Nielsen realisation problem asks whether any finite group of the mapping class group has a global fixed point (a point fixed by all group elements) in Teichmüller space. In more classical terms the question is: can every finite subgroup of MCG(S) be realised as a group of isometries of some complete hyperbolic metric on S (or equivalently as a group of holomorphic diffeomorphisms of some complex structure). This was solved by Steven Kerckhoff.
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For heteronuclear coupling, the magnitude of J is related to the nuclear magnetic moments of the coupling partners. 19F, with a high nuclear magnetic moment, gives rise to large coupling to protons. 103Rh, with a very small nuclear magnetic moment, gives only small couplings to 1H. To correct for the effect of the nuclear magnetic moment (or equivalently the gyromagnetic ratio γ), the "reduced coupling constant" K is often discussed, where :K = .
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An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). Note that this σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets. On the Euclidean space Rn, another σ-algebra is of importance: that of all Lebesgue measurable sets.
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It is also important to understand the relationship between odds and implied probabilities: Fractional odds of a-b (with corresponding decimal odds D) represent an implied probability of =, e.g. 6-4 corresponds to = = 0.4 (40%). An implied probability of x is represented by fractional odds of (1-x)/x, e.g. 0.2 is (1-0.2)/0.2 = 0.8/0.2 = 4/1 (4-1, 4 to 1) (equivalently, to 1), and decimal odds of D=.
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Equivalently, it is a relatively open subset of its closure. ;Locally compact: A space is locally compact if every point has a compact neighbourhood: the alternative definition that each point has a local base consisting of compact neighbourhoods is sometimes used: these are equivalent for Hausdorff spaces. Every locally compact Hausdorff space is Tychonoff. ;Locally connected: A space is locally connected if every point has a local base consisting of connected neighbourhoods.
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Hydroelectric dams with reservoirs can be operated to provide electricity at times of peak demand. Water is stored in the reservoir during periods of low demand and released when demand is high. The net effect is similar to pumped storage, but without the pumping loss. While a hydroelectric dam does not directly store energy from other generating units, it behaves equivalently by lowering output in periods of excess electricity from other sources.
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There is a space CP∞ which, in a sense, is the inductive limit of CPn as n → ∞. It is BU(1), the classifying space of U(1), in the sense of homotopy theory, and so classifies complex line bundles; equivalently it accounts for the first Chern class. See, for instance, and . The space CP∞ is also the same as the infinite-dimensional projective unitary group; see that article for additional properties and discussion.
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A solar calendar is a calendar whose dates indicate the season or almost equivalently the apparent position of the Sun relative to the stars. The Gregorian calendar, widely accepted as standard in the world, is an example of a solar calendar. The main other type of calendar is a lunar calendar, whose months correspond to cycles of Moon phases. The months of the Gregorian calendar do not correspond to cycles of Moon phase.
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A closely related problem, the minimax path problem, asks for the path that minimizes the maximum weight of any of its edges. It has applications that include transportation planning. Any algorithm for the widest path problem can be transformed into an algorithm for the minimax path problem, or vice versa, by reversing the sense of all the weight comparisons performed by the algorithm, or equivalently by replacing every edge weight by its negation.
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Equivalently, in terms of the indirect utility function (v(\cdot,\cdot)), v(p_0,w+EV) = u_1 This can be shown to be equivalent to the above by taking the expenditure function of both sides at p_0 e(p_0,v(p_0,w+EV)) = e(p_0,u_1) w+EV = e(p_0,u_1) EV = e(p_0,u_1) -w One of the three identical equations above. Compensating variation (CV) is a closely related measure of welfare change.
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In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. It can also be shown to be equivalent that every open subset of such a space is compact, and in fact the seemingly stronger statement that every subset is compact.
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In this case if is a function then : dh = h_x \, dx + h_y \, dy. The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to x and y. The gradient theorem asserts that a 1-form is exact if and only if the line integral of the form depends only on the endpoints of the curve, or equivalently, if the integral around any smooth closed curve is zero.
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Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions. The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in each application by requirements like time and frequency resolution.
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The ping- pong argument goes back to late 19th century and is commonly attributed to Felix Klein who used it to study subgroups of Kleinian groups, that is, of discrete groups of isometries of the hyperbolic 3-space or, equivalently Möbius transformations of the Riemann sphere. The ping-pong lemma was a key tool used by Jacques Tits in his 1972 paperJ. Tits. Free subgroups in linear groups. Journal of Algebra, vol. 20 (1972), pp.
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The degree of inactivation by ultraviolet radiation is directly related to the UV dose applied to the water. The dosage, a product of UV light intensity and exposure time, is usually measured in microjoules per square centimeter, or equivalently as microwatt seconds per square centimeter (μW·s/cm2). Dosages for a 90% kill of most bacteria and viruses range between 2,000 and 8,000 μW·s/cm2. Larger parasites such as cryptosporidium require a lower dose for inactivation.
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This result can easily be seen by adding the corresponding sides of the inequalities for pairs (i) and (ii) and cancelling common variables. Again, this result reflects the transitivity property: (i) 121≻112 and (ii) 112≻211 implies (iii) 121≻211; equivalently, 122≻221 and 221≻212 implies 122≻212. As a result of two explicit pairwise comparisons – i.e. explicitly performed by the decision-maker – five of the six undominated pairs have been ranked.
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So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so pq = qp. Equivalently, X, Y, Z are collinear. The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus's theorem implies Desargues's theorem.
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Two disjoint sets. In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of more than two sets is called disjoint if any two distinct sets of the collection are disjoint.
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Problems 48–55 show how to compute an assortment of areas. Problem 48 is notable in that it succinctly computes the area of a circle by approximating π. Specifically, problem 48 explicitly reinforces the convention (used throughout the geometry section) that "a circle's area stands to that of its circumscribing square in the ratio 64/81." Equivalently, the papyrus approximates π as 256/81, as was already noted above in the explanation of problem 41.
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This is different from confluence in that b and c must be reduced from a in one step. In analogy with this, confluence is sometimes referred to as global confluence. The relation , introduced as a notation for reduction sequences, may be viewed as a rewriting system in its own right, whose relation is the reflexive-transitive closure of →. Since a sequence of reduction sequences is again a reduction sequence (or, equivalently, since forming the reflexive-transitive closure is idempotent), = .
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A subset S of M is said to be a generator of M if M is the smallest set containing S that is closed under the monoid operation, or equivalently M is the result of applying the finitary closure operator to S. If there is a generator of M that has finite cardinality, then M is said to be finitely generated. Not every set S will generate a monoid, as the generated structure may lack an identity element.
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Reciprocally, according to the Nash-Kuiper theorem, any Riemannian surface with boundary can be embedded in Euclidean space preserving the lengths and area specified by the Riemannian metric. Thus the filling problem can be stated equivalently as a question about Riemannian surfaces, that are not placed in Euclidean space in any particular way. :Conjecture (Gromov's filling area conjecture, 1983): The hemisphere has minimum area among the orientable compact Riemannian surfaces that fill isometrically their boundary curve, of given length.
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First, coordinates of cluster peaks (e.g. the voxels where the differences between patients and healthy controls were highest), and statistical maps if available, are selected according to SDM inclusion criteria. Second, coordinates are used to recreate statistical maps, and effect-sizes maps and their variances are derived from t-statistics (or equivalently from p-values or z-scores). Finally, individual study maps are meta-analyzed using different tests to complement the main outcome with sensitivity and heterogeneity analyses.
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Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with n + 1 sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s. Revised from a 1961 Ph.D. thesis, Princeton University, . after earlier work on them by Dov Tamari..
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A subfield of a field is a subset of that is a field with respect to the field operations of . Equivalently is a subset of that contains , and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that , that for all both and are in , and that for all in , both and are in . Field homomorphisms are maps between two fields such that , , and , where and are arbitrary elements of .
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In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier transform) of the structure repeat per unit of distance. The SI unit of spatial frequency is cycles per m. In image- processing applications, spatial frequency is often expressed in units of cycles per mm or equivalently line pairs per mm.
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A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is (a difference of) one hundredth of a percent or equivalently one percent of one percent or one ten thousandth. A very rarely used term, permyriad, means parts per ten thousand, differing in meaning only in that basis points are normally used to express differences in parts per ten thousand. Figures are commonly quoted in basis points in finance, especially in fixed income markets.
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These principles can also be equivalently expressed more abstractly by Hamiltonian mechanics or Lagrangian mechanics. The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a Turing machine can be regarded as the time evolution of the machine's control state together with the state of the tape (or possibly multiple tapes) including the position of the machine's read-write head (or heads). In this case, time is discrete.
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There are higher-dimensional versions of this fact using n-cubes of spaces. These examples suggest that crossed modules may be thought of as "2-dimensional groups". In fact, this idea can be made precise using category theory. It can be shown that a crossed module is essentially the same as a categorical group or 2-group: that is, a group object in the category of categories, or equivalently a category object in the category of groups.
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The distance between two consecutive points of a point process on the real line will be an exponential random variable with parameter \textstyle \lambda (or equivalently, mean \textstyle 1/\lambda). This implies that the points have the memoryless property: the existence of one point existing in a finite interval does not affect the probability (distribution) of other points existing, but this property has no natural equivalence when the Poisson process is defined on a space with higher dimensions.
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Two constraints may be conjoined as a single constraint, called a local conjunction, which gives only one violation each time both constraints are violated within a given domain, such as a segment, syllable or word. For example, []segment is violated once per voiced obstruent in a coda ("VOP" stands for "voiced obstruent prohibition"), and may be equivalently written as . Local conjunctions are used as a way of circumventing the problem of phonological opacity that arises when analyzing chain shifts.
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A k-Sperner poset is a graded poset in which no union of k antichains is larger than the union of the k largest rank levels, or, equivalently, the poset has a maximum k-family consisting of k rank levels. A strict Sperner poset is a graded poset in which all maximum antichains are rank levels. A strongly Sperner poset is a graded poset which is k-Sperner for all values of k up to the largest rank value.
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A ring is called Noetherian (in honor of Emmy Noether, who developed this concept) if every ascending chain of ideals :0 ⊆ I0 ⊆ I1 ... ⊆ In ⊆ In + 1 ⊆ ... becomes stationary, i.e. becomes constant beyond some index n. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. Being Noetherian is a highly important finiteness condition, and the condition is preserved under many operations that occur frequently in geometry.
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A sphere is simply connected because every loop can be contracted (on the surface) to a point. The definition rules out only handle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of any dimension, is called contractibility.
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Simple models include mass balances only. When adding thermodynamic constraints such as energy balances to the model, its scope and the level of redundancy increases. Indeed, as we have seen above, the level of redundancy is defined as p-m, where p is the number of equations. Including energy balances means adding equations to the system, which results in a higher level of redundancy (provided that enough measurements are available, or equivalently, not too many variables are unmeasured).
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A de Bruijn graph. Every four-digit sequence occurs exactly once if one traverses every edge exactly once and returns to one's starting point (an Eulerian cycle). Every three-digit sequence occurs exactly once if one visits every vertex exactly once (a Hamiltonian path). The de Bruijn sequences can be constructed by taking a Hamiltonian path of an n-dimensional de Bruijn graph over k symbols (or equivalently, an Eulerian cycle of an (n − 1)-dimensional de Bruijn graph).
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In binary (base-2), a full space can be used between groups of four digits, corresponding to a nibble, or equivalently to a hexadecimal digit. For integer numbers, dots are used as well to separate groups of four bits.As an example, the DR-DOS DEBUG `H` command displays the entered number in hexadecimal, decimal, octal and binary notation: -h 1234 1234 #4660 \011064 %0001.0010.0011.0100 Alternatively, binary digits may be grouped by threes, corresponding to an octal digit.
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A pair of random variables X and Y are independent if and only if the random vector (X, Y) with joint cumulative distribution function (CDF) F_{X,Y}(x,y) satisfies :F_{X,Y}(x,y) = F_X(x) F_Y(y), or equivalently, their joint density f_{X,Y}(x,y) satisfies :f_{X,Y}(x,y) = f_X(x) f_Y(y). That is, the joint distribution is equal to the product of the marginal distributions. Definition 2.5.1, page 109.
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In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters. Equivalently, a suitably defined dual set is relatively dense in the Pontryagin dual of the group. This notion was introduced by Yves Meyer in 1970 and later turned out to play an important role in the mathematical theory of quasicrystals. Some related concepts are model sets, Meyer sets, and cut- and-project sets.
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Items that compare equal receive the same ranking number, which is the mean of what they would have under ordinal rankings. Equivalently, the ranking number of 1 plus the number of items ranked above it plus half the number of items equal to it. This strategy has the property that the sum of the ranking numbers is the same as under ordinal ranking. For this reason, it is used in computing Borda counts and in statistical tests (see below).
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They can be treated by plane surveying and mapped by scale drawings in which any two points at the same distance on the drawing are at the same distance on the ground. True ground distances are calculated by measuring the distance on the map and then multiplying by the inverse of the scale fraction or, equivalently, simply using dividers to transfer the separation between the points on the map to a bar scale on the map.
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The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two-dimensional aspherical CW complex is aspherical. A group presentation G=(S\mid R) is called aspherical if the two-dimensional CW complex K(S\mid R) associated with this presentation is aspherical or, equivalently, if \pi_2(K(S\mid R))=0.
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In one of the breakthrough events in the field of polymer science, Szwarc elucidated that electron transfer occurred from radical anion of naphthalene to styrene. The results in the formation of a dianion (or equivalently disodio-) species, which rapidly added styrene to form a "two – ended living polymer." An important aspect of his work, Szwarc employed the aprotic solvent tetrahydrofuran. Being a physical chemist, Szwarc elucidated the kinetics and the thermodynamics of the process in considerable detail.
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Some diagonals of a concave polygon lie partly or wholly outside the polygon. Some sidelines of a concave polygon fail to divide the plane into two half- planes one of which entirely contains the polygon. None of these three statements holds for a convex polygon. As with any simple polygon, the sum of the internal angles of a concave polygon is ×(n − 2) radians, equivalently 180×(n − 2) degrees (°), where n is the number of sides.
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A., 217, 295–305) or the torus (the bivariate von Mises distribution). The matrix von Mises–Fisher distribution is a distribution on the Stiefel manifold, and can be used to construct probability distributions over rotation matrices. The Bingham distribution is a distribution over axes in N dimensions, or equivalently, over points on the (N − 1)-dimensional sphere with the antipodes identified. For example, if N = 2, the axes are undirected lines through the origin in the plane.
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A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns yes. These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as a Gödel numbering, the strings can be encoded as natural numbers.
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For example, if the bidding goes 1 - (P) - P - (1NT), the 1NT bid is a balancing action. The balancing bid is often made with a hand of substandard strength in order to prevent the opponents from securing a low-level contract. ;Balanced distribution # Narrowly, a balanced distribution of a hand is 4-3-3-3, 4-4-3-2 or 5-3-3-2. Equivalently, there are no voids, no singletons, and at most one doubleton.
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Iterative impedance is the input impedance of one port of a two-port network when the other port is connected to an infinite chain of identical networks.Iyer, p. 340 Equivalently, iterative impedance is that impedance that when connected to port 2 of a two-port network is equal to the impedance measured at port 1. This can be seen to be equivalent by considering the infinite chain of identical networks connected to port 2 in the first definition.
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A Rayl, rayl or Rayleigh is one of two units of specific acoustic impedance or, equivalently, characteristic acoustic impedance; one an MKS unit, and the other a CGS unit. The units are named after John William Strutt, 3rd Baron Rayleigh, and not to be confused with the rayleigh unit of photon flux, used to measure airglow, and named after his son, Robert John Strutt, 4th Baron Rayleigh. It has the same dimensions as momentum per volume.
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Schematic picture of a 2-bridge knot. In the mathematical field of knot theory, a 2-bridge knot is a knot which can be regular isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points. Equivalently, these are the knots with bridge number 2, the smallest possible bridge number for a nontrivial knot. Other names for 2-bridge knots are rational knots', 4-plats, and ''''' ().
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Suppose is a semidirect product of the normal subgroup and the subgroup . If is also normal in , or equivalently, if there exists a homomorphism that is the identity on with kernel , then is the direct product of and . The direct product of two groups and can be thought of as the semidirect product of and with respect to for all in . Note that in a direct product, the order of the factors is not important, since is isomorphic to .
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The r = 2 case can be rephrased as stating that every complete graph with an even number of vertices has an edge coloring whose number of colors equals its degree, or equivalently that its edges may be partitioned into perfect matchings. It may be used to schedule round-robin tournaments, and its solution was already known in the 19th century. The case that k = 2r is also easy. The r = 3 case was established by R. Peltesohn in 1936.
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After the preliminary step of interest point detection, feature generation and clustering, we have a large set of candidate parts over the training images. To learn the model, Weber & Welling first perform a greedy search over possible model configurations, or equivalently, over potential subsets of the candidate parts. This is done in an iterative fashion, starting with random selection. At subsequent iterations, parts in the model are randomly substituted, the model parameters are estimated, and the performance is assessed.
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If X is a Riemannian manifold and G its full group of isometry, then a (G,X)-structure is complete if and only if the underlying Riemannian manifold is geodesically complete (equivalently metrically complete). In particular, in this case if the underlying space of a (G,X)-manifold is compact then the latter is automatically complete. In the case where X is the hyperbolic plane the developing map is the same map as given by the Uniformisation Theorem.
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Edge coloring Tietze's graph requires four colors; that is, its chromatic index is 4. Equivalently, the edges of Tietze's graph can be partitioned into four matchings, but no fewer. Tietze's graph matches part of the definition of a snark: it is a cubic bridgeless graph that is not 3-edge- colorable. However, some authors restrict snarks to graphs without 3-cycles and 4-cycles, and under this more restrictive definition Tietze's graph is not a snark.
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When the surface of a sphere is divided by finitely many great arcs (equivalently, by planes passing through the center of the sphere), the result is called a spherical polyhedron. Many convex polytopes having some degree of symmetry (for example, all the Platonic solids) can be projected onto the surface of a concentric sphere to produce a spherical polyhedron. However, the reverse process is not always possible; some spherical polyhedra (such as the hosohedra) have no flat-faced analogue..
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A saeculum is a length of time roughly equal to the potential lifetime of a person or, equivalently, of the complete renewal of a human population. The term was first used by the Etruscans. Originally it meant the period of time from the moment that something happened (for example the founding of a city) until the point in time that all people who had lived at the first moment had died. At that point a new saeculum would start.
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The cognitive capabilities of current architectures are very limited, using only a simplified version of what intelligence is really capable of. For instance, the human mind has come up with ways to reason beyond measure and logical explanations to different occurrences in life. What would have been otherwise straightforward, an equivalently difficult problem may be challenging to solve computationally as opposed to using the human mind. This gives rise to two classes of models: structuralist and functionalist.
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It is now believed to be an intractable computational problem to find a Brouwer fixed point or equivalently a Sperner coloring, even in the plane, in the general case. The problem is PPAD-complete, a complexity class invented by Christos Papadimitriou. According to the Soviet Mathematical Encyclopaedia (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had also become known as the Sperner lemma – this point is discussed in the English translation (ed.
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It is possible for the result of global title translation to be route on SSN. This means that, instead of the Global Title routing, lower level MTP routing will be used for this message from this point on. Equivalently, in a system using SS7 over IP (for example, SIGTRAN), the result from Global Title Translation may be to route to an IP server, though the exact details depend greatly on which variant of SS7 over IP is being used.
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So there are both statistical mechanical and quantum mechanical superselection rules. In a theory where the vacuum is invariant under a symmetry, the conserved charge leads to superselection sectors in the case that the charge is conserved. Electric charge is conserved in our universe, so it seems at first like a trivial example. But when a superconductor fills space, or equivalently in a Higgs phase, electric charge is still globally conserved but no longer defines the superselection sectors.
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Equivalently to the above, a discrete random variable can be defined as a random variable whose cumulative distribution function (cdf) increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.
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A Halin graph is a graph formed from an undirected plane tree (with no degree- two nodes) by connecting its leaves into a cycle, in the order given by the plane embedding of the tree. Equivalently, it is a polyhedral graph in which one face is adjacent to all the others. Every Halin graph is planar. Like outerplanar graphs, Halin graphs have low treewidth, making many algorithmic problems on them more easily solved than in unrestricted planar graphs..
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Starting with B0 = B1 = 1, the first few Bell numbers are :1, 1, 2, 5, 15, 52, 203, 877, 4140, ... . The Bell number Bn counts the number of different ways to partition a set that has exactly n elements, or equivalently, the number of equivalence relations on it. Bn also counts the number of different rhyme schemes for n-line poems. As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions.
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There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively. # Regularity. A summation method is regular if, whenever the sequence s converges to x, Equivalently, the corresponding series-summation method evaluates # Linearity. A is linear if it is a linear functional on the sequences where it is defined, so that for sequences r, s and a real or complex scalar k.
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Fig. 3a: Barrel distortion Fig. 3b: Pincushion distortion Even if the image is sharp, it may be distorted compared to ideal pinhole projection. In pinhole projection, the magnification of an object is inversely proportional to its distance to the camera along the optical axis so that a camera pointing directly at a flat surface reproduces that flat surface. Distortion can be thought of as stretching the image non-uniformly, or, equivalently, as a variation in magnification across the field.
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The two right triangles with leg and hypotenuse (7,13) and (13,17) have equal third sides of length . The square of this side, 120, is a congruum: it is the difference between consecutive values in the arithmetic progression of squares 72, 132, 172. Equivalently, the two annuli between the three yellow circles have equal areas, times the congruum. In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares.
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It can be CRCW, CREW, or EREW. See PRAM for descriptions of those models. Equivalently, NC can be defined as those decision problems decidable by a uniform Boolean circuit (which can be calculated from the length of the input, for NC, we suppose we can compute the Boolean circuit of size n in logarithmic space in n) with polylogarithmic depth and a polynomial number of gates. RNC is a class extending NC with access to randomness.
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When the Christoffel symbols are considered as being defined by the first fundamental form, the Gauss and Codazzi equations represent certain constraints between the first and second fundamental forms. The Gauss equation is particularly noteworthy, as it shows that the Gaussian curvature can be computed directly from the first fundamental form, without the need for any other information; equivalently, this says that can actually be written as a function of , even though the individual components cannot.
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Dakar 18 is a racing game set in an open world environment. It features both offline and online single-player and multiplayer modes. The world of the game is over 5,791 square miles (15,000 km²) in size, which is roughly the size of the Santiago Metropolitan Region (equivalently, the U.S. state of Connecticut or the Moquegua Region of Peru), and completely explorable. Five categories of vehicles will be available, including cars, trucks, motorcycles, quads and UTVs.
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Continuously differentiable functions are sometimes said to be of class C1. A function is of class C2 if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of class Ck if the first k derivatives '(x), '(x), ..., f (k)(x) all exist and are continuous. If derivatives f (n) exist for all positive integers n, the function is smooth or equivalently, of class C∞.
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The S-matrix is the quantity that describes how a collection of incoming particles turn into outgoing ones. Heisenberg proposed to study the S-matrix directly, without any assumptions about space-time structure. But when transitions from the far-past to the far-future occur in one step with no intermediate steps, it becomes difficult to calculate anything. In quantum field theory, the intermediate steps are the fluctuations of fields or equivalently the fluctuations of virtual particles.
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A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3, and searches for them have been made up to 5 \times 10^7 as of 1970. It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound.
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K correction converts measurements of astronomical objects into their respective rest frames. The correction acts on that object's observed magnitude (or equivalently, its flux). Because astronomical observations often measure through a single filter or bandpass, observers only measure a fraction of the total spectrum, redshifted into the frame of the observer. For example, to compare measurements of stars at different redshifts viewed through a red filter, one must estimate K corrections to these measurements in order to make comparisons.
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Homomorphic images and subgroups of Noetherian groups are Noetherian, and an extension of a Noetherian group by a Noetherian group is Noetherian. Analogous results hold for Artinian groups. Noetherian groups are equivalently those such that every subgroup is finitely generated, which is stronger than the group itself being finitely generated: the free group on 2 or finitely more generators is finitely generated, but contains free groups of infinite rank. Noetherian groups need not be finite extensions of polycyclic groups.
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The latter can be proved by contradiction. The previous example employed the contrapositive of a definition to prove a theorem. One can also prove a theorem by proving the contrapositive of the theorem's statement. To prove that if a positive integer N is a non-square number, its square root is irrational, we can equivalently prove its contrapositive, that if a positive integer N has a square root that is rational, then N is a square number.
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One of the simplest ways to represent a board is to create an 8x8 two-dimensional array (or, equivalently, a 64 element one-dimensional array). Each array element would identify what piece occupied the given square, or alternatively, if the square is empty. A common encoding is to consider 0 as empty, positive as white, and negative as black, e.g., white pawn +1, black pawn −1, white knight +2, black knight −2, white bishop +3, and so on.
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The complete graph K4 has the ten matchings shown, so its Hosoya index is ten, the maximum for any four-vertex graph. The Hosoya index, also known as the Z index, of a graph is the total number of matchings in it. The Hosoya index is always at least one, because the empty set of edges is counted as a matching for this purpose. Equivalently, the Hosoya index is the number of non-empty matchings plus one.
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This baseline can be as long as the major axis of the Earth's orbit or, equivalently, two astronomical units (AU). The distance to a star with a parallax angle of only one arcsecond measured on a baseline of one AU is a unit known as the parsec (pc) in astronomy and is equal to about 3.26 light years.Abell et al., inside front cover There is an inverse relationship between the distance in parsecs and the angle in arcseconds.
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An Apollonian network The Goldner–Harary graph, a non-Hamiltonian Apollonian network In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.
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Furthermore, the n-by-n invertible matrices are a dense open set in the topological space of all n-by-n matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of n-by-n matrices. In practice however, one may encounter non-invertible matrices. And in numerical calculations, matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be ill- conditioned.
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Using functional principal component analysis and the Karhunen-Loève expansion, these processes can be equivalently expressed as a countable sequence of their functional principal component scores (FPCs) and eigenfunctions. In the FAM the responses (scalar or functional) conditional on the predictor functions are modeled as function of the functional principal component scores of the predictor function in an additive structure. This model can be categorized as a Frequency Additive Model since it is additive in the predictor FPC scores.
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The XQuery language allows cursors to be created using the subsequence() function. The format is: let $displayed-sequence := subsequence($result, $start, $item- count) Where $result is the result of the initial XQuery, $start is the item number to start and $item-count is the number of items to return. Equivalently this can also be done using a predicate: let $displayed-sequence := $result[$start to $end] Where `$end` is the end sequence. For complete examples see the .
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Equivalently, any division algebra of period dividing n is Brauer equivalent to a tensor product of cyclic algebras of degree n. Even for a prime number p, there are examples showing that a division algebra of period p need not be actually isomorphic to a tensor product of cyclic algebras of degree p.Gille & Szamuely (2006), Remark 2.5.8. It is a major open problem (raised by Albert) whether every division algebra of prime degree over a field is cyclic.
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A percentile (or a centile) is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations falls. For example, the 20th percentile is the value (or score) below which 20% of the observations may be found. Equivalently, 80% of the observations are found above the 20th percentile. The term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests.
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In particle physics, the quenched approximation is an approximation often used in lattice gauge theory in which the quantum loops of fermions in Feynman diagrams are neglected. Equivalently, the corresponding one-loop determinants are set to one. This approximation is often forced upon the physicists because the calculation with the Grassmann numbers is computationally very difficult in lattice gauge theory. In particular, quenched QED is QED without dynamical electrons, and quenched QCD is QCD without dynamical quarks.
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Charles A. Weibel, Robert W. Thomason (1952–1995). Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem. Equivalently, K_i^G(X) may be defined as the K_i of the category of coherent sheaves on the quotient stack [X/G]. (Hence, the equivariant K-theory is a specific case of the K-theory of a stack.) A version of the Lefschetz fixed point theorem holds in the setting of equivariant (algebraic) K-theory.
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Midday (חֲצוֹת הַיּוֹם, Chatzot Hayom or just Chatzot) means the midpoint between sunrise and sunset, or equivalently between daybreak and sundown. The absolute latest time for the Shacharit Amidah, ex post facto, is this time. On the Shabbat and on holidays, one is supposed to eat before this time (except on Rosh Hashanah). On Tish'a Ba'av one may sit on a chair at this time, and those who fast on Erev Rosh Hashanah usually eat at this time.
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Drilling holes does not provide direct evidence against the hypothesis. The deepest hole drilled to date is the Kola Superdeep Borehole, with a true vertical drill-depth of more than 7.5 miles (12 kilometers). However, the distance to the center of the Earth is nearly 4,000 miles (6,400 kilometers). Oil wells with longer depths are not vertical wells; the total depths quoted are measured depth (MD) or equivalently, along-hole depth (AHD) as these wells are deviated to horizontal.
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Of course, the operating system is also able to disallow such programming to the other programs. This technique is called supervisor mode; compared to more crude approaches (such as storing the TCB in ROM, or equivalently, using the Harvard architecture), it has the advantage of allowing the security- critical software to be upgraded in the field, although allowing secure upgrades of the trusted computing base poses bootstrap problems of its own.A Secure and Reliable Bootstrap Architecture, op. cit.
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Infrared light is guided through an interferometer and then through the sample (or vice versa). A moving mirror inside the apparatus alters the distribution of infrared light that passes through the interferometer. The signal directly recorded, called an "interferogram", represents light output as a function of mirror position. A data-processing technique called Fourier transform turns this raw data into the desired result (the sample's spectrum): Light output as a function of infrared wavelength (or equivalently, wavenumber).
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In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: # C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear); # C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts; # given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is by definition the kernel of f), as does the coequaliser (this is by definition the cokernel of f). Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2.
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This closure has as its feasible sets the unions of prefixes of strings in L. In terms of this closure operation, the join is the closure of the union of the languages of A and B. Every antimatroid can be represented as a join of a family of chain antimatroids, or equivalently as the closure of a set of basic words; the convex dimension of an antimatroid A is the minimum number of chain antimatroids (or equivalently the minimum number of basic words) in such a representation. If F is a family of chain antimatroids whose basic words all belong to A, then F generates A if and only if the feasible sets of F include all paths of A. The paths of A belonging to a single chain antimatroid must form a chain in the path poset of A, so the convex dimension of an antimatroid equals the minimum number of chains needed to cover the path poset, which by Dilworth's theorem equals the width of the path poset.; Korte et al., Theorem 6.9.
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Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. (Here the term κ-additive means that, for any sequence Aα, α<λ of cardinality λ < κ, Aα being pairwise disjoint sets of ordinals less than κ, the measure of the union of the Aα equals the sum of the measures of the individual Aα.) Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since V is a proper class, a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick. Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter.
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The folded cube graph of order k (containing 2k − 1 vertices) may be formed by adding edges between opposite pairs of vertices in a hypercube graph of order k − 1\. (In a hypercube with 2n vertices, a pair of vertices are opposite if the shortest path between them has length n.) It can, equivalently, be formed from a hypercube graph (also) of order k, which has twice as many vertices, by identifying together (or contracting) every opposite pair of vertices.
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A 1900 script reform ordained that only one selected character be used for each mora, with the rest deemed hentaigana. Although not normally used in publication, hentaigana are still used in shop signs and brand names to create a traditional or antiquated air. Hiragana originate in man'yōgana, a system where kanji were used to write sounds without regard to their meaning. There was more than one kanji that could be used equivalently for each syllable (at the time, a syllable was a mora).
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Suramin is not orally bioavailable and must be given intravenously. Intramuscular and subcutaneous administration could result in local tissue inflammation or necrosis. Suramin is approximately 99-98% protein bound in the serum and has a half-life of 41–78 days average of 50 days; however, the pharmacokinetics of suramin can vary substantially between individual patients. Suramin does not distribute well into cerebral spinal fluid and its concentration in the tissues is equivalently lower than its concentration in the plasma.
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The reciprocal of permeability is magnetic reluctivity. In SI units, permeability is measured in henries per meter (H/m), or equivalently in newtons per ampere squared (N/A2). The permeability constant μ0, also known as the magnetic constant or the permeability of free space, is a measure of the amount of resistance encountered when forming a magnetic field in a classical vacuum. Until 20 May 2019, the magnetic constant had the exact (defined) value μ0 = 4 × 10−7 H/m ≈ .
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The Simple Dietz methodDietz, Peter O. Pension Funds: Measuring Investment Performance. Free Press, 1966. applies a simple rate of interest principle, as opposed to the compounding principle underlying the internal rate of return method, and further assumes that flows occur at the midpoint within the time interval (or equivalently that they are distributed evenly throughout the time interval). However, the Simple Dietz method is unsuitable when such assumptions are invalid, and will produce different results to other methods in such a case.
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The Schläfli symbol is a recursive description, starting with {p} for a p-sided regular polygon that is convex. For example, {3} is an equilateral triangle, {4} is a square, {5} a convex regular pentagon and so on. Regular star polygons are not convex, and their Schläfli symbols {p/q} contain irreducible fractions p/q, where p is the number of vertices, and q is their turning number. Equivalently, {p/q} is created from the vertices of {p}, connected every q.
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Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve.
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Like other bugle calls, "recall" is a short tune that originated as a military signal announcing scheduled and certain non-scheduled events on a military installation, battlefield, or ship. Historically bugle calls indicated the change in daily routines of camp. A defining feature of a bugle call is that it consists only of notes from a single overtone series. This is in fact a requirement if it is to be playable on a bugle or equivalently on a trumpet without moving the valves.
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The mountain systems of the earth are characterized by a tree structure, where mountain ranges can contain sub-ranges. The sub-range relationship is often expressed as a parent-child relationship. For example, the White Mountains of New Hampshire and the Blue Ridge Mountains are sub-ranges of the Appalachian Mountains. Equivalently, the Appalachians are the parent of the White Mountains and the Blue Ridge Mountains, and the White Mountains and the Blue Ridge Mountains are children of the Appalachians.
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In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field- free" space. In quantum mechanics, it means a region of uniform potential, usually set to zero in the region of interest since potential can be arbitrarily set to zero at any point (or surface in three dimensions) in space.
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Chandrasekhar's most notable work is on the astrophysical Chandrasekhar limit. The limit gives the maximum mass of a white dwarf star, ~1.44 solar masses, or equivalently, the minimum mass that must be exceeded for a star to collapse into a neutron star or black hole (following a supernova). The limit was first calculated by Chandrasekhar in 1930 during his maiden voyage from India to Cambridge, England for his graduate studies. In 1979, NASA named the third of its four "Great Observatories" after Chandrasekhar.
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In geometry, the Chebyshev center of a bounded set Q having non-empty interior is the center of the minimal-radius ball enclosing the entire set Q, or alternatively (and non-equivalently) the center of largest inscribed ball of Q. In the field of parameter estimation, the Chebyshev center approach tries to find an estimator \hat x for x given the feasibility set Q , such that \hat x minimizes the worst possible estimation error for x (e.g. best worst case).
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Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the great dirhombicosidodecahedron in three dimensions and the grand antiprism in four dimensions. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson. Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension.
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On Catholics who dissent from Catholic teachings Levada wrote that "Catholic theology does not recognize the right to dissent, if by that we mean adopting conclusions which are contrary to the clear teachings of the authoritative, infallible magisterium and which are presented to the public in such a way as to constitute equivalently an alternative personal magisterium"."Dissent and the Catholic Religion Teacher", Speech to National Catholic Educational Association, April 2, 1986. Printed in Origins, v. 16 (1986), pp. 195–200.
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After using two hours worth of fuel it could maintain height on one engine alone, or equivalently hold altitude on half power with two engines. There was great pressure on all teams in early 1919 and the company's management had assembled a large group to observe the first flight which probably led to the disastrous first flight of the Atlantic in April. The Boulton & Paul test pilot Frank Courtney had run up each engine to full power, but separately, not together.
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Three of the above-mentioned commas, namely the diaschisma, the diesis and the greater diesis, meet the definition of the diminished second, being the difference between the sizes in cents of a diatonic and a chromatic semitone (or equivalently the ratio between their frequency ratios). On the contrary, the syntonic comma is defined either as the difference in cents between two chromatic semitones (S2 and S1), or between two diatonic semitones (S4 and S3), and cannot be considered a diminished second.
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For example, in the R transformation, a single voice moves by whole step; in the N or S transformation, two voices move by semitone. When common-tone maximization is prioritized, R is more efficient; when voice-leading efficiency is measured by summing the motions of the individual voices, the transformations are equivalently efficient. Early neo- Riemannian theory conflated these two conceptions. More recent work has disentangled them, and measures distance unilaterally by voice-leading proximity independently of common-tone preservation.
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The propellant in just the S-IC made up about three-quarters of Saturn V's entire launch mass, and it was consumed at . Newton's second law of motion states that force is equal to mass multiplied by acceleration, or equivalently that acceleration is equal to force divided by mass, so as the mass decreased (and the force increased somewhat), acceleration rose. Including gravity, launch acceleration was only g, i.e., the astronauts felt g while the rocket accelerated vertically at g.
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Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit. Thus, in case the action is transitive, the normal core of any isotropy subgroup is precisely the kernel of the action. A core-free subgroup is a subgroup whose normal core is the trivial subgroup. Equivalently, it is a subgroup that occurs as the isotropy subgroup of a transitive, faithful group action.
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A family of subsets of is said to be an algebra of sets if and for all , the complement , the intersection , and the union all belong to . Equivalently, is an algebra of sets if and only if and for all , and . The condition "" may be replaced with "". Every algebra of sets is also a ring of sets and moreover, a family of sets is an algebra of sets if and only if it is a ring of sets that contains (or alternatively, ).
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Polish does not regularly place nouns together to form compound noun expressions. Equivalents to such expressions are formed using noun-derived adjectives (as in , "orange juice", where is an adjective derived from "orange"), or using prepositional phrases or (equivalently) a noun in the genitive or other case. A group of nouns connected by a word for "and" is treated as plural. It is masculine personal plural if it contains any male person (in fact, if it contains any person and any masculine noun).
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The resonance frequency is defined as the frequency at which the impedance of the circuit is at a minimum. Equivalently, it can be defined as the frequency at which the impedance is purely real (that is, purely resistive). This occurs because the impedances of the inductor and capacitor at resonance are equal but of opposite sign and cancel out. Circuits where L and C are in parallel rather than series actually have a maximum impedance rather than a minimum impedance.
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A Euclidean graph (a graph embedded in some Euclidean space) is periodic if there exists a basis of that Euclidean space whose corresponding translations induce symmetries of that graph (i.e., application of any such translation to the graph embedded in the Euclidean space leaves the graph unchanged). Equivalently, a periodic Euclidean graph is a periodic realization of an abelian covering graph over a finite graph. A Euclidean graph is uniformly discrete if there is a minimal distance between any two vertices.
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A Whitney triangulation or clean triangulation of a surface is an embedding of a graph onto the surface in such a way that the faces of the embedding are exactly the cliques of the graph. Equivalently, every face is a triangle, every triangle is a face, and the graph is not itself a clique. The clique complex of the graph is then homeomorphic to the surface. The 1-skeletons of Whitney triangulations are exactly the locally cyclic graphs other than K4.
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However, lattices in which every two elements have a supremum and an infimum, and in particular total orders, are F-algebras. This is because they can equivalently be defined in terms of the algebraic operations: x∨y = inf(x,y) and x∧y = sup(x,y), subject to certain axioms (commutativity, associativity, absorption and idempotency). Thus they are F-algebras of signature P x P + P x P. It is often said that lattice theory draws on both order theory and universal algebra.
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The four component values of the network provide four degrees of freedom in the design. It is required from image theory (see Zobel network) that the L/C branch and the L'/C' branch are the dual of each other (ignoring the transformer action) which provides two parameters for calculating component values. Equivalently, every transmission pole, sp in the s-domain left half-plane must have a matching zero, sz in the right half-plane such that sp=−sz.Chaloupka & Kolesov, p.
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First assume K is a field of characteristic different from 2. Let E be an elliptic curve over K of the form: : y^2 = x^3 + a_2 x^2 +a_4 x + a_6. \, Given d eq 0 not quadratic residue, the quadratic twist of E is the curve E^d, defined by the equation: : dy^2 = x^3 + a_2 x^2 + a_4 x + a_6. \, or equivalently : y^2 = x^3 + d a_2 x^2 + d^2 a_4 x + d^3 a_6.
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The unicity distance can equivalently be defined as the minimum amount of ciphertext required to permit a computationally unlimited adversary to recover the unique encryption key. The expected unicity distance can then be shown to be: : U = H(k) / D where U is the unicity distance, H(k) is the entropy of the key space (e.g. 128 for 2128 equiprobable keys, rather less if the key is a memorized pass-phrase). D is defined as the plaintext redundancy in bits per character.
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Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.
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The braid group is the universal central extension of the modular group. The braid group is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group . Further, the modular group has a trivial center, and thus the modular group is isomorphic to the quotient group of modulo its center; equivalently, to the group of inner automorphisms of . The braid group in turn is isomorphic to the knot group of the trefoil knot.
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As for the Wadge lemma, this holds for any pointclass Γ, assuming the axiom of determinacy. If we associate with each set A the collection of all sets strictly below A on the Wadge hierarchy, this forms a pointclass. Equivalently, for each ordinal α ≤ θ the collection Wα of sets that show up before stage α is a pointclass. Conversely, every pointclass is equal to some Wα. A pointclass is said to be self-dual if it is closed under complementation.
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The area of the standard fundamental polygon is 4\pi(g-1) where g is the genus of the Riemann surface (equivalently, where 4g is the number of the sides of the polygon). Since the standard polygon is a representative of H/Γ, the total area of the Riemann surface is equal to the area of the standard polygon. The area formula follows from the Gauss–Bonnet theorem and is in a certain sense generalized through the Riemann–Hurwitz formula.
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For each I, we let the nodes in I be active for x_I time units; meanwhile, each node not in I is inactive. In more concrete terms, each node of G might represent a radio transmission in a wireless communication network; the edges of G represent interference between radio transmissions. Each radio transmission needs to be active for 1 time unit in total; an optimal fractional graph coloring provides a minimum-length schedule (or, equivalently, a maximum-bandwidth schedule) that is conflict- free.
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In the theory of semigroups, a weak inverse of an element x in a semigroup is an element y such that . If every element has a weak inverse, the semigroup is called an E-inversive or E-dense semigroup. An E-inversive semigroup may equivalently be defined by requiring that for every element , there exists such that and are idempotents. preprint An element x of S for which there is an element y of S such that is called regular.
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The neutron flux is a scalar quantity used in nuclear physics and nuclear reactor physics. It is the total length travelled by all free neutrons per unit time and volume.Rudi J. J. Stamm'ler, Máximo Julio Abbate, Methods of steady-state reactor physics in nuclear design. Equivalently, it can be defined as the number of neutrons travelling through a small sphere of radius R in a time interval, divided by \pi R^2 (the cross section of the sphere) and by the time interval.
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The linear arboricity of a graph is the minimum number of linear forests (a collection of paths) into which the edges of the graph can be partitioned. The linear arboricity of a graph is closely related to its maximum degree and its slope number. The pseudoarboricity of a graph is the minimum number of pseudoforests into which its edges can be partitioned. Equivalently, it is the maximum ratio of edges to vertices in any subgraph of the graph, rounded up to an integer.
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Here, "greatest" means "having a maximal degree" or, equivalently, being maximal for the preorder defined by the degree. Given a greatest common divisor of two polynomials, the other greatest common divisors are obtained by multiplication by a nonzero constant (that is, all greatest common divisors of and are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to ). The extended Euclidean algorithm allows computing (and proving) Bézout's identity.
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One of the few general approaches is through the Hasse principle. Infinite descent is the traditional method, and has been pushed a long way. The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as equivalently described as recursively enumerable. In other words, the general problem of Diophantine analysis is blessed or cursed with universality, and in any case is not something that will be solved except by re-expressing it in other terms.
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In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that , or equivalently if the map from to that sends to is not injective (one to one). Similarly, an element of a ring is called a right zero divisor if there exists a nonzero such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.
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A zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.
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The temperature of the small and large horizon in the de Sitter–Schwarzschild can be calculated as the period in imaginary time of the solution, or equivalently as the surface gravity near the horizon. The temperature of the smaller black hole is relatively larger, so there is heat flow from the smaller to the larger horizon. The quantity which is the temperature of the black hole is hard to define, because there is no asymptotically flat space to measure it relative to.
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Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient is zero). Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i.e., parallel to the -axis).
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The lune of Hippocrates is the upper left shaded area. It has the same area as the lower right shaded triangle. In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex plane region bounded by one 180-degree circular arc and one 90-degree circular arc.
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In general, a linear Diophantine equation has no solutions, or an infinite number of solutions. To find the latter, consider two solutions, (x1, y1) and (x2, y2), where : or equivalently : Therefore, the smallest difference between two x solutions is b/g, whereas the smallest difference between two y solutions is a/g. Thus, the solutions may be expressed as : : . By allowing u to vary over all possible integers, an infinite family of solutions can be generated from a single solution (x1, y1).
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The plasmon injection scheme can be implemented either physically or equivalently through deconvolution post-processing method. However, the physical implementation has shown to be more effective than the deconvolution. Physical construction of convolution and selective amplification of the spatial frequencies within a narrow bandwidth are the keys to the physical implementation of the plasmon injection scheme. This loss compensation scheme is ideally suited especially for metamaterial lenses since it does not require gain medium, nonlinearity, or any interaction with phonons.
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The idea of unparticles comes from conjecturing that there may be "stuff" that does not necessarily have zero mass but is still scale-invariant, with the same physics regardless of a change of length (or equivalently energy). This stuff is unlike particles, and described as unparticle. The unparticle stuff is equivalent to particles with a continuous spectrum of mass. Such unparticle stuff has not been observed, which suggests that if it exists, it must couple with normal matter weakly at observable energies.
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In Carl Jung's school of analytical psychology, the anima and animus are the two primary anthropomorphic archetypes of the unconscious mind. The anima and animus are described by Jung as elements of his theory of the collective unconscious, a domain of the unconscious that transcends the personal psyche. In the unconscious of the male, it finds expression as a feminine inner personality: anima; equivalently, in the unconscious of the female, it is expressed as a masculine inner personality: animus.Jung, Carl.
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Equivalently, RE is the class of decision problems for which a Turing machine can list all the 'yes' instances, one by one (this is what 'enumerable' means). Each member of RE is a recursively enumerable set and therefore a Diophantine set. Similarly, co-RE is the set of all languages that are complements of a language in RE. In a sense, co-RE contains languages of which membership can be disproved in a finite amount of time, but proving membership might take forever.
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An n × n matrix A is said to be skew- centrosymmetric if its entries satisfy Ai,j = -An−i+1,n−j+1 for 1 ≤ i,j ≤ n. Equivalently, A is skew-centrosymmetric if AJ = -JA, where J is the exchange matrix defined above. The centrosymmetric relation AJ = JA lends itself to a natural generalization, where J is replaced with an involutory matrix K (i.e., K2 = I) or, more generally, a matrix K satisfying Km = I for an integer m > 1.
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Hadwiger's conjecture is that this set consists of a single forbidden minor, Kk. The Hadwiger number h(G) of a graph G is the size k of the largest complete graph Kk that is a minor of G (or equivalently can be obtained by contracting edges of G). It is also known as the contraction clique number of G.. The Hadwiger conjecture can be stated in the simple algebraic form χ(G) ≤ h(G) where χ(G) denotes the chromatic number of G.
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Every open subset of is sequentially open and every closed set is sequentially closed. The converses are not generally true. The spaces for which the converse is true are called sequential spaces; that is, a sequential space is a topological space in which every sequentially open subset is necessarily open (or equivalently, a space in which every sequentially closed subset is necessarily closed). Every Fréchet-Urysohn space is a sequential space but there are sequential spaces that are not Fréchet-Urysohn spaces.
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In terms of possible processes, Rudolf Clausius stated that heat does not spontaneously pass from a colder body to a warmer body. Equivalently, perpetual motion machines of the second kind (machines that spontaneously convert thermal energy into mechanical work) are impossible. In terms of entropy, in a natural thermodynamic process, the sum of the entropies of interacting thermodynamic systems increases. The third law of thermodynamics states that a system's entropy approaches a constant value as the temperature approaches absolute zero.
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Also known as non- local gravity. A few physicists think that Einstein's theory of gravity will have to be modified – not at small scales, but at large distances, or equivalently, at small accelerations. This would change the gravity force to a non-local force. They point out that dark matter and dark energy are unexplained by the Standard Model of particle physics and suggest that some modification of gravity is necessary, possibly arising from modified Newtonian dynamics or the holographic principle.
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The Immerman–Szelepcsényi theorem was proven independently by Neil Immerman and Róbert Szelepcsényi in 1987, for which they shared the 1995 Gödel Prize. In its general form the theorem states that NSPACE(s(n)) = co-NSPACE(s(n)) for any function s(n) ≥ log n. The result is equivalently stated as NL = co-NL; although this is the special case when s(n) = log n, it implies the general theorem by a standard padding argument. The result solved the second LBA problem.
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Ukaan (also Ikan, Anyaran, Auga, or Kakumo) is a poorly described Niger–Congo language or dialect cluster of uncertain affiliation. Roger Blench suspects, based on wordlists, that it might be closest to the (East) Benue–Congo languages (or, equivalently, the most divergent of the Benue–Congo languages). Blench (2012) states that "noun-classes and concord make it look Benue-Congo, but evidence is weak."Roger Blench, Niger-Congo: an alternative view Speakers refer to their language as Ùkãã or Ìkã.
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The p-value does not provide the probability that either hypothesis is correct (a common source of confusion). If the p-value is less than the chosen significance threshold (equivalently, if the observed test statistic is in the critical region), then we say the null hypothesis is rejected at the chosen level of significance. Rejection of the null hypothesis is a conclusion. This is like a "guilty" verdict in a criminal trial: the evidence is sufficient to reject innocence, thus proving guilt.
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In mathematics, particularly topology, a cosmic space is any topological space that is a continuous image of some separable metric space. Equivalently (for regular T1 spaces but not in general), a space is cosmic if and only if it has a countable network; namely a countable collection of subsets of the space such that any open set is the union of a subcollection of these sets. Cosmic spaces have several interesting properties. There are a number of unsolved problems about them.
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The greatest length of any of these paths is the diameter of the graph. A central vertex in a graph of radius r is one whose eccentricity is r--that is, a vertex that achieves the radius or, equivalently, a vertex v such that \epsilon(v) = r. A peripheral vertex in a graph of diameter d is one that is distance d from some other vertex--that is, a vertex that achieves the diameter. Formally, v is peripheral if \epsilon(v) = d.
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Thus AK/AB = CD/BD, and CK/BC = DA/BD; equivalently, AK·BD = AB·CD, and CK·BD = BC·DA. By adding two equalities we have AK·BD + CK·BD = AB·CD + BC·DA, and factorizing this gives (AK+CK)·BD = AB·CD + BC·DA. But AK+CK = AC, so AC·BD = AB·CD + BC·DA, Q.E.D.. The proof as written is only valid for simple cyclic quadrilaterals. If the quadrilateral is self-crossing then K will be located outside the line segment AC. But in this case, AK−CK=±AC, giving the expected result.
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They decompose slowly at room temperature. Thorium forms the monocapped trigonal prismatic anion [Th(CH3)7]3−, heptamethylthorate, which forms the salt [Li(tmeda)]3[ThMe7] (tmeda= Me2NCH2CH2NMe2). Although one methyl group is only attached to the thorium atom (Th–C distance 257.1 pm) and the other six connect the lithium and thorium atoms (Th–C distances 265.5–276.5 pm), they behave equivalently in solution. Tetramethylthorium, Th(CH3)4, is not known, but its adducts are stabilised by phosphine ligands.
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The Gospels refer to him not only as a Messiah but in the absolute form as "the Messiah" or, equivalently, "the Christ". In early Judaism, this absolute form of the title is not found, but only phrases such as "his Messiah". The tradition is ambiguous enough to leave room for debate as to whether Jesus defined his eschatological role as that of the Messiah. The Jewish messianic tradition included many different forms, some of them focused on a Messiah figure and others not.
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Equivalently, by the Pythagorean theorem, they are the odd prime numbers p for which is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers p for which p itself is the hypotenuse of a primitive Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; is the hypotenuse of a right triangle with legs 1 and 2, and 5 itself is the hypotenuse of a right triangle with legs 3 and 4.
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A well-covered graph, the intersection graph of the nine diagonals of a hexagon. The three red vertices form one of its 14 equal-sized maximal independent sets, and the six blue vertices form the complementary minimal vertex cover. In graph theory, a well-covered graph is an undirected graph in which every minimal vertex cover has the same size as every other minimal vertex cover. Equivalently, these are the graphs in which every maximal independent set has the same size.
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Equivalently, every dependent set contains a finite dependent set. Examples are linear dependence of arbitrary subsets of infinite-dimensional vector spaces (but not infinite dependencies as in Hilbert and Banach spaces), and algebraic dependence in arbitrary subsets of field extensions of possibly infinite transcendence degree. Again, the class of finitary matroid is not self-dual, because the dual of a finitary matroid is not finitary. Finitary infinite matroids are studied in model theory, a branch of mathematical logic with strong ties to algebra.
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An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety. For example, in the nineteenth century there was much interest in hyperelliptic integrals that may be expressed in terms of elliptic integrals. This comes down to asking that J is a product of elliptic curves, up to an isogeny.
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Conversely, every instance of the longest decreasing subsequence problem can be described equivalently as a problem of finding a maximum clique in a permutation graph., p. 159. provide an alternative quadratic-time algorithm for maximum cliques in comparability graphs, a broader class of perfect graphs that includes the permutation graphs as a special case. In chordal graphs, the maximal cliques can be found by listing the vertices in an elimination ordering, and checking the clique neighborhoods of each vertex in this ordering.
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Any monadic Boolean algebra can be considered to be an interior algebra where the interior operator is the universal quantifier and the closure operator is the existential quantifier. The monadic Boolean algebras are then precisely the variety of interior algebras satisfying the identity xIC = xI. In other words, they are precisely the interior algebras in which every open element is closed or equivalently, in which every closed element is open. Moreover, such interior algebras are precisely the semisimple interior algebras.
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A sufficient sample-rate is therefore anything larger than 2B samples per second. Equivalently, for a given sample rate f_s, perfect reconstruction is guaranteed possible for a bandlimit B < f_s/2. When the bandlimit is too high (or there is no bandlimit), the reconstruction exhibits imperfections known as aliasing. Modern statements of the theorem are sometimes careful to explicitly state that x(t) must contain no sinusoidal component at exactly frequency , or that must be strictly less than the sample rate.
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The kernel useful in classifying the properties of prefilters and other families of sets. If then for any point , if and only if . Example: The set of all cofinite subsets of (meaning those sets whose complement in is finite) is proper if and only if is infinite (or equivalently, is infinite), in which case is a filter on known as the _Fréchet_ or _cofinite filter_ on . If finite then is equal to the dual ideal , which is not a filter.
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The value of the electrode potential under non-equilibrium depends on the nature and composition of the contacting phases, and on the kinetics of electrode reactions at the interface (see Butler–Volmer equation). An operational assumption for determinations of the electrode potentials with the standard hydrogen electrode involves this reference electrode with hydrogen ion in an ideal solution having is "zero potential at all temperatures" equivalently to standard enthalpy of formation of hydrogen ion is also "zero at all temperatures".
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A chart to convert between degrees and radians In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions have simpler and more "natural" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2' radians, so 180° is equal to radians, or equivalently, the degree is a mathematical constant: 1° = .
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The convex hull or lower convex envelope of a function f on a real vector space is the function whose epigraph is the lower convex hull of the epigraph of f. It is the unique maximal convex function majorized by f. The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of the union of their epigraphs, or equivalently from their pointwise minimum) and, in this form, is dual to the convex conjugate operation.
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The Hilbert space for n particles is given by the tensor product \otimes_n H . The permutation group of S_n acts on this space by permuting the entries. By definition the expectation values for an observable a of n indistinguishable particles should be invariant under these permutation. This means that for all \psi \in H and \sigma \in S_n : (\sigma \Psi )^t a (\sigma \Psi) = \Psi^t a \Psi, or equivalently for each \sigma \in S_n : \sigma^t a \sigma = a .
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Earth's energy balance may be measured by Argo floats by measuring the temperature anomaly or equivalently, the accumulation of ocean heat content. Ocean heat content was unchanged in the northern extra-tropical ocean and in the tropical ocean during the 2005-2014 time frame. Ocean heat content increased only in the extra-tropical southern ocean. There is no known reason that the extra-tropical southern ocean will experience ocean heat content increases while ocean heat content remains constant over the bulk of the measured ocean.
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In computational complexity theory, the complexity class TFNP is the class of total function problems which can be solved in nondeterministic polynomial time. That is, it is the class of function problems that are guaranteed to have an answer, and this answer can be checked in polynomial time, or equivalently it is the subset of FNP where a solution is guaranteed to exist. The abbreviation TFNP stands for "Total Function Nondeterministic Polynomial". TFNP contains many natural problems that are of interest to computer scientists.
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With doses that usually range from 10 to 150 mg, users are likely to experience effects similar to heroin, morphine, and fentanyl such as euphoria and respiratory depression. When an overdose occurs users often experience tachycardia, hypertension, and seizures. Mice, dogs, and monkeys, have been used in tests which showed the drug was almost equivalently potent to morphine, and had a very steep dose response curve. Rats given 20 mg doses three times a day for five days, experienced withdrawal symptoms similar to other opioids.
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An Archimedean field is an ordered field such that for each element there exists a finite expression : whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no infinitesimals (elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of . Each bounded real set has a least upper bound. An ordered field is Dedekind-complete if all upper bounds, lower bounds (see Dedekind cut) and limits, which should exist, do exist.
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Absolute risk aversion is decreasing if A'(W) < 0 (equivalently T '(W) > 0), which occurs if and only if \gamma is finite and less than 1; this is considered the empirically plausible case, since it implies that an investor will put more funds into risky assets the more funds are available to invest. Constant absolute risk aversion occurs as \gamma goes to positive or negative infinity, and the particularly implausible case of increasing absolute risk aversion occurs if \gamma is greater than one and finite.
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In other words, a scale invariant theory is one without any fixed length scale (or equivalently, mass scale) in the theory. For a scalar field theory with spacetime dimensions, the only dimensionless parameter satisfies = . For example, in = 4, only is classically dimensionless, and so the only classically scale-invariant scalar field theory in = 4 is the massless 4 theory. Classical scale invariance, however, normally does not imply quantum scale invariance, because of the renormalization group involved – see the discussion of the beta function below.
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If so, Paul would be thinking here of Christ as the ideal Adam, with his humanity perfectly expressing the divine image. But this exegesis is not fully convincing.As the divine eikōn or image (2 Cor. 4:4), Christ reveals God. The "glory" which becomes visible on the face of Christ is his own glory or, equivalently, "the glory of God" (2 Cor. 4:6). Cf. J.A. Fitzmyer, "Glory Reflected on the Face of Christ and a Palestinian Jewish Motif", Theological Studies, 42 (1981), pp.
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Seeking a guitar-tuning that would facilitate improvisation using twelve-tones, he introduced major-thirds tuning by 1964, perhaps in 1963. To achieve the E-E open-string range of standard (Spanish) tuning, Patt started using seven-string guitars in 1963, before settling on eight-string guitars with high G (equivalently A) as their highest open-notes. Patt used major-thirds tuning during all of his work as a session musician after 1965 in New York. Patt developed a webpage with extensive information about major- thirds tuning.
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There are many widely used formulas having terms involving natural-number exponents that require to be evaluated to . For example, regarding as an empty product assigns it the value , even when . Alternatively, the combinatorial interpretation of is the number of empty tuples of elements from a set with elements; there is exactly one empty tuple, even if . Equivalently, the set-theoretic interpretation of is the number of functions from the empty set to the empty set; there is exactly one such function, the empty function.
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The check digit is chosen so that the sum of all digits in the bar code is a multiple of 10. Equivalently, the modulo-10 sum is 0. To calculate the check digit: #Add up the digits. For example, if a letter is sent to Young America, Minnesota, it might be sent to 55555-1237, which would have the sum of 38. #Find the remainder of this number when it is divided by 10, in this case 8. This is also known as the sum modulo 10.
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Gentzen's proof is the first example of what is called proof-theoretical ordinal analysis. In ordinal analysis one gauges the strength of theories by measuring how large the (constructive) ordinals are that can be proven to be well-ordered, or equivalently for how large a (constructive) ordinal can transfinite induction be proven. A constructive ordinal is the order type of a recursive well-ordering of natural numbers. Laurence Kirby and Jeff Paris proved in 1982 that Goodstein's theorem cannot be proven in Peano arithmetic.
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In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}. Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable.
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The "cyclic order = arranging in a circle" idea works because any subset of a cycle is itself a cycle. In order to use this idea to impose cyclic orders on sets that are not actually subsets of the unit circle in the plane, it is necessary to consider functions between sets. A function between two cyclically ordered sets, , is called a monotonic function or a homomorphism if it pulls back the ordering on : whenever , one has . Equivalently, is monotone if whenever and , and are all distinct, then .
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At optical frequencies the electromagnetic analogs to quantum tunneling involve wave propagation in photonic bandgap structures and frustrated total internal reflection at the interface between two prisms in close contact. Spielmann, et al sent 12 fs (FWHM) laser pulses through the stop band of a multilayer dielectric structure. They found that the measured group delay was independent of the number of layers, or equivalently, the thickness of the photonic barrier, thus confirming the Hartman effect for tunneling light waves. In another optical experiment, Longhi, et al.
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There is an alternative definition of fold change, although this has generally fallen out of use. Here, fold change is defined as the ratio of the difference between final value and the initial value divided by the initial value. For quantities A and B, the fold change is given as (B − A)/A, or equivalently B/A − 1. This formulation has appealing properties such as no change being equal to zero, a 100% increase is equal to 1, and a 100% decrease is equal to −1.
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In some dialects of English there is a distinction between two vowel heights of reduced vowels: in addition to schwa, there is a distinct near-close central unrounded vowel (or equivalently ). In the British phonetic tradition, the latter vowel is represented with the symbol , and in the American tradition . An example of a minimal pair contrasting these two reduced vowels is Rosas vs. roses: the a in Rosa's is a schwa, while the e in roses (for speakers who make the distinction) is the near-close vowel.
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In 1983 Guillermo Calvo wrote "Staggered Prices in a Utility-Maximizing Framework". The original article was written in a continuous time mathematical framework, but nowadays is mostly used in its discrete time version. The Calvo model has become the most common way to model nominal rigidity in new Keynesian models. There is a probability that the firm can reset its price in any one period h (the hazard rate), or equivalently the probability (1-h) that the price will remain unchanged in that period (the survival rate).
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Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two. The description of photon polarization contains many of the physical concepts and much of the mathematical machinery of more involved quantum descriptions, such as the quantum mechanics of an electron in a potential well.
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The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The i^{th} principal component can be taken as a direction orthogonal to the first i-1 principal components that maximizes the variance of the projected data. From either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix.
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In mathematics, a taut foliation is a codimension 1 foliation of a 3-manifold with the property that there is a single transverse circle intersecting every leaf. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Taut foliations were brought to prominence by the work of William Thurston and David Gabai.
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Each permutation in the sequence that it generates differs from the previous permutation by swapping two adjacent elements of the sequence. Equivalently, this algorithm finds a Hamiltonian cycle in the permutohedron. This method was known already to 17th-century English change ringers, and calls it "perhaps the most prominent permutation enumeration algorithm". As well as being simple and computationally efficient, it has the advantage that subsequent computations on the permutations that it generates may be sped up because these permutations are so similar to each other..
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A smooth plane curve is a curve in a real Euclidean plane R2 and is a one-dimensional smooth manifold. This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function. Equivalently, a smooth plane curve can be given locally by an equation , where is a smooth function, and the partial derivatives and are never both 0 at a point of the curve.
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The complete list of all trees on 2,3,4 labeled vertices: 2^{2-2}=1 tree with 2 vertices, 3^{3-2}=3 trees with 3 vertices and 4^{4-2}=16 trees with 4 vertices. In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n, the number of trees on n labeled vertices is n^{n-2}. The formula equivalently counts the number of spanning trees of a complete graph with labeled vertices .
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In the latter case, there are two types: A Fuchsian group of the first type is a group for which the limit set is the closed real line R ∪ ∞. This happens if the quotient space H/Γ has finite volume, but there are Fuchsian groups of the first kind of infinite covolume. Otherwise, a Fuchsian group is said to be of the second type. Equivalently, this is a group for which the limit set is a perfect set that is nowhere dense on R ∪ ∞.
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To detect the mode water we can use the minimum value in the vertical gradient of potential density, or equivalently in the Brunt–Väisälä frequency. Since temperature profiles are more abundant and both salinity and temperature are relatively homogeneous in mode water, vertical temperature gradients are sometimes used instead of potential vorticity or the vertical gradient of potential density to identify the core of the mode water. There is no specific values of those gradients to define the boundaries of a given mode water.
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In category theory, a branch of mathematics, an antiisomorphism (or anti- isomorphism) between structured sets A and B is an isomorphism from A to the opposite of B (or equivalently from the opposite of A to B). If there exists an antiisomorphism between two structures, they are said to be antiisomorphic. Intuitively, to say that two mathematical structures are antiisomorphic is to say that they are basically opposites of one another. The concept is particularly useful in an algebraic setting, as, for instance, when applied to rings.
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Two paths of type a = b resp. b = c can be concatenated, forming a path of type a = c; this corresponds to the transitive property of equality. Most importantly, given a path p:a=b, and a proof of some property P(a), the proof can be "transported" along the path p to yield a proof of the property P(b). (Equivalently stated, an object of type P(a) can be turned into an object of type P(b).) This corresponds to the substitution property of equality.
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A taut foliation is a codimension 1 foliation of a 3-manifold with the property that there is a single transverse circle intersecting every leaf. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Taut foliations were brought to prominence by the work of William Thurston and David Gabai.
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The fundamental theorem of algebra states that every non-constant single- variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.
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Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequency do not change as a result of the scattering. This limit is valid as long as the photon energy is much smaller than the mass energy of the particle: u\ll mc^2/h , or equivalently, if the wavelength of the light is much greater than the Compton wavelength of the particle.
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If a sample is randomly constituted from a population, the sample ATE (abbreviated SATE) is also an estimate of the population ATE (abbreviated PATE). While an experiment ensures, in expectation, that potential outcomes (and all covariates) are equivalently distributed in the treatment and control groups, this is not the case in an observational study. In an observational study, units are not assigned to treatment and control randomly, so their assignment to treatment may depend on unobserved or unobservable factors. Observed factors can be statistically controlled (e.g.
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The Turing degrees are partially ordered by Turing reducibility. The notation a ≤T b indicates there is a set in degree b that computes a set in degree a. Equivalently, a ≤T b holds if and only if every set in b computes every set in a. A function f from the natural numbers to the natural numbers is said to be diagonally nonrecursive (DNR) if, for all n, f(n) ot = \phi_n(n) (here inequality holds by definition if \phi_n(n) is undefined).
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A continuous map f\colon X \to Y of simply connected topological spaces is called a rational homotopy equivalence if it induces an isomorphism on homotopy groups tensored with the rational numbers \Q. Equivalently: f is a rational homotopy equivalence if and only if it induces an isomorphism on singular homology groups with rational coefficients.Félix, Halperin & Thomas (2001), Theorem 8.6. The rational homotopy category (of simply connected spaces) is defined to be the localization of the category of simply connected spaces with respect to rational homotopy equivalences.
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Living polymerization was demonstrated by Michael Szwarc in 1956 in the anionic polymerization of styrene with an alkali metal / naphthalene system in tetrahydrofuran (THF). Szwarc showed that electron transfer occurred from radical anion of naphthalene to styrene. The initial radical anion of styrene converts to a dianion (or equivalently disodio-) species, which rapidly added styrene to form a "two – ended living polymer." An important aspect of his work, Szwarc employed the aprotic solvent tetrahydrofuran, which dissolves but is otherwise unreactive toward the organometallic intermediates.
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In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. The triangle-free graphs with the most edges for their vertices are balanced complete bipartite graphs. Many triangle-free graphs are not bipartite, for example any cycle graph Cn for odd n > 3\.
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For instance, a cubic crystal may have low-energy planes on the faces of the cube or on the diagonals. The planes are low-energy in the sense that if the crystal is cleaved along these planes, there will be relatively few broken bonds and a relatively small increase in energy over the unbroken crystal. Equivalently, these planes have a low surface energy. The planes with the lowest energy will form the largest facets, in order to minimize the overall thermodynamic free energy of the crystal.
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For an n-bit multiplier, this will take n clock cycles (where each cycle does either a shift or a shift-and-add). To convert this into an algorithm for modular multiplication, with a modulus r, it is necessary to subtract r conditionally at each stage: #Double the contents of the accumulator. #If the result is greater than or equal to r, subtract r. (Equivalently, subtract r from the accumulator and store the result back into the accumulator if and only if it is non-negative).
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Silver forcing (after Jack Howard Silver) is the set of all those partial functions from the natural numbers into } whose domain is coinfinite; or equivalently the set of all pairs , where A is a subset of the natural numbers with infinite complement, and p is a function from A into a fixed 2-element set. A condition q is stronger than a condition p if q extends p. Silver forcing satisfies Fusion, the Sacks property, and is minimal with respect to reals (but not minimal).
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Price–sales ratio, P/S ratio, or PSR, is a valuation metric for stocks. It is calculated by dividing the company's market capitalization by the revenue in the most recent year; or, equivalently, divide the per-share stock price by the per-share revenue. Also, justified p/s is calculated as (profit margin × payout × (1 + g)/(r − g)). Unless otherwise stated, P/S is "trailing twelve months" (TTM), the reported sales for the four previous quarters, although of course longer time periods can be examined.
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The example above was tested in Quicken 2007, Quicken 2008, Quicken 2010, Quicken 2012, Quicken 2015 and an equivalently formatted text file using "TCCard" instead of "TInvst" under Quicken 2011. If the transactions are being imported into an existing account, make sure to reconcile the account in the old data file before creating the QIF export file. Before accepting an import, validate any 'Match' transactions and change to 'New' where there isn't really a match. Quicken can match on amounts even when the dates are significantly different.
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This consists in avoiding to round to midpoints for the final rounding (except when the midpoint is exact). In binary arithmetic, the idea is to round the result toward zero, and set the least significant bit to 1 if the rounded result is inexact; this rounding is called sticky rounding. Equivalently, it consists in returning the intermediate result when it is exactly representable, and the nearest floating-point number with an odd significand otherwise; this is why it is also known as rounding to odd.
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Straight-line grammars are widely used in the development of algorithms that execute directly on compressed structures (without prior decompression). SLGs are of interest in fields like Kolmogorov complexity, Lossless data compression, Structure discovery and Compressed data structures. The problem of finding a context-free grammar (equivalently: an SLG) of minimal size that generates a given string is called the smallest grammar problem. Straight-line grammars (more precisely: straight-line context-free string grammars) can be generalized to Straight-line context-free tree grammars.
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Research into paraconsistent logic has also led to the establishment of the philosophical school of dialetheism (most notably advocated by Graham Priest), which asserts that true contradictions exist in reality, for example groups of people holding opposing views on various moral issues. Being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise embracing trivialism, i.e. accepting that all contradictions (and equivalently all statements) are true. However, the study of paraconsistent logics does not necessarily entail a dialetheist viewpoint.
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Equivalently, f is a quotient map if and only if it is the transfinite composition of maps X\rightarrow X/Z, where Z\subset X is a subset. Note that this doesn't imply that f is an open function. ;Quotient space: If X is a space, Y is a set, and f : X → Y is any surjective function, then the quotient topology on Y induced by f is the finest topology for which f is continuous. The space X is a quotient space or identification space.
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A negative cross elasticity denotes two products that are complements, while a positive cross elasticity denotes two substitute products. For example, if products A and B are complements, an increase in the price of B leads to a decrease in the quantity demanded for A. Equivalently, if the price of product B decreases, the demand curve for product A shifts to the right reflecting an increase in A's demand, resulting in a negative value for the cross elasticity of demand. The exact opposite reasoning holds for substitutes.
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Equivalently, the relative isotopic mass of an isotope or nuclide is the mass of the isotope relative to 1/12 of the mass of a carbon-12 atom. For example, the relative isotopic mass of a carbon-12 atom is exactly 12. For comparison, the atomic mass of a carbon-12 atom is exactly 12 daltons. Alternately, the atomic mass of a carbon-12 atom may be expressed in any other mass units: for example, the atomic mass of a carbon-12 atom is about .
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Seeking a guitar tuning that would facilitate jazz improvisation, Ralph Patt invented major-thirds tuning in 1963. Patt's tuning is a regular tuning, in the sense that all of the intervals between its successive open strings are major thirds; in contrast, standard guitar tuning has one major-third amid four fourths. Seven-string guitars are needed for major-thirds tuning to have the E-e' range of the standard tuning.: Having an eight-string instrument allowed Patt's guitar to have G (equivalently A) as its open note.
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Alias and alibi rotations The interpretation of a rotation matrix can be subject to many ambiguities. In most cases the effect of the ambiguity is equivalent to the effect of a rotation matrix inversion (for these orthogonal matrices equivalently matrix transpose). ; Alias or alibi (passive or active) transformation : The coordinates of a point may change due to either a rotation of the coordinate system (alias), or a rotation of the point (alibi). In the latter case, the rotation of also produces a rotation of the vector representing .
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In general, let M be a manifold. A line field on M is a function μ that assigns to each point p of M a line μ(p) through the origin in the tangent space Tp(M). Equivalently, one may say that μ(p) is an element of the projective tangent space PTp(M), or that μ is a section of the projective tangent bundle PT(M). In the study of complex dynamical systems, the manifold M is taken to be a Hersee surface.
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An atomic formula is a formula that contains no logical connectives nor quantifiers, or equivalently a formula that has no strict subformulas. The precise form of atomic formulas depends on the formal system under consideration; for propositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. According to some terminology, an open formula is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers.
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The Fibonacci cube of order n is the simplex graph of the complement graph of an n-vertex path graph., p.3. That is, each vertex in the Fibonacci cube represents a clique in the path complement graph, or equivalently an independent set in the path itself; two Fibonacci cube vertices are adjacent if the cliques or independent sets that they represent differ by the addition or removal of a single element. Therefore, like other simplex graphs, Fibonacci cubes are median graphs and more generally partial cubes.
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In mathematics, a free abelian group or free Z-module is an abelian group with a basis, or, equivalently, a free module over the integers. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}.
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In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives. The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, the atomic formulas are the propositional variables.
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Let w ∈ F(A) be such that w = 1 in G. Then the area of w, denoted Area(w), is defined as the minimum of the areas of all van Kampen diagrams with boundary labels w (van Kampen's lemma says that at least one such diagram exists). One can show that the area of w can be equivalently defined as the smallest n≥0 such that there exists a representation (♠) expressing w as a product in F(A) of n conjugates of the defining relators.
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As a homotopy theory, the notion of covering spaces works well when the deck transformation group is discrete, or, equivalently, when the space is locally path-connected. However, when the deck transformation group is a topological group whose topology is not discrete, difficulties arise. Some progress has been made for more complex spaces, such as the Hawaiian earring; see the references there for further information. A number of these difficulties are resolved with the notion of semicovering due to Jeremy Brazas, see the paper cited below.
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The proposal applies machine learning for anomaly detection, providing energy-efficiency to a Decision Tree, Naive-Bayes, and k-Nearest Neighbors classifiers implementation in an Atom CPU and its hardware-friendly implementation in a FPGA. In the literature, this was the first work that implement each classifier equivalently in software and hardware and measures its energy consumption on both. Additionally, it was the first time that was measured the energy consumption for extracting each features used to make the network packet classification, implemented in software and hardware.
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Systems that have nonzero energy but zero rest mass (such as photons moving in a single direction, or equivalently, plane electromagnetic waves) do not have COM frames, because there is no frame in which they have zero net momentum. Due to the invariance of the speed of light, a massless system must travel at the speed of light in any frame, and always possesses a net momentum. Its energy is—for each reference frame—equal to the magnitude of momentum multiplied by the speed of light: : E = p c .
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A polytree is a directed graph formed by orienting the edges of a free tree.. Every polytree is a DAG. In particular, this is true of the arborescences formed by directing all edges outwards from the roots of a tree. A multitree (also called a strongly unambiguous graph or a mangrove) is a directed graph in which there is at most one directed path (in either direction) between any two vertices; equivalently, it is a DAG in which, for every vertex , the subgraph reachable from forms a tree..
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Presbyopia The cause of presbyopia is lens hardening by decreasing levels of -crystallin, a process which may be sped up by higher temperatures. It results in a near point greater than 25 cm (or equivalently, less than 4 diopters). In optics, the closest point at which an object can be brought into focus by the eye is called the eye's near point. A standard near point distance of 25 cm is typically assumed in the design of optical instruments, and in characterizing optical devices such as magnifying glasses.
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Spironolactone and its metabolite canrenone are highly plasma protein bound, with percentages of 88.0% and 99.2%, respectively. Spironolactone is bound equivalently to albumin and α1-acid glycoprotein, while canrenone is bound only to albumin. Spironolactone and its metabolite 7α-thiospironolactone show very low or negligible affinity for sex hormone-binding globulin (SHBG). In accordance, a study of high-dosage spironolactone treatment found no change in steroid binding capacity related to SHBG or to corticosteroid-binding globulin (CBG), suggesting that spironolactone does not displace steroid hormones from their carrier proteins.
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The bad scenario is that our traversal leads us to two different pieces of the same partner, or equivalently to the other piece of partner i from whom we started. Such a path, leading from one piece of partner i to another piece of the same partner, is called a pair path. If the implication graph contains no pair paths, then the selection algorithm just described returns a collection of n non-overlapping final pieces and we are done. Now it remains to calculate the probability that the implication graph contains a pair path.
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Most machines of this type are designed to be operated on three-phase electric power and are generally of more rugged construction than equivalently sized belt-driven units. Virtually all examples have geared racks for adjusting the table and head position on the column. Geared head drill presses are commonly found in tool rooms and other commercial environments where a heavy duty machine capable of production drilling and quick setup changes is required. In most cases, the spindle is machined to accept Morse taper tooling for greater flexibility.
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Since the tails are hydrophobic, their exposure to air is favoured over that to water. Similarly, since the heads are hydrophilic, the head–water interaction is more favourable than air–water interaction. The overall effect is reduction in the surface energy (or equivalently, surface tension of water). File:Surfactant.jpg Figure 1: Surfactant molecules arranged on an air–water interface For very small concentrations, far from the surface density compatible with the collapse of the monolayer (which leads to polylayers structures) the surfactant molecules execute a random motion on the water–air interface.
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In the spacial case where the dualities are the canonical dualities and , the transpose of a linear map is always well- defined. This transpose is called the algebraic adjoint of F and it will be denoted by ; that is, . In this case, for all , where the defining condition for is: : for all , or equivalently, for all . ;Examples If for some integer , is a basis for with dual basis , is a linear operator, and the matrix representation of with respect to is , then the transpose of is the matrix representation with respect to ℰ of .
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In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint. If the constraint (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.
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Strongly chordal graphs are graphs that are chordal and contain no n-sun (for n ≥ 3) as an induced subgraph. Here an n-sun is an n-vertex chordal graph G together with a collection of n degree-two vertices, adjacent to the edges of a Hamiltonian cycle in G. K-trees are chordal graphs in which all maximal cliques and all maximal clique separators have the same size.. Apollonian networks are chordal maximal planar graphs, or equivalently planar 3-trees. Maximal outerplanar graphs are a subclass of 2-trees, and therefore are also chordal.
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Every Hermitian symmetric space is a Kähler manifold. They can be defined equivalently as Riemannian symmetric spaces with a parallel complex structure with respect to which the Riemannian metric is Hermitian. The complex structure is automatically preserved by the isometry group H of the metric, and so any Hermitian symmetric space M is a homogeneous complex manifold. Some examples are complex vector spaces and complex projective spaces, with their usual Hermitian metrics and Fubini–Study metrics, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric.
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The Prime Pages, The Prime Glossary: megaprime. All three were the first known prime of any kind of that size. The number of digits in the decimal representation of equals , where denotes the floor function (or equivalently ). In September 2008, mathematicians at UCLA participating in the Great Internet Mersenne Prime Search (GIMPS) won part of a $100,000 prize from the Electronic Frontier Foundation for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits.
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W. T. Tutte conjectured that every snark has the Petersen graph as a minor. That is, he conjectured that the smallest snark, the Petersen graph, may be formed from any other snark by contracting some edges and deleting others. Equivalently (because the Petersen graph has maximum degree three) every snark has a subgraph that can be formed from the Petersen graph by subdividing some of its edges. This conjecture is a strengthened form of the four color theorem, because any graph containing the Petersen graph as a minor must be nonplanar.
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In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of Aronszajn and Panitchpakdi (1956; see e.g. ) that these two different types of definitions are equivalent.
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The set of the elements of L that are algebraic over K form a subextension, which is called the algebraic closure of K in L. This results from the preceding characterization: if s and t are algebraic, the extensions and are finite. Thus is also finite, as well as the sub extensions , and (if ). It follows that , st and 1/s are all algebraic. An algebraic extension is an extension such that every element of L is algebraic over K. Equivalently, an algebraic extension is an extension that is generated by algebraic elements.
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The vector calculus operations of grad, curl, and div are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and , these all being 3-dimensional spaces.
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In fact, only single rules over extensional predicate symbols can be easily rewritten as an equivalent conjunctive query. The problem of deciding whether for a given Datalog program there is an equivalent nonrecursive program (corresponding to a positive relational algebra query, or, equivalently, a formula of positive existential first-order logic, or, as a special case, a conjunctive query) is known as the Datalog boundedness problem and is undecidable.Gerd G. Hillebrand, Paris C. Kanellakis, Harry G. Mairson, Moshe Y. Vardi: Undecidable Boundedness Problems for Datalog Programs. J. Log. Program.
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The Finite Field Diffie-Hellman algorithm has roughly the same key strength as RSA for the same key sizes. The work factor for breaking Diffie-Hellman is based on the discrete logarithm problem, which is related to the integer factorization problem on which RSA's strength is based. Thus, a 2048-bit Diffie-Hellman key has about the same strength as a 2048-bit RSA key. Elliptic-curve cryptography (ECC) is an alternative set of asymmetric algorithms that is equivalently secure with shorter keys, requiring only approximately twice the bits as the equivalent symmetric algorithm.
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The white vertex sets are maximal nonblockers In graph theory, a nonblocker is a subset of vertices in an undirected graph, all of which are adjacent to vertices outside of the subset. Equivalently, a nonblocker is the complement of a dominating set. The computational problem of finding the largest nonblocker in a graph was formulated by , who observed that it belongs to MaxSNP. Although computing a dominating set is not fixed-parameter tractable under standard assumptions, the complementary problem of finding a nonblocker of a given size is fixed-parameter tractable.
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Some properties of free groups follow readily from the definition: #Any group G is the homomorphic image of some free group F(S). Let S be a set of generators of G. The natural map f: F(S) → G is an epimorphism, which proves the claim. Equivalently, G is isomorphic to a quotient group of some free group F(S). The kernel of φ is a set of relations in the presentation of G. If S can be chosen to be finite here, then G is called finitely generated.
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A formal language that can be described by a context-sensitive grammar, or, equivalently, by a noncontracting grammar or a linear bounded automaton, is called a context-sensitive language. Some textbooks actually define CSGs as non-contracting, although this is not how Noam Chomsky defined them in 1959. This choice of definition makes no difference in terms of the languages generated (i.e. the two definitions are weakly equivalent), but it does make a difference in terms of what grammars are structurally considered context-sensitive; the latter issue was analyzed by Chomsky in 1963.
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An icon for the amazon used in diagrams An amazon (also known as a queen+knight compound) is a fairy chess piece that can move like a queen or a knight (or, equivalently, like a rook, bishop, or knight). It may thus be considered the sum of all orthodox chess pieces (not including pawns, nor the royal powers of a king). It cannot jump over other pieces when moving as a queen, but may do so when moving as a knight. Chess moves in this article use A as notation for the amazon.
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141); Parry (1974: p.149) It was popularized by Martín Cortés de Albacar in his 1551 Breve compendio la esfera y del arte de navegar. The "Regiment of the Leagues" is not very different from the rule of marteloio. The Regiment of the Leagues always considers the west-east bearing as the "intended course" and measures set deviations from it. More specifically, the league table considers a fixed value of alargar – set at 1 latitude degree (or, in the measurements of the time, 17.5 (Portuguese) leagues, or equivalently 70 (Italian) miles).
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Methods based on statistical inference attempt to fit a generative model to the network data, which encodes the community structure. The overall advantage of this approach compared to the alternatives is its more principled nature, and the capacity to inherently address issues of statistical significance. Most methods in the literature are based on the stochastic block model as well as variants including mixed membership, degree-correction, and hierarchical structures. Model selection can be performed using principled approaches such as minimum description length (or equivalently, Bayesian model selection) and likelihood-ratio test.
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The Football Association Challenge Vase, usually referred to as the FA Vase, is an annual football competition for teams playing in Steps 5 and 6 of the English National League System (or equivalently, tier 9 or 10 of the overall English football league system). For the 2017–18 season 619 entrants were accepted, with two qualifying rounds preceding the six proper rounds, semi- finals (played over two legs) and final to be played at Wembley Stadium. The 2019 winners were Chertsey Town, who beat Cray Valley Paper Mills 3–1 at Wembley Stadium.
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Yet, there are infinitely many nonhypotenuse numbers, and the number of nonhypotenuse numbers not exceeding a value x scales asymptotically with x/.. This review of a manuscript of Beiler's (which was later published in J. Rec. Math. 7 (1974) 120–133, ) attributes this bound to Landau. The nonhypotenuse numbers are those numbers that have no prime factors of the form 4k+1.. Equivalently, they are the number that cannot be expressed in the form K(m^2+n^2) where K, m, and n are all positive integers.
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A typical example of a monotone function is the following function on the cycle with 6 elements: : : : A function is called an embedding if it is both monotone and injective. Equivalently, an embedding is a function that pushes forward the ordering on : whenever , one has . As an important example, if is a subset of a cyclically ordered set , and is given its natural ordering, then the inclusion map is an embedding. Generally, an injective function from an unordered set to a cycle induces a unique cyclic order on that makes an embedding.
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One can understand the Cayley–Bacharach theorem, and why it arises for degree 3, by dimension counting. Simply stated, nine points determine a cubic, but in general define a unique cubic. Thus if the nine points lie on more than one cubic, equivalently on the intersection of two cubics (as ), they are not in general position – they are overdetermined by one dimension – and thus cubics passing through them satisfying one additional constraint, as reflected in the "eight implies nine" property. The general phenomenon is called superabundance; see Riemann–Roch theorem for surfaces.
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The nine blue vertices form a maximum independent set for the Generalized Petersen graph GP(12,4). In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. The size of an independent set is the number of vertices it contains.
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Round Mountain Oil Field Geologic Cross Section Oil in the Round Mountain Field comes from four primary pools, the Freeman-Jewett, Pyramid Hill, Vedder, and Walker. Each resides in a sedimentary formation equivalently named, and are listed from top to bottom in stratigraphic sequence. The Freeman-Jewett and Pyramid Hill are of Miocene Age; the Vedder of Oligocene; and the lowest-lying productive unit, the Walker formation, was deposited during the Eocene and Oligocene. Of these units, the Vedder has been by far the most productive, with over of oil being extracted.
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Some PSL groups arise as automorphism groups of Hurwitz surfaces, i.e., as quotients of the (2,3,7) triangle group, which is the symmetries of the order-3 bisected heptagonal tiling. PSL groups arise as Hurwitz groups (automorphism groups of Hurwitz surfaces – algebraic curves of maximal possibly symmetry group). The Hurwitz surface of lowest genus, the Klein quartic (genus 3), has automorphism group isomorphic to PSL(2, 7) (equivalently GL(3, 2)), while the Hurwitz surface of second-lowest genus, the Macbeath surface (genus 7), has automorphism group isomorphic to PSL(2, 8).
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The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, then the temperature distribution of the rod t = 1 time units later will be given by the function F. By using values of t different from 1, we can define the generalized Weierstrass transform of . The generalized Weierstrass transform provides a means to approximate a given integrable function arbitrarily well with analytic functions.
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In competition ranking, items that compare equal receive the same ranking number, and then a gap is left in the ranking numbers. The number of ranking numbers that are left out in this gap is one less than the number of items that compared equal. Equivalently, each item's ranking number is 1 plus the number of items ranked above it. This ranking strategy is frequently adopted for competitions, as it means that if two (or more) competitors tie for a position in the ranking, the position of all those ranked below them is unaffected (i.e.
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In dense ranking, items that compare equally receive the same ranking number, and the next item(s) receive the immediately following ranking number. Equivalently, each item's ranking number is 1 plus the number of items ranked above it that are distinct with respect to the ranking order. Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 2 ("joint second"), C also gets ranking number 2 ("joint second") and D gets ranking number 3 ("Third").
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Duffie and Rahi also devote a considerable section to examining the utility and efficiency implications of financial innovation. This is also the topic of many of the papers in the special edition of the Journal of Economic Theory in which theirs is the lead article. The usefulness of spanning the market appears to be limited (or, equivalently, the disutility of incomplete markets is not great). Allen and Gale (1988) is one of the first papers to endogenize security issuance contingent on financial regulation—specifically, bans on short sales.
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In economics, hedonic regression or hedonic demand theory is a revealed preference method of estimating the demand for a good, or equivalently its value to consumers. It breaks down the item being researched into its constituent characteristics, and obtains estimates of the contributory value of each characteristic. This requires that the composite good being valued can be reduced to its constituent parts and the market values of those constituent parts. Hedonic models are most commonly estimated using regression analysis, although more generalized models exist, such as sales adjustment grids.
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In the primality example, PRIME is the set of strings representing natural numbers that, when input into a computer running an algorithm that correctly tests for primality, the algorithm answers "yes, this number is prime". This "yes-no" format is often equivalently stated as "accept-reject"; that is, an algorithm "accepts" an input string if the answer to the decision problem is "yes" and "rejects" if the answer is "no". While some problems cannot easily be expressed as decision problems, they nonetheless encompass a broad range of computational problems.Arora and Barak p.
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The category of topological spaces, Top, admits a standard model category structure with the usual (Serre) fibrations and with weak equivalences as weak homotopy equivalences. The cofibrations are not the usual notion found here, but rather the narrower class of maps that have the left lifting property with respect to the acyclic Serre fibrations. Equivalently, they are the retracts of the relative cell complexes, as explained for example in Hovey's Model Categories. This structure is not unique; in general there can be many model category structures on a given category.
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Extension operators are the most natural way to define H^s(\Omega) for non-integer s (we cannot work directly on \Omega since taking Fourier transform is a global operation). We define H^s(\Omega) by saying that u \in H^s(\Omega) if and only if Au \in H^s(\R^n). Equivalently, complex interpolation yields the same H^s(\Omega) spaces so long as \Omega has an extension operator. If \Omega does not have an extension operator, complex interpolation is the only way to obtain the H^s(\Omega) spaces.
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Since chromatic number is an upper bound on the order of the maximum clique, the latter invariant is also at most degeneracy plus one. By using a greedy coloring algorithm on an ordering with optimal coloring number, one can graph color a k-degenerate graph using at most k + 1 colors.; . A k-vertex-connected graph is a graph that cannot be partitioned into more than one component by the removal of fewer than k vertices, or equivalently a graph in which each pair of vertices can be connected by k vertex-disjoint paths.
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In computational complexity theory, the Immerman–Szelepcsényi theorem states that nondeterministic space complexity classes are closed under complementation. It was proven independently by Neil Immerman and Róbert Szelepcsényi in 1987, for which they shared the 1995 Gödel Prize. In its general form the theorem states that NSPACE(s(n)) = co-NSPACE(s(n)) for any function s(n) ≥ log n. The result is equivalently stated as NL = co-NL; although this is the special case when s(n) = log n, it implies the general theorem by a standard padding argument.
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They proved that every irreducible belonged to an infinite family G(m, p, n) depending on 3 positive integer parameters (with p dividing m) or was one of 34 exceptional cases, which they numbered from 4 to 37.Lehrer and Taylor, p. 271. The group G(m, 1, n) is the generalized symmetric group; equivalently, it is the wreath product of the symmetric group Sym(n) by a cyclic group of order m. As a matrix group, its elements may be realized as monomial matrices whose nonzero elements are mth roots of unity.
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In functional analysis and related areas of mathematics, a topological vector spaces (TVS) is complete if every net (or equivalently, every filter) that is Cauchy with respect to the space's canonical uniformity always converges to some point. Said differently, a TVS is complete if its canonical uniformity is complete. The canonical uniformity on a TVS is the uniqueCompleteness of normed spaces and metrizable TVSs are defined in terms of norms and metrics. In contrast to the uniqueness of this translation-invariant uniformity, in general, many different norms (e.g.
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In a lattice, an element x is join-irreducible if x is not the join of a finite set of other elements. Equivalently, x is join-irreducible if it is neither the bottom element of the lattice (the join of zero elements) nor the join of any two smaller elements. For instance, in the lattice of divisors of 120, there is no pair of elements whose join is 4, so 4 is join-irreducible. An element x is join-prime if, whenever x ≤ y ∨ z, either x ≤ y or x ≤ z.
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The term irrumare is likewise a transitive verb, meaning to "insert one's penis into another person's mouth for suckling", and derives from the Latin word, rūma meaning "udder" (as in: "to give something to suck on"). A male who suckles a penis is denoted as a fellator or, equivalently, a pathicus (line 2). Catullus neither confirms nor denies the claim of Aurelius and Furius that he is "not a man", since sexual slang "irrumare" and "pedicare" while having sexual slang meaning of homosexuality, could also mean as little as "go to hell".
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In computer graphics and geography, the illumination angle of a surface with a light source (such as the Earth's surface and the Sun) is the angle between the inward surface normal and the direction of light. It can also be equivalently described as the angle between the tangent plane of the surface and another plane at right angles to the light rays. This means that the illumination angle of a certain point on Earth's surface is 0° if the Sun is precisely overhead and that it is 90° at sunset or sunrise.
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The same is true for simple polytopes of arbitrary dimension (Blind & Mani-Levitska 1987, proving a conjecture of Micha Perles).. Kalai (1988). gives a simple proof based on unique sink orientations. Because these polytopes' face lattices are determined by their graphs, the problem of deciding whether two three-dimensional or simple convex polytopes are combinatorially isomorphic can be formulated equivalently as a special case of the graph isomorphism problem. However, it is also possible to translate these problems in the opposite direction, showing that polytope isomorphism testing is graph-isomorphism complete.
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One can prove that, for all n, there exists one and only one pattern clearable in exactly n moves, or equivalently, one and only one clearable pattern consisting of exactly n seeds. These patterns require arbitrarily long rows of pits and n increases. For example, it can be seen on the right that the unique 5-seed pattern requires only 3 pits, but the 17-seed pattern requires 6 pits. The relationship between the required number of pits and the number of seeds can be described in the following way.
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For metric () gravity (as opposed to Palatini or metric- affine () gravity), the quadratic term is best constrained by fifth force measurements, since it leads to a Yukawa correction to the gravitational potential. The best current bounds are or equivalently The parameterized post-Newtonian formalism is designed to be able to constrain generic modified theories of gravity. However, () gravity shares many of the same values as General Relativity, and is therefore indistinguishable using these tests. In particular light deflection is unchanged, so () gravity, like General Relativity, is entirely consistent with the bounds from Cassini tracking.
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This curve has total curvature 6, and turning number 3, though it only has winding number 2 about p. Immersed plane curves have a well-defined turning number, which can be defined as the total curvature divided by 2\. This is invariant under regular homotopy, by the Whitney–Graustein theorem – topologically, it is the degree of the Gauss map, or equivalently the winding number of the unit tangent (which does not vanish) about the origin. Further, this is a complete set of invariants – any two plane curves with the same turning number are regular homotopic.
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The unit of measure of acceleration in the International System of Units (SI) is m/s2. However, to distinguish acceleration relative to free fall from simple acceleration (rate of change of velocity), the unit g (or g) is often used. One g is the force per unit mass due to gravity at the Earth's surface and is the standard gravity (symbol: gn), defined as metres per second squared,BIPM: Declaration on the unit of mass and on the definition of weight; conventional value of gn. or equivalently newtons of force per kilogram of mass.
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A simplicial object X in a category C is a contravariant functor :X : Δ -> C or equivalently a covariant functor :X: Δop -> C, where Δ still denotes the simplex category. When C is the category of sets, we are just talking about the simplicial sets that were defined above. Letting C be the category of groups or category of abelian groups, we obtain the categories sGrp of simplicial groups and sAb of simplicial abelian groups, respectively. Simplicial groups and simplicial abelian groups also carry closed model structures induced by that of the underlying simplicial sets.
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In contrast, extensive formations associated with large areas may be equivalently represented at many locations. Such formations may be listed either without a site, with a site or sites that represent the type locality, or with multiple sites of note. When a type locality is listed as the site for a formation with many good outcrops, the site is flagged with a note (). When a particular site of note is listed for an extensive fossil- bearing formation, but that site is somehow atypical, it is also flagged with a note ().
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Binary union is an associative operation; that is, for any sets A, B, and C, :A \cup (B \cup C) = (A \cup B) \cup C. The operations can be performed in any order, and the parentheses may be omitted without ambiguity (i.e., either of the above can be expressed equivalently as A ∪ B ∪ C). Similarly, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union. That is, A ∪ ∅ = A, for any set A. This follows from analogous facts about logical disjunction.
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Busch-Vishniac, pp. 18, 21 There is a limit to the applicability of this lumped element model. The model works well if the components are small enough that the time taken for a wave to cross them is insignificant, or equivalently, if there is no significant phase difference in the wave either side of the component. What amounts to significant depends on how accurate the model is required to be, but a common rule of thumb is to require components to be smaller than one sixteenth of a wavelength.
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For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal P of R is called a prime ideal if for any elements x, y\in R we have that xy \in P implies either x \in P or y\in P. Equivalently, P is prime if for any ideals I, J we have that IJ \subseteq P implies either I \subseteq P or J \subseteq P. This latter formulation illustrates the idea of ideals as generalizations of elements.
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3D model of a (uniform) triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides. Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes).
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The stretched octave, for example 2.01:1, sounds out of tune when played with true harmonic overtones, but in tune when played with tones whose overtones are stretched equivalently. In piano tuning, stretched octaves are commonly encountered, where the inharmonicity caused by string thickness and tension makes it necessary to widen every interval very slightly. See: stretched tuning. The octaves of Balinese gamelans are never tuned 2:1, but instead are stretched or compressed in a consistent manner throughout the range of each individual gamelan, due to the physical characteristics of their instruments.
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For people with PKU, a recommendation for children up to age 10 years is 200 to 500 mg/d; for older children and adults 220 to 1200 mg/day. Where in the range depends on body weight and age, and on monitoring blood concentration. Optimal health ranges (or "target ranges") are between 120 and 360 µmol/L or equivalently 2 to 6 mg/dL, and aimed to be achieved during at least the first 10 years,Chapter 55, page 255 in: to allow the brain to develop normally.
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It provides a method for constructing band- limited functions to overcome the effects of intersymbol interference. When consecutive symbols are transmitted over a channel by a linear modulation (such as ASK, QAM, etc.), the impulse response (or equivalently the frequency response) of the channel causes a transmitted symbol to be spread in the time domain. This causes intersymbol interference because the previously transmitted symbols affect the currently received symbol, thus reducing tolerance for noise. The Nyquist theorem relates this time-domain condition to an equivalent frequency-domain condition.
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Observing the previous equation, a trivial solution is found for the case dc/dξ = 0, that is when concentration is constant over ξ. This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (x \propto \sqrt t), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the distance (t \propto x^2); the square term gives the name parabolic law.See an animation of the parabolic law.
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When oil production first began in the mid-nineteenth century, the largest oil fields recovered fifty barrels of oil for every barrel used in the extraction, transportation, and refining. This ratio is often referred to as the Energy Return on Energy Investment (EROI or EROEI). Currently, between one and five barrels of oil are recovered for each barrel-equivalent of energy used in the recovery process. As the EROEI drops to one, or equivalently the Net energy gain falls to zero, the oil production is no longer a net energy source.
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A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph in which every pair of distinct vertices is connected by a directed edge with any one of the two possible orientations. Many of the important properties of tournaments were first investigated by in order to model dominance relations in flocks of chickens. Current applications of tournaments include the study of voting theory and social choice theory among other things.
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The Baer radical of a ring is the intersection of the prime ideals of the ring R. Equivalently it is the smallest semiprime ideal in R. The Baer radical is the lower radical of the class of nilpotent rings. Also called the "lower nilradical" (and denoted Nil∗R), the "prime radical", and the "Baer-McCoy radical". Every element of the Baer radical is nilpotent, so it is a nil ideal. For commutative rings, this is just the nilradical and closely follows the definition of the radical of an ideal.
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10-CHO-THF is produced from methylenetetrahydrofolate (CH2H4F) via a two step process. The first step generates 5,10-methenyltetrahydrofolate: :CH2H4F + NAD+ \rightleftharpoons CH2H2F + NADH + H+ In the second step 5,10-methenyltetrahydrofolate undergoes hydrolysis: :CH2H2F + H2O \rightleftharpoons CHO-H4F + The latter is equivalently written: :5,10-methenyltetrahydrofolate + H2O \rightleftharpoons 10-formyltetrahydrofolate 10-CHO-THF is also produced by the reaction :ATP + formate + tetrahydrofolate \rightleftharpoons ADP + phosphate + 10-formyltetrahydrofolate This reaction is catalyzed by formate- tetrahydrofolate ligase. It can be converted back into tetrahydrofolate (THF) by formyltetrahydrofolate dehydrogenase or THF and formate by formyltetrahydrofolate deformylase.
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A differentiable map that is a submersion at each point p\in M is called a submersion. Equivalently, is a submersion if its differential Df_p has constant rank equal to the dimension of . A word of warning: some authors use the term critical point to describe a point where the rank of the Jacobian matrix of at is not maximal.. Indeed, this is the more useful notion in singularity theory. If the dimension of is greater than or equal to the dimension of then these two notions of critical point coincide.
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An annulus Illustration of Mamikon's visual calculus method showing that the areas of two annuli with the same chord length are the same regardless of inner and outer radii. In mathematics, an annulus (the Latin word for "little ring" is anulus / annulus, with plural anuli / annuli) is a ring-shaped object, a region bounded by two concentric circles; equivalently, it is the set difference between two concentric disks. The adjectival form is annular (as in annular eclipse). The open annulus is topologically equivalent to both the open cylinder and the punctured plane.
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In crystallography, a lattice plane of a given Bravais lattice is a plane (or family of parallel planes) whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices) and intersect the Bravais lattice; equivalently, a lattice plane is any plane containing at least three noncollinear Bravais lattice points.Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: New York, 1976). All lattice planes can be described by a set of integer Miller indices, and vice versa (all integer Miller indices define lattice planes).
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The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions :f(z_1,z_2, \ldots, z_n) on the space of -tuples of complex numbers. As in complex analysis, which is the case but of a distinct character, these are not just any functions: they are supposed to be holomorphic or complex analytic, so that locally speaking they are power series in the variables . Equivalently, as it turns out, they are locally uniform limits of polynomials; or local solutions to the -dimensional Cauchy–Riemann equations.
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Open and closed maps are not necessarily continuous. Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact remains true even if one restricts oneself to metric spaces. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function is continuous if the preimage of every open set of is open in X. (Equivalently, if the preimage of every closed set of is closed in ).
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Voluntarism is "any metaphysical or psychological system that assigns to the will (Latin: voluntas) a more predominant role than that attributed to the intellect",Voluntarism (philosophy) – Britannica.com or equivalently "the doctrine that will is the basic factor, both in the universe and in human conduct". Voluntarism has appeared at various points throughout the history of philosophy, seeing application in the areas of metaphysics, psychology, political philosophy and theology. The term voluntarism was introduced by Ferdinand Tönnies into the philosophical literature and particularly used by Wilhelm Wundt and Friedrich Paulsen.
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The action of a group G on a set M is said to be transitive if, for every two elements s, t of M, there is some group element g such that g(s) = t. Equivalently, the set M forms a single orbit under the action of G. Of the examples above, the group {e, (1 2), (3 4), (1 2)(3 4)} of permutations of {1, 2, 3, 4} is not transitive (no group element takes 1 to 3) but the group of symmetries of a square is transitive on the vertices.
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A model due to West, Enquist, and Brown (hereafter WEB) suggests that -scaling arises because of efficiency in nutrient distribution and transport throughout an organism. In most organisms, metabolism is supported by a circulatory system featuring branching tubules (i.e., plant vascular systems, insect tracheae, or the human cardiovascular system). WEB claim that (1) metabolism should scale proportionally to nutrient flow (or, equivalently, total fluid flow) in this circulatory system and (2) in order to minimize the energy dissipated in transport, the volume of fluid used to transport nutrients (i.e.
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Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "reals" in many programming languages). In this context, the quadratic formula is not completely stable. This occurs when the roots have different order of magnitude, or, equivalently, when and are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root.
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Victor Love has stated that his earliest inspiration for starting the band came from Finnish rock bands where they derive their original glam rock poise from. Specifically, Love named Hanoi Rocks and HIM as inspiration. Musically, Dope Stars insert the catchy choruses of 1980s pop-metal bands such as Bon Jovi (or, equivalently, 1990s alternative pop bands like Weezer) into the industrial-metal and cyberpunk frameworks of White Zombie, Foetus, Frontline Assembly and KMFDM. Love has stated that the lyrical content has little to do with 1980s music.
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Over a thousand characters from the Latin script are encoded in the Unicode Standard, grouped in several basic and extended Latin blocks. The extended ranges contain mainly precomposed letters plus diacritics that are equivalently encoded with combining diacritics, as well as some ligatures and distinct letters, used for example in the orthographies of various African languages (including click symbols in Latin Extended-B) and the Vietnamese alphabet (Latin Extended Additional). Latin Extended-C contains additions for Uighur and the Claudian letters. Latin Extended-D comprises characters that are mostly of interest to medievalists.
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Etendue or étendue (; ) is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. From the source point of view, it is the product of the area of the source and the solid angle that the system's entrance pupil subtends as seen from the source. Equivalently, from the system point of view, the etendue equals the area of the entrance pupil times the solid angle the source subtends as seen from the pupil.
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In linear and homological algebra, a monad is a 3-term complex : A → B → C of objects in some abelian category whose middle term B is projective and whose first map A → B is injective and whose second map B → C is surjective. Equivalently a monad is a projective object together with a 3-step filtration (B ⊃ ker(B → C) ⊃ im(A → B)). In practice A, B, and C are often vector bundles over some space, and there are several minor extra conditions that some authors add to the definition. Monads were introduced by .
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The empty set cannot belong to any collection with the finite intersection property. The condition is trivially satisfied if the intersection over the entire collection is non-empty (in particular, if the collection itself is empty), and it is also trivially satisfied if the collection is nested, meaning that the collection is totally ordered by inclusion (equivalently, for any finite subcollection, a particular element of the subcollection is contained in all the other elements of the subcollection), e.g. the nested sequence of intervals (0, 1/n). However, these are not the only possibilities.
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A similar formula has been derived for 2D networks where the prefactor is 1/18 instead of 1/30 . Hence, based on the Zaccone-Scossa-Romano expression for G , upon setting G= 0 , one obtains z= 2d =6 , or equivalently in different notation, n_c = 3 , which defines the Maxwell isostatic condition. A similar analysis can be done for 3D networks with bond-bending interactions (on top of bond- stretching), which leads to the isostatic condition z= 2.4 , with a lower threshold due to the angular constraints imposed by bond-bending.
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Two subspaces and of the same dimension in a Euclidean space are parallel if they have the same direction. Equivalently, they are parallel, if there is a translation vector that maps one to the other: :T= S+v. Given a point and a subspace , there exists exactly one subspace that contains and is parallel to , which is P + \overrightarrow S. In the case where is a line (subspace of dimension one), this property is Playfair's axiom. It follows that in a Euclidean plane, two lines either meet in one point or are parallel.
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Hidden Markov models are generative models, in which the joint distribution of observations and hidden states, or equivalently both the prior distribution of hidden states (the transition probabilities) and conditional distribution of observations given states (the emission probabilities), is modeled. The above algorithms implicitly assume a uniform prior distribution over the transition probabilities. However, it is also possible to create hidden Markov models with other types of prior distributions. An obvious candidate, given the categorical distribution of the transition probabilities, is the Dirichlet distribution, which is the conjugate prior distribution of the categorical distribution.
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The Ranger B had twin headlights in place of the Rekord/Commodore's single rectangular lights, repositioned front direction indicators and a reworked front grill with a horizontal bar. There was a stylish raised section on the bonnet/hood which was different from the one used on diesel-engined Rekords, and the rear panel also incorporates detail differences. The Rangers were not strikingly more expensive than equivalently powered Rekords and Commodores, but nor were they perceived as sufficiently differentiated, and Rangers were quietly withdrawn from the showrooms by late 1975.
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According to the Church–Turing thesis, computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space. Equivalently, this thesis states that a function is computable if and only if it has an algorithm. Note that an algorithm in this sense is understood to be a sequence of steps a person with unlimited time and an unlimited supply of pen and paper could follow. The Blum axioms can be used to define an abstract computational complexity theory on the set of computable functions.
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Once again, the LM curve plays a passive role, and the outcomes are determined by the IS-BoP interaction. Under perfect capital mobility, the new BoP curve will be horizontal at the new world interest rate, so the equilibrium domestic interest rate will equal the world interest rate. If the global interest rate declines below the domestic rate, the opposite occurs. The BoP curve shifts down, foreign money flows in and the home currency is pressured to appreciate, so the central bank offsets the pressure by selling domestic currency (equivalently, buying foreign currency).
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This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that do include an equals sign. Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group. A closely related but different concept is that of an absolute presentation of a group.
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The interpolant is linear along lines parallel to either the x or the y direction, equivalently if x or y is set constant. Along any other straight line, the interpolant is quadratic. However, even if the interpolation is not linear in the position (x and y), it is linear in the amplitude, as it is apparent from the equations above: all the coefficients aj, j = 1–4, are proportional to the value of the function f. The result of bilinear interpolation is independent of which axis is interpolated first and which second.
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The Knudsen paradox has been observed in experiments of channel flow with varying channel width or equivalently different pressures.Martin Knudsen, "Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren", Annalen der Physik 333 (1909) 75–130. If the normalized mass flux through the channel is plotted over the Knudsen number based on the channel width a distinct minimum is observed around Kn= 0.8. This is a paradoxical behaviour because, based on the Navier–Stokes equations, one would expect the mass flux to decrease with increasing the Knudsen number.
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Diagram of a counter automaton. Each cell of its stack either contains an "A" or a space symbol. In computer science, more particular in the theory of formal languages, a counter automaton, or counter machine, is a pushdown automaton with only two symbols, A and the initial symbol in \Gamma, the finite set of stack symbols. Equivalently, a counter automaton is a nondeterministic finite automaton with an additional memory cell that can hold one nonnegative integer number (of unlimited size), which can be incremented, decremented, and tested for being zero.
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However, proved a complex-number analogue of the Sylvester–Gallai theorem: whenever the points of a Sylvester–Gallai configuration are embedded into a complex projective space, the points must all lie in a two-dimensional subspace. Equivalently, a set of points in three- dimensional complex space whose affine hull is the whole space must have an ordinary line, and in fact must have a linear number of ordinary lines. Similarly, showed that whenever a Sylvester–Gallai configuration is embedded into a space defined over the quaternions, its points must lie in a three- dimensional subspace.
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It says that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. The first law of thermodynamics says that when energy passes into or out of a system (as work, heat, or matter), the system's internal energy changes in accord with the law of conservation of energy. Equivalently, perpetual motion machines of the first kind (machines that produce work with no energy input) are impossible. The second law of thermodynamics can be expressed in two main ways.
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In mathematics, a sofic group is a group whose Cayley graph is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite- rank symmetric groups such that every two elements of the group have distance 1.Ceccherini-Silberstein & Coornaert (2010) p. 276 They were introduced by as a common generalization of amenable and residually finite groups. The name "sofic", from the Hebrew word meaning "finite", was later applied by , following Weiss's earlier use of the same word to indicate a generalization of finiteness in sofic subshifts.
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The accuracy of a quadrature rule of the Newton- Cotes type is generally a function of the number of evaluation points. The result is usually more accurate as the number of evaluation points increases, or, equivalently, as the width of the step size between the points decreases. It is natural to ask what the result would be if the step size were allowed to approach zero. This can be answered by extrapolating the result from two or more nonzero step sizes, using series acceleration methods such as Richardson extrapolation.
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A surjunctive group is a group with the property that, when its elements are used as the cells of cellular automata, every injective transition function of a cellular automaton is also surjective. Equivalently, summarizing the definitions above, a group G is surjunctive if, for every finite set S, every continuous equivariant injective function f:S^G \to S^G is also surjective.Ceccherini-Silberstein & Coornaert (2010) p.57 The implication from injectivity to surjectivity is a form of the Garden of Eden theorem, and the cellular automata defined from injective and surjective transition functions are reversible.
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In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring R.
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The algorithm simplifies in the degenerate case that all partners have the same preference function because the partner that optimally first cuts a slice will also be its last diminisher. Equivalently, each partner 1, 2, ..., n−1 in turn cuts a slice from the remaining cake. Then in reverse order, each partner n, n−1, ..., 1 in turn selects a slice that has not yet been claimed. The first partner who cut a slice other than of value 1/n would be envious of another partner who ended up with more than they did.
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We might accept the alternative hypothesis (and the research hypothesis). If the p-value is not less than the chosen significance threshold (equivalently, if the observed test statistic is outside the critical region), then the evidence is insufficient to support a conclusion. (This is similar to a "not guilty" verdict.) The researcher typically gives extra consideration to those cases where the p-value is close to the significance level. Some people find it helpful to think of the hypothesis testing framework as analogous to a mathematical proof by contradiction.
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Spiric sections as planar sections of a torus In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form :(x^2+y^2)^2=dx^2+ey^2+f. \, Equivalently, spiric sections can be defined as bicircular quartic curves that are symmetric with respect to the x and y-axes. Spiric sections are included in the family of toric sections and include the family of hippopedes and the family of Cassini ovals. The name is from σπειρα meaning torus in ancient Greek.
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In graph theory, an arborescence is a directed graph in which, for a vertex u called the root and any other vertex v, there is exactly one directed path from u to v. An arborescence is thus the directed-graph form of a rooted tree, understood here as an undirected graph. Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of other equivalent characterizations exist. Every arborescence is a directed acyclic graph (DAG), but not every DAG is an arborescence.
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Chang's (1958, 1959) completeness theorem states that any MV-algebra equation holding in the standard MV-algebra over the interval [0,1] will hold in every MV-algebra. Algebraically, this means that the standard MV-algebra generates the variety of all MV-algebras. Equivalently, Chang's completeness theorem says that MV- algebras characterize infinite-valued Łukasiewicz logic, defined as the set of [0,1]-tautologies. The way the [0,1] MV-algebra characterizes all possible MV- algebras parallels the well-known fact that identities holding in the two- element Boolean algebra hold in all possible Boolean algebras.
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In natural optical activity, the difference between the LCP light and the RCP light is caused by the asymmetry of the molecules. Because of the handedness of the molecule, the absorption of the LCP light would be different from the RCP light. However, in MCD in the presence of a magnetic field, LCP and RCP no longer interact equivalently with the absorbing medium. Thus there is not the same direct relation between magnetic optical activity and molecular stereochemistry which would be expected, because it is found in natural optical activity.
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Chemical potential is also referred to as partial molar Gibbs energy (see also partial molar property). Chemical potential is measured in units of energy/particle or, equivalently, energy/mole. In his 1873 paper A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, Gibbs introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying the interactions of homogeneous substances in contact, i.e.
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With the proliferation of the Internet, cloud computing, grid computing, and small, portable, powerful computing devices (e.g., smartphones,) the need for effective distributed serializability techniques to ensure correctness in and among distributed applications seems to increase. Distributed serializability is achieved by implementing distributed versions of the known centralized techniques. Typically, all such distributed versions require utilizing conflict information (of either materialized or non- materialized conflicts, or, equivalently, transaction precedence or blocking information; conflict serializability is usually utilized) that is not generated locally, but rather in different processes, and remote locations.
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The mass (equivalently, rest energy) of a particle can be converted, not "to energy" (it already is energy), but rather to other forms of energy that require motion, such as kinetic energy, thermal energy, or radiant energy. Similarly, kinetic or radiant energy can be converted to other kinds of particles that have mass. This view requires that if either kinetic energy or mass disappears from a system, it is always found to have simply changed forms. In an isolated system, where no energy is allowed to escape, the rest mass is also conserved.
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After the September 11, 2001, attacks on the United States, the small town, with its equivalently small police force, lost a sizable contract to patrol nearby Marshall Creek Park. Instead, the Army Corps of Engineers awarded that contract to nearby Trophy Club, which had a larger and more experienced force. In addition, in 2002, police officers who had written traffic tickets for the city on nearby US Highway 377 were revealed to have expired peace officer certifications, thus rendering those citations invalid. The town was forced to repay approximately US$4,300 in traffic fines.
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Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach that is particularly useful in functional analysis (with pseudometrics provided by seminorms). More precisely, let f: X × X → R be a pseudometric on a set X. The inverse images Ua = f−1([0,a]) for a > 0 can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the Ua is the uniformity defined by the single pseudometric f. Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces.
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In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime. It is a vector space that allows not only vectors, but also bivectors (directed quantities associated with particular planes, such as areas, or rotations) or blades (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted. It is also the natural parent algebra of spinors in special relativity.
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Additionally, every well-covered graph is a critical graph for vertex covering in the sense that, for every vertex , deleting from the graph produces a graph with a smaller minimum vertex cover. The independence complex of a graph is the simplicial complex having a simplex for each independent set in . A simplicial complex is said to be pure if all its maximal simplices have the same cardinality, so a well-covered graph is equivalently a graph with a pure independence complex.For examples of papers using this terminology, see and .
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Patt purchased six-string archtop hollow-body guitars that were then modified by luthiers to have wider necks, wider pickups, and eight strings. Patt's Gibson ES-150 was modified by Vincent "Jimmy" DiSerio c. 1965. Luthier Saul Koll modified a sequence of guitars: a 1938 Gibson Cromwell, a Sears Silvertone, a c. 1922 Mango archtop, a 1951 Gibson L-50, and a 1932 Epiphone Broadway; for Koll's modifications, custom pick-ups accommodated Patt's wide necks and high G (equivalently A); custom pick-ups were manufactured by Seymour Duncan and by Bill Lawrence.
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An ordinal that is both admissible and a limit of admissibles, or equivalently such that \alpha is the \alpha-th admissible ordinal, is called recursively inaccessible. There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals. For example, we can define recursively Mahlo ordinals: these are the \alpha such that every \alpha-recursive closed unbounded subset of \alpha contains an admissible ordinal (a recursive analog of the definition of a Mahlo cardinal). But note that we are still talking about possibly countable ordinals here.
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The notion of a Morse function can be generalized to consider functions that have nondegenerate manifolds of critical points. A Morse–Bott function is a smooth function on a manifold whose critical set is a closed submanifold and whose Hessian is non-degenerate in the normal direction. (Equivalently, the kernel of the Hessian at a critical point equals the tangent space to the critical submanifold.) A Morse function is the special case where the critical manifolds are zero-dimensional (so the Hessian at critical points is non- degenerate in every direction, i.e., has no kernel).
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In mathematics, a supersolvable arrangement is a hyperplane arrangement which has a maximal flag with only modular elements. Equivalently, the intersection semilattice of the arrangement is a supersolvable lattice, in the sense of Richard P. Stanley. As shown by Hiroaki Terao, a complex hyperplane arrangement is supersolvable if and only if its complement is fiber-type. Examples include arrangements associated with Coxeter groups of type A and B. It is known that the Orlik–Solomon algebra of a supersolvable arrangement is a Koszul algebra; whether the converse is true is an open problem.
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An oriented graph is a finite directed graph obtained from a simple undirected graph by assigning an orientation to each edge. Equivalently, it is a directed graph that has no self-loops, no parallel edges, and no two-edge cycles. The first neighborhood of a vertex v (also called its open neighborhood) consists of all vertices at distance one from v, and the second neighborhood of v consists of all vertices at distance two from v. These two neighborhoods form disjoint sets, neither of which contains v itself.
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V-belts may be homogeneously rubber or polymer throughout, or there may be fibers embedded in the rubber or polymer for strength and reinforcement. The fibers may be of textile materials such as cotton, polyamide (such as Nylon) or polyester or, for greatest strength, of steel or aramid (such as Technora, Twaron or Kevlar). When an endless belt does not fit the need, jointed and link V-belts may be employed. Most models offer the same power and speed ratings as equivalently-sized endless belts and do not require special pulleys to operate.
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In virtue of their governing and executive powers, the Congregations grant privileges and dispensations from ecclesiastical laws, or issue ordinances to safeguard their observance; in virtue of their power of interpreting laws, they give authentic declarations; in virtue of their judicial power they give decisions between contending parties. All these powers, however, do not belong to each Congregation. Again, their decrees are particular or universal, according as they are directed to individuals or to the whole Church. Particular decrees, containing simply an authentic interpretation of a universal law, are called equivalently universal.
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Radiative transfer refers to energy transfer through an atmosphere or other medium by means of electromagnetic waves or (equivalently) photons. The simplest form of radiative transfer involves a collinear beam of radiation traveling through a sample to a detector. That flux can be reduced by absorption, scattering or reflection, resulting in energy transmission over a path of less than 100%. The concept of radiative transfer extends beyond simple laboratory phenomena to include thermal emission of radiation by the medium - which can result in more photons arriving at the end of a path than entering it.
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In cellular biology, dependence receptors are proteins that mediate programmed cell death by monitoring the absence of certain trophic factors (or, equivalently, the presence of anti-trophic factors) that otherwise serve as ligands (interactors) for the dependence receptors. A trophic ligand is a molecule whose protein binding stimulates cell growth, differentiation, and/or survival. Cells depend for their survival on stimulation that is mediated by various receptors and sensors, and integrated via signaling within the cell and between cells. The withdrawal of such trophic support leads to a form of cellular suicide.
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For a general positive real number, the binary logarithm may be computed in two parts.. First, one computes the integer part, \lfloor\log_2 x\rfloor (called the characteristic of the logarithm). This reduces the problem to one where the argument of the logarithm is in a restricted range, the interval [1, 2), simplifying the second step of computing the fractional part (the mantissa of the logarithm). For any , there exists a unique integer such that , or equivalently . Now the integer part of the logarithm is simply , and the fractional part is .
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Then the wave height H at a certain location can be expressed as: :H = K_S\; H_0, with K_S the shoaling coefficient and H_0 the wave height in deep water. The shoaling coefficient K_S depends on the local water depth h and the wave frequency f (or equivalently on h and the wave period T=1/f). Deep water means that the waves are (hardly) affected by the sea bed, which occurs when the depth h is larger than about half the deep- water wavelength L_0=gT^2/(2\pi).
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The primal graph G(H) of a hypergraph is the graph on the same vertex set that has as its edges the pairs of vertices appearing together in the same hyperedge. A hypergraph is said to be conformal if every maximal clique of its primal graph is a hyperedge, or equivalently, if every clique of its primal graph is contained in some hyperedge.; . If the hypergraph is required to be downward-closed (so it contains all hyperedges that are contained in some hyperedge) then the hypergraph is conformal precisely when it is a flag complex.
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Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing. Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together.
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Midnight (חֲצוֹת הַלַּילָה, Chatzot Halaylah or just Chatzot) is the midpoint between nightfall and daybreak, or equivalently between sunset and sunrise. The evening Shema should be recited by now, and the Afikoman on Passover should be eaten by this time. The Talmud in Berachot rules that all "night" mitzvot should be performed by Chatzot, at least ab initio, in case the person would otherwise fall asleep and then fail to perform the mitzvot. Some rise at this time and recite Tikkun Chatzot, a series of supplications for the rebuilding of the Temple.
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Many algorithms with bad worst-case performance have good average-case performance. For problems we want to solve, this is a good thing: we can hope that the particular instances we care about are average. For cryptography, this is very bad: we want typical instances of a cryptographic problem to be hard. Here methods like random self-reducibility can be used for some specific problems to show that the worst case is no harder than the average case, or, equivalently, that the average case is no easier than the worst case.
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In some games, enemies may be respawned (or, equivalently, new ones spawned) to keep players on their toes and create tension, or force players to move on, making it too costly (in resources) and/or too dangerous to stay in one place for too long. Enemies may visibly spawn or, in more realistic games, spawn outside the player's line of sight and move towards the player. Early games including monster respawning are Joust, Doom and its sequel Doom II: Hell on Earth. The enemies in these games had the ability to spawn from their teammates.
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In functional analysis, the girth of a Banach space is the infimum of lengths of centrally symmetric simple closed curves in the unit sphere of the space. Equivalently, it is twice the infimum of distances between opposite points of the sphere, as measured within the sphere... See in particular p. 16. Every finite-dimensional Banach space has a pair of opposite points on the unit sphere that achieves the minimum distance, and a centrally symmetric simple closed curve that achieves the minimum length. However, such a curve may not always exist in infinite-dimensional spaces.
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In addition to the isomorphisms :L2(2) ≅ S3, L2(3) ≅ A4, and PGL(2, 3) ≅ S4, there are other exceptional isomorphisms between projective special linear groups and alternating groups (these groups are all simple, as the alternating group over 5 or more letters is simple): :L_2(4) \cong A_5 :L_2(5) \cong A_5 (see here for a proof) :L_2(9) \cong A_6 :L_4(2) \cong A_8. The isomorphism L2(9) ≅ A6 allows one to see the exotic outer automorphism of A6 in terms of field automorphism and matrix operations. The isomorphism L4(2) ≅ A8 is of interest in the structure of the Mathieu group M24. The associated extensions SL(n, q) → PSL(n, q) are covering groups of the alternating groups (universal perfect central extensions) for A4, A5, by uniqueness of the universal perfect central extension; for L2(9) ≅ A6, the associated extension is a perfect central extension, but not universal: there is a 3-fold covering group. The groups over F5 have a number of exceptional isomorphisms: :PSL(2, 5) ≅ A5 ≅ I, the alternating group on five elements, or equivalently the icosahedral group; :PGL(2, 5) ≅ S5, the symmetric group on five elements; :SL(2, 5) ≅ 2 ⋅ A5 ≅ 2I the double cover of the alternating group A5, or equivalently the binary icosahedral group.
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Equivalently, a DAG is a directed graph that has a topological ordering, a sequence of the vertices such that every edge is directed from earlier to later in the sequence. DAGs can model many different kinds of information. For example, a spreadsheet can be modeled as a DAG, with a vertex for each cell and an edge whenever the formula in one cell uses the value from another; a topological ordering of this DAG can be used to update all cell values when the spreadsheet is changed. Similarly, topological orderings of DAGs can be used to order the compilation operations in a makefile.
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Unit fractions play an important role in modular arithmetic, as they may be used to reduce modular division to the calculation of greatest common divisors. Specifically, suppose that we wish to perform divisions by a value x, modulo y. In order for division by x to be well defined modulo y, x and y must be relatively prime. Then, by using the extended Euclidean algorithm for greatest common divisors we may find a and b such that :\displaystyle ax + by = 1, from which it follows that :\displaystyle ax \equiv 1 \pmod y, or equivalently :a \equiv \frac1x \pmod y.
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Although the terms direct and indirect bilirubin are used equivalently with conjugated and unconjugated bilirubin, this is not quantitatively correct, because the direct fraction includes both conjugated bilirubin and δ bilirubin. Delta bilirubin is albumin-bound conjugated bilirubin. In the other words, delta bilirubin is the kind of bilirubin covalently bound to albumin, which appears in the serum when hepatic excretion of conjugated bilirubin is impaired in patients with hepatobiliary disease. Furthermore, direct bilirubin tends to overestimate conjugated bilirubin levels due to unconjugated bilirubin that has reacted with diazosulfanilic acid, leading to increased azobilirubin levels (and increased direct bilirubin).
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The rules are equivalent--if one divides both sides of inequality TR > VC (total revenue exceeds variable costs) by the output quantity Q one obtains P > AVC (price exceeds average variable cost). If the firm decides to operate it will produce where marginal revenue equals marginal costs because these conditions insure profit maximization (or equivalently, when profit is negative, loss minimization).Samuelson, W & Marks, S (2006) p.286. Another way to state the rule is that a firm should compare the profits from operating to those realized if it shut down, and select the option that produces the greater profit (positive or negative).
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A cluster graph with clusters (complete subgraphs) of sizes 1, 2, 3, 4, 4, 5, and 6 In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P3-free graphs. They are the complement graphs of the complete multipartite graphsCluster graphs, Information System on Graph Classes and their Inclusions, accessed 2016-06-26. and the 2-leaf powers..
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That would also be a rule of the bucket view of science. Popper's main methodological rule is that scientists must try to guess and corroborate (or equivalently falsify) bold and useful conjectures and take any falsification as a problem that can be used to start a critical discussion. In other words, the usefulness of falsifiability is that falsifiable conjectures say more, because they prohibit more and, in the case of their falsification, they lead to useful problems, which steer the creative process of science. For Popper, who knew most of section , this is exactly what we should expect from a scientific theory.
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A basic controversy in loss aversion research is whether losses are actually experienced more negatively than equivalent gains or merely predicted to be more painful but actually experienced equivalently. Neuroeconomic research has attempted to distinguish between these hypotheses by measuring different physiological changes in response to both loss and gain. Studies have found that skin conductance, pupil dilation and heart rate are all higher in response to monetary loss than to equivalent gain. All three measures are involved in stress responses, so one might argue that losing a particular amount of money is experienced more strongly than gaining the same amount.
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Inclusions are quite common for Lebanese amber, which is rich in organisms of the Barremian epoch. Next to Jordanian amber, Lebanese amber is the oldest amber to have yielded significant invertebrate inclusions alongside the Wealden amber from the equivalently aged Wessex Formation of the UK, which is much less productive. Organisms preserved in Lebanese amber are dated back to the period prior to the angiosperm radiation, which was the period of massive extinction of old groups of arthropods, as well as the emergence of the new ones, some of which co-evolved with angiosperms. The organisms are preserved in good condition and shape.
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In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs.
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Equivalently, a set of vectors is linearly independent if the only way to express the zero vector as a linear combination of elements of is to take zero for every coefficient a_i. A set of vectors that spans a vector space is called a spanning set or generating set. If a spanning set is linearly dependent (that is not linearly independent), then some element of is in the span of the other elements of , and the span would remain the same if one remove from . One may continue to remove elements of until getting a linearly independent spanning set.
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All of these are observables except for the implied volatility, which can computed from the observable price using the Black–Scholes formula. In order for this function to reflect moneyness – i.e., for moneyness to increase as spot and strike move relative to each other – it must be monotone in both spot S and in strike K (equivalently forward F, which is monotone in S), with at least one of these strictly monotone, and have opposite direction: either increasing in S and decreasing in K (call moneyness) or decreasing in S and increasing in K (put moneyness). Somewhat different formalizations are possible.
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Nucleic acid aptamers are nucleic acid species (next-gen antibody mimics) having selectivity at par of antibodies for a given target generated via in-vitro selection or equivalently, SELEX (systematic evolution of ligands by exponential enrichment) ranging from small entities such as heavy metal ions to large entities like cells. On the molecular level, aptamers bind to its cognate target through various non-covalent interactions viz., electrostatic interactions, hydrophobic interactions, and induced fitting. Aptamers are useful in biotechnological and therapeutic applications as they offer molecular recognition properties that rival that of the commonly used biomolecule, antibodies.
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A graph with 16 vertices and 6 bridges (highlighted in red) An undirected connected graph with no bridge edges In graph theory, a bridge, isthmus, cut- edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components.. Equivalently, an edge is a bridge if and only if it is not contained in any cycle. For a connected graph, a bridge can uniquely determine a cut. A graph is said to be bridgeless or isthmus-free if it contains no bridges. Another meaning of "bridge" appears in the term bridge of a subgraph.
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A point x of a subset A of a topological space X is called a limit point of A (in X) if every neighbourhood of x also contains a point of A other than x itself, and an isolated point of A otherwise. A subset without isolated points is said to be dense-in-itself. A subset A of a topological space X is called nowhere dense (in X) if there is no neighborhood in X on which A is dense. Equivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty.
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Their early service was in the Midwest, between Chicago, Illinois and Colorado; later, some were assigned to service in the San Joaquin Valley of California between Bakersfield and Oakland. They were smaller and less powerful locomotives than the later 3460 class, but were capable of equivalently high speeds. The first locomotive built, No. 3450, was donated by the Santa Fe in 1955 to the Railway and Locomotive Historical Society's Southern California chapter, and is preserved at the Society's museum in the Los Angeles County Fairgrounds at Pomona, California. The locomotive sits inside the RailGiants Train Museum.
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Such symmetries can be discrete or continuous; they materialize in the correlation functions for the random variables (discussed below). Thus a symmetry in the Hamiltonian becomes a symmetry of the correlation function (and vice versa). This symmetry has a critically important interpretation in probability theory: it implies that the Gibbs measure has the Markov property; that is, it is independent of the random variables in a certain way, or, equivalently, the measure is identical on the equivalence classes of the symmetry. This leads to the widespread appearance of the partition function in problems with the Markov property, such as Hopfield networks.
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The resolution of an incremental encoder is a measure of the precision of the position information it produces. Encoder resolution is typically specified in terms of the number of A (or B) pulses per unit displacement or, equivalently, the number of A (or B) square wave cycles per unit displacement. In the case of rotary encoders, resolution is specified as the number of pulses per revolution (PPR) or cycles per revolution (CPR), whereas linear encoder resolution is typically specified as the number of pulses issued for a particular linear traversal distance (e.g., 1000 pulses per mm).
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SFC is dependent on engine design, but differences in the SFC between different engines using the same underlying technology tend to be quite small. Increasing overall pressure ratio on jet engines tends to decrease SFC. In practical applications, other factors are usually highly significant in determining the fuel efficiency of a particular engine design in that particular application. For instance, in aircraft, turbine (jet and turboprop) engines are typically much smaller and lighter than equivalently powerful piston engine designs, both properties reducing the levels of drag on the plane and reducing the amount of power needed to move the aircraft.
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The Boolean algebras we have seen so far have all been concrete, consisting of bit vectors or equivalently of subsets of some set. Such a Boolean algebra consists of a set and operations on that set which can be shown to satisfy the laws of Boolean algebra. Instead of showing that the Boolean laws are satisfied, we can instead postulate a set X, two binary operations on X, and one unary operation, and require that those operations satisfy the laws of Boolean algebra. The elements of X need not be bit vectors or subsets but can be anything at all.
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A unique advantage of MRI is that it provides not only the phase image but also the magnitude image. In principle, the contrast change, or equivalently the edge, on a magnitude image arises from the underlying change of tissue type, which is the same cause for the change of susceptibility. This observation is translated into mathematics in MEDI, where edges in a QSM which do not exist in the corresponding magnitude image are sparsified by solving a weighted l_1 norm minimization problem. MEDI has also been validated extensively in phantom, in vitro and ex vivo experiments.
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In the geometry of the Euclidean plane, axiality is a measure of how much axial symmetry a shape has. It is defined as the ratio of areas of the largest axially symmetric subset of the shape to the whole shape. Equivalently it is the largest fraction of the area of the shape that can be covered by a mirror reflection of the shape (with any orientation). A shape that is itself axially symmetric, such as an isosceles triangle, will have an axiality of exactly one, whereas an asymmetric shape, such as a scalene triangle, will have axiality less than one.
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In mathematics, a Prym differential of a Riemann surface is a differential form on the universal covering space that transforms according to some complex character of the fundamental group. Equivalently it is a section of a certain line bundle on the Riemann surface in the same component as the canonical bundle. Prym differentials were introduced by . The space of Prym differentials on a compact Riemann surface of genus g has dimension g – 1, unless the character of the fundamental group is trivial, in which case Prym differentials are the same as ordinary differentials and form a space of dimension g.
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Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers a and b, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O, and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A,\; B, respectively and the fourth point of the parallelogram X, the triangles OAB and XBA are congruent. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend.
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A circle is a shape consisting of all points in a plane that are a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior.
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In this theory, the class P consists of all those decision problems (defined below) that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; the class NP consists of all those decision problems whose positive solutions can be verified in polynomial time given the right information, or equivalently, whose solution can be found in polynomial time on a non-deterministic machine.Sipser, Michael: Introduction to the Theory of Computation, Second Edition, International Edition, page 270. Thomson Course Technology, 2006. Definition 7.19 and Theorem 7.20.
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In the early period, between the early 9th century and late 7th century BCE, mat Kaldi was the name of a small sporadically independent migrant-founded territory under the domination of the Neo-Assyrian Empire (911-605 BCE) in southeastern Babylonia, extending to the western shores of the Persian Gulf. The expression mat Bit Yâkin is also used, apparently synonymously. Bit Yâkin was the name of the largest and most powerful of the five tribes of the Chaldeans, or equivalently, their territory.bit is the "house of" tribal denominator, Yâkin (Ia-kin) is presumably the name of a king of the Arabian Sealand.
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The outcomes in a lottery can themselves be lotteries between other outcomes, and the expanded expression is considered an equivalent lottery: 0.5(0.5A + 0.5B) + 0.5C = 0.25A + 0.25B + 0.50C. If lottery M is preferred over lottery L, we write L \prec M, or equivalently, M \succ L. If the agent is indifferent between L and M, we write the indifference relationKreps, David M. Notes on the Theory of Choice. Westview Press (May 12, 1988), chapters 2 and 5. L\sim M. If M is either preferred over or viewed with indifference relative to L, we write L \preceq M.
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In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean. Equivalently, every high- dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids. A new proof found by Vitali Milman in the 1970s was one of the starting points for the development of asymptotic geometric analysis (also called asymptotic functional analysis or the local theory of Banach spaces).
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Additionally, if some bidders are known to have a higher valuation for the lot, techniques such as price- discriminating against such bidders will yield higher returns. In other words, if a bidder is known to value the lot at $X more than the next highest bidder, the seller can increase their profits by charging that bidder $X – Δ (a sum just slightly inferior to the sum is willing to pay) more than any other bidder (or equivalently a special bidding fee of $X – Δ). This bidder will still win the lot, but will pay more than would otherwise be the case.
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Macroeconomics is the study of the factors applying to an economy as a whole. Influential economic factors include the overall price level, the interest rate, and the level of employment (or equivalently, of income/output measured in real terms). The classical tradition of partial equilibrium theory had been to split the economy into separate markets, each of whose equilibrium conditions could be stated as a single equation determining a single variable. The theoretical apparatus of supply and demand curves developed by Fleeming Jenkin and Alfred Marshall provided a unified mathematical basis for this approach, which the Lausanne School generalized to general equilibrium theory.
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Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattices). They play an important role in number theory and topology. An integral quadratic form has integer coefficients, such as ; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning if . This is the current use of the term; in the past it was sometimes used differently, as detailed below.
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In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operators in the product. The process of putting a product into normal order is called normal ordering (also called Wick ordering). The terms antinormal order and antinormal ordering are analogously defined, where the annihilation operators are placed to the left of the creation operators. Normal ordering of a product quantum fields or creation and annihilation operators can also be defined in many other ways.
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An MDS matrix (Maximum Distance Separable) is a matrix representing a function with certain diffusion properties that have useful applications in cryptography. Technically, an m×n matrix A over a finite field K is an MDS matrix if it is the transformation matrix of a linear transformation f(x)=Ax from Kn to Km such that no two different (m+n)-tuples of the form (x,f(x)) coincide in n or more components. Equivalently, the set of all (m+n)-tuples (x,f(x)) is an MDS code, i.e. a linear code that reaches the Singleton bound.
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In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles. It is built on the earlier Borel–Weil theorem of Armand Borel and André Weil, dealing just with the space of sections (the zeroth cohomology group), the extension to higher cohomology groups being provided by Raoul Bott. One can equivalently, through Serre's GAGA, view this as a result in complex algebraic geometry in the Zariski topology.
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Sepp Hochreiter developed the long short-term memory (LSTM) for which the first results were reported in his diploma thesis in 1991. The main LSTM paper appeared in 1997 and is considered as a discovery that is a milestone in the timeline of machine learning. LSTM overcomes the problem of recurrent neural networks (RNNs) and deep networks to forget information over time or, equivalently, through layers (vanishing or exploding gradient). LSTM learns from training sequences to process new sequences in order to produce an output (sequence classification) or generate an output sequence (sequence to sequence mapping).
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Dunham stated the game's visuals as "top of the line" by virtue of its textures, character designs, pyrotechnic effects, and a "silky smooth" frame rate. Navarro equivalently touted the visuals as its "strongest asset". GameSpot contributor Alex Navarro argued that despite not being the most graphically-advanced game available for the PS2 during its release, the game will impress players with Kawamoto's character designs and the protagonist Lau's cinematic flair. Joe Juba of Game Informer similarly enjoyed the game's character designs and action sequences, discovering within it a "unique sense of style, reminiscent of Hong Kong cinema, [permeating] every aspect".
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Surface waves in water Unless the volume of a liquid exactly matches the volume of its container, one or more surfaces are observed. The presence of a surface introduces new phenomena which are not present in a bulk liquid. This is because a molecule at a surface possesses bonds with other liquid molecules only on the inner side of the surface, which implies a net force pulling surface molecules inward. Equivalently, this force can be described in terms of energy: there is a fixed amount of energy associated with forming a surface of a given area.
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Placing a replicate on this candidate could degrade performance for all clients accessing this candidate server. Therefore, placement of replicas with this method is done by evaluating each candidate node for access load to find a suitable node for the placement of the replica. If all candidate nodes are equivalently rated for access load, none or less accessed than the other, then the candidate node with the lowest storage load will be chosen to host the replicas. Similar methods to the other described replication methods are used to remove unused or lower requested replicates if needed.
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In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
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New Keynesian economists agree with New Classical economists that in the long run, the classical dichotomy holds: changes in the money supply are neutral. However, because prices are sticky in the New Keynesian model, an increase in the money supply (or equivalently, a decrease in the interest rate) does increase output and lower unemployment in the short run. Furthermore, some New Keynesian models confirm the non-neutrality of money under several conditions.Benchimol, J., Fourçans, A. (2012), Money and risk in a DSGE framework : A Bayesian application to the Eurozone, Journal of Macroeconomics, vol. 34, pp. 95-111.
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Affine shape adaptation is a methodology for iteratively adapting the shape of the smoothing kernels in an affine group of smoothing kernels to the local image structure in neighbourhood region of a specific image point. Equivalently, affine shape adaptation can be accomplished by iteratively warping a local image patch with affine transformations while applying a rotationally symmetric filter to the warped image patches. Provided that this iterative process converges, the resulting fixed point will be affine invariant. In the area of computer vision, this idea has been used for defining affine invariant interest point operators as well as affine invariant texture analysis methods.
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If G is a semisimple algebraic group (or Lie group) and V is a (finite dimensional) highest weight representation of G, then the highest weight space is a point in the projective space P(V) and its orbit under the action of G is a projective algebraic variety. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for G arises in this way. Armand Borel showed that this characterizes the flag varieties of a general semisimple algebraic group G: they are precisely the complete homogeneous spaces of G, or equivalently (in this context), the projective homogeneous G-varieties.
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The rank of a finitely generated group G can be equivalently defined as the smallest cardinality of a set X such that there exists an onto homomorphism F(X) → G, where F(X) is the free group with free basis X. There is a dual notion of co-rank of a finitely generated group G defined as the largest cardinality of X such that there exists an onto homomorphism G → F(X). Unlike rank, co-rank is always algorithmically computable for finitely presented groups,John R. Stallings. Problems about free quotients of groups. Geometric group theory (Columbus, OH, 1992), pp.
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The aerobic threshold (AeT or AerT) is sometimes defined as the exercise intensity at which blood lactate concentrations rise above resting levels. Anaerobic threshold (AnT) is sometimes defined equivalently to the lactate threshold(LT); as the exercise intensity beyond which blood lactate concentration is no longer linearly related to exercise intensity, but increases with both exercise intensity and duration. The blood lactate concentration at the anaerobic threshold is called the "maximum steady-state lactate concentration" (MLSS). AeT is the exercise intensity at which anaerobic energy pathways start to operate, considered to be around 65-85% of an individual's maximum heart rate.
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From the referenced studies on the lack of correlation between the intelligence and adult family income of adopted children and their adoptive parents, Clark infers that family environment cannot explain the transmittal of social status from one generation to the next. Clark's hypothesis is that the unexpectedly high persistence of social status in families—or, equivalently, of the unexpectedly low degree of social mobility—is that high-status people are more likely to have genes that are beneficial to them achieving high status, and are therefore more likely to pass such genes on to their children.
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This takes place for three days from Wednesday to Friday. Wednesday night is the most important day of the festival, during that day they bring silk cloth covered goddess statue from Thevarapanpatti and removing this silk cloth and eye opening (prana prathisdai) and decorate the goddess in Alappa gowder's temple house and then only it goes to the temple where festival begins and final Friday aarathi was also taken by Alappa gowder family to the goddess as send of function. Sadayandi swami petti festival is also equivalently important. Periya Ayyanar temple on the bank of Marudanadi river is also a worth seeing place.
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They define it as the difference between an estimated upper limit of public debt (beyond which action would have to be taken to avoid default) and actual public debt, expressed as a percentage of GDP or equivalently as the difference between the debt-limit-to-GDP percentage and the actual-debt-to-GDP percentage. The World Bank's 2019 World Development Report on the future of work World Bank World Development Report 2019: The Changing Nature of Work. suggests ways governments increase fiscal space in order to prepare for the future of work through investments in human capital and social protection.
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The best theoretical explanation of the experimental results was the Rayleigh–Jeans law, which agrees with experimental results well at large wavelengths (or, equivalently, low frequencies), but strongly disagrees at short wavelengths (or high frequencies). In fact, at short wavelengths, classical physics predicted that energy will be emitted by a hot body at an infinite rate. This result, which is clearly wrong, is known as the ultraviolet catastrophe. However, classical physics led to the Rayleigh–Jeans law, which, as shown in the figure, agrees with experimental results well at low frequencies, but strongly disagrees at high frequencies.
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Cardi B is Catholic; she has mentioned her "strong relationship" with God in interviews, often saying that she directly communicates with God. Cardi B's younger sister, Hennessy Carolina, also has a strong following on social media and has accompanied her to award shows, such as the 2017 Grammy Awards. In an interview in 2018, Cardi talked about being Afro-Latina and Afro-Caribbean: She has been a resident of Edgewater, New Jersey, renting an apartment for $3,000 a month that she says would be twice as much in Manhattan for an equivalently sized unit.Shamsian, Jacob; and Singh, Olivia.
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We then apply a specific filter, which will allow only the desired signal to pass. Thus the noise will be removed completely. Then we use the fractional Fourier transform again to rotate the signal back and we can get the desired signal. Fractional Fourier transform in DSP Thus, using just truncation in the time domain, or equivalently low-pass filters in the frequency domain, one can cut out any convex set in time–frequency space; just using time domain or frequency domain methods without fractional Fourier transforms only allow cutting out rectangles parallel to the axes.
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A k-core of a graph G is a maximal connected subgraph of G in which all vertices have degree at least k. Equivalently, it is one of the connected components of the subgraph of G formed by repeatedly deleting all vertices of degree less than k. If a non- empty k-core exists, then, clearly, G has degeneracy at least k, and the degeneracy of G is the largest k for which G has a k-core. A vertex u has coreness c if it belongs to a c-core but not to any (c+1)-core.
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At the end of each flight, the appropriate scattering mechanism must be chosen in order to determine the final energy of the scattered carrier, or equivalently, its new momentum and scattering angle. In this sense, one will distinguish two broad types of scattering mechanisms which naturally derive form the classic kinetic theory of collision between two bodies: Elastic scattering, where the energy of the particle is conserved after being scattered. Elastic scattering will hence only change the direction of the particle's momentum. Impurity scattering and surface scattering are, with a fair approximation, two good examples of elastic scattering processes.
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Points in space and the real numbers under Euclidean geometry are another structure, with relations such as "the distance between point P1 and point P2 is real number R1"; equivalently, the "distance" relation includes the element (P1, P2, R1). Other structures include the Riemann space of general relativity and the Hilbert space of quantum mechanics. The entities in a mathematical structure do not have any independent existence outside their participation in relations. Two descriptions of a structure are considered equivalent, and to be describing the same underlying structure, if there is a correspondence between the descriptions that preserves all relations.
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M3 CRT The M3 CRT (Carbon Racing Technology) was a special edition of the M3 sedan that was produced in 2011. The CRT was powered by the same 4.4 litre version of the S65 engine as the GTS, however it retained a higher level of luxury features compared to the track-focussed GTS. The CRT used a carbon fibre bonnet and front seats to reduce weight, resulting in a kerb weight approximately lower than an equivalently specified version of the regular M3 sedan. Production was limited to 67 cars, all numbered with a plaque on the dashboard.
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Following this, three chapters concern lattice tilings by polycubes. The question here is to determine, from the shape of the polycube, whether all cubes in the tiling meet face-to-face or, equivalently, whether the lattice of symmetries must be a subgroup of the integer lattice. After a chapter on the general version of this problem, two chapters consider special classes of cross and "semicross"-shaped polycubes, both with regard to tiling and then, when these shapes do not tile, with regard to how densely they can be packed. In three dimensions, this is the notorious tripod packing problem.
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Principal component analysis (PCA) is often used for dimension reduction. Given an unlabeled set of n input data vectors, PCA generates p (which is much smaller than the dimension of the input data) right singular vectors corresponding to the p largest singular values of the data matrix, where the kth row of the data matrix is the kth input data vector shifted by the sample mean of the input (i.e., subtracting the sample mean from the data vector). Equivalently, these singular vectors are the eigenvectors corresponding to the p largest eigenvalues of the sample covariance matrix of the input vectors.
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Birkhoff's representation theorem may also be generalized to finite structures other than distributive lattices. In a distributive lattice, the self-dual median operation. :m(x,y,z)=(x\vee y)\wedge(x\vee z)\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)\vee(y\wedge z) gives rise to a median algebra, and the covering relation of the lattice forms a median graph. Finite median algebras and median graphs have a dual structure as the set of solutions of a 2-satisfiability instance; formulate this structure equivalently as the family of initial stable sets in a mixed graph.
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The effect equivalently states that more ancient packages of sediment in the stratigraphic record will record slower sedimentation rates stretched over longer periods of time. For instance, it explains the fact that in general, the more ancient geological periods of the Phanerozoic are longer than the more recent ones; i.e., the periods of the Palaeozoic are much longer than those in the Cenozoic. Conversely, it also explains that the maximum sediment accumulation rates seen in the Cambrian at the start of the Phanerozoic are almost two orders of magnitude lower than those observed in the Quaternary, at its end.
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In Mithraic images, Mithras either represents the sun, or is a close friend of the sun god Helios or Sol Invictus (Latin: the invincible sun) with whom Mithras dines. So attendants Cautes and Cautopates are supposed to represent the stations of sunrise and sunset respectively, or perhaps the spring and autumn equinoxes, or equivalently the ascending (spring) and descending (autumnal) nodes of the Sun's apparent path on the celestial sphere. If eclipses of the sun and moon formed part of Mithraic symbolism, they could also represent the ascending and descending nodes where the Moon crosses the ecliptic.
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The dimensions hp,q of spaces of harmonic (p,q)-differential forms (equivalently, the cohomology, i.e., closed forms modulo exact forms) are conventionally arranged in a diamond shape called the Hodge Diamond. These (p,q)-betti numbers can be computed for complete intersections using a generating function described by Friedrich Hirzebruch. For a three-dimensional manifold, for example, the Hodge diamond has p and q ranging from 0 to 3: Mirror symmetry translates the dimension number of the (p, q)-th differential form hp,q for the original manifold into hn-p,q of that for the counter pair manifold.
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For points in Euclidean space, a set X is a Meyer set if it is relatively dense and its difference set X − X is uniformly discrete. Equivalently, X is a Meyer set if both X and X − X are Delone. Meyer sets are named after Yves Meyer, who introduced them (with a different but equivalent definition based on harmonic analysis) as a mathematical model for quasicrystals. They include the point sets of lattices, Penrose tilings, and the Minkowski sums of these sets with finite sets.. The Voronoi cells of symmetric Delone sets form space-filling polyhedra called plesiohedra..
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The standard contact structure on R3. Each point in R3 has a plane associated to it by the contact structure, in this case as the kernel of the one-form These planes appear to twist along the y-axis. In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non- integrability condition translates into a maximal non-degeneracy condition on the form.
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Islamic Community of Kosovo (Albanian:Bashkësia Islame e Kosovës) is an institution or a community, which is independent for Kosovar Muslims. This community aims to organize the life of Muslims in Kosovo on the basis of Islamic law, which under their responsibility itself has mosques and imams of these mosques, madrases and an Islamic science faculty in Kosovo. Madras, as a high school, is recognized equivalently with all Islamic universities around the world, as well as other non-Islamic universities in the country and world. Graduates of this school, how havedoctorate degrees, and masters, and are different scientific workers, theologians, doctors, engineers, etc.
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Cadmium hydroxide adopts the same structure as Mg(OH)2, consisting of slabs of octahedral metal centers surrounded by octahedral of hydroxide ligands.Hemmingsen, L.; Bauer, R.; Bjerrum, M. J.; Schwarz, K.; Blaha, P.; Andersen, P., "Structure, Chemical Bonding, and Nuclear Quadrupole Interactions of β-Cd(OH)2: Experiment and First Principles Calculations", Inorganic Chemistry 1999, volume 38, 2860-2867. It is produced by treating cadmium nitrate with sodium hydroxide: : Cd(NO3)2 \+ 2 NaOH → Cd(OH)2 \+ 2 NaNO3 Attempted preparation from other cadmium salts is more complicated. Cd(OH)2 and cadmium oxide react equivalently.
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In graph theory, a moral graph is used to find the equivalent undirected form of a directed acyclic graph. It is a key step of the junction tree algorithm, used in belief propagation on graphical models. The moralized counterpart of a directed acyclic graph is formed by adding edges between all pairs of non- adjacent nodes that have a common child, and then making all edges in the graph undirected. Equivalently, a moral graph of a directed acyclic graph is an undirected graph in which each node of the original is now connected to its Markov blanket.
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The ratio of energy extracted to the energy expended in the process is often referred to as the Energy Return on Energy Investment (EROI or EROEI). Should the EROEI drops to one, or equivalently the Net energy gain falls to zero, the oil production is no longer a net energy source. There is a difference between a barrel of oil, which is a measure of oil, and a barrel of oil equivalent (BOE), which is a measure of energy. Many sources of energy, such as fission, solar, wind, and coal, are not subject to the same near-term supply restrictions that oil is.
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For an R-module A, a maximal submodule M of A is a submodule M≠A satisfying the property that for any other submodule N, M⊆N⊆A implies N=M or N=A. Equivalently, M is a maximal submodule if and only if the quotient module A/M is a simple module. The maximal right ideals of a ring R are exactly the maximal submodules of the module RR. Unlike rings with unity, a nonzero module does not necessarily have maximal submodules. However, as noted above, finitely generated nonzero modules have maximal submodules, and also projective modules have maximal submodules.
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94 Transducers have (at leastPiezoelectric transducers are frequently modelled as three-port devices, one electrical and two mechanical, because mechanical vibrations are induced on both sides of the crystal (Cheeke, pp. 213-214).) two ports, one port in the mechanical domain and one in the electrical domain, and are analogous to electrical two-port networks. This is to be compared to the elements discussed so far which are all one-ports. Two-port networks can be represented as a 2×2 matrix, or equivalently, as a network of two dependent generators and two impedances or admittances.
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The special case in which the integers whose reciprocals are taken must be square numbers appears in two ways in the context of right triangles. First, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse. This holds whether or not the numbers are integers; there is a formula (see here) that generates all integer cases.Voles, Roger, "Integer solutions of a−2+b−2=d−2," Mathematical Gazette 83, July 1999, 269-271.
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A completely normal space' or a ' is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods. Also, X is completely normal if and only if every open subset of X is normal with the subspace topology. A completely T4 space, or T5 space is a completely normal T1 space topological space X, which implies that X is Hausdorff; equivalently, every subspace of X must be a T4 space.
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The application of semicircles to edge layout in book embeddings was already made by , but the explicit connection of arc diagrams with two-page book embeddings seems to be due to . For instance, a maximal planar graph has such an embedding if and only if it contains a Hamiltonian cycle. Therefore, a non-Hamiltonian maximal planar graph such as the Goldner–Harary graph cannot have a planar embedding with one semicircle per edge. Testing whether a given graph has a crossing- free arc diagram of this type (or equivalently, whether it has pagenumber two) is NP-complete.
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With an altazimuth mount, those directions are rotated by the amount of the parallactic angle. The cardinal points referred to here are the points on the limb located such that a line from the center of the disk through them will point to one of the celestial poles or 90° away from them; these are not the cardinal points defined by the object's axis of rotation. The orientation of the disk of the Moon, as related to the horizon, changes throughout its diurnal motion and the parallactic angle changes equivalently. This is also the case with other celestial objects.
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Routine monitoring and dose adjustments of DOACs is less important than for warfarin, as they have better predictable anticoagulation activity. Both DOACs and warfarin are equivalently effective but compared to warfarin, DOACs have fewer drug interactions, no known dietary interactions, wider therapeutic index, and have conventional dosing that do not require dose adjustments with constant monitoring. However, there is presently no countermeasure for most DOACs unlike in warfarin; nonetheless, the short half-lives of DOACs will result in its effects to swiftly recede. A reversal agent for dabigatran, idarucizumab, is currently available and approved for use by the FDA.
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The Fano plane is the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines. In the figure at right, the seven points are shown as small black balls, and the seven lines are shown as six line segments and a circle. However, one could equivalently consider the balls to be the "lines" and the line segments and circle to be the "points" – this is an example of duality in the projective plane: if the lines and points are interchanged, the result is still a projective plane (see below).
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The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1. An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently—by clearing denominators—with integer coefficients). All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as and , are called transcendental numbers.
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Bosons are quantum mechanical particles that follow Bose–Einstein statistics, or equivalently, that possess integer spin. These particles can be classified as elementary: these are the Higgs boson, the photon, the gluon, the W/Z and the hypothetical graviton; or composite like the atom of hydrogen, the atom of 16O, the nucleus of deuterium, mesons etc. Additionally, some quasiparticles in more complex systems can also be considered bosons like the plasmons (quanta of charge density waves). The first model that treated a gas with several bosons, was the photon gas, a gas of photons, developed by Bose.
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Minerva as a symbol of enlightened wisdom protects the believers of all religions (Daniel Chodowiecki, 1791) Religious toleration has been described as a "remarkable feature" of the Achaemenid Empire of Persia. As reported in the Old Testament, king Cyrus the Great was believed to have released the Jews from captivity in 539–530 BCE, and permitted their return to their homeland. Cyrus the Great assisted in the restoration of the sacred places of various cities. The Hellenistic city of Alexandria, founded 331 BCE, contained a large Jewish community which lived in peace with equivalently sized Greek and Egyptian populations.
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The complement graphs of tolerance graphs are perfectly orderable. Another class of perfectly orderable graphs is given by the graphs G such that, in every subset of five vertices from G, at least one of the five has a closed neighborhood that is a subset of (or equal to) the closed neighborhood of another of the five vertices. Equivalently, these are the graphs in which the partial order of closed neighborhoods, ordered by set inclusion, has width at most four. The 5-vertex cycle graph has a neighborhood partial order of width five, so four is the maximum width that ensures perfect orderability.
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From the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples. Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set W is a subspace if and only if every linear combination of finitely many elements of W also belongs to W. The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time. In a topological vector space X, a subspace W need not be topologically closed, but a finite-dimensional subspace is always closed.
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In mathematics, a k-ultrahomogeneous graph is a graph in which every isomorphism between two of its induced subgraphs of at most k vertices can be extended to an automorphism of the whole graph. A k-homogeneous graph obeys a weakened version of the same property in which every isomorphism between two induced subgraphs implies the existence of an automorphism of the whole graph that maps one subgraph to the other (but does not necessarily extend the given isomorphism). A homogeneous graph is a graph that is k-homogeneous for every k, or equivalently k-ultrahomogeneous for every k.
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In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called the eliminant. The resultant is widely used in number theory, either directly or through the discriminant, which is essentially the resultant of a polynomial and its derivative. The resultant of two polynomials with rational or polynomial coefficients may be computed efficiently on a computer.
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The dual notion of a uniform module is that of a hollow module: a module M is said to be hollow if, when N1 and N2 are submodules of M such that N_1+N_2=M, then either N1 = M or N2 = M. Equivalently, one could also say that every proper submodule of M is a superfluous submodule. These modules also admit an analogue of uniform dimension, called co-uniform dimension, corank, hollow dimension or dual Goldie dimension. Studies of hollow modules and co-uniform dimension were conducted in , , , and . The reader is cautioned that Fleury explored distinct ways of dualizing Goldie dimension.
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The Paley graph of order 9, colored with three colors and showing a clique of three vertices. In this graph and each of its induced subgraphs the chromatic number equals the clique number, so it is a perfect graph. In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph (clique number). Equivalently stated in symbolic terms an arbitrary graph G=(V,E) is perfect if and only if for all S\subseteq V we have \chi(G[S])=\omega(G[S]).
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In the latter case, where the ply of a search is even, the effective branching factor is reduced to its square root, or, equivalently, the search can go twice as deep with the same amount of computation. The explanation of b×1×b×1×... is that all the first player's moves must be studied to find the best one, but for each, only the second player's best move is needed to refute all but the first (and best) first player move—alpha–beta ensures no other second player moves need be considered. When nodes are considered in a random order (i.e.
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These suffice to uniquely calculate the determinant of any square matrix. Provided the underlying scalars form a field (more generally, a commutative ring), the definition below shows that such a function exists, and it can be shown to be unique.Serge Lang, Linear Algebra, 2nd Edition, Addison-Wesley, 1971, pp 173, 191. Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is −1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.
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If is an arbitrary graph, an interval completion of is an interval graph on the same vertex set that contains as a subgraph. The parameterized version of interval completion (find an interval supergraph with additional edges) is fixed parameter tractable, and moreover, is solvable in parameterized subexponential time. The pathwidth of an interval graph is one less than the size of its maximum clique (or equivalently, one less than its chromatic number), and the pathwidth of any graph G is the same as the smallest pathwidth of an interval graph that contains G as a subgraph..
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The Buckingham π theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of dimensionless parameters, where m is the rank of the dimensional matrix. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables. A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature. This gives insight into the fundamental properties of the system, as illustrated in the examples below.
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Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at the weak homotopy equivalences, the theorem can equivalently be stated for functors on a category defined in this way. However, the theorem is false without the restriction to connected pointed spaces, and an analogous statement for unpointed spaces is also false. A similar statement does, however, hold for spectra instead of CW complexes. Brown also proved a general categorical version of the representability theorem, which includes both the version for pointed connected CW complexes and the version for spectra.
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An object with angular momentum of can be reduced to zero rotation (all of the rotational energy can be transferred out of it) by an angular impulse of or equivalently, by torque or work of for one second, or energy of for one second. The plane perpendicular to the axis of angular momentum and passing through the center of mass, p. 97 is sometimes called the invariable plane, because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System.
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Define a circulant numbering of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to in such a way that, if some two vertices numbered and are adjacent, then every two vertices numbered and are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix. Let be an integer that is relatively prime to , and let be any integer. Then the linear function that takes a number to transforms a circulant numbering to another circulant numbering.
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The nuclear force is not a fundamental force of nature, but a consequence of the residual effects of the strong force that surround the nucleons. One consequence of these complications is that although deuterium, a bound state of a proton (p) and a neutron (n) is stable, exotic nuclides such as diproton or dineutron are unbound. The nuclear force is not sufficiently strong to form either p-p or n-n bound states, or equivalently, the nuclear force does not form a potential well deep enough to bind these identical nucleons. Stable nuclides require approximately equal numbers of protons and neutrons.
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Both the proton and the neutron are composite particles, meaning each is composed of smaller parts, namely three quarks each; although once thought to be so, neither is an elementary particle. A proton is composed of two up quarks and one down quark, while the neutron has one up quark and two down quarks. Quarks are held together by the strong force, or equivalently, by gluons, which mediate the strong force at the quark level. An up quark has electric charge e, and a down quark has charge e, so the summed electric charges of proton and neutron are +e and 0, respectively.
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Therefore, the Gibbs measure applies to widespread problems outside of physics, such as Hopfield networks, Markov networks, Markov logic networks, and boundedly rational potential games in game theory and economics. A Gibbs measure in a system with local (finite- range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density. The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique. The existence of more than one Gibbs measure is associated with statistical phenomena such as symmetry breaking and phase coexistence.
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Despite its name, an antenna "tuner" does not actually tune the antenna. It matches the complex impedance of the transmitter to that of the input end of the feedline. The input impedance of the transmission line will be different from the characteristic impedance of the feedline if the impedance of the antenna on the other end of the line does not match the line's characteristic impedance. The consequence of the mismatch is that the line's impedance (voltage to current ratio and phase) will oscillate along the line, or equivalently, raise out-of-phase voltage standing waves and current standing waves along the feedline.
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A Riemann surface is a 1-dimensional complex manifold, and so its codimension-1 submanifolds have dimension 0. The group of divisors on a compact Riemann surface X is the free abelian group on the points of X. Equivalently, a divisor on a compact Riemann surface X is a finite linear combination of points of X with integer coefficients. The degree of a divisor on X is the sum of its coefficients. For any nonzero meromorphic function f on X, one can define the order of vanishing of f at a point p in X, ordp(f).
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Magazines – Anschutz – Anschutz .22LR bolt action – RifleMags.co.uk However, magazines specifically made for biathlon differ by having a special base plate for ease of use with gloves. The sight adjustment Fortner biathlon rifles is most commonly clocwise (CW) with a click value of 0.06 mrad, which translates to 3 mm at the 50 meter distance used in competitions (or equivalently, 6 mm when training outdoor at 100 m and 0.9 mm when training indoor at 15 m).Pilotprosjekt skyting Norges Skiskytterforbund – Det frivillige Skyttervesen Diopter, English translation: One click on the diopter at [...] 100 m moves the point of impact approximately 6 mm.
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As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one for each object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G : C → D from every object of D, then G has a left adjoint. However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of D (equivalently, every object of C).
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In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space.
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This means the speaker can be driven past safe mechanical limits at frequencies below the tuning frequency with much less power than in an equivalently-sized sealed enclosure. For this reason, high-powered systems using a bass reflex design are often protected by a high-pass filter that removes signals below a certain frequency. Unfortunately, electrical filtering adds further frequency- dependent group delay. Even if such filtering can be adjusted not to remove musical content, it may interfere with sonic information connected with the size and ambiance of the recording venue, information which often exists in the low bass spectrum.
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In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices are row equivalent if and only if they have the same row space. The concept is most commonly applied to matrices that represent systems of linear equations, in which case two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space. Because elementary row operations are reversible, row equivalence is an equivalence relation.
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Equivalently, a claw- free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph. Claw-free graphs were initially studied as a generalization of line graphs, and gained additional motivation through three key discoveries about them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for finding maximum independent sets in claw-free graphs, and the characterization of claw-free perfect graphs., p. 88. They are the subject of hundreds of mathematical research papers and several surveys.
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If the true value of the project is only known to the borrower, the lender must incur a monitoring or auditing cost in order to reveal the true project returns and receive full re- payment. The size of the external finance premium that results from these market frictions may be affected by monetary policy actions. The credit channel—or, equivalently, changes in the external finance premium—can occur through two conduits: the balance sheet channel and the bank lending channel. The balance sheet channel refers to the notion that changes in interest rates affect borrowers' balance sheets and income statements.
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Thus this difference must be zero, and, thus ; that is This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This is the Archimedean property, that is verified for rational numbers and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers (infinitesimals) and infinitely large numbers (infinite numbers). When using such systems, notation 0.999... is generally not used, as there is no smallest number that is no less than all .
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As outlined in Bridgman's The Logic of Modern Physics if two researchers had different operational definitions, they had different concepts. There was no “surplus meaning.” If, for example, two researchers had different measures of “Anomia” or “Intelligence,” they had different concepts. Behaviorists focussed on Stimulus-Response laws and were deeply skeptical of "unscientific" explanations in terms of unobservable psychological processes. Behaviorists and operationists would have rejected as unscientific any notion that there was some general thing called “intelligence” that existed inside a person’s head and that might be reflected almost-equivalently in Stanford- Binet I.Q. tests or Weschler tests.
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The spectrum of a standard illuminant, like any other profile of light, can be converted into tristimulus values. The set of three tristimulus coordinates of an illuminant is called a white point. If the profile is normalized, then the white point can equivalently be expressed as a pair of chromaticity coordinates. If an image is recorded in tristimulus coordinates (or in values which can be converted to and from them), then the white point of the illuminant used gives the maximum value of the tristimulus coordinates that will be recorded at any point in the image, in the absence of fluorescence.
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In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures. Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors (the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice). The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets.
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A point p in a Riemannian submanifold is umbilical if, at p, the (vector-valued) Second fundamental form is some normal vector tensor the induced metric (First fundamental form). Equivalently, for all vectors U, V at p, II(U, V) = gp(U, V) u, where u is the mean curvature vector at p. A submanifold is said to be umbilic (or all- umbilic) if this condition holds at every point "p". This is equivalent to saying that the submanifold can be made totally geodesic by an appropriate conformal change of the metric of the surrounding ("ambient") manifold.
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It is a consequence of Helmholtz's theorems (or equivalently, of Kelvin's circulation theorem) that in an inviscid fluid the 'strength' of the vortex tube is also constant with time. Viscous effects introduce frictional losses and time dependence. In a three-dimensional flow, vorticity (as measured by the volume integral of the square of its magnitude) can be intensified when a vortex line is extended — a phenomenon known as vortex stretching.Batchelor, section 5.2 This phenomenon occurs in the formation of a bathtub vortex in outflowing water, and the build-up of a tornado by rising air currents.
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Memory for previous interactions is critical for successful long-term social relationships. This is largely due to a human expectation of reciprocity, which requires humans to have a better memory for favours or debts owed. These memories play an important role in deciding whether or not a human interacts with another in the future, as they contribute to the construction of a person's reputation. Milinski, Semmann & Krambeck (2002) have demonstrated, through the use of public goods games and indirect reciprocity games, that humans are less likely to interact with those who have a reputation for not equivalently returning the favour.
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Equivalently, some German speakers pronounce the beginning of the word "Cent" , but since they are familiar with the English pronunciation of the American unit cent, most people pronounce it [s]. As these are nouns, both Euro and Cent are capitalised in German. Slang terms: In Austria and Germany, the euro has also been called Teuro, a play on the word teuer, meaning 'expensive'. The Deutsche Mark was worth half as much as the euro (a ratio of approximately 2:1) and some grocers and restaurants were accused of taking advantage of the smaller numbers to increase their actual prices with the changeover.
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Winlink networking started by providing interconnection services for amateur radio (also known as ham radio). It is well known for its central role in emergency and contingency communications worldwide. The system used to employ multiple central message servers around the world for redundancy, but in 2017–2018 upgraded to Amazon Web Services that provides a geographically-redundant cluster of virtual servers with dynamic load balancers and global content- distribution. Gateway stations have operated on sub-bands of HFHigh frequency (HF) is the band of radio frequencies from 3–30 megaHertz, or equivalently wavelengths from 100–10 meters.
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In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM). A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian).
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Equivalently, it is the inverse of a permutation formed by choosing independently at random for each card whether to put it into one of two piles (maintaining the original order of the cards within each pile), and then stacking the two piles on top of each other. Gilbert tessellations are a mathematical model of crack formation introduced by Gilbert in 1967. In this model, fractures begin at a set of random points, with random orientations, chosen according to a Poisson process, and then grow at a constant rate until they terminate by running into previously formed cracks.
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However, it is impossible to define orthogonal (perpendicular) lines, or to single out circles among ellipses, because in a linear space there is no structure like a scalar product that could be used for measuring angles. The dimension of a linear space is defined as the maximal number of linearly independent vectors or, equivalently, as the minimal number of vectors that span the space; it may be finite or infinite. Two linear spaces over the same field are isomorphic if and only if they are of the same dimension. A complex linear space is also a real linear space.
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The averaged Lagrangian approach applies to wave motion – possibly superposed on a mean motion – that can be described in a Lagrangian formulation. Using an ansatz on the form of the wave part of the motion, the Lagrangian is phase averaged. Since the Lagrangian is associated with the kinetic energy and potential energy of the motion, the oscillations contribute to the Lagrangian, although the mean value of the wave's oscillatory excursion is zero (or very small). The resulting averaged Lagrangian contains wave characteristics like the wavenumber, angular frequency and amplitude (or equivalently the wave's energy density or wave action).
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Born rigidity is satisfied if the orthogonal spacetime distance between infinitesimally separated curves or worldlines is constant, or equivalently, if the length of the rigid body in momentary co-moving inertial frames measured by standard measuring rods (i.e. the proper length) is constant and is therefore subjected to Lorentz contraction in relatively moving frames.Gron (1981) Born rigidity is a constraint on the motion of an extended body, achieved by careful application of forces to different parts of the body. A body rigid in itself would violate special relativity, as its speed of sound would be infinite.
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Tractor units typically have large displacement diesel engines for power, durability, and economy; several axles; and a multi-ratio transmission (10, 13, or 18 gears) for maximum flexibility in gearing. The tractor-trailer combination distributes a load across multiple axles while being more maneuverable than an equivalently sized rigid truck. The most common trailer attachment system is a fifth wheel coupling, allowing a rapid shift between trailers performing different functions, such as a bulk tipper and box trailer. Trailers containing differing cargos can be rapidly swapped between tractors, eliminating downtime while a trailer is unloaded or loaded.
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The space of uniform almost periodic functions on G can be identified with the space of all continuous functions on the Bohr compactification of G. More generally the Bohr compactification can be defined for any topological group G, and the spaces of continuous or Lp functions on the Bohr compactification can be considered as almost periodic functions on G. For locally compact connected groups G the map from G to its Bohr compactification is injective if and only if G is a central extension of a compact group, or equivalently the product of a compact group and a finite-dimensional vector space.
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In 1990, Paul Seymour conjectured that, in every oriented graph, there always exists at least one vertex v whose second neighborhood is at least as large as its first neighborhood. Equivalently, in the square of the graph, the degree of v is at least doubled. The problem was first published by Nathaniel Dean and Brenda J. Latka in 1995, in a paper that studied the problem on a restricted class of oriented graphs, the tournaments (orientations of complete graphs). Dean had previously conjectured that every tournament obeys the second neighborhood conjecture, and this special case became known as Dean's conjecture.
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It is possible to construct an optimal layout in time , where n denotes the number of vertices of the input graph and m denotes the number of edges., Theorem 2.3. Graphs of bounded queue number also have bounded expansion, meaning that their shallow minors are sparse graphs with a ratio of edges to vertices (or equivalently degeneracy or arboricity) that is bounded by a function of the queue number and the depth of the minor. As a consequence, several algorithmic problems including subgraph isomorphism for pattern graphs of bounded size have linear time algorithms for these graphs.
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Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the squaring function x^2 and the exponential function e^x. In simple terms, a convex function refers to a function that is in the shape of a cup \cup, and a concave function is in the shape of a cap \cap.
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Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (which could be confused with ZFC with urelements). This axiomatic system is useful to prove for example that every category has an appropriate Yoneda embedding. This is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i.e.
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There is more than one upper bound on the achievable code rate of linear block codes for multiple phased-burst correction (MPBC). One such bound is constrained to a maximum correctable cyclic burst length within every subblock, or equivalently a constraint on the minimum error free length or gap within every phased-burst. This bound, when reduced to the special case of a bound for single burst correction, is the Abramson bound (a corollary of the Hamming bound for burst-error correction) when the cyclic burst length is less than half the block length.Ling, San, and Chaoping Xing.
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Because the exponent of depends on , this algorithm is not fixed-parameter tractable. Although it can be improved by fast matrix multiplication the running time still has an exponent that is linear in Thus, although the running time of known algorithms for the clique problem is polynomial for any fixed , these algorithms do not suffice for fixed-parameter tractability. defined a hierarchy of parametrized problems, the W hierarchy, that they conjectured did not have fixed-parameter tractable algorithms. They proved that independent set (or, equivalently, clique) is hard for the first level of this hierarchy, W[1].
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In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L2 is evident because the Fourier transform converts them into multiplication operators. Continuity on Lp spaces was first established by Marcel Riesz.
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In physical oceanography, the significant wave height (SWH or Hs) is defined traditionally as the mean wave height (trough to crest) of the highest third of the waves (H1/3). Nowadays it is usually defined as four times the standard deviation of the surface elevation – or equivalently as four times the square root of the zeroth-order moment (area) of the wave spectrum. The symbol Hm0 is usually used for that latter definition. The significant wave height may thus refer to Hm0 or Hs; the difference in magnitude between the two definitions is only a few percent.
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In fact, the definition of a neighborhood base can be equivalently restated as: "a neighborhood base is any prefilter that is equivalent the neighborhood filter." Neighborhood bases at points are examples of prefilters that are fixed but may or may not be principle. If has its usual topology and if , then any neighborhood filter base of is fixed by (in fact, it's even true that }) but is _not_ principal since . In contrast, a topological space has the discrete topology if and only if the neighborhood filter of every point is a principal filter generated by exactly one point.
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Potential of a cell assembled of two electrodes can be determined from the two individual electrode potentials using :ΔVcell = Ered,cathode − Ered,anode or, equivalently, :ΔVcell = Ered,cathode \+ Eoxy,anode. This follows from the IUPAC definition of the electric potential difference of a galvanic cell, according to which the electric potential difference of a cell is the difference of the potentials of the electrodes on the right and the left of the galvanic cell. When ΔVcell is positive, then positive electrical charge flows through the cell from the left electrode (anode) to the right electrode (cathode).
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The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. Alternatively, and equivalently, it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.
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The convex hull of the red set is the blue and red convex set. In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
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In combinatorics, a Sperner family (or Sperner system; named in honor of Emanuel Sperner), or clutter, is a family F of subsets of a finite set E in which none of the sets contains another. Equivalently, a Sperner family is an antichain in the inclusion lattice over the power set of E. A Sperner family is also sometimes called an independent system or irredundant set. Sperner families are counted by the Dedekind numbers, and their size is bounded by Sperner's theorem and the Lubell–Yamamoto–Meshalkin inequality. They may also be described in the language of hypergraphs rather than set families, where they are called clutters.
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In mathematics, a trigonometric number is an irrational number produced by taking the sine or cosine of a rational multiple of a full circle, or equivalently, the sine or cosine of an angle which in radians is a rational multiple of , or the sine or cosine of a rational number of degrees. One of the simplest examples is \cos \frac \pi 4=\frac \sqrt 2 2. A real number different from is a trigonometric number if and only if it is the real part of a root of unity (see Niven's theorem). Thus every trigonometric number is half the sum of two complex conjugate roots of unity.
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Discussion of unbounded nondeterminism tends to get involved with discussions of fairness. The basic concept is that all computation paths must be "fair" in the sense that if the machine enters a state infinitely often, it must take every possible transition from that state. This amounts to requiring that the machine be guaranteed to service a request if it can, since an infinite sequence of states will only be allowed if there is no transition that leads to the request being serviced. Equivalently, every possible transition must occur eventually in an infinite computation, although it may take an unbounded amount of time for the transition to occur.
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The transverse orbital speed is inversely proportional to the distance to the central body because of the law of conservation of angular momentum, or equivalently, Kepler's second law. This states that as a body moves around its orbit during a fixed amount of time, the line from the barycenter to the body sweeps a constant area of the orbital plane, regardless of which part of its orbit the body traces during that period of time. This law implies that the body moves slower near its apoapsis than near its periapsis, because at the smaller distance along the arc it needs to move faster to cover the same area.
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So the oscillator stays in its ground state, storing next to no energy at all. This means that at very cold temperatures, the change in energy with respect to beta, or equivalently the change in energy with respect to temperature, is also exponentially small. The change in energy with respect to temperature is the specific heat, so the specific heat is exponentially small at low temperatures, going to zero like :: \exp(-\hbar\omega/kT) At small values of \beta, at high temperatures, the average energy U is equal to 1/\beta = kT. This reproduces the equipartition theorem of classical thermodynamics: every harmonic oscillator at temperature T has energy kT on average.
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Limits and colimits are defined as universal cones. That is, cones through which all other cones factor. A cone φ from L to F is a universal cone if for any other cone ψ from N to F there is a unique morphism from ψ to φ. center Equivalently, a universal cone to F is a universal morphism from Δ to F (thought of as an object in CJ), or a terminal object in (Δ ↓ F). Dually, a cone φ from F to L is a universal cone if for any other cone ψ from F to N there is a unique morphism from φ to ψ.
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Intensifier (abbreviated ') is a linguistic term (but not a proper lexical category) for a modifier that makes no contribution to the propositional meaning of a clause but serves to enhance and give additional emotional context to the word it modifies. Intensifiers are grammatical expletives, specifically expletive attributives (or, equivalently, attributive expletives or attributive-only expletives; they also qualify as expressive attributives), because they function as semantically vacuous filler. Characteristically, English draws intensifiers from a class of words called degree modifiers, words that quantify the idea they modify. More specifically, they derive from a group of words called adverbs of degree, also known as degree adverbs.
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