A lengthy term for reciprocal in this case is MULTIPLICATIVE INVERSE.
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So in MULTIPLICATIVE INVERSE, for example, that second T is the beginning of TWO.
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The number 1 in its multiplicative identity is practically bedridden, leaving other numbers unchanged: 6 times 1 equals 6.
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It's not additive, it's multiplicative and if any of them are zero, it takes the entire equation down to zero.
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"The point is, it's a multiplicative process and that creates an exponential growth or an exponential decline," Bar-Yam told Motherboard.
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The Pokémon brand offers a multiplicative effect for the potential growth of an app, but the game still has to be good.
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Ever since the wildly popular Cookie Clicker , idle clicker games have been about hockey stick curves, about exponential growth unleashed by multiplicative advances in productivity.
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A recent example that highlights the positive, multiplicative effect of our collaboration approach is Cologuard®, a noninvasive, multi-target stool DNA screening test for colorectal cancer.
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"We believe the project is of strategic importance to Slovenia ... as it will have multiplicative effects on the economy," Infrastructure Minister Peter Gaspersic told parliament before the vote.
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" Repeating this wrenching process by pressing snooze frequently puts your cardiovascular system through such a shock again and again, causing what Walker says is "multiplicative abuse to your heart and nervous system.
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This means if you can identify the "brokers" in any social network—the people who are well-connected and respected—training them to communicate anti-conflict messages can have a multiplicative effect.
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Then in 1971 Arnold Schönhage and Volker Strassen published a method capable of multiplying large numbers in n × log n × log(log n) multiplicative steps, where log n is the logarithm of n.
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King noted that a previous study looking at over half a million scientific papers concluded that each self-citation leads to nearly three additional total citations of an author's work over the next few years, due to increased visibility—meaning that pointing people to your own work can have a multiplicative effect.
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Directed by George Nolfi from a long-gestating script by a team of screenwriters, "The Banker" is a handsome-looking if occasionally dull affair: As gratifying as it is to see Mackie given the kind of showcase he's long deserved, even he can't make windy explanations of cap rates, multiplicative inverses and markups exciting.
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In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article.
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A multiplicative scrambler used in V.34 recommendation A multiplicative descrambler used in V.34 recommendation Multiplicative scramblers (also known as feed-through) are called so because they perform a multiplication of the input signal by the scrambler's transfer function in Z-space. They are discrete linear time- invariant systems. A multiplicative scrambler is recursive, and a multiplicative descrambler is non-recursive. Unlike additive scramblers, multiplicative scramblers do not need the frame synchronization, that is why they are also called self-synchronizing.
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In number theory, the multiplicative digital root of a natural number n in a given number base b is found by multiplying the digits of n together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of n. Multiplicative digital roots are the multiplicative equivalent of digital roots.
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"Multiplicative weights" implies the iterative rule used in algorithms derived from the multiplicative weight update method. It is given with different names in the different fields where it was discovered or rediscovered.
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The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a−1. The rational numbers, the real numbers and the complex numbers are all examples of fields.
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A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı. Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 2008. Luc Florack and Hans van Assen."Multiplicative calculus in biomedical image analysis", Journal of Mathematical Imaging and Vision, 2011.
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Later, proved more generally that all circular cliques Kn/k with n/k < 4 are multiplicative. In terms of the circular chromatic number χc, this means that if , then . has shown that square-free graphs are multiplicative. Examples of non- multiplicative graphs can be constructed from two graphs G and H that are not comparable in the homomorphism order (that is, neither G→H nor H→G holds).
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In photography, filter factor refers to the multiplicative amount of light a filter blocks.
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Window functions have also been constructed as multiplicative or additive combinations of other windows.
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In number theory, a multiplicative partition or unordered factorization of an integer n is a way of writing n as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors. The number n is itself considered one of these products. Multiplicative partitions closely parallel the study of multipartite partitions, discussed in , which are additive partitions of finite sequences of positive integers, with the addition made pointwise. Although the study of multiplicative partitions has been ongoing since at least 1923, the name "multiplicative partition" appears to have been introduced by .
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The additive-increase/multiplicative-decrease (AIMD) algorithm is a feedback control algorithm best known for its use in TCP congestion control. AIMD combines linear growth of the congestion window with an exponential reduction when congestion is detected. Multiple flows using AIMD congestion control will eventually converge to use equal amounts of a shared link. The related schemes of multiplicative-increase/multiplicative-decrease (MIMD) and additive- increase/additive-decrease (AIAD) do not reach stability.
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AdaBoost Algorithm formulated by Yoav Freund and Robert Schapire also employed the Multiplicative Weight Update Method.
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By contrast, zero has no multiplicative inverse, but it has a unique quasi-inverse, "0" itself.
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A quadratic form q over a field F is multiplicative if, for vectors of indeterminates x and y, we can write q(x).q(y) = q(z) for some vector z of rational functions in the x and y over F. Isotropic quadratic forms are multiplicative.
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The extended Euclidean algorithm is particularly useful when a and b are coprime. With that provision, x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly, the polynomial extended Euclidean algorithm allows one to compute the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non prime order. It follows that both extended Euclidean algorithms are widely used in cryptography.
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In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity, and every non-zero element has a multiplicative inverse.
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In mathematics, the necklace ring is a ring introduced by to elucidate the multiplicative properties of necklace polynomials.
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In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself.
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As with classical colorings, the reverse implication always holds: if G (or H, symmetrically) is K-colorable, then G × H is easily K-colored by using the same values independently of H. Hedetniemi's conjecture is then equivalent to the statement that each complete graph is multiplicative. The above known cases are equivalent to saying that K1, K2, and K3 are multiplicative. The case of K4 is widely open. On the other hand, the proof of has been generalized by to show that all cycle graphs are multiplicative.
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Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors.
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A rng is the same as a ring, except that the existence of a multiplicative identity is not assumed.Jacobson 2009.
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The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9\. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4\. Also, 39 is the smallest number of multiplicative persistence 3\.
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The set S is assumed to be a submonoid of the multiplicative monoid of R, i.e. 1 is in S and for s and t in S we also have st in S. A subset of R with this property is called a multiplicatively closed set, multiplicative set or multiplicative system. This requirement on S is natural and necessary to have since its elements will be turned into units of the localization, and units must be closed under multiplication. It is standard practice to assume that S is multiplicatively closed.
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In computer science, multiplicative binary search is a variation of binary search that uses a specific permutation of keys in an array instead of the sorted order used by regular binary search. Multiplicative binary search was first described by Thomas Standish in 1980. This algorithm was originally proposed to simplify the midpoint index calculation on small computers without efficient division or shift operations. On modern hardware, the cache-friendly nature of multiplicative binary search makes it suitable for out-of-core search on block-oriented storage as an alternative to B-trees and B+ trees.
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If the near- ring has a multiplicative identity, then distributivity on both sides is sufficient, and commutativity of addition follows automatically.
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Therefore, if an element possesses both a left and right quasi- inverse, they are equal.Since 0 is the multiplicative identity, if x \cdot y = 0 = y' \cdot x, then y = (y' \cdot x) \cdot y = y' \cdot (x \cdot y) = y'. Quasiregularity does not require the ring to have a multiplicative identity. Note that some authors use different definitions.
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Multiplicative number theory is a subfield of analytic number theory that deals with prime numbers and with factorization and divisors. The focus is usually on developing approximate formulas for counting these objects in various contexts. The prime number theorem is a key result in this subject. The Mathematics Subject Classification for multiplicative number theory is 11Nxx.
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As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring ).See Euler's theorem.
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Since and implies , the set of classes coprime to n is closed under multiplication. Integer multiplication respects the congruence classes, that is, and implies . This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying .
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Ordinal, multiplicative and multiple numerals declined and still decline in the same way as adjectives so they will not be discussed here.
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Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.
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This is particularly useful with rings and fields to distinguish the additive underlying group from the multiplicative group of the invertible elements.
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The multiplicative digital root can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Since addition and multiplication of matrices have all needed properties for field operations except for commutativity of multiplication and existence of multiplicative inverses, one way to verify if a set of matrices is a field with the usual operations of matrix sum and multiplication is to check whether # the set is closed under addition, subtraction and multiplication; # the neutral element for matrix addition (that is, the zero matrix) is included; # multiplication is commutative; # the set contains a multiplicative identity (note that this does not have to be the identity matrix); and # each matrix that is not the zero matrix has a multiplicative inverse.
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Andrianov was an Invited Speaker at the ICM in 1970 in Nice with talk On the zeta function of the general linear group and in 1983 in Warsaw with talk Integral representation of quadratic forms by quadratic forms: multiplicative properties.Andrianov, A. N. "Integral representations of quadratic forms by quadratic forms: multiplicative properties." In Proc. Intern. Congress of Mathematicians, Warsaw (1983), vol.
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These various systems do not have a single unifying trait or feature. The most common structure is ciphered-additive with a decimal base, with or without the use of multiplicative-additive structuring for the higher numbers. Exceptions include the Armenian notation of Shirakatsi, which is multiplicative-additive and sometimes uses a base 1,000, and the Greek and Arabic astronomical notation systems.
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Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplicative inverses are not required to exist. A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field. The additive group of a ring is the ring equipped just with the structure of addition.
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Any conserved quantum number is a symmetry of the Hamiltonian of the system (see Noether's theorem). Symmetry groups which are examples of the abstract group called Z2 give rise to multiplicative quantum numbers. This group consists of an operation, P, whose square is the identity, P2 = 1. Thus, all symmetries which are mathematically similar to parity (physics) give rise to multiplicative quantum numbers.
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For optimal performance, the branching factor of a B-tree or B+-tree must match the block size of the file system that it is stored on. The permutation used by multiplicative binary search places the optimal number of keys in the first (root) block, regardless of block size. Multiplicative binary search is used by some optimizing compilers to implement switch statements.
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Since these numbers hi are the multiplicative inverses of the Mi, they may be found using Euclid's algorithm as described in the previous subsection.
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Vector spaces endowed with an additional multiplicative structure are called algebras. The tensor product of such algebras is described by the Littlewood–Richardson rule.
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Also, foveation encoding may be applied to the image before other types of image compression are applied and therefore can result in a multiplicative reduction.
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The condition that A,B are both non-empty is clearly necessary. This condition is not part of the multiplicative versions of BM stated below.
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In good situations, the algebraic character is multiplicative, i.e., the character of the tensor product of two weight modules is the product of their characters.
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A maximum likely fit of a multiplicative cascade to the dataset not only estimates the complete spectrum but also gives reasonable estimates of the errors.
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Therefore, Newton's iteration needs only two multiplications and one subtraction. This method is also very efficient to compute the multiplicative inverse of a power series.
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Modern computers compute division by methods that are faster than long division, with the more efficient ones relying on approximation techniques from numerical analysis. For division with remainder, see Division algorithm. In modular arithmetic (modulo a prime number) and for real numbers, nonzero numbers have a multiplicative inverse. In these cases, a division by may be computed as the product by the multiplicative inverse of .
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The easiest example of a completely multiplicative function is a monomial with leading coefficient 1: For any particular positive integer n, define f(a) = an. Then f(bc) = (bc)n = bncn = f(b)f(c), and f(1) = 1n = 1. The Liouville function is a non-trivial example of a completely multiplicative function as are Dirichlet characters, the Jacobi symbol and the Legendre symbol.
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An ideal multiplicative mixer produces an output signal equal to the product of the two input signals. In communications, a multiplicative mixer is often used together with an oscillator to modulate signal frequencies. A multiplicative mixer can be coupled with a filter to either up-convert or down-convert an input signal frequency, but they are more commonly used to down-convert to a lower frequency to allow for simpler filter designs, as done in superheterodyne receivers. In many typical circuits, the single output signal actually contains multiple waveforms, namely those at the sum and difference of the two input frequencies and harmonic waveforms.
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Thus, the trivial ring, consisting of a single element, is not a field. Every finite subgroup of the multiplicative group of a field is cyclic (see ).
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In particular, the computation of the modular multiplicative inverse is an essential step in the derivation of key-pairs in the RSA public- key encryption method.
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Adding the prefix wəj- to a base numeral results in a multiplicative numeral. Fifth is expressed by wəj- rəŋə, the prefix to the base word for 5.
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While performing the decryption, the step (the inverse of ) is used, which requires first taking the inverse of the affine transformation and then finding the multiplicative inverse.
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The multiplicative factor represents the present value of the expected length (number of periods) of the customer relationship. When retention equals 0, the customer will never be retained, and the multiplicative factor is zero. When retention equals 1, the customer is always retained, and the firm receives the margin in perpetuity. The present value of the margin in perpetuity turns out to be the Margin divided by the Discount Rate.
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Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion. In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field Z/pZ. (In other words, every congruence class except zero modulo p has a multiplicative inverse.
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In the multiplicative model, the expected phenotype resulting from the mutation of two independent genes is the product of the phenotypes due to the individual mutations. Which model is best depends on the situation. It turns out in the case that fitness is used as the phenotype, the multiplicative model is best option. Methods exist to measure genetic interactions even when one of the genes is essential to an organism.
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In an integral domain, every nonzero element a has the cancellation property, that is, if , an equality implies . "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity.B.L. van der Waerden, Algebra Erster Teil, p.
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In order to assemble local data pertaining to all local fields attached to F, the adele ring is set up. A multiplicative variant is referred to as ideles.
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"Habituality, Pluractionality, and Imperfectivity." In The Oxford Handbook of Tense and Aspect, edited by Robert I. Binnick, 852-880. Oxford: Oxford University Press. or "multiplicative",Tatevosov, Sergej. 2002.
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Similarly, the set of integers in W is itself a multiplicative semigroup. It is called the wild integer semigroup and members of this semigroup are called wild numbers.
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Algebraic groups can be decomposed into unipotent groups, multiplicative groups, and abelian varieties, but the statement of how they decompose depends upon the characteristic of their base field.
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Even without knowledge that we are working in the multiplicative group of integers modulo n, we can show that a actually has an order by noting that the powers of a can only take a finite number of different values modulo n, so according to the pigeonhole principle there must be two powers, say s and t and without loss of generality s > t, such that as ≡ at (mod n). Since a and n are coprime, this implies that a has an inverse element a−1 and we can multiply both sides of the congruence with a−t, yielding as−t ≡ 1 (mod n). The concept of multiplicative order is a special case of the order of group elements. The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n.
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Tukey's F-statistic for testing interaction has a distribution based on the randomized assignment of treatments to experimental units. When Mandel's multiplicative model holds, the F-statistics randomization distribution is closely approximated by the distribution of the F-statistic assuming a normal distribution for the error, according to the 1975 paper of Robinson. The rejection of multiplicative interaction need not imply the rejection of non- multiplicative interaction, because there are many forms of interaction. Generalizing earlier models for Tukey's test are the “bundle-of-straight lines” model of Mandel (1959) and the functional model of Milliken and Graybill (1970), which assumes that the interaction is a known function of the block and treatment main-effects.
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Many arithmetic functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers.
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Finally the fact that it vanishes (to order 1) at enough points of this form implies using Vandermonde determinants that there is a multiplicative relation between the numbers ai.
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In mathematics, a multiplicative sequence or m-sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.
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This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit.
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A cobordism (W; M, N). In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e. up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.
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A minor but often confusing point is as follows: most conserved quantum numbers are additive, so in an elementary particle reaction, the sum of the quantum numbers should be the same before and after the reaction. However, some, usually called a parity, are multiplicative; i.e., their product is conserved. All multiplicative quantum numbers belong to a symmetry (like parity) in which applying the symmetry transformation twice is equivalent to doing nothing (involution).
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These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of a group for example, the identity element is sometimes simply denoted by the symbol e. The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domains, and fields. The multiplicative identity is often called unity in the latter context (a ring with unity).
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Even the inherent statistical fluctuations of neutron multiplication within a chain reaction have implications with regard to implosion speed and symmetry. In November 1944, David Hawkins and Ulam addressed this problem in a report entitled "Theory of Multiplicative Processes". This report, which invokes probability- generating functions, is also an early entry in the extensive literature on statistics of branching and multiplicative processes. In 1948, its scope was extended by Ulam and Everett.
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It is the product of the multiplicative group Gm by a unipotent group of dimension ni−1, which in characteristic 0 is isomorphic to a product of ni−1 additive groups.
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Multiplicative scrambler/descrambler is defined similarly by a polynomial (for the scrambler on the picture it is 1+z^{-18}+z^{-23}), which is also a transfer function of the descrambler.
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The corresponding result for the multiplicative group is known as Hilbert's Theorem 90, and was known before 1900. Kummer theory was another such early part of the theory, giving a description of the connecting homomorphism coming from the m-th power map. In fact for a while the multiplicative case of a 1-cocycle for groups that are not necessarily cyclic was formulated as the solubility of Noether's equations, named for Emmy Noether; they appear under this name in Emil Artin's treatment of Galois theory, and may have been folklore in the 1920s. The case of 2-cocycles for the multiplicative group is that of the Brauer group, and the implications seem to have been well known to algebraists of the 1930s.
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CSV expounded the theory in a series of remarkable research papers which soon became a classic and CSV was awarded the Ph.D. degree by the University of Madras in 1952 for his "Contributions to the Theory of Multiplicative Functions". During the work leading to the award of Ph.D. degree, he derived a new identity for multiplicative functions of two variables. Vaidyanathaswami’s identity for multiplicative functions which appeared in Transactions of the American Mathematical Society in 1931 could be deduced from that of CSV's. In view of his contributions to the theory of numbers, CSV was nominated for a visiting professorship at the University of North Carolina, Durham, NC, US. This was for possible collaboration with Leonard Carlitz, who came to know about CSV's research interest.
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It follows that the matrices over a ring form a ring, which is noncommutative except if and the ground ring is commutative. A square matrix may have a multiplicative inverse, called an inverse matrix. In the common case where the entries belong to a commutative ring , a matrix has an inverse if and only if its determinant has a multiplicative inverse in . The determinant of a product of square matrices is the product of the determinants of the factors.
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In principle, multiplicative quantum numbers can be defined for any abelian group. An example would be to trade the electric charge, Q, (related to the abelian group U(1) of electromagnetism), for the new quantum number exp(2iπ Q). Then this becomes a multiplicative quantum number by virtue of the charge being an additive quantum number. However, this route is usually followed only for discrete subgroups of U(1), of which Z2 finds the widest possible use.
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The multiplicative version of the Jordan-Chevalley decomposition generalizes to a decomposition in a linear algebraic group, and the additive version of the decomposition generalizes to a decomposition in a Lie algebra.
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A subset S of R is said to be a subring if it can be regarded as a ring with the addition and the multiplication restricted from R to S. Equivalently, S is a subring if it is not empty, and for any x, y in S, xy, x+y and -x are in S. If all rings have been assumed, by convention, to have a multiplicative identity, then to be a subring one would also require S to share the same identity element as R.In the unital case, like addition and multiplication, the multiplicative identity must be restricted from the original ring. The definition is also equivalent to requiring the set- theoretic inclusion is a ring homomorphism. So if all rings have been assumed to have a multiplicative identity, then a proper ideal is not a subring. For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X] (in both cases, Z contains 1, which is the multiplicative identity of the larger rings).
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One notion of derivative in this setting is the H-derivative of a function on an abstract Wiener space. Multiplicative calculus replaces addition with multiplication, and hence rather than dealing with the limit of a ratio of differences, it deals with the limit of an exponentiation of ratios. This allows the development of the geometric derivative and bigeometric derivative. Moreover, just like the classical differential operator has a discrete analog, the difference operator, there are also discrete analogs of these multiplicative derivatives.
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This procedure should be avoided unless the errors are multiplicative and log-normally distributed because it can give misleading results. This comes from the fact that whatever the experimental errors on y might be, the errors on log y are different. Therefore, when the transformed sum of squares is minimized different results will be obtained both for the parameter values and their calculated standard deviations. However, with multiplicative errors that are log-normally distributed, this procedure gives unbiased and consistent parameter estimates.
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In quantum field theory, multiplicative quantum numbers are conserved quantum numbers of a special kind. A given quantum number q is said to be additive if in a particle reaction the sum of the q-values of the interacting particles is the same before and after the reaction. Most conserved quantum numbers are additive in this sense; the electric charge is one example. A multiplicative quantum number q is one for which the corresponding product, rather than the sum, is preserved.
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It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0\. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 returns that number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number.
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To obtain the empirical formula from these names, the stoichiometry can be deduced from the charges on the ions, and the requirement of overall charge neutrality. If there are multiple different cations and/or anions, multiplicative prefixes (di-, tri-, tetra-, ...) are often required to indicate the relative compositions, and cations then anions are listed in alphabetical order. For example, KMgCl3 is named magnesium potassium trichloride to distinguish it from K2MgCl4, magnesium dipotassium tetrachloride (note that in both the empirical formula and the written name, the cations appear in alphabetical order, but the order varies between them because the symbol for potassium is K). When one of the ions already has a multiplicative prefix within its name, the alternate multiplicative prefixes (bis-, tris-, tetrakis-, ...) are used. For example, Ba(BrF4)2 is named barium bis(tetrafluoridobromate).
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The result was proven by sieve methods by Montgomery and Vaughan; an earlier result of Brun and Titchmarsh obtained a weaker version of this inequality with an additional multiplicative factor of 1+o(1).
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Increased limit factors or ILFs are multiplicative factors that are applied to premiums for "basic" limits of coverage to determine premiums for higher limits of coverage. They are commonly used in casualty insurance pricing.
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The name MultiSwap comes from the cipher's multiplications and swaps. WMDRM uses this algorithm only as a MAC, never for encryption. Borisov, et al. applied a multiplicative form of differential cryptanalysis to break MultiSwap.
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However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist.
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The goal is to reduce the total loss suffered for a particular allocation. The allocation for the following iteration is then revised, based on the total loss suffered in the current iteration using multiplicative update.
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In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that provides both is the classic approximation algorithm of Lenstra, Shmoys and Tardos for scheduling on unrelated parallel machines.
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If is a positive integer, the ring may be identified with the set } of the remainders of Euclidean division by , the addition and the multiplication consisting in taking the remainder by of the result of the addition and the multiplication of integers. An element of has a multiplicative inverse (that is, it is a unit) if it is coprime to . In particular, if is prime, has a multiplicative inverse if it is not zero (modulo ). Thus is a field if and only if is prime.
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This is symmetric in the βi and homogeneous of weight j: so can be expressed as a polynomial Kj(p1, ..., pj) in the elementary symmetric functions p of the β. Then Kj defines a multiplicative sequence.
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In addition to the regioselectivity of these variants of the genus Trichophyton, combinations of varying species of the genus can have multiplicative effects that are invisible to the host immune system, resulting in potentially chronic infection.
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Multiplicative characters are linearly independent, i.e. if \chi_1, \chi_2, \ldots, \chi_n are different characters on a group G then from a_1\chi_1 + a_2\chi_2 + \cdots + a_n\chi_n = 0 it follows that a_1 = a_2 = \cdots = a_n = 0.
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The classical Möbius function is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832. It is a special case of a more general object in combinatorics.
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For the same reason, the empty product is taken to be the multiplicative identity. For sums of other objects (such as vectors, matrices, polynomials), the value of an empty summation is taken to be its additive identity.
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RSMs were also studied in a worst-case scenario in which the market is small. In such cases, we want to get an absolute, multiplicative approximation factor, that does not depend on the size of the market.
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In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets (also spelled "Oseledec") in 1965 and reported at the International Mathematical Congress in Moscow in 1966. A conceptually different proof of the multiplicative ergodic theorem was found by M. S. Raghunathan. The theorem has been extended to semisimple Lie groups by V. A. Kaimanovich and further generalized in the works of David Ruelle, Grigory Margulis, Anders Karlsson, and François Ledrappier.
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In mathematics, and more specifically in abstract algebra, a rng (or pseudo- ring or non-unital ring) is an algebraic structure satisfying the same properties as a ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element". There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the ring axioms (see the history section of the article on rings).
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Non- replicated experiments are used by knowledgeable experimentalists when replications have prohibitive costs. When the block-design lacks replicates, interactions have been modeled. For example, Tukey's F-test for interaction (non-additivity) has been motivated by the multiplicative model of Mandel (1961); this model assumes that all treatment-block interactions are proportion to the product of the mean treatment-effect and the mean block- effect, where the proportionality constant is identical for all treatment- block combinations. Tukey's test is valid when Mandel's multiplicative model holds and when the errors independently follow a normal distribution.
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Triple exponential smoothing applies exponential smoothing three times, which is commonly used when there are three high frequency signals to be removed from a time series under study. There are different types of seasonality: 'multiplicative' and 'additive' in nature, much like addition and multiplication are basic operations in mathematics. If every month of December we sell 10,000 more apartments than we do in November the seasonality is additive in nature. However, if we sell 10% more apartments in the summer months than we do in the winter months the seasonality is multiplicative in nature.
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As explained in below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. Authors who follow this convention sometimes refer to a structure satisfying all the axioms except the requirement that there exists a multiplicative identity element as a rng (commonly pronounced rung) and sometimes as a pseudo-ring. For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring.
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Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication of free random variables (see below; in the classical case, what would be the analog of free multiplicative convolution can be reduced to additive convolution by passing to logarithms of random variables). These operations have some interpretations in terms of empirical spectral measures of random matrices.Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010).
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Finding multiplicative inverses is an essential step in the RSA algorithm, which is widely used in electronic commerce; specifically, the equation determines the integer used to decrypt the message. Although the RSA algorithm uses rings rather than fields, the Euclidean algorithm can still be used to find a multiplicative inverse where one exists. The Euclidean algorithm also has other applications in error-correcting codes; for example, it can be used as an alternative to the Berlekamp–Massey algorithm for decoding BCH and Reed–Solomon codes, which are based on Galois fields.
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Perhaps the most famous, still open problem of the theory of topological algebras is whether all linear multiplicative functionals on an m-convex Frechet algebra are continuous. The statement that this be the case is known as Michael's Conjecture (, ).
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Roughly speaking, the condition of multiplicative reduction amounts to saying that the singular point is a double point, rather than a cusp.Husemoller (1987) pp.116-117 Deciding whether this condition holds is effectively computable by Tate's algorithm.Husemöller (1987) pp.
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The extended Euclidean algorithm is the essential tool for computing multiplicative inverses in modular structures, typically the modular integers and the algebraic field extensions. A notable instance of the latter case are the finite fields of non-prime order.
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Modulo 8, the product of the nonresidues 3 and 5 is the nonresidue 7, and likewise for permutations of 3, 5 and 7. In fact, the multiplicative group of the non-residues and 1 form the Klein four-group.
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The gap gn between the nth and (n + 1)st prime numbers is an example of an arithmetic function. In this context it is usually denoted dn and called the prime difference function. The function is neither multiplicative nor additive.
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This generalized Reichardt model allows arbitrary filters before the multiplicative nonlinearity as well as filters post-nonlinearity. Phi Phenomenon is often regarded as first-order motion, but reversed phi could be both first-order and second-order, according to this model.
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The earliest known version of this technique was in an algorithm named "fictitious play" which was proposed in game theory in the early 1950s. Grigoriadis and Khachiyan applied a randomized variant of "fictitious play" to solve two- player zero-sum games efficiently using the multiplicative weights algorithm. In this case, player allocates higher weight to the actions that had a better outcome and choose his strategy relying on these weights. In machine learning, Littlestone applied the earliest form of the multiplicative weights update rule in his famous winnow algorithm, which is similar to Minsky and Papert's earlier perceptron learning algorithm.
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In signal processing, the term multiplicative noise refers to an unwanted random signal that gets multiplied into some relevant signal during capture, transmission, or other processing. An important example is the speckle noise commonly observed in radar imagery. Examples of multiplicative noise affecting digital photographs are proper shadows due to undulations on the surface of the imaged objects, shadows cast by complex objects like foliage and Venetian blinds, dark spots caused by dust in the lens or image sensor, and variations in the gain of individual elements of the image sensor array. Maria Petrou, Costas Petrou (2010) Image Processing: The Fundamentals.
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Formally, we start with a non-zero algebra D over a field. We call D a division algebra if for any element a in D and any non-zero element b in D there exists precisely one element x in D with a = bx and precisely one element y in D such that . For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1 and every non-zero element a has a multiplicative inverse (i.e. an element x with ).
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Such problems cannot be approximated to any multiplicative factor by a bounded-error probabilistic algorithm unless NP = RP, because any multiplicative approximation would distinguish the values 0 and 1, effectively solving the decision version in bounded-error probabilistic polynomial time. In particular, under the same assumption, this rules out the possibility of a fully polynomial time randomised approximation scheme (FPRAS). For other points, more complicated arguments are needed, and the question is the focus of active research. , it is known that there is no FPRAS for computing P(G, x) for any x > 2, unless NP = RP holds.
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In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent.
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In mathematics, a multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of G. Sometimes only unitary characters are considered (characters whose image is in the unit circle); other such homomorphisms are then called quasi-characters. Dirichlet characters can be seen as a special case of this definition.
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In mathematics, the persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point at which the operation no longer alters the number. Usually, this involves additive or multiplicative persistence of an integer, which is how often one has to replace the number by the sum or product of its digits until one reaches a single digit. Because the numbers are broken down into their digits, the additive or multiplicative persistence depends on the radix. In the remainder of this article, base ten is assumed.
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Examples and applications of groups abound. A starting point is the group Z of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra.
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Gilfoyle, p. 265. The viewer physically enters the art when he walks underneath it into its "navel". The omphalos is a "warped dimension of fluid space". In this dimension, solid is transformed into fluid in a disorienting multiplicative manner that intensifies the experience.
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The global class field theory for A-fields is developed using the pairings of Chapter XII, replacing multiplicative groups of local fields with idèle class groups of A-fields. The pairing is constructed as a product over places of local Hasse invariants.
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For instance, he required every non-zero- divisor to have a multiplicative inverse.Fraenkel, p. 144, axiom R8). In 1921, Emmy Noether gave the modern axiomatic definition of (commutative) ring and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.
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The Star Charm: Increases the chance of success by 15%. Whether this is multiplicative or additive remains to be confirmed. 2\. The Serenity Jade: Prevents the item from breaking. Instead, should the refinery process fail, the refine level will be reduced by 1.
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A division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ringIn this article, rings have a 1. in which every nonzero element a has a multiplicative inverse, i.e., an element x with .
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A semifield is a quasifield which also satisfies the left distributive law: x \otimes (y + z) = x \otimes y + x \otimes z. A planar nearfield is a quasifield whose multiplicative loop is associative (and hence a group). Not all nearfields are planar nearfields.
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Orbital angular momentum operators have the ladder operators: :L_\pm = L_x \pm i L_y which raise or lower the orbital magnetic quantum number m by one unit. This has almost exactly the same form as the spherical basis, aside from constant multiplicative factors.
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The BDDC method uses the same corner basis functions as, but in an additive rather than multiplicative fashion.J. Mandel and C. R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl., 10 (2003), pp. 639–659.
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It exists precisely when a is coprime to n, because in that case and by Bézout's lemma there are integers x and y satisfying . Notice that the equation implies that x is coprime to n, so the multiplicative inverse belongs to the group.
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The multiplicative weights algorithm is also widely applied in computational geometry, such as Clarkson's algorithm for linear programming (LP) with a bounded number of variables in linear time. Later, Bronnimann and Goodrich employed analogous methods to find Set Covers for hypergraphs with small VC dimension.
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For the majority of homomorphic encryption schemes, the multiplicative depth of circuits is the main practical limitation in performing computations over encrypted data. Homomorphic encryption schemes are inherently malleable. In terms of malleability, homomorphic encryption schemes have weaker security properties than non-homomorphic schemes.
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Homomorphic filtering is sometimes used for image enhancement. It simultaneously normalizes the brightness across an image and increases contrast. Here homomorphic filtering is used to remove multiplicative noise. Illumination and reflectance are not separable, but their approximate locations in the frequency domain may be located.
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An exponential E on an ordered field K is a strictly increasing isomorphism of the additive group of K onto the multiplicative group of positive elements of K. The ordered field K\, together with the additional function E\, is called an ordered exponential field.
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From any additive function f(n) it is easy to create a related multiplicative function g(n) i.e. with the property that whenever a and b are coprime we have: :g(ab) = g(a) × g(b). One such example is g(n) = 2f(n).
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From the definition of , it follows that . For example, given , and , the solution is the remainder of dividing by . Modular exponentiation can be performed with a negative exponent by finding the modular multiplicative inverse of modulo using the extended Euclidean algorithm. That is: :, where and .
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Consider the multiplicative monoid of positive integers as a category with one object. In this category, the pullback of two positive integers and is just the pair , where the numerators are both the least common multiple of and . The same pair is also the pushout.
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By the uniqueness of the multiplicative identity, "unitarity" is often treated like a property. If one drops the requirement for the associativity, then one obtains a non- associative algebra. If A itself is commutative (as a ring) then it is called a commutative R-algebra.
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Let I be an ideal and let M be a multiplicative system (i.e. M is closed under multiplication) in a ring R, and suppose I \cap M = \varnothing. Then there exists a prime ideal P satisfying I \subseteq P and P \cap M = \varnothing.
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Pascual-Leone, J., Johnson, J., & Agostino, A. (2010). "Mental attention, multiplicative structures, and the causal problem of cognitive development". In M. Ferrari & L. Vuletic (Eds.), Developmental interplay between mind, brain and education: Essays in honor of Robbie Case (pp. 49-82). New York, NY: Springer.
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Macedonian numerals are words that are used in the Macedonian language for expressing quantity. The Macedonian numerals have three grammatical genders (masculine, feminine and neutral) and they can have articles. There are several types of numerals: cardinal numerals, ordinal numerals, collective numerals and multiplicative numerals.
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In 2008, Clozel, Harris, Shepherd-Barron, and Taylor published a proof of the Sato–Tate conjecture for elliptic curves over totally real fields satisfying a certain condition: of having multiplicative reduction at some prime,That is, for some p where E has bad reduction (and at least for elliptic curves over the rational numbers there are some such p), the type in the singular fibre of the Néron model is multiplicative, rather than additive. In practice this is the typical case, so the condition can be thought of as mild. In more classical terms, the result applies where the j-invariant is not integral. in a series of three joint papers.
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In computational complexity theory, the class APX (an abbreviation of "approximable") is the set of NP optimization problems that allow polynomial- time approximation algorithms with approximation ratio bounded by a constant (or constant-factor approximation algorithms for short). In simple terms, problems in this class have efficient algorithms that can find an answer within some fixed multiplicative factor of the optimal answer. An approximation algorithm is called an f(n)-approximation algorithm for input size n if it can be proven that the solution that the algorithm finds is at most a multiplicative factor of f(n) times worse than the optimal solution. Here, f(n) is called the approximation ratio.
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A twiddle factor, in fast Fourier transform (FFT) algorithms, is any of the trigonometric constant coefficients that are multiplied by the data in the course of the algorithm. This term was apparently coined by Gentleman & Sande in 1966, and has since become widespread in thousands of papers of the FFT literature. More specifically, "twiddle factors" originally referred to the root-of-unity complex multiplicative constants in the butterfly operations of the Cooley–Tukey FFT algorithm, used to recursively combine smaller discrete Fourier transforms. This remains the term's most common meaning, but it may also be used for any data-independent multiplicative constant in an FFT.
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Computational complexity varies with the number of instructions required and latency of individual instructions, with the simplest being the bitwise methods (folding), followed by the multiplicative methods, and the most complex (slowest) are the division-based methods. Because collisions should be infrequent, and cause a marginal delay but are otherwise harmless, it's usually preferable to choose a faster hash function over one that needs more computation but saves a few collisions. Division-based implementations can be of particular concern, because division is microprogrammed on nearly all chip architectures. Divide (modulo) by a constant can be inverted to become a multiply by the word-size multiplicative-inverse of the constant.
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While the notation might be misunderstood, certainly denotes the multiplicative inverse of and has nothing to do with the inverse function of . In keeping with the general notation, some English authors use expressions like to denote the inverse of the sine function applied to (actually a partial inverse; see below) Other authors feel that this may be confused with the notation for the multiplicative inverse of , which can be denoted as . To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin arcus). For instance, the inverse of the sine function is typically called the arcsine function, written as .
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The split-octonions, like the octonions, are noncommutative and nonassociative. Also like the octonions, they form a composition algebra since the quadratic form N is multiplicative. That is, :N(xy) = N(x)N(y). The split-octonions satisfy the Moufang identities and so form an alternative algebra.
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Lorenz (2008) p. 31 The Witt ring is a Jacobson ring.Lorenz (2008) p. 35 It is a Noetherian ring if and only if there are finitely many square classes; that is, if the squares in k form a subgroup of finite index in the multiplicative group.
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"Multiplicative calculus and its applications", Journal of Mathematical Analysis and Applications, 2008.Diana Andrada Filip and Cyrille Piatecki. "A non-Newtonian examination of the theory of exogenous economic growth", CNCSIS – UEFISCSU (project number PNII IDEI 2366/2008) and LEO , 2010.Luc Florack and Hans van Assen.
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The additive increase/multiplicative decrease (AIMD) algorithm is a closed-loop control algorithm. AIMD combines linear growth of the congestion window with an exponential reduction when a congestion takes place. Multiple flows using AIMD congestion control will eventually converge to use equal amounts of a contended link.
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1\. Euler's theorem can be proven using concepts from the theory of groups:Ireland & Rosen, corr. 1 to prop 3.3.2 The residue classes modulo that are coprime to form a group under multiplication (see the article Multiplicative group of integers modulo n for details). The order of that group is .
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The formula most widely used for finding the distances is: d = ks + c Here, s is the stadia interval (top intercept minus bottom intercept); k and c are multiplicative and additive constants. Generally, the instrument is made so that k = 100 and c = 0 exactly, to simplify calculations.
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This follows from elementary group theory, because the exponent of any finite group must divide the order of the group. is the exponent of the multiplicative group of integers modulo while is the order of that group. We can thus view Carmichael's theorem as a sharpening of Euler's theorem.
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Any abelian category, in particular the category Ab of abelian groups, is pseudo-abelian. Indeed, in an abelian category, every morphism has a kernel. The category of associative rngs (not rings!) together with multiplicative morphisms is pseudo-abelian. A more complicated example is the category of Chow motives.
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For this reason, one may multiply formal power series without worrying about the usual questions of absolute, conditional and uniform convergence which arise in dealing with power series in the setting of analysis. Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows.
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Suppose A is an algebraic structure. We might ask, does A have the zero-product property? In order for this question to have meaning, A must have both additive structure and multiplicative structure.There must be a notion of zero (the additive identity) and a notion of products, i.e.
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This algorithm tries to find the maximum cwnd by searching in three parts: binary search increase, additive increase, and slow start. When a network failure occurs, the BIC uses multiplicative decrease in correcting the cwnd. BIC TCP is implemented and used by default in Linux kernels 2.6.8 and above.
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The Fano plane, as a block design, is a Steiner triple system. As such, it can be given the structure of a quasigroup. This quasigroup coincides with the multiplicative structure defined by the unit octonions e1, e2, ..., e7 (omitting 1) if the signs of the octonion products are ignored .
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The addition in is the addition of polynomials. The multiplication in is the remainder of the Euclidean division by of the product of polynomials. Thus, to complete the arithmetic in , it remains only to define how to compute multiplicative inverses. This is done by the extended Euclidean algorithm.
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Key or hash function should avoid clustering, the mapping of two or more keys to consecutive slots. Such clustering may cause the lookup cost to skyrocket, even if the load factor is low and collisions are infrequent. The popular multiplicative hash is claimed to have particularly poor clustering behaviour.
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"Executive functions underlying multiplicative reasoning: Problem type matters". Journal of Experimental Child Psychology, 105, 286-305.Balioussis, C., Johnson, J., & Pascual-Leone, J. (2012) "Fluency and complexity in children's writing: The role of mental attention and executive function". Rivista di Psicolinguistica Applicata / Journal of Applied Psycholinguistics, 12, 33-45.
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Any deviation from the above assumptions—a nonlinear state equation, a non-quadratic objective function, noise in the multiplicative parameters of the model, or decentralization of control—causes the certainty equivalence property not to hold. For example, its failure to hold for decentralized control was demonstrated in Witsenhausen's counterexample.
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Kendall shows an example of a decomposition into smooth, seasonal and irregular factors for a set of data containing values of the monthly aircraft miles flown by UK airlines. In policy analysis, forecasting future production of biofuels is key data for making better decisions, and statistical time series models have recently been developed to forecast renewable energy sources, and a multiplicative decomposition method was designed to forecast future production of biohydrogen. The optimum length of the moving average (seasonal length) and start point, where the averages are placed, were indicated based on the best coincidence between the present forecast and actual values. An example of using multiplicative decomposition in biohydrogen production forecast.
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The logarithm and square root transformations are commonly used for positive data, and the multiplicative inverse (reciprocal) transformation can be used for non-zero data. The power transformation is a family of transformations parameterized by a non-negative value λ that includes the logarithm, square root, and multiplicative inverse as special cases. To approach data transformation systematically, it is possible to use statistical estimation techniques to estimate the parameter λ in the power transformation, thereby identifying the transformation that is approximately the most appropriate in a given setting. Since the power transformation family also includes the identity transformation, this approach can also indicate whether it would be best to analyze the data without a transformation.
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Discrete logarithm records are the best results achieved to date in solving the discrete logarithm problem, which is the problem of finding solutions x to the equation gx = h given elements g and h of a finite cyclic group G. The difficulty of this problem is the basis for the security of several cryptographic systems, including Diffie–Hellman key agreement, ElGamal encryption, the ElGamal signature scheme, the Digital Signature Algorithm, and the elliptic curve cryptography analogs of these. Common choices for G used in these algorithms include the multiplicative group of integers modulo p, the multiplicative group of a finite field, and the group of points on an elliptic curve over a finite field.
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The product and the multiplicative inverse of two roots of unity are also roots of unity. In fact, if and , then , and , where is the least common multiple of and . Therefore, the roots of unity form an abelian group under multiplication. This group is the torsion subgroup of the circle group.
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The reciprocity law of local class field theory over a local field in the context of a pairing of the multiplicative group of a field and the character group of the absolute Galois group of the algebraic closure of the field is proved. Ramification theory for abelian extensions is developed.
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In computer architecture, instructions per cycle (IPC), commonly called instructions per clock is one aspect of a processor's performance: the average number of instructions executed for each clock cycle. It is the multiplicative inverse of cycles per instruction. John L. Hennessy, David A. Patterson. "Computer architecture: a quantitative approach". 2007.
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In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps :R × R → R, where R × R carries the product topology. That means R is an additive topological group and a multiplicative topological semigroup.
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AIMD requires a binary signal of congestion. Most frequently, packet loss serves as the signal; the multiplicative decrease is triggered when a timeout or acknowledgement message indicates a packet was lost. It is also possible for in-network mechanisms to mark congestion (without discarding packets) as in Explicit Congestion Notification (ECN).
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A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.
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Richert made contributions to additive number theory, Dirichlet series, Riesz summability, the multiplicative analog of the Erdős–Fuchs theorem, estimates of the number of non-isomorphic abelian groups, and bounds for exponential sums. He proved the exponent 15/46 for the Dirichlet divisor problem, a record that stood for many years.
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Let A = , the polynomial ring over the finite field with q elements. A is a principal ideal domain and therefore A is a unique factorization domain. A complex-valued function \lambda on A is called multiplicative if \lambda(fg)=\lambda(f)\lambda(g) whenever f and g are relatively prime.
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One way to thwart these attacks is to ensure that the decryption operation takes a constant amount of time for every ciphertext. However, this approach can significantly reduce performance. Instead, most RSA implementations use an alternate technique known as cryptographic blinding. RSA blinding makes use of the multiplicative property of RSA.
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Let be a root of this polynomial (in the polynomial representation this would be ), that is, . Now , so is not a primitive element of GF(28) and generates a multiplicative subgroup of order 51. However, is a primitive polynomial. Consider the field element (in the polynomial representation this would be + 1).
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The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory. In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.
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Let be a root of the primitive polynomial . The traditional representation of elements of this field is as polynomials in α of degree 2 or less. A table of Zech logarithms for this field are , , , , , , , and . The multiplicative order of α is 7, so the exponential representation works with integers modulo 7.
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For instance, until 2004 France counted its killed at six days, but in an effort to enable comparison with neighbor countries a multiplicative coefficient was used up to 2004 and since 2005 to convert the killed at six days into killed at thirty days, before France adopted the international definition in 2005.
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These are often based on linear regression, and may adopt a multiplicative or additive dispersion parameter to adjust for the presence of between-study heterogeneity. Some approaches may even attempt to compensate for the (potential) presence of publication bias, which is particularly useful to explore the potential impact on meta-analysis results.
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Hinkelmann and Kempthorne (2008, Volume 1, Section 6.10: Completely randomized design; Transformations) Also, a statistician may specify that logarithmic transforms be applied to the responses, which are believed to follow a multiplicative model.Bailey (2008) According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.
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Kimberlé Crenshaw coined the term "intersectionality" in 1989 in response to the white, middle-class views that dominated second-wave feminism. Intersectionality describes the way systems of oppression (i.e. sexism, racism) have multiplicative, not additive, effects, on those who are multiply marginalized. It has become a core tenet of third- wave feminism.
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When (Z/nZ)× is cyclic, its generators are called primitive roots modulo n. For a prime number p, the group (Z/pZ)× is always cyclic, consisting of the non-zero elements of the finite field of order p. More generally, every finite subgroup of the multiplicative group of any field is cyclic..
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Multiplicative binary search operates on a permuted sorted array. Keys are stored in the array in level-order sequence of the corresponding balanced binary search tree. This places the first pivot of a binary search as the first element in the array. The second pivots are placed at the next two positions.
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The RSA cryptosystem has the following multiplicative property: \sigma(m_1) \cdot \sigma(m_2) = \sigma (m_1 \cdot m_2). This property can be exploited by creating a message m' = m_1 \cdot m_2 with a signature \sigma(m') = \sigma (m_1 \cdot m_2). A common defense to this attack is to hash the messages before signing them.
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F1 cannot be a field because by definition all fields must contain two distinct elements, the additive identity zero and the multiplicative identity one. Even if this restriction is dropped (for instance by letting the additive and multiplicative identities be the same element), a ring with one element must be the zero ring, which does not behave like a finite field. For instance, all modules over the zero ring are isomorphic (as the only element of such a module is the zero element). However, one of the key motivations of F1 is the description of sets as "F1-vector spaces"—if finite sets were modules over the zero ring, then every finite set would be the same size, which is not the case.
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Both fitness and herbivory can be measured or analyzed using an absolute (additive) scale or a relative (multiplicative) scale (Wise and Carr 2008b). The absolute scale may refer to number of fruits produced or total area of leaf eaten, while the relative scale may refer to proportion of fruits damaged or proportion of leaves eaten. Wise and Carr (2008b) suggested that it is best to keep the measure of fitness and the measure of damage on the same scale when analyzing tolerance since having them on different scales may result is misleading outcomes. Even if the data were measured using different scales, data on the absolute scale can be log-transformed to be more similar to data on a relative (multiplicative) scale (Wise and Carr 2008b).
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Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm. The factors it finds are ones for which the number preceding the factor, p − 1, is powersmooth; the essential observation is that, by working in the multiplicative group modulo a composite number N, we are also working in the multiplicative groups modulo all of N's factors. The existence of this algorithm leads to the concept of safe primes, being primes for which p − 1 is two times a Sophie Germain prime q and thus minimally smooth.
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In most Tomo-PIV experiments, the multiplicative algebraic reconstruction technique (MART) is used. The advantage of this pixel-by-pixel reconstruction technique is that it avoids the need to identify individual particles. Reconstructing the discretized 3-D intensity field is computationally intensive and, beyond MART, several developments have sought to significantly reduce this computational expense, for example the multiple line-of-sight simultaneous multiplicative algebraic reconstruction technique (MLOS-SMART) which takes advantage of the sparsity of the 3-D intensity field to reduce memory storage and calculation requirements. As a rule of thumb, at least four cameras are needed for acceptable reconstruction accuracy, and best results are obtained when the cameras are placed at approximately 30 degrees normal to the measurement volume.
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The multiplicative weights method is usually used to solve a constrained optimization problem. Let each expert be the constraint in the problem, and the events represent the points in the area of interest. The punishment of the expert corresponds to how well its corresponding constraint is satisfied on the point represented by an event.
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Kost's method extends Brown's to allow one to combine p-values when the covariance matrix is known only up to a scalar multiplicative factor. The harmonic mean p-value offers an alternative to Fisher's method for combining p-values when the dependency structure is unknown but the tests cannot be assumed to be independent.
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Any abelian group can be made a rng of square zero by defining the multiplication so that for all x and y;Bourbaki, p. 102. thus every abelian group is the additive group of some rng. The only rng of square zero with a multiplicative identity is the zero ring {0}.Bourbaki, p. 102.
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The ring Z and its quotients Z/nZ have no subrings (with multiplicative identity) other than the full ring. Every ring has a unique smallest subring, isomorphic to some ring Z/nZ with n a nonnegative integer (see characteristic). The integers Z correspond to in this statement, since Z is isomorphic to Z/0Z.
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The Digital Signature Algorithm (DSA) is a variant of the ElGamal signature scheme, which should not be confused with ElGamal encryption. ElGamal encryption can be defined over any cyclic group G, like multiplicative group of integers modulo n. Its security depends upon the difficulty of a certain problem in G related to computing discrete logarithms.
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The other suffix used to denote a lactone is -olide, used in substance class names like butenolide, macrolide, cardenolide or bufadienolide. To obtain the preferred IUPAC names, lactones are named as heterocyclic pseudoketones by adding the suffix ‘one’, ‘dione’, ‘thione’, etc. and the appropriate multiplicative prefixes to the name of the heterocyclic parent hydride.
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Criticism related to the Drake equation focuses not on the equation itself, but on the fact that the estimated values for several of its factors are highly conjectural, the combined multiplicative effect being that the uncertainty associated with any derived value is so large that the equation cannot be used to draw firm conclusions.
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If the base a is coprime to the exponent p then Fermat's little theorem says that qp(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp(a) will be a cyclic number, and p will be a full reptend prime.
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Later, he generalized the winnow algorithm to weighted majority algorithm. Freund and Schapire followed his steps and generalized the winnow algorithm in the form of hedge algorithm. The multiplicative weights algorithm is also widely applied in computational geometry such as Clarkson's algorithm for linear programming (LP) with a bounded number of variables in linear time.KENNETH L. CLARKSON.
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For instance, in a chapter in an edited volume on achievement, IQ researcher Arthur Jensen proposed a multiplicative model of genius consisting of high ability, high productivity, and high creativity.Jensen, A. R. (1996). “Giftedness and genius: Crucial differences”. In C. P. Benbow and D. Lubinski (Eds.), Intellectual talent: Psychometric and social issues, Baltimore: Johns Hopkins University Press.
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The product and the multiplicative inverse of two th roots of unity are also th roots of unity. Therefore, the th roots of unity form a group under multiplication. Given a primitive th root of unity , the other th roots are powers of . This means that the group of the th roots of unity is a cyclic group.
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Ryser's conjecture is that, if H is not only r-uniform but also r-partite (i.e., its vertices can be partitioned into r sets so that every edge contains exactly one element of each set), then: > \tau(H) \leq (r-1)\cdot u(H) I.e., the multiplicative factor in the above inequality can be decreased by 1.
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Strange particles appear copiously due to "associated production" of a strange and an antistrange particle together. It was soon shown that this could not be a multiplicative quantum number, because that would allow reactions which were never seen in the new synchrotrons which were commissioned in Brookhaven National Laboratory in 1953 and in the Lawrence Berkeley Laboratory in 1955.
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Kannan Soundararajan (born December 27, 1973) is a mathematician and a professor of mathematics at Stanford University. Before moving to Stanford in 2006, he was a faculty member at University of Michigan where he pursued his undergraduate studies. His main research interest is in analytic number theory, particularly in the subfields of automorphic L-functions, and multiplicative number theory.
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P. Safonov, V.M. Shalaev, V.A. Markel, Yu.E. Danilova, N.N. Lepeshkin, W.Kim, S.G. Rautian, and R.L. Armstrong, Spectral Dependence of Selective Photomodification in Fractal Aggregates of Colloidal Particles, Physical Review Letters, v. 80, pp. 1102–1105 (1998)W. Kim, V.P. Safonov, V.M. Shalaev, and R.L. Armstrong, Fractals in Microcavities: Giant Coupled Multiplicative Enhancement of Optical Responses, Physical Review Letters, v.
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Weil references Pierre de Fermat's approach to calculus, as well as the jets of Charles Ehresmann. For an extended treatment, see O. O. Luciano, Categories of multiplicative functors and Weil's infinitely near points, Nagoya Math. J. 109 (1988), 69–89 (online here) for a full discussion. though these are not the same as infinitely near points in algebraic geometry.
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The term "rng" was coined to alleviate this ambiguity when people want to refer explicitly to a ring without the axiom of multiplicative identity. A number of algebras of functions considered in analysis are not unital, for instance the algebra of functions decreasing to zero at infinity, especially those with compact support on some (non-compact) space.
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Similarly, the quaternions and the octonions are respectively four- and eight- dimensional real vector spaces, and Cn is a 2n-dimensional real vector space. The vector space Fn has a standard basis: :e_1 = (1, 0, \ldots, 0) :e_2 = (0, 1, \ldots, 0) :\vdots :e_n = (0, 0, \ldots, 1) where 1 denotes the multiplicative identity in F.
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Many authors do not require rings to have a multiplicative identity element and, accordingly, do not require ring homomorphism to preserve the identity (should it exist). This leads to a rather different category. For distinction we call such algebraic structures rngs and their morphisms rng homomorphisms. The category of all rngs will be denoted by Rng.
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The risk of developing lung cancer depends on the level, duration, and frequency of asbestos exposure (cumulative exposure). Smoking and individual susceptibility are other contributing factors towards lung cancer. Smokers who have been exposed to asbestos are at far greater risk of lung cancer. Smoking and asbestos exposure have a multiplicative (synergistic) effect on the risk of lung cancer.
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Seeker's navigation system consists of two core FSW applications and six applications that provide appropriately processed sensor information. The system leverages Project Morpheus architecture and code components. The core of the navigation system is a propagator that integrates the vehicle state at 50 Hz and a multiplicative extended kalman filter that updates the state at 5Hz.
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A gap reduction is a type of reduction that, while useful in proving some inapproximability results, does not resemble the other reductions shown here. Gap reductions deal with optimization problems within a decision problem container, generated by changing the problem goal to distinguishing between the optimal solution and solutions some multiplicative factor worse than the optimum.
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That type of directional modulation is really a subset of Negi and Goel's additive artificial noise encryption scheme. Another scheme using pattern-reconfigurable transmit antennas for Alice called reconfigurable multiplicative noise (RMN) complements additive artificial noise. The two work well together in channel simulations in which nothing is assumed known to Alice or Bob about the eavesdroppers.
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That intertwiner is then unique up to a multiplicative factor (a non-zero scalar from ). These properties hold when the image of is a simple algebra, with centre (by what is called Schur's Lemma: see simple module). As a consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same..
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Extrapolated Runs (XR) is a baseball statistic invented by sabermetrician Jim Furtado to estimate the number of runs a hitter contributes to his team. XR measures essentially the same thing as Bill James' Runs Created, but it is a linear weights formula that assigns a run value to each event, rather than a multiplicative formula like James' creation.
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When they carry out the experiment and see that the answer is 3 units, this establishes some cognitive dissonance. This is a prime time for the teacher to move the lesson into the second stage of the learning cycle. Students using the Water Triangle. It is important that the students not over apply the multiplicative strategies they learn.
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Therefore, some of the hands- on activities might not be based on a multiplicative relation. Here is a picture of two students working with an apparatus where the constant sum relation is correct. The constant sum relation works here. It is not always possible or feasible to put carefully designed hands-on activities into the hands of students.
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This formula yields the decomposition for n = 101 in the table. Ahmes was suggested to have converted 2/p (where p was a prime number) by two methods, and three methods to convert 2/pq composite denominators.. Others have suggested only one method was used by Ahmes which used multiplicative factors similar to least common multiples.
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S. Winograd, "On Computing the Discrete Fourier Transform", Mathematics of Computation, 32(141), 175-199 (1978). and today Rader's algorithm is sometimes described as a special case of Winograd's FFT algorithm, also called the multiplicative Fourier transform algorithm (Tolimieri et al., 1997),R. Tolimieri, M. An, and C.Lu, Algorithms for Discrete Fourier Transform and Convolution, Springer- Verlag, 2nd ed.
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S-box is fixed for 8-bit input and 8-bit output, noted as Sbox(). As with AES, the S-box is based on the multiplicative inverse over . The affine transforms and polynomial bases are different from that of AES, but due to affine isomorphism it can be calculated efficiently given an AES Rijndael S-box.
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Given R and G, there is a ring homomorphism sending each r to r1 (where 1 is the identity element of G), and a monoid homomorphism (where the latter is viewed as a monoid under multiplication) sending each g to 1g (where 1 is the multiplicative identity of R). We have that α(r) commutes with β(g) for all r in R and g in G. The universal property of the monoid ring states that given a ring S, a ring homomorphism , and a monoid homomorphism to the multiplicative monoid of S, such that α'(r) commutes with β'(g) for all r in R and g in G, there is a unique ring homomorphism such that composing α and β with γ produces α' and β '.
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If n is composite, there exists a subgroup of the multiplicative group, called the "group of false witnesses", in which the elements, when raised to the power , are congruent to 1 modulo n (since the residue 1, to any power, is congruent to 1 modulo n, the set of such elements is nonempty). One could say, because of Fermat's Little Theorem, that such residues are "false positives" or "false witnesses" for the primality of n. The number 2 is the residue most often used in this basic primality check, hence is famous since 2340 is congruent to 1 modulo 341, and 341 is the smallest such composite number (with respect to 2). For 341, the false witnesses subgroup contains 100 residues and so is of index 3 inside the 300 element multiplicative group mod 341.
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According to Galois theory, the existence of the Klein four-group (and in particular, the permutation representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals, as established by Lodovico Ferrari: the map corresponds to the resolvent cubic, in terms of Lagrange resolvents. In the construction of finite rings, eight of the eleven rings with four elements have the Klein four-group as their additive substructure. If R× denotes the multiplicative group of non-zero reals and R+ the multiplicative group of positive reals, R× × R× is the group of units of the ring , and is a subgroup of (in fact it is the component of the identity of ). The quotient group is isomorphic to the Klein four-group.
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One way to precise the meaning of "large scale" is to require invariance under quasi- isometry. This is true of hyperbolicity. :If a geodesic metric space Y is quasi-isometric to a \delta-hyperbolic space X then there exists \delta' such that Y is \delta'-hyperbolic. The constant \delta' depends on \delta and on the multiplicative and additive constants for the quasi-isometry.
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Jia Xian's additive multiplicative method of cubic root extraction Jiuzhang suanshu vol iv "shaoguang" provided algorithm for extraction of cubic root. problem 19: We have a 1860867 cubic chi, what is the length of a side ? Answer:123 chi. North Song dynasty mathematician Jia Xian invented a method similar to simplified form of Horner scheme for extraction of cubic root.
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Let R be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, or 0 = 1. Let r be any element of R. Then : r = r × 1 = r × 0 = 0 proving that R is trivial, that is, R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.
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For multiplication it fails because 0 does not have a multiplicative inverse. An ad hoc attempt to deal with this would be to define 0−1 = 0\. (This attempt fails, essentially because with this definition 0 × 0−1 = 1 is not true.) Therefore, one is naturally led to allow partial functions, i.e., functions that are defined only on a subset of their domain.
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In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of divisibility for rings is richer. An element a of ring R is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors, i.e. an element a such that there exists a non-zero element b of the ring such that .
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A target neuron that takes only two types of direct input can only combine them additively. However mathematical models show that when also receiving recursive input from neighboring neurons, the resulting transformation to the target neurons firing rate is multiplicative. In this model, neurons with overlapping receptive fields excite each other, multiplying the strength. Likewise, neurons with non-overlapping receptive fields are inhibitory.
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Nathan Jacobson described the automorphisms of composition algebras in 1958. The classical composition algebras over and are unital algebras. Composition algebras without a multiplicative identity were found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others.Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in The Book of Involutions, pp.
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The \zeta(q) are statistically interpreted, as they characterize the evolution of the distributions of the T_X(a) as a goes from larger to smaller scales. This evolution is often called statistical intermittency and betrays a departure from Gaussian models. Modelling as a multiplicative cascade also leads to estimation of multifractal properties. This methods works reasonably well, even for relatively small datasets.
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The number of primitive elements in a finite field is , where is Euler's totient function, which counts the number of elements less than or equal to which are relatively prime to . This can be proved by using the theorem that the multiplicative group of a finite field is cyclic of order , and the fact that a finite cyclic group of order contains generators.
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There are several properties that may be familiar from ring theory, or from associative algebras, which are not always true for non-associative algebras. Unlike the associative case, elements with a (two- sided) multiplicative inverse might also be a zero divisor. For example, all non-zero elements of the sedenions have a two-sided inverse, but some of them are also zero divisors.
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"Multiplicative calculus in biomedical image analysis", Journal of Mathematical Imaging and Vision, DOI: 10.1007/s10851-011-0275-1, 2011. The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). Interested readers are encouraged to improve the table by inserting citations for verification, and by inserting more functions and more calculi.
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The multiplicative structure of an H-space adds structure to its homology and cohomology groups. For example, the cohomology ring of a path-connected H-space with finitely generated and free cohomology groups is a Hopf algebra. Also, one can define the Pontryagin product on the homology groups of an H-space. The fundamental group of an H-space is abelian.
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In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A_\infty-ring is called an E_\infty-ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory.
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Rather than , we express the property of the vending machine as a linear implication . From and this fact, we can conclude , but not . In general, we can use the linear logic proposition to express the validity of transforming resource into resource . Running with the example of the vending machine, consider the "resource interpretations" of the other multiplicative and additive connectives.
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These numbers occur also in other, related areas. In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real n×n matrices, for which each non-zero matrix is a similarity transformation, i.e. a product of an orthogonal matrix and a scalar matrix. In quadratic forms, the Hurwitz problem asks for multiplicative identities between quadratic forms.
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This technique is named after Friedrich Bessel. In estimating the population variance from a sample when the population mean is unknown, the uncorrected sample variance is the mean of the squares of deviations of sample values from the sample mean (i.e. using a multiplicative factor 1/n). In this case, the sample variance is a biased estimator of the population variance.
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Formally, let be a set S with a closed binary operation • on it (known as a magma). A zero element is an element z such that for all s in S, . A refinement are the notions of left zero, where one requires only that , and right zero, where . Absorbing elements are particularly interesting for semigroups, especially the multiplicative semigroup of a semiring.
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When the characteristic of the base field is p there is an analogous statement for an algebraic group G: there exists a smallest subgroup H such that # G/H is a unipotent group # H is an extension of an abelian variety A by a group M of multiplicative type. # M is unique upto Commensurability in G and A is unique up to Isogeny.
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Per the epidemiology of AKI, the risk of AKI increases in multiplicative fashion with increased risk factors. The incidence of AKI demonstrates fold-increases (5 to 10 to 50%) for higher risk patients. This same increase is seen retrospectively in the case of fluid overload. Risk of mortality in patients with AKI demonstrates similar fold-increases for increasing AKI severity.
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Another theory of learning multiplication derives from those studying embodied cognition, which examined the underlying metaphors for multiplication. Together these investigations have inspired curricula with "inherently multiplicative" tasks for young children. Examples of these tasks include: elastic stretching, zoom, folding, projecting shadows, or dropping shadows. These tasks don't depend on counting, and cannot be easily conceptualized in terms of repeated addition.
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The original MurmurHash was created as an attempt to make a faster function than Lookup3. Although successful, it hadn't been tested thoroughly and wasn't capable of providing 64-bit hashes as in Lookup3. It had a rather elegant design, that would be later built upon in MurmurHash2, combining a multiplicative hash (similar to Fowler–Noll–Vo hash function) with a Xorshift.
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A basic example of functional decomposition is expressing the four binary arithmetic operations of addition, subtraction, multiplication, and division in terms of the two binary operations of addition a + b and multiplication a \times b, and the two unary operations of additive inversion -a and multiplicative inversion 1/a. Subtraction can then be realized as the composition of addition and additive inversion, a - b = a + (-b), and division can be realized as the composition of multiplication and multiplicative inverse, a \div b = a \times (1/b). This simplifies the analysis of subtraction and division, and also makes it easier to axiomatize these operations in the notion of a field, as there are only two binary and two unary operations, rather than four binary operations. Extending these primitive operations, there is a rich literature on the topic of polynomial decomposition.
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Since illumination and reflectance combine multiplicatively, the components are made additive by taking the logarithm of the image intensity, so that these multiplicative components of the image can be separated linearly in the frequency domain. Illumination variations can be thought of as a multiplicative noise, and can be reduced by filtering in the log domain. To make the illumination of an image more even, the high-frequency components are increased and low-frequency components are decreased, because the high- frequency components are assumed to represent mostly the reflectance in the scene (the amount of light reflected off the object in the scene), whereas the low-frequency components are assumed to represent mostly the illumination in the scene. That is, high-pass filtering is used to suppress low frequencies and amplify high frequencies, in the log-intensity domain.
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Again, over a perfect field, such a decomposition exists, the decomposition is unique, and xs and xu are polynomials in x. The multiplicative version of the decomposition follows from the additive one since, as x_s is easily seen to be invertible, :x = x_s + x_n = x_s(1 + x_s^{-1}x_n) and 1 + x_s^{-1}x_n is unipotent. (Conversely, by the same type of argument, one can deduce the additive version from the multiplicative one.) If x is written in Jordan normal form (with respect to some basis) then xs is the endomorphism whose matrix contains just the diagonal terms of x, and xn is the endomorphism whose matrix contains just the off-diagonal terms; xu is the endomorphism whose matrix is obtained from the Jordan normal form by dividing all entries of each Jordan block by its diagonal element.
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In three dimensions the effect is the same, the structure is again reduced by a multiplicative factor, and this factor is often called the Debye–Waller factor. Note that the Debye–Waller factor is often ascribed to thermal motion, i.e., the \delta x are due to thermal motion, but any random displacements about a perfect lattice, not just thermal ones, will contribute to the Debye–Waller factor.
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Euler's proof is short and depends on the fact that the sum of divisors function is multiplicative; that is, if and are any two relatively prime integers, then . For this formula to be valid, the sum of divisors of a number must include the number itself, not just the proper divisors. A number is perfect if and only if its sum of divisors is twice its value.
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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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The addition and the subtraction are those of polynomials over . The product of two elements is the remainder of the Euclidean division by of the product in . The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm; see Extended Euclidean algorithm § Simple algebraic field extensions. Except in the construction of , there are several possible choices for , which produce isomorphic results.
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Logunov made a notable contribution to theory of gravity. He studied quantum field theory. In 1956 he built generalized finite multiplicative renormalization groups and functional and differential renormalization group equations of electrodynamics in case of arbitrary calibration. Jointly with Piotr Isayev (Russian: Петр Степанович Исаев), Lev Soloviov (Russian: Лев Дмитриевич Соловьев), Albert Tavkhelidze (Russian: Альберт Никифорович Тавхелидзе) and Ivan Todorov (Bulgarian: Иван Тодоров) et al.
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Hammondia hammondi is a species of obligate heteroxenous parasitic alveolates of domestic cats (final host). Intracellular cysts develop mainly in striated muscle. After the ingestion of cysts by cats, a multiplicative cycle precedes the development of gametocytes in the epithelium of the small intestine (each oocyst of the species averaging 11×13 μm). Oocyst shedding persists for 10 to 28 days followed by immunity.
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Bicircular matroids, like all other transversal matroids, can be represented by vectors over any infinite field. However, unlike graphic matroids, they are not regular: they cannot be represented by vectors over an arbitrary finite field. The question of the fields over which a bicircular matroid has a vector representation leads to the largely unsolved problem of finding the fields over which a graph has multiplicative gains. See .
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Titanium foams are characterized structurally by their pore topology (relative percentage of open vs. closed pores), porosity (the multiplicative inverse of relative density), pore size and shape, and anisotropy. Microstructures are most often examined by optical microscopy, scanning electron microscopy and X-ray tomography. Categorizing titanium foams in terms of pore structure (as either open- or close-celled) is the most basic form of differentiation.
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Grossman has worked in several fields. His early work (1984–1990) was in mathematics, where he developed algorithms in symbolic and numeric computing. In 1989, working with Richard Larson, he showed that trees have a natural multiplicative structure and are in fact a Hopf algebra.Robert L. Grossman and Richard G. Larson, Hopf algebraic structures of families of trees, Journal Algebra, Volume 26, 1989, pages 184-210.
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The subring test is a theorem that states that for any ring R, a subset S of R is a subring if and only if it is closed under multiplication and subtraction, and contains the multiplicative identity of R. As an example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X].
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An alphabetic numeral system employs the letters of a script in the specific order of the alphabet in order to express numerals. In Greek, letters are assigned to respective numbers in the following sets: 1 through 9, 10 through 90, 100 through 900, and so on. Decimal places are represented by a single symbol. As the alphabet ends, higher numbers are represented with various multiplicative methods.
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An additive function f(n) is said to be completely additive if f(ab) = f(a) + f(b) holds for all positive integers a and b, even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0. Every completely additive function is additive, but not vice versa.
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H. Lehmer (1956) Review of "Calculus of Approximations" from Mathematical Reviews with the split-complex number plane by M. Warmus and D. H. Lehmer through the identification : z = (x + y)/2 + j (x − y)/2. This linear mapping of the plane, which amounts of a ring isomorphism, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as polar decomposition.
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The multiplicative decomposition states that if g is an element of the corresponding connected semisimple Lie group G with corresponding Iwasawa decomposition G = KAN, then g can be written as the product of three commuting elements g = sdu with s, d and u conjugate to elements of K, A and N respectively. In general the terms in the Iwasawa decomposition g = kan do not commute.
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In many cases, a statistician may specify that logarithmic transforms be applied to the responses, which are believed to follow a multiplicative model. Pre-publication chapters are available on-line. The assumption of unit treatment additivity was enunciated in experimental design by Kempthorne and Cox. Kempthorne's use of unit treatment additivity and randomization is similar to the design-based analysis of finite population survey sampling.
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This composition is explicitly (fg)(x) := f(g(x)). The multiplicative identity is the identity homomorphism on A. If the set A does not form an abelian group, then the above construction is not necessarily additive, as then the sum of two homomorphisms need not be a homomorphism. This set of endomorphisms is a canonical example of a near-ring that is not a ring.
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Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction. Hanski's model predicted that the power law exponent would be constrained to range closely about the value of 2, which seemed inconsistent with many reported values. Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function. The Lewontin Cohen growth model.
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It follows immediately that, if is an integral domain, then so is .Herstein p.162 It follows also that, if is an integral domain, a polynomial is a unit (that is, it has a multiplicative inverse) if and only if it is constant and is a unit in . Two polynomials are associated if either one is the product of the other by a unit.
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Multiplicative mixers have been implemented in many ways. The most popular are Gilbert cell mixers, diode mixers, diode ring mixers (ring modulation) and switching mixers. Diode mixers take advantage of the non-linearity of diode devices to produce the desired multiplication in the squared term. They are very inefficient as most of the power output is in other unwanted terms which need filtering out.
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In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ringIn this article, rings have a 1. in which every nonzero element has a multiplicative inverse, i.e., an element with Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements.
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The lattice condition for μ is also called multivariate total positivity, and sometimes the strong FKG condition; the term (multiplicative) FKG condition is also used in older literature. The property of μ that increasing functions are positively correlated is also called having positive associations, or the weak FKG condition. Thus, the FKG theorem can be rephrased as "the strong FKG condition implies the weak FKG condition".
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The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Adolf Piltz in 1884. Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers. The formal statement of the hypothesis follows. A Dirichlet character is a completely multiplicative arithmetic function χ such that there exists a positive integer k with for all n and whenever .
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Nimber multiplication (nim-multiplication) is defined recursively by :. Except for the fact that nimbers form a proper class and not a set, the class of nimbers determines an algebraically closed field of characteristic 2. The nimber additive identity is the ordinal 0, and the nimber multiplicative identity is the ordinal 1. In keeping with the characteristic being 2, the nimber additive inverse of the ordinal is itself.
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If the repeated sequence is all 0s, both the zeros and the quote can be omitted. The radix point has its usual function; moving it left divides by the base; moving it right multiplies by the base. When the radix point is at the right end, the multiplicative factor is 1, and the point can be omitted. This gives the natural numbers their familiar form.
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Another concrete realization would be obtained by thinking of as the 3×3 symmetric matrix which represents it. If and have such concrete realizations then every member of the above pencil will as well. Since the setting uses homogeneous coordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic if they differ by a non-zero multiplicative constant.
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Over characteristic 0 there is a nice decomposition theorem of an algebraic group G relating its structure to the structure of a linear algebraic group and an Abelian variety. There is a short exact sequence of groupspage 8 > 0 \to M\times U \to G \to A \to 0 where A is an abelian variety, M is of multiplicative type, meaning, and U is a unipotent group.
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The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name—for example, = . Nevertheless, certain authors advise against using it for its ambiguity. Another convention used by a few authors is to use an uppercase first letter, along with a superscript: , , , etc. This potentially avoids confusion with the multiplicative inverse, which should be represented by , , etc.
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The j-invariant of the Tate curve is given by a power series in q with leading term q−1.Silverman (1994) p.423 Over a p-adic local field, therefore, j is non- integral and the Tate curve has semistable reduction of multiplicative type. Conversely, every semistable elliptic curve over a local field is isomorphic to a Tate curve (up to quadratic twist).
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In algebra, a primitive element of a co-algebra C (over an element g) is an element x that satisfies :\mu(x) = x \otimes g + g \otimes x where \mu is the co-multiplication and g is an element of C that maps to the multiplicative identity 1 of the base field under the co-unit (g is called group-like). If C is a bi-algebra, i.e., a co-algebra that is also an algebra (with certain compatibility conditions satisfied), then one usually takes g to be 1, the multiplicative identity of C. The bi-algebra C is said to be primitively generated if it is generated by primitive elements (as an algebra). If C is a bi-algebra, then the set of primitive elements form a Lie algebra with the usual commutator bracket [x, y] = xy - yx (graded commutator if C is graded).
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The Lucas–Lehmer–Riesel test is a particular case of group-order primality testing; we demonstrate that some number is prime by showing that some group has the order that it would have were that number prime, and we do this by finding an element of that group of precisely the right order. For Lucas-style tests on a number N, we work in the multiplicative group of a quadratic extension of the integers modulo N; if N is prime, the order of this multiplicative group is N2 − 1, it has a subgroup of order N + 1, and we try to find a generator for that subgroup. We start off by trying to find a non- iterative expression for the u_i. Following the model of the Lucas–Lehmer test, put u_i = a^{2^i} + a^{-2^i}, and by induction we have u_i = u_{i-1}^2 - 2.
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The determinant of a matrix product of square matrices equals the product of their determinants: :\det(AB) = \det (A) \times \det (B) Thus the determinant is a multiplicative map. This property is a consequence of the characterization given above of the determinant as the unique n-linear alternating function of the columns with value 1 on the identity matrix, since the function that maps can easily be seen to be n-linear and alternating in the columns of M, and takes the value det(A) at the identity. The formula can be generalized to (square) products of rectangular matrices, giving the Cauchy–Binet formula, which also provides an independent proof of the multiplicative property. The determinant det(A) of a matrix A is non-zero if and only if A is invertible or, yet another equivalent statement, if its rank equals the size of the matrix.
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Run of Megiddo's algorithm phase, discarding from point set A,B,...,U needless points E, T. Megiddo's algorithm. is based on the technique called prune and search reducing size of the problem by removal of n/16 of unnecessary points. That leads to the recurrence t(n)≤ t(15n/16)+cn giving t(n)=16cn. The algorithm is rather complicated and it is reflected in big multiplicative constant.
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The formal definition does not use isometries, but almost isometries. A Banach space Y is finitely representableJames, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. 24:896-904. in a Banach space X if for every finite-dimensional subspace Y0 of Y and every , there is a subspace X0 of X such that the multiplicative Banach-Mazur distance between X0 and Y0 satisfies :d(X_0, Y_0) < 1 + \varepsilon.
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The exponential function uniformizes the exponential map of the multiplicative group . Therefore, we can reformulate the six exponential theorem more abstractly as follows: :Let and take to be a non-zero complex-analytic group homomorphism. Define to be the set of complex numbers for which is an algebraic point of . If a minimal generating set of over has more than two elements then the image is an algebraic subgroup of .
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Shifting all characters by one position to the left requires multiplying the entire sum H by a. Shifting all characters by one position to the right requires dividing the entire sum H by a. Note that in modulo arithmetic, a can be chosen to have a multiplicative inverse a^{-1} by which H can be multiplied to get the result of the division without actually performing a division.
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In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group. In contrast to the classical theory of p-adic analytic manifolds, rigid analytic spaces admit meaningful notions of analytic continuation and connectedness.
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The methods belong primarily to analytic number theory, but elementary methods, especially sieve methods, are also very important. The large sieve and exponential sums are usually considered part of multiplicative number theory. The distribution of prime numbers is closely tied to the behavior of the Riemann zeta function and the Riemann hypothesis, and these subjects are studied both from a number theory viewpoint and a complex analysis viewpoint.
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The elements of the natural basis are multiplicative, namely, Tyw=Ty Tw whenever l(yw)=l(y)+l(w), where l denotes the length function on the Coxeter group W. 3\. Elements of the natural basis are invertible. For example, from the quadratic relation we conclude that T = q Ts \+ (q-1). 4\. Suppose that W is a finite group and the ground ring is the field C of complex numbers.
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In this approach, models are developed using different data mining, machine learning algorithms to predict users' rating of unrated items. There are many model- based CF algorithms. Bayesian networks, clustering models, latent semantic models such as singular value decomposition, probabilistic latent semantic analysis, multiple multiplicative factor, latent Dirichlet allocation and Markov decision process based models.Xiaoyuan Su, Taghi M. Khoshgoftaar, A survey of collaborative filtering techniques, Advances in Artificial Intelligence archive, 2009.
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The Berlekamp–Massey algorithm is an algorithm that will find the shortest linear feedback shift register (LFSR) for a given binary output sequence. The algorithm will also find the minimal polynomial of a linearly recurrent sequence in an arbitrary field. The field requirement means that the Berlekamp–Massey algorithm requires all non-zero elements to have a multiplicative inverse. Reeds and Sloane offer an extension to handle a ring.
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In hydrodynamics, the Perrin friction factors are multiplicative adjustments to the translational and rotational friction of a rigid spheroid, relative to the corresponding frictions in spheres of the same volume. These friction factors were first calculated by Jean-Baptiste Perrin. These factors pertain to spheroids (i.e., to ellipsoids of revolution), which are characterized by the axial ratio p = (a/b), defined here as the axial semiaxis a (i.e.
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In mathematics, a pseudoreflection is an invertible linear transformation of a finite-dimensional vector space such that it is not the identity transformation, has a finite (multiplicative) order, and fixes a hyperplane. The concept of pseudoreflection generalizes the concepts of reflection and complex reflection, and is simply called reflection by some mathematicians. It plays an important role in Invariant theory of finite groups, including the Chevalley-Shephard-Todd theorem.
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The equation of the Steiner inellipse in trilinear coordinates for a triangle with side lengths a, b, c (with these parameters having a different meaning than previously) is :a^2x^2+b^2y^2+c^2z^2-2abxy-2bcyz-2cazx = 0 where x is an arbitrary positive constant times the distance of a point from the side of length a, and similarly for b and c with the same multiplicative constant.
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Brian Christian and Tom Griffiths have suggested the additive increase/multiplicative decrease algorithm as a solution to the Peter principle less severe than firing employees who fail to advance. They propose a dynamic hierarchy in which employees are regularly either promoted or reassigned to a lower level, so that any worker who is promoted to their point of failure is soon moved to an area where they are productive.
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Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital. One example of a non- unital associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero. Another example is the vector space of continuous periodic functions, together with the convolution product.
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Variable expressivity refers to the degree in which a genotype is phenotypically expressed. For example, multiple people with the same disease can have the same genotype but one may express more severe symptoms, while another carrier may appear normal. This variation in expression can be affected by modifier genes, epigenetic factors or the environment. Modifier genes can alter the expression of other genes in either an additive or multiplicative way.
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Combinations of such variants could lead to multiplicative effects. Sporadic cancers likely result from the complex interplay between the expression of low penetrance gene(s) (risk variants) and environmental factors. However, the suspected impact of most of these variants on breast cancer risk should, in most cases, be confirmed in large populations studies. Indeed, low penetrance genes cannot be easily tracked through families, as is true for dominant high-risk genes.
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Some find that untreated dry leaf produces unnoticeable or only light effects. Concentrated preparations or extracts which may be smoked in place of untreated leaves, have become widely available. This enhanced (or "fortified") leaf is described by a number followed by an x (e.g. 5x, 10x), the multiplicative factors being generally indicative of the relative amounts of leaf concentrate, though there is no accepted standard for these claims.
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Linear fractional transformations are shown to be conformal maps by consideration of their generators: multiplicative inversion z → 1/z and affine transformations z → a z + b. Conformality can be confirmed by showing the generators are all conformal. The translation z → z + b is a change of origin and makes no difference to angle. To see that z → az is conformal, consider the polar decomposition of a and z.
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UDT uses an AIMD (additive increase multiplicative decrease) style congestion control algorithm. The increase parameter is inversely proportional to the available bandwidth (estimated using the packet pair technique), thus UDT can probe high bandwidth rapidly and can slow down for better stability when it approaches maximum bandwidth. The decrease factor is a random number between 1/8 and 1/2. This helps reduce the negative impact of loss synchronization.
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The modulus operation may provide some additional mixing; this is especially useful with a poor hash function. For open addressing schemes, the hash function should also avoid clustering, the mapping of two or more keys to consecutive slots. Such clustering may cause the lookup cost to skyrocket, even if the load factor is low and collisions are infrequent. The popular multiplicative hash is claimed to have particularly poor clustering behavior.
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The hardness proof of the exact boson sampling problem can be achieved following two distinct paths. Specifically, the first one uses the tools of the computational complexity theory and combines the following two facts: # Approximating the probability p(t_1, t_2, ..., t_N) of a specific measurement outcome at the output of a linear interferometer to within a multiplicative constant is a #P-hard problem (due to the complexity of the permanent) # If a polynomial-time classical algorithm for exact boson sampling existed, then the above probability p(t_1, t_2, ..., t_N) could have been approximated to within a multiplicative constant in the BPPNPcomplexity class, i.e. within the third level of the polynomial hierarchy When combined together these two facts along with the Toda's theorem result in the collapse of the polynomial hierarchy, which as mentioned above is highly unlikely to occur. This leads to the conclusion that there is no classical polynomial-time algorithm for the exact boson sampling problem.
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Highly structured ring spectra have better formal properties than multiplicative cohomology theories – a point utilized, for example, in the construction of topological modular forms, and which has allowed also new constructions of more classical objects such as Morava K-theory. Beside their formal properties, E_\infty-structures are also important in calculations, since they allow for operations in the underlying cohomology theory, analogous to (and generalizing) the well-known Steenrod operations in ordinary cohomology. As not every cohomology theory allows such operations, not every multiplicative structure may be refined to an E_\infty-structure and even in cases where this is possible, it may be a formidable task to prove that. The rough idea of highly structured ring spectra is the following: If multiplication in a cohomology theory (analogous to the multiplication in singular cohomology, inducing the cup product) fulfills associativity (and commutativity) only up to homotopy, this is too lax for many constructions (e.g.
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Thus quoting an average value containing three significant digits in the output with just one significant digit in the input data could be recognized as an example of false precision. With the implied accuracy of the data points of ±0.5, the zeroth order approximation could at best yield the result for y of ~3.7±2.0 in the interval of x from -0.5 to 2.5, considering the standard deviation. If the data points are reported as :x=[0.00,1.00,2.00]\, :y=[3.00,3.00,5.00]\, the zeroth-order approximation results in :y\sim f(x)=3.67\, The accuracy of the result justifies an attempt to derive a multiplicative function for that average, for example, :y \sim\ x+2.67 One should be careful though because the multiplicative function will be defined for the whole interval. If only three data points are available, one has no knowledge about the rest of the interval, which may be a large part of it.
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However, principal constructions of such more detailed theories for small algebraic number fields are not extendable to the general case of algebraic number fields, and different conceptual principles are in use in the general class field theory. The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension (this is done inside local class field theory) and then prove that the product of all such local reciprocity maps when defined on the idele group of the global field is trivial on the image of the multiplicative group of the global field. The latter property is called the global reciprocity law and is a far reaching generalization of the Gauss quadratic reciprocity law. One of the methods to construct the reciprocity homomorphism uses class formation which derives class field theory from axioms of class field theory.
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Corrado Segre (1912) continued the development with that ring. Arthur Conway, one of the early adopters of relativity via biquaternion transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study.Arthur Conway (1911) "On the application of quaternions to some recent developments of electrical theory", Proceedings of the Royal Irish Academy 29:1–9, particularly page 9 In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in Ireland.
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Informally, a field is a set, along with two operations defined on that set: an addition operation written as , and a multiplication operation written as , both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse for all elements , and of a multiplicative inverse for every nonzero element . This allows one to also consider the so- called inverse operations of subtraction, , and division, , by defining: :, :.
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Abadie, A., Angrist, J., Imbens, G. (2002). Instrumental variables estimates of the effects of subsidized training on the quantiles of trainee earnings. Econometrica, 70(1), pp. 91-117. With regard to limited dependent variable models with binary endogenous regressors, Angrist argues in favour of using 2SLS, multiplicative models for conditional means, linear approximation of non-linear causal models, models for distribution effects, and quantile regression with an endogenous binary regressor.Angrist, J.D. (2001).
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The th roots of unity form under multiplication a cyclic group of order , and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive th root of unity. The th roots of unity form an irreducible representation of any cyclic group of order . The orthogonality relationship also follows from group-theoretic principles as described in character group.
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Many other improvements have been made to the package-merge algorithm to reduce the multiplicative constant and to make it faster in special cases, such as those problems having repeated pis. The package-merge approach has also been adapted to related problems such as alphabetic coding. Methods involving graph theory have been shown to have better asymptotic space complexity than the package-merge algorithm, but these have not seen as much practical application.
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Furthermore, techniques such as partial summation and Tauberian theorems can be used to get information about the coefficients from analytic information about the Dirichlet series. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series (or a product of simpler Dirichlet series using convolution identities), examine this series as a complex function and then convert this analytic information back into information about the original function.
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The approach taken is to increase the transmission rate (window size), probing for usable bandwidth, until loss occurs. The policy of additive increase may, for instance, increase the congestion window by a fixed amount every round trip time. When congestion is detected, the transmitter decreases the transmission rate by a multiplicative factor; for example, cut the congestion window in half after loss. The result is a saw-tooth behavior that represents the probe for bandwidth.
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However, it is strictly weaker. For example, is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as :6 = 2 \cdot 3 = (-2) \cdot (-3). In general, if is a unit, meaning a number with a multiplicative inverse in , and if is a prime element, then is also a prime element.
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Deitmar's construction of monoid schemes has been called "the very core of F1-geometry", as most other theories of F1-geometry contain descriptions of monoid schemes. Morally, it mimicks the theory of schemes developed in the 1950s and 1960s by replacing commutative rings with monoids. The effect of this is to "forget" the additive structure of the ring, leaving only the multiplicative structure. For this reason, it is sometimes called "non-additive geometry".
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A coordinate grid prior to a special conformal transformation The same grid after a special conformal transformation In projective geometry, a special conformal transformation is a linear fractional transformation that is not an affine transformation. Thus the generation of a special conformal transformation involves use of multiplicative inversion, which is the generator of linear fractional transformations that is not affine. In mathematical physics, certain conformal maps known as spherical wave transformations are special conformal transformations.
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Still, this assumption is not essential for the results. If electrons are free to move along the direction, the wave function acquires an additional multiplicative term exp(); the energy corresponding to this free motion, , is added to the discussed. This term then fills in the separation in energy of the different Landau levels, blurring the effect of the quantization. Nevertheless, the motion in the --plane, perpendicular to the magnetic field, is still quantized.
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Their life cycle is much simpler than that of digenean trematodes, including a mollusc and a facultative or compulsory vertebrate host. There are no multiplicative larval stages in the mollusc host, as known from all digeneans. Host specificity of most aspidogastreans is very low, i.e., a single species of aspidogastrean can infect a wide range of host species, whereas a typical digenean trematode is restricted to few species (at least of molluscs).
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If both numbers are positive, then the inequality relation between the multiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbers a and b that are both positive (or both negative): : If a ≤ b, then ≥ . All of the cases for the signs of a and b can also be written in chained notation, as follows: : If 0 < a ≤ b, then ≥ > 0. : If a ≤ b < 0, then 0 > ≥ .
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As was the custom for bright students from landed families those days, Menon had to travel to Madras city and join the Madras Christian College for his higher studies. There, he completed his MA in Mathematics and was awarded a scholarship to pursue research under the guidance of Prof R Vaidyanathaswamy. In 1941, the University of Madras awarded him a MSc Degree with a thesis entitled "Contributions to the theory of multiplicative arithmetic functions".
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The Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that for and , the intersection . A domain that satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.
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The independence number of a lexicographic product may be easily calculated from that of its factors : :. The clique number of a lexicographic product is as well multiplicative: :. The chromatic number of a lexicographic product is equal to the b-fold chromatic number of G, for b equal to the chromatic number of H: :, where b χ(H). The lexicographic product of two graphs is a perfect graph if and only if both factors are perfect .
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Mathematically, SALs are probabilities, often very small but (by definition) always lying between zero and one. So when they are expressed in scientific notation their exponents are negative, as for instance, "The SAL of this process is 10−6". But the term SAL is sometimes also used to refer to a sterilization's efficacy. This usage (technically the multiplicative inverse) results in positive exponents, as in "The SAL of this process is 106".
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In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionallyGłazek (2002) p.7—this originated as a joke, suggesting that rigs are rings without negative elements, similar to using rng to mean a ring without a multiplicative identity. Tropical semirings are an active area of research, linking algebraic varieties with piecewise linear structures.
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Since the kernel is only defined to within a multiplicative constant, if the above dimensionless constant is raised to any arbitrary power, it will yield another equivalent dimensionless constant. This example is easy because three of the dimensional quantities are fundamental units, so the last (g) is a combination of the previous. Note that if a2 were non-zero, there would be no way to cancel the M value; therefore a2 must be zero.
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These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of the prime number theorem. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive. (x), the prime counting function, is the number of primes not exceeding x. It is the summation function of the characteristic function of the prime numbers.
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Georg Rasch is best known for his contributions to psychometrics. His work in this field began when he used the Poisson distribution to model the number of errors made by students when reading texts. He referred to the model as the multiplicative Poisson model. He later developed the Rasch model for dichotomous data, which he applied to response data derived from intelligence and attainment tests including data collected by the Danish military.
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These Hodge theaters use two main symmetries of IUT: multiplicative arithmetic and additive geometric. On one hand, Hodge theaters generalize such classical objects in number theory as the adeles and ideles in relation to their global elements. On the other hand, they generalize certain structures appearing in the previous Hodge-Arakelov theory of Mochizuki. The links between theaters are not compatible with ring or scheme structures and are performed outside conventional arithmetic geometry.
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Prime powers are powers of prime numbers. Every prime power (except powers of 2) has a primitive root; thus the multiplicative group of integers modulo pn (i.e. the group of units of the ring Z/pnZ) is cyclic. The number of elements of a finite field is always a prime power and conversely, every prime power occurs as the number of elements in some finite field (which is unique up to isomorphism).
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In physics, the C parity or charge parity is a multiplicative quantum number of some particles that describes their behavior under the symmetry operation of charge conjugation. Charge conjugation changes the sign of all quantum charges (that is, additive quantum numbers), including the electrical charge, baryon number and lepton number, and the flavor charges strangeness, charm, bottomness, topness and Isospin (I3). In contrast, it doesn't affect the mass, linear momentum or spin of a particle.
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A problem is said to have a polynomial-time approximation scheme (PTAS) if for every multiplicative factor of the optimum worse than 1 there is a polynomial-time algorithm to solve the problem to within that factor. Unless P = NP there exist problems that are in APX but without a PTAS, so the class of problems with a PTAS is strictly contained in APX. One such problem is the bin packing problem.
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The L2 regularization has the intuitive interpretation of heavily penalizing peaky weight vectors and preferring diffuse weight vectors. Due to multiplicative interactions between weights and inputs this has the useful property of encouraging the network to use all of its inputs a little rather than some of its inputs a lot. L1 regularization is another common form. It is possible to combine L1 with L2 regularization (this is called Elastic net regularization).
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Complex n-dimensional matrices can be characterized as real 2n-dimensional matrices that preserve a linear complex structure -- concretely, that commute with a matrix J such that , where J corresponds to multiplying by the imaginary unit i. The Lie algebra corresponding to consists of all complex matrices with the commutator serving as the Lie bracket. Unlike the real case, is connected. This follows, in part, since the multiplicative group of complex numbers C∗ is connected.
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The Gaunt factor (or Kramers-Gaunt factor) is used as a multiplicative correction to the continuous absorption or emission results when calculated using classical physics techniques. In cases where classical physics provides a close approximation, the Gaunt factor can be set to 1.0. It varies from this value in cases where quantum mechanics becomes important. The Gaunt factor was named after the physicist John Arthur Gaunt, based on his work on the quantum mechanics of continuous absorption.
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Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties. Division by zero is, by definition, excluded.. In order to avoid existential quantifiers, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two nullary operations (the constants and ). These operations are then subject to the conditions above.
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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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If G is the group of invertible 3 × 3 real matrices, and N is the subgroup of 3 × 3 real matrices with determinant 1, then N is normal in G (since it is the kernel of the determinant homomorphism). The cosets of N are the sets of matrices with a given determinant, and hence G/N is isomorphic to the multiplicative group of non- zero real numbers. The group N is known as the special linear group SL(3).
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Alice chooses a ring of prime order p, with multiplicative generator g. Alice randomly picks a secret value x from 0 to p − 1 to commit to and calculates c = gx and publishes c. The discrete logarithm problem dictates that from c, it is computationally infeasible to compute x, so under this assumption, Bob cannot compute x. On the other hand, Alice cannot compute a x' <> x, such that gx' = c, so the scheme is binding.
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The Conway polynomial is chosen to be primitive, so that each of its roots generates the multiplicative group of the associated finite field. The subfields of Fpn are fields Fpm with m dividing n. The cyclic group formed from the non-zero elements of Fpm is a subgroup of the cyclic group of Fpn. If α generates the latter, then the smallest power of α that generates the former is αr where r = (pn − 1)/(pm − 1).
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Every logical matrix a = ( a i j ) has an transpose aT = ( a j i ). Suppose a is a logical matrix with no columns or rows identically zero. Then the matrix product, using Boolean arithmetic, aT a contains the m × m identity matrix, and the product a aT contains the n × n identity. As a mathematical structure, the Boolean algebra U forms a lattice ordered by inclusion; additionally it is a multiplicative lattice due to matrix multiplication.
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Consequently, the overall insertion loss of a cascade of constant resistance sections is simply to sum total of the individual sections. Conversely, a given complicated transfer impedance may be decomposed into multiplicative factors, whose individual lattice realizations, when connected in cascade, represent a synthesis of that transfer impedance. So, although it is possible to synthesize a single lattice with complicated impedances Za and Zb, it is practically easier to construct and align a cascade of simpler circuits.
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The importance of these structures in all mathematics, and specially in linear algebra and homological algebra, may explain the coexistence of two non-equivalent definitions. Algebraic structures for which there exist non- surjective epimorphisms include semigroups and rings. The most basic example is the inclusion of integers into rational numbers, which is an homomorphism of rings and of multiplicative semigroups. For both structures it is a monomorphism and a non-surjective epimorphism, but not an isomorphism.
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A wide generalization of this example is the localization of a ring by a multiplicative set. Every localization is a ring epimorphism, which is not, in general, surjective. As localizations are fundamental in commutative algebra and algebraic geometry, this may explain why in these areas, the definition of epimorphisms as right cancelable homomorphisms is generally preferred. A split epimorphism is a homomorphism that has a right inverse and thus it is itself a left inverse of that other homomorphism.
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Wuvulu is an Austronesian Island located in the Manus Province of Papua New Guinea. The languages numbering system is multiplicative construction, where each number is based on multiplying pre-existing numbers smaller than five. Wuvulu is most similar to most Oceanic languages, and their numbering system is representative of some systems found in the Marshall Islands. For examples, the number two in Wuvulu is roa and the number four in both Proto-Oceanic language and Wuvulu is fa.
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A cup product gives a ring structure to a cohomology group, turning it into a cohomology ring. Thus, it is natural to consider a spectral sequence with a ring structure as well. Let E^{p, q}_r be a spectral sequence of cohomological type. We say it has multiplicative structure if (i) E_r are (doubly graded) differential graded algebras and (ii) the multiplication on E_{r+1} is induced by that on E_r via passage to cohomology.
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It had a multiplicative purpose to the Picayune Rancheria. Fire was used to cut hair, drive rabbits and squirrels out of their holes when hunting, and reduce grassland and vegetation that may otherwise cause greater fires. Because the brush from the grassland would be cleared by the fire, hunting and gathering were made easier as it attracted more wildlife and produced the ability to sow crops. Fire also increased the number of livable conditions within their environment.
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Let P\colon V\rightarrow R be a potential function defined on the graph. Note that P can be considered to be a multiplicative operator acting diagonally on \phi :(P\phi)(v)=P(v)\phi(v). Then H=\Delta+P is the discrete Schrödinger operator, an analog of the continuous Schrödinger operator. If the number of edges meeting at a vertex is uniformly bounded, and the potential is bounded, then H is bounded and self- adjoint.
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An associative algebra A over a commutative ring R is defined to be a nilpotent algebra if and only if there exists some positive integer n such that 0=y_1\ y_2\ \cdots\ y_n for all y_1, \ y_2, \ \ldots,\ y_n in the algebra A. The smallest such n is called the index of the algebra A. In the case of a non-associative algebra, the definition is that every different multiplicative association of the n elements is zero.
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Let be an abelian group and we consider the group homomorphisms from A into A. Then addition of two such homomorphisms may be defined pointwise to produce another group homomorphism. Explicitly, given two such homomorphisms f and g, the sum of f and g is the homomorphism [f + g](x) := f(x) + g(x). Under this operation End(A) is an abelian group. With the additional operation of composition of homomorphisms, End(A) is a ring with multiplicative identity.
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His hypothesis was initially qualitative, but as it evolved it became semi- quantitative and was supported by simulations. In proposing that animal behavior was the principal mechanism behind the clustering of organisms, Taylor though appeared to have ignored his own report of clustering seen with tobacco necrosis virus plaques. Following Taylor's initial publications several alternative hypotheses for the power law were advanced. Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction.
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In the step, each byte in the state is replaced with its entry in a fixed 8-bit lookup table, S; bij = S(aij). In the step, each byte a_{i,j} in the state array is replaced with a S(a_{i,j}) using an 8-bit substitution box. This operation provides the non-linearity in the cipher. The S-box used is derived from the multiplicative inverse over , known to have good non-linearity properties.
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Wieferich primes can be defined by other equivalent congruences. If p is a Wieferich prime, one can multiply both sides of the congruence by 2 to get . Raising both sides of the congruence to the power p shows that a Wieferich prime also satisfies , and hence for all . The converse is also true: for some implies that the multiplicative order of 2 modulo p2 divides gcd, φ, that is, and thus p is a Wieferich prime.
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There are also additional restrictions on the first two digits. Based on these restrictions, the number of candidates for n-digit numbers with record-breaking persistence is only proportional to the square of n, a tiny fraction of all possible n-digit numbers. However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist.
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We can express this situation as . The constant 0 represents a product that cannot be made, and thus serves as the unit of ⊕ (a machine that might produce or is as good as a machine that always produces because it will never succeed in producing a 0). So unlike above, we cannot deduce from this. Multiplicative disjunction is more difficult to gloss in terms of the resource interpretation, although we can encode back into linear implication, either as or .
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However the efficiencies of any two "reasonable" implementations of a given algorithm are related by a constant multiplicative factor called a hidden constant. Exact (not asymptotic) measures of efficiency can sometimes be computed but they usually require certain assumptions concerning the particular implementation of the algorithm, called model of computation. A model of computation may be defined in terms of an abstract computer, e.g., Turing machine, and/or by postulating that certain operations are executed in unit time.
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Adolf Hurwitz (1855–1919), whose work on composition algebras was published posthumously in 1923. The classification of simple Euclidean Jordan algebras was accomplished by , with details of the one exceptional algebra provided in the article immediately following theirs by . Using the Peirce decomposition, they reduced the problem to an algebraic problem involving multiplicative quadratic forms already solved by Hurwitz. The presentation here, following , using composition algebras or Euclidean Hurwitz algebras, is a shorter version of the original derivation.
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Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja, is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule (see Hebbian learning) that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons.
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Given an array A of n elements with values A0 ... An−1, and target value T, the following subroutine uses multiplicative binary search to find the index of T in A. # Set i to 0 # if i ≥ n, the search terminates unsuccessful. # if Ai = T, the search is done; return i. # if Ai < T, set i to 2×i + 1 and go to step 2. # if Ai > T, set i to 2×i + 2 and go to step 2.
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The cyclotomic fast Fourier transform is a type of fast Fourier transform algorithm over finite fields.S.V. Fedorenko and P.V. Trifonov, This algorithm first decomposes a DFT into several circular convolutions, and then derives the DFT results from the circular convolution results. When applied to a DFT over GF(2^m), this algorithm has a very low multiplicative complexity. In practice, since there usually exist efficient algorithms for circular convolutions with specific lengths, this algorithm is very efficient.
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The magnetic field produced by the magnet then is the net magnetic field of these dipoles; any net force on the magnet is a result of adding up the forces on the individual dipoles. There are two competing models for the nature of these dipoles. These two models produce two different magnetic fields, and . Outside a material, though, the two are identical (to a multiplicative constant) so that in many cases the distinction can be ignored.
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However, the perceptron algorithm uses an additive weight-update scheme, while Winnow uses a multiplicative scheme that allows it to perform much better when many dimensions are irrelevant (hence its name winnow). It is a simple algorithm that scales well to high-dimensional data. During training, Winnow is shown a sequence of positive and negative examples. From these it learns a decision hyperplane that can then be used to label novel examples as positive or negative.
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Thus, multiplying by 2 is calculated in base-2 by an arithmetic shift. The factor (2−1) is a right arithmetic shift, a (0) results in no operation (since 20 = 1 is the multiplicative identity element), and a (21) results in a left arithmetic shift. The multiplication product can now be quickly calculated using only arithmetic shift operations, addition and subtraction. The method is particularly fast on processors supporting a single-instruction shift-and-addition-accumulate.
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Multiple jeopardy is the theory that the various factors of one's identity that lead to discrimination or oppression, such as gender, class, or race, have a multiplicative effect on the discrimination that person experiences. The term was coined by scholar Deborah K. King in 1988 to account for the limitations of the double or triple jeopardy models of discrimination, which assert that every unique prejudice has an individual effect on one's status, and that the discrimination one experiences is the additive result of all of these prejudices. Under the model of multiple jeopardy, it is instead believed that these prejudices are interdependent and have a multiplicative relationship; for this reason, the "multiple" in its name refers not only to the various forms of prejudices that factor into one's discrimination but also to the relationship between these prejudices. King used the term in relation to multiple consciousness, or the ability of a victim of multiple forms of discrimination to perceive how those forms work together, to support the validity of the black feminist and other intersectional causes.
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Extending the commutative theory of Benz, the existence of a right or left multiplicative inverse of a ring element is related to P(R) and GL(2,R). The Dedekind-finite property is characterized. Most significantly, representation of P(R) in a projective space over a division ring K is accomplished with a (K,R)-bimodule U that is a left K-vector space and a right R-module. The points of P(R) are subspaces of isomorphic to their complements.
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Every linearly ordered field contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of , which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered monoid. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in . The following are equivalent characterizations of Archimedean fields in terms of these substructures. 1\. The natural numbers are cofinal in .
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In neuroscience, synaptic scaling (or homeostatic scaling) is a form of homeostatic plasticity, in which the brain responds to chronically elevated activity in a neural circuit with negative feedback, allowing individual neurons to reduce their overall action potential firing rate. Where Hebbian plasticity mechanisms modify neural synaptic connections selectively, synaptic scaling normalizes all neural synaptic connections by decreasing the strength of each synapse by the same factor (multiplicative change), so that the relative synaptic weighting of each synapse is preserved.
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If S is not multiplicatively closed, it suffices to replace it by its multiplicative closure, consisting of the set of the products of elements of S (including the empty product 1). This does not change the result of the localization. The fact that we talk of "a localization with respect to the powers of an element" instead of "a localization with respect to an element" is an example of this. Therefore, we shall suppose S to be multiplicatively closed in what follows.
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The biggest technical change after 1950 has been the development of sieve methods, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.
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Cubes occasionally have the surjective property in other fields, such as in for such prime that ,The multiplicative group of is cyclic of order , and if it is not divisible by 3, then cubes define a group automorphism. but not necessarily: see the counterexample with rationals above. Also in only three elements 0, ±1 are perfect cubes, of seven total. −1, 0, and 1 are perfect cubes anywhere and the only elements of a field equal to the own cubes: .
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By combining this upper bound with Katok's optimal lower bound in terms of the volume, one obtains a simpler alternative proof of Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, such an approach yields an improved multiplicative constant in Gromov's theorem. As an application, this method implies that every metric on a surface of genus at least 20 satisfies Loewner's torus inequality. This improves the best earlier estimate of 50 which followed from an estimate of Gromov's.
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Despite a growing inclination of the IRS and Tax Courts to challenge valuation discounts, Shannon Pratt suggested in a scholarly presentation recently that valuation discounts are actually increasing as the differences between public and private companies is widening . Publicly traded stocks have grown more liquid in the past decade due to rapid electronic trading, reduced commissions, and governmental deregulation. These developments have not improved the liquidity of interests in private companies, however. Valuation discounts are multiplicative, so they must be considered in order.
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When two compact Hausdorff spaces and are homeomorphic, the Banach spaces and are isometric. Conversely, when is not homeomorphic to , the (multiplicative) Banach–Mazur distance between and must be greater than or equal to , see above the results by Amir and Cambern. Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:Milyutin, Alekseĭ A. (1966), "Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum". (Russian) Teor.
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Ellipsoids usually provide tighter fitting than a sphere. Intersections with ellipsoids are done by scaling the other object along the principal axes of the ellipsoid by an amount equal to the multiplicative inverse of the radii of the ellipsoid, thus reducing the problem to intersecting the scaled object with a unit sphere. Care should be taken to avoid problems if the applied scaling introduces skew. Skew can make the usage of ellipsoids impractical in certain cases, for example collision between two arbitrary ellipsoids.
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Noncommutative logic is an extension of linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties (a family of cyclic orders which may be viewed as a species of structure), and the correctness criterion for its proof nets is given in terms of partial permutations. It also has a denotational semantics in which formulas are interpreted by modules over some specific Hopf algebras.
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In more detail: an affine group scheme G of finite type over a field k is called linearly reductive if its finite-dimensional representations are completely reducible. For k of characteristic zero, G is linearly reductive if and only if the identity component Go of G is reductive.Milne (2017), Corollary 22.43. For k of characteristic p>0, however, Masayoshi Nagata showed that G is linearly reductive if and only if Go is of multiplicative type and G/Go has order prime to p.
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Neglecting losses, the gain demonstrates a linear dependence on the number of devices (stages). Unlike the multiplicative nature of a cascade of conventional amplifiers, the DA demonstrates an additive quality. It is this synergistic property of the DA architecture that makes it possible for it to provide gain at frequencies beyond that of the unity-gain frequency of the individual stages. In practice, the number of stages is limited by the diminishing input signal resulting from attenuation on the input line.
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Top trees have been implemented in a variety of ways, some of them include implementation using a Multilevel Partition (Top-trees and dynamic graph algorithms Jacob Holm and Kristian de Lichtenberg. Technical Report), and even by using Sleator-Tarjan s-t trees (typically with amortized time bounds), Frederickson's Topology Trees (with worst case time bounds) (Alstrup et al. Maintaining Information in Fully Dynamic Trees with Top Trees). Amortized implementations are more simple, and with small multiplicative factors in time complexity.
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In computer programming languages, the inverse trigonometric functions are usually called by the abbreviated forms asin, acos, atan. The notations , , , etc., as introduced by John Herschel in 1813, are often used as well in English-language sources—conventions consistent with the notation of an inverse function. This might appear to conflict logically with the common semantics for expressions such as , which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse or reciprocal and compositional inverse.
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Fuzzy-set theorists Fuzzy sets were introduced by Zadeh (1965) as an extension of the classical notion of sets. This idea is used in many MCDM algorithms to model and solve fuzzy problems. Multi-attribute utility theorists Multi-attribute utility or value functions are elicited and used to identify the most preferred alternative or to rank order the alternatives. Elaborate interview techniques, which exist for eliciting linear additive utility functions and multiplicative nonlinear utility functions, are used (Keeney and Raiffa, 1976).
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There are four beavers on the left side of a river and six beavers on the right side of the river. At a later time with the same group of beavers there are eight beavers on the right side of the river. How many beavers will there be on the left side? So there are situations where the additive relations (constant difference and constant sum) are correct and other situations where the multiplicative relations (constant ratio and constant product) are correct.
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A hash function can be designed to exploit existing entropy in the keys. If the keys have leading or trailing zeros, or particular fields that are unused, always zero or some other constant, or generally vary little, then masking out only the volatile bits and hashing on those will provide a better and possibly faster hash function. Selected divisors or multipliers in the division and multiplicative schemes may make more uniform hash functions if the keys are cyclic or have other redundancies.
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Let be a set equipped with a binary operation ∗. Then an element of is called a left identity if for all in , and a right identity if for all in . If is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity. An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1).
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György Elekes ( – ) was a Hungarian mathematician and computer scientist who specialized in Combinatorial geometry and Combinatorial set theory. He may be best known for his work in the field that would eventually be called Additive Combinatorics. Particularly notable was his "ingenious" application of the Szemerédi–Trotter theorem to improve the best known lower bound for the sum- product problem. He also proved that any polynomial-time algorithm approximating the volume of convex bodies must have a multiplicative error, and the error grows exponentially on the dimension.
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" Billboard wrote that "Thug uses this multiplicative vocal delivery to his advantage: where another rapper might lapse into repetition, he finds a new way to distress and warp his tone, to burrow resourcefully into rhythmic cracks and crevices." Complex noted his aptitude for creating catchy, melodic hooks. XXL called him a "rap weirdo", stating that "Thug's charisma, unhinged flow and hooks make his music intriguing." Critic Sheldon Pearce wrote that "Thug understands the modern pop song construction better than anyone: anything and everything can be a hook.
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It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5.
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The gauge (or commonly bore in British English) of a firearm is a unit of measurement used to express the inner diameter (bore diameter) of the barrel. Gauge is determined from the weight of a solid sphere of lead that will fit the bore of the firearm and is expressed as the multiplicative inverse of the sphere's weight as a fraction of a pound, e.g., a one-twelfth pound lead ball fits a 12-gauge bore. Thus there are twelve 12-gauge balls per pound, etc.
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This project was set up in order to try to solve the Erdős discrepancy problem. It was active for much of 2010 and had a brief revival in 2012, but did not end up solving the problem. However, in September 2015, Terence Tao, one of the participants of Polymath5, solved the problem in a pair of papers. One paper proved an averaged form of the Chowla and Elliott conjectures, making use of recent advances in analytic number theory concerning correlations of values of multiplicative functions.
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Fixed-ratio studies require a predefined number of operant responses to dispense one unit of reinforcer. Standard fixed ratio reinforcement schedules include FR5 and FR10, requiring 5 and 10 operant responses to dispense a unit of reinforcer, respectively. Progressive ratio reinforcement schedules utilize a multiplicative increase in the number of operant responses required to dispense a unit of reinforcer. For example, successive trials might require 5 operant responses per unit of reward, then 10 responses per unit of reward, then 15, and so on.
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In computer science, the Aharonov–Jones–Landau algorithm is an efficient quantum algorithm for obtaining an additive approximation of the Jones polynomial of a given link at an arbitrary root of unity. Finding a multiplicative approximation is a #P-hard problem, so a better approximation is considered unlikely. However, it is known that computing an additive approximation of the Jones polynomial is a BQP-complete problem. The algorithm was published in 2009 in a paper written by Dorit Aharonov, Vaughan Jones and Zeph Landau.
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Neurons involved in gain field encoding work multiplicatively, taking the input from several together to form the gain field. It is this process that allows the complexity of motor control. Instead of simply encoding the motion of the limb in which a specific motion is desired, the multiplicative nature of the gain field ensures that the positioning of the rest of the body is taken in to consideration. This process allows for motor coordination of flexible bimanual actions as opposed to restricting the individual to unimanual motion.
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In physics, the Landé g-factor is a particular example of a g-factor, namely for an electron with both spin and orbital angular momenta. It is named after Alfred Landé, who first described it in 1921. In atomic physics, the Landé g-factor is a multiplicative term appearing in the expression for the energy levels of an atom in a weak magnetic field. The quantum states of electrons in atomic orbitals are normally degenerate in energy, with these degenerate states all sharing the same angular momentum.
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A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type.Husemöller (1987) pp.116-117 Suppose E is an elliptic curve defined over the rational number field Q. It is known that there is a finite, non-empty set S of prime numbers p for which E has bad reduction modulo p. The latter means that the curve Ep obtained by reduction of E to the prime field with p elements has a singular point.
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Exponentiation of a non-zero real number can be extended to negative integers. We make the definition that x−1 = , meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition is then extended to negative integers, preserving the exponential law xaxb = x(a + b) for real numbers a and b. Exponentiation to negative integers can be extended to invertible elements of a ring, by defining x−1 as the multiplicative inverse of x.
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In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric standard deviation is a multiplicative factor, and thus is dimensionless, rather than having the same dimension as the input values. Thus, the geometric standard deviation may be more appropriately called geometric SD factor GraphPad GuideKirkwood, T.B.L. (1993).
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The common disease-common variant (often abbreviated CD-CV) hypothesis predicts that common disease-causing alleles, or variants, will be found in all human populations which manifest a given disease. Common variants (not necessarily disease-causing) are known to exist in coding and regulatory sequences of genes. According to the CD-CV hypothesis, some of those variants lead to susceptibility to complex polygenic diseases. Each variant at each gene influencing a complex disease will have a small additive or multiplicative effect on the disease phenotype.
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Safe primes are also important in cryptography because of their use in discrete logarithm-based techniques like Diffie–Hellman key exchange. If is a safe prime, the multiplicative group of numbers modulo has a subgroup of large prime order. It is usually this prime- order subgroup that is desirable, and the reason for using safe primes is so that the modulus is as small as possible relative to p. A prime number p = 2q + 1 is called a safe prime if q is prime.
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Lighting and reflection calculations (shown here in the first-person shooter OpenArena) use the fast inverse square root code to compute angles of incidence and reflection. Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates , the reciprocal (or multiplicative inverse) of the square root of a 32-bit floating-point number in IEEE 754 floating-point format. This operation is used in digital signal processing to normalize a vector, i.e., scale it to length 1.
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In abstract algebra, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map 0: x \mapsto 0 as additive identity and the identity map 1: x \mapsto x as multiplicative identity.
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In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression (including variables such as a, b and c). In the latter case, the variables appearing in the coefficients are often called parameters, and must be clearly distinguished from the other variables. For example, in :7x^2-3xy+1.5+y, the first two terms have the coefficients 7 and −3, respectively. The third term 1.5 is a constant coefficient.
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During 1976–1992 he was with LOMI, Leningrad Department of the Steklov Mathematical Institute of the USSR Academy of Sciences. In 1979 he earned PhD (Candidate of Sciences) in Physics and Mathematics with thesis "Multiplicative Complexity of a Family of Bilinear Forms" (from LOMI, under the direction of Anatol Slissenko). In 1985 he earned Doctor of Science (higher doctorate) with thesis "Computational Complexity in Polynomial Algebra". Since 1988 till 1992 he was the head of Laboratory of algorithmic methods Leningrad Department of the Steklov Mathematical Institute.
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They are appropriate to use when past numerical data is available and when it is reasonable to assume that some of the patterns in the data are expected to continue into the future. These methods are usually applied to short- or intermediate-range decisions. Examples of quantitative forecasting methods are last period demand, simple and weighted N-Period moving averages, simple exponential smoothing, poisson process model based forecasting and multiplicative seasonal indexes. Previous research shows that different methods may lead to different level of forecasting accuracy.
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Saks research in computational complexity theory, combinatorics, and graph theory has contributed to the study of lower bounds in order theory, randomized computation, and space–time tradeoff. In Kahn and Saks (1984) it was shown there exist a tight information-theoretical lower bound for sorting under partially ordered information up to a multiplicative constant. In the first super-linear lower bound for the noisy broadcast problem was proved. In a noisy broadcast model, n+1 processors P_0,P_1,\ldots,P_n are assigned a local input bit x_i.
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In algebra, the 3x + 1 semigroup is a special subsemigroup of the multiplicative semigroup of all positive rational numbers. The elements of a generating set of this semigroup are related to the sequence of numbers involved in the still open Collatz conjecture or the "3x + 1 problem". The 3x + 1 semigroup has been used to prove a weaker form of the Collatz conjecture. In fact, it was in such context the concept of the 3x + 1 semigroup was introduced by H. Farkas in 2005.
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There is evidence that the combined effects of such a policy are not only additive but even multiplicative. For instance, by simultaneously addressing customer wishes in addition to employee and stockholder interests, both of the latter two groups also benefit from increased sales. # Supporters also take issue with the preeminent role given to stockholders by many business thinkers, especially in the past. The argument is that debt holders, employees, and suppliers also make contributions and thus also take risks in creating a successful firm.
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The logarithm of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a log-normal distribution. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different random factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called Gibrat's law.
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In mathematics, the Hurwitz problem, named after Adolf Hurwitz, is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables. There are well-known multiplicative relationships between sums of squares in two variables : (x^2+y^2)(u^2+v^2) = (xu-yv)^2 + (xv+yu)^2 \ , (known as the Brahmagupta–Fibonacci identity), and also Euler's four-square identity and Degen's eight-square identity. These may be interpreted as multiplicativity for the norms on the complex numbers, quaternions and octonions respectively.Charles W. Curtis (1963) "The Four and Eight Square Problem and Division Algebras" in Studies in Modern Algebra edited by A.A. Albert, pages 100–125, Mathematical Association of America, Solution of Hurwitz’s Problem on page 115 The Hurwitz problem for the field K is to find general relations of the form : (x_1^2+\cdots+x_r^2) \cdot (y_1^2+\cdots+y_s^2) = (z_1^2 + \cdots + z_n^2) \ , with the z being bilinear forms in the x and y: that is, each z is a K-linear combination of terms of the form xiyj.
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Groups for which the commutativity equation always holds are called abelian groups (in honor of Niels Henrik Abel). The symmetry group described in the following section is an example of a group that is not abelian. The identity element of a group G is often written as 1 or 1G, a notation inherited from the multiplicative identity. If a group is abelian, then one may choose to denote the group operation by + and the identity element by 0; in that case, the group is called an additive group.
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The 6th complex roots of unity form a cyclic group. z is a primitive element, but z2 is not, because the odd powers of z are not a power of z2. A cyclic group is a group all of whose elements are powers of a particular element a. In multiplicative notation, the elements of the group are: :..., a−3, a−2, a−1, a0 = e, a, a2, a3, ..., where a2 means a ⋅ a, and a−3 stands for a−1 ⋅ a−1 ⋅ a−1 = (a ⋅ a ⋅ a)−1 etc.
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If the members of the Fibonacci sequence are taken mod n, the resulting sequence is periodic with period at most 6n. The lengths of the periods for various n form the so-called Pisano periods . Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular n, the Pisano period may be found as an instance of cycle detection.
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A commutative ring is a set, equipped with an addition and multiplication operation, satisfying all the axioms of a field, except for the existence of multiplicative inverses . For example, the integers form a commutative ring, but not a field: the reciprocal of an integer is not itself an integer, unless . In the hierarchy of algebraic structures fields can be characterized as the commutative rings in which every nonzero element is a unit (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct ideals, and .
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Module homomorphisms between finitely generated free modules may be represented by matrices. The theory of matrices over a ring is similar to that of matrices over a field, except that determinants exist only if the ring is commutative, and that a square matrix over a commutative ring is invertible only if its determinant has a multiplicative inverse in the ring. Vector spaces are completely characterized by their dimension (up to an isomorphism). In general, there is not such a complete classification for modules, even if one restricts oneself to finitely generated modules.
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A first publication on this topic was made by Hans Kropf in 1899. Edwin Mueller in his Handbook of Austria and Lombardy-Venetia Cancellations on the Postage Stamp Issues 1850-1864, published in 1961, described all postmarks used in the Austrian empire, Lombardy, Venetia and in the Austria post-offices in the Ottoman empire. The relative valuation of those postmarks contains a popularity index, which is a multiplicative factor on top of the rarity. Closely related article is Postage stamps and postal history of Austria for a better understanding of the historical context.
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By adjoining a primitive th root of unity to \Q, one obtains the th cyclotomic field \Q(\exp(2\pi i/n)).This field contains all th roots of unity and is the splitting field of the th cyclotomic polynomial over \Q. The field extension \Q(\exp(2\pi i /n))/\Q has degree φ(n) and its Galois group is naturally isomorphic to the multiplicative group of units of the ring \Z/n\Z. As the Galois group of \Q(\exp(2\pi i /n))/\Q is abelian, this is an abelian extension.
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Since for any two VNM-agents X and Y, their VNM-utility functions uX and uY are only determined up to additive constants and multiplicative positive scalars, the theorem does not provide any canonical way to compare the two. Hence expressions like uX(L) + uY(L) and uX(L) − uY(L) are not canonically defined, nor are comparisons like uX(L) < uY(L) canonically true or false. In particular, the aforementioned "total VNM- utility" and "average VNM-utility" of a population are not canonically meaningful without normalization assumptions.
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Early hypotheses of gain field encoding suggested that the gain field works as a model for motion additively. This would mean that if two limbs needed to move, models for each would be called separately but at the same time. However, more recent studies in which more complex motor movements are observed have found that the gain field is created multiplicatively in order to allow the body to adapt to the constantly changing frames of reference experienced in everyday life. This multiplicative property is an effect of recurrent neural circuitry.
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Other than its application to the frequency ratios of intervals (for example, Just intonation, and the twelfth root of two in equal temperament), it has been used in other ways for twelve-tone technique, and musical set theory. Additionally ring modulation is an electrical audio process involving multiplication that has been used for musical effect. A multiplicative operation is a mapping in which the argument is multiplied . Multiplication originated intuitively in interval expansion, including tone row order number rotation, for example in the music of Béla Bartók and Alban Berg .
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Proportional hazards models are a class of survival models in statistics. Survival models relate the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. In a proportional hazards model, the unique effect of a unit increase in a covariate is multiplicative with respect to the hazard rate. For example, taking a drug may halve one's hazard rate for a stroke occurring, or, changing the material from which a manufactured component is constructed may double its hazard rate for failure.
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The Krohn–Rhodes complexity (also called group complexity or just complexity) of a finite semigroup S is the least number of groups in a wreath product of finite groups and finite aperiodic semigroups of which S is a divisor. All finite aperiodic semigroups have complexity 0, while non-trivial finite groups have complexity 1. In fact, there are semigroups of every non-negative integer complexity. For example, for any n greater than 1, the multiplicative semigroup of all (n+1)×(n+1) upper-triangular matrices over any fixed finite field has complexity n (Kambites, 2007).
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The output sequence is strictly periodic if and only if u is between -q and 0. It is possible to express u as a simple quadratic polynomial involving the initial state and the qi. There is also an exponential representation of FCSRs: if g is the inverse of N \pmod q, and the output sequence is strictly periodic, then a_i = (A g_i \bmod q) \bmod N, where A is an integer. It follows that the period is at most the order of in the multiplicative group of units modulo .
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Prior to introducing the measurement model he is best known for, Rasch had applied the Poisson distribution to reading data as a measurement model, hypothesizing that in the relevant empirical context, the number of errors made by a given individual was governed by the ratio of the text difficulty to the person's reading ability. Rasch referred to this model as the multiplicative Poisson model. Rasch's model for dichotomous data – i.e. where responses are classifiable into two categories – is his most widely known and used model, and is the main focus here.
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A complete set of MOLS() exists whenever is a prime or prime power. This follows from a construction that is based on a finite field GF(), which only exist if is a prime or prime power. The multiplicative group of GF() is a cyclic group, and so, has a generator, λ, meaning that all the non-zero elements of the field can be expressed as distinct powers of λ. Name the elements of GF() as follows: ::α0 = 0, α1 = 1, α2 = λ, α3 = λ2, ..., α-1 = λ-2.
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The second follows from the definition of −1 as additive inverse of 1: it is precisely that number that when added to 1 gives 0. Now, using the distributive law, we see that :0 =-1\cdot [1+(-1)]=-1\cdot1+(-1)\cdot(-1)=-1+(-1)\cdot(-1) The second equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies :(-1) \cdot (-1) = 1 The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers.
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Kaisa Sofia Matomäki (born April 30, 1985) is a Finnish mathematician specializing in number theory. Since September 2015, she has been working as an Academic Research Fellow in the Department of Mathematics and Statistics, University of Turku, Turku, Finland. Her research includes results on the distribution of multiplicative functions over short intervals of numbers; for instance, she showed that the values of the Möbius function are evenly divided between +1 and −1 over short intervals. These results, in turn, were among the tools used by Terence Tao to prove the Erdős discrepancy problem.
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Absolute, or thermodynamic, temperature is conventionally measured in kelvins (Celsius-scaled increments) and in the Rankine scale (Fahrenheit- scaled increments) with increasing rarity. Absolute temperature measurement is uniquely determined by a multiplicative constant which specifies the size of the degree, so the ratios of two absolute temperatures, T2/T1, are the same in all scales. The most transparent definition of this standard comes from the Maxwell–Boltzmann distribution. It can also be found in Fermi–Dirac statistics (for particles of half-integer spin) and Bose–Einstein statistics (for particles of integer spin).
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In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced by André Weil (1940) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function.
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Transmission Control Protocol (TCP) uses a network congestion-avoidance algorithm that includes various aspects of an additive increase/multiplicative decrease (AIMD) scheme, along with other schemes including slow start and congestion window, to achieve congestion avoidance. The TCP congestion- avoidance algorithm is the primary basis for congestion control in the Internet. Per the end-to-end principle, congestion control is largely a function of internet hosts, not the network itself. There are several variations and versions of the algorithm implemented in protocol stacks of operating systems of computers that connect to the Internet.
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When a connection is set up, the congestion window, a value maintained independently at each host, is set to a small multiple of the MSS allowed on that connection. Further variance in the congestion window is dictated by an additive increase/multiplicative decrease (AIMD) approach. This means that if all segments are received and the acknowledgments reach the sender on time, some constant is added to the window size. When the window reaches ssthresh, the congestion window increases linearly at the rate of 1/(congestion window) segment on each new acknowledgement received.
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Different permutations may be related by transformation, through the application of zero or more operations, such as transposition, inversion, retrogradation, circular permutation (also called rotation), or multiplicative operations (such as the cycle of fourths and cycle of fifths transforms). These may produce reorderings of the members of the set, or may simply map the set onto itself. Order is particularly important in the theories of composition techniques originating in the 20th century such as the twelve-tone technique and serialism. Analytical techniques such as set theory take care to distinguish between ordered and unordered collections.
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For a radix of 10, there is thought to be no number with a multiplicative persistence > 11: this is known to be true for numbers up to 1020000. The smallest numbers with persistence 0, 1, ... are: :0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899. The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be sorted, and, except for the first two digits, all digits must be 7, 8, or 9.
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Compare the values 6 and 4 for Euler's totient function, the multiplicative group of integers modulo n for n = 9 and 10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms as isometries (keeping the order of the rotations the same or reversing the order). The only values of n for which φ(n) = 2 are 3, 4, and 6, and consequently, there are only three dihedral groups that are isomorphic to their own automorphism groups, namely (order 6), (order 8), and (order 12).
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In physical settings, Legendre's differential equation arises naturally whenever one solves Laplace's equation (and related partial differential equations) by separation of variables in spherical coordinates. From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the spherical harmonics, of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis. The polynomials appear as P_n(\cos\theta) where \theta is the polar angle. This approach to the Legendre polynomials provides a deep connection to rotational symmetry.
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Thus every module direct summand of R is generated by an idempotent. If a is a central idempotent, then the corner ring is a ring with multiplicative identity a. Just as idempotents determine the direct decompositions of R as a module, the central idempotents of R determine the decompositions of R as a direct sum of rings. If R is the direct sum of the rings R1,...,Rn, then the identity elements of the rings Ri are central idempotents in R, pairwise orthogonal, and their sum is 1.
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In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital. Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An R-algebra is an R-module with an associative R-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if S is any ring with center C, then S is an associative C-algebra.
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The function Λ which gives t_1^2 is a multiplicative character, or homomorphism from the group's torus to the underlying field R. The function λ giving θ is a weight of the Lie Algebra with weight space given by the span of the matrices. It is satisfying to show the multiplicativity of the character and the linearity of the weight. It can further be proved that the differential of Λ can be used to create a weight. It is also educational to consider the case of SL(3, R).
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These secondary, replicative spores are globose and elongate in physiology. Once the spore has been discharged, all subsequent developmental events are triggered, including germination. Sporangial germination, either through secondary spore formation or vegetative germ tube formation, seems to be increasingly dependent on the time elapsed since discharge, rather than on the external environmental factors, however these external factors do still play a role. The spores formed by C. coronatus during asexual reproduction are globose, villose and multiplicative in some isolates, and have at least seven nuclei per spore.
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This presence of villose and multiplicative spores is what differentiates C. coronatus from the genus Entomophthora. Though C.coronatus is classified under Zygomycota, it does not produce zygospores and therefore does not undergo sexual reproduction. It has been demonstrated that C. coronatus produces lipolytic, chitinolytic and proteolytic enzymes, especially extracellular proteinases, namely serine proteases which are optimally active at pH 10 and 40 °C. Serine proteases are a diverse group of bacterial, fungal and animal enzymes whose common element is an active site composed of serine, histidine and aspartic acid.
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In order to measure the interaction between two genes, one must have some standard for the expected phenotype if the genes do not interact. Some common models for how the phenotypes of independent genes combine include the min, additive, and multiplicative models. In the min model, the expected fitness resulting from the mutation of two independent genes is the same as the fitness of the least-fit single mutant. In the additive model, the expected phenotype resulting from the mutation of two independent genes is the sum of the phenotypes due to the individual mutations.
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However, ALDH2-deficient people who do not carry this ADH variant are at the highest risk of cancer as these risk factors act in a multiplicative manner through increasing exposure time to salivary acetaldehyde. The idea that acetaldehyde is the cause of the flush is also shown by the clinical use of disulfiram (Antabuse), which blocks the removal of acetaldehyde from the body via ALDH inhibition. The high acetaldehyde concentrations described share similarity to symptoms of the flush (flushing of the skin, accelerated heart rate, shortness of breath, throbbing headache, mental confusion and blurred vision).
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In his career, Teichmüller wrote 34 papers in the space of around 6 years. His early algebraic investigations dealt with the valuation theory of fields and the structure of algebras. In valuation theory, he introduced multiplicative systems of representatives of the residue field of valuation rings, which led to a characterisation of the structure of the whole field in terms of the residue field. In the theory of algebras, he started to generalise Emmy Noether's concept of crossed products from fields to certain kind of algebras, gaining new insights into the structure of p-algebras.
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In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that for and , the intersection . A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.
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Cancelling out is a mathematical process used for removing subexpressions from a mathematical expression, when this removal does not change the meaning or the value of the expression because the subexpressions have equal and opposing effects. For example, a fraction is put in lowest terms by cancelling out the common factors of the numerator and the denominator. As another example, if a×b=a×c, then the multiplicative term a can be canceled out if a≠0, resulting in the equivalent expression b=c; this is equivalent to dividing through by a.
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These are the graphs K such that a product G × H has a homomorphism to K only when one of G or H also does. Identifying multiplicative graphs lies at the heart of Hedetniemi's conjecture.. Graph homomorphisms also form a category, with graphs as objects and homomorphisms as arrows. The initial object is the empty graph, while the terminal object is the graph with one vertex and one loop at that vertex. The tensor product of graphs is the category-theoretic product and the exponential graph is the exponential object for this category.
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H-TCP is a loss-based algorithm, using additive- increase/multiplicative-decrease (AIMD) to control TCP's congestion window. It is one of many TCP congestion avoidance algorithms which seeks to increase the aggressiveness of TCP on high bandwidth-delay product (BDP) paths, while maintaining "TCP friendliness" for small BDP paths. H-TCP increases its aggressiveness (in particular, the rate of additive increase) as the time since the previous loss increases. This avoids the problem encountered by HSTCP and BIC TCP of making flows more aggressive if their windows are already large.
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He worked on the deformation theory for mappings to groups, which led to the solution of the Novikov problem on multiplicative subgroups in operator doubles, and to construction of the quantum group of complex cobordisms. He went on to treat problems related both with algebraic geometry and integrable systems. He is also well known for his work on sigma-functions on universal spaces of Jacobian varieties of algebraic curves that give effective solutions of important integrable systems. Buchstaber created an algebro-functional theory of symmetric products of spaces and described algebraic varieties of polysymmetric polynomials.
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Unlike multiplicative fluctuations, additive fluctuations do not lead to Benford's law: They lead instead to normal probability distributions (again by the central limit theorem), which do not satisfy Benford's law. For example, the "number of heartbeats that I experience on a given day" can be written as the sum of many random variables (e.g. the sum of heartbeats per minute over all the minutes of the day), so this quantity is unlikely to follow Benford's law. By contrast, that hypothetical stock price described above can be written as the product of many random variables (i.e.
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It is critically important that students on their own recognize that their current mode of reasoning, say that it is additive, is inappropriate for a multiplicative problem they are trying to solve. Robert Karplus developed a model of learning he called the learning cycle that facilitates the acquisition of new reasoning skills. # The first phase is one of exploration in which students learn through their own actions and reactions with minimal guidance. The learning environment has to be carefully designed to focus student’s attention on the relevant issues.
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Population genetics must either model this complexity in detail, or capture it by some simpler average rule. Empirically, beneficial mutations tend to have a smaller fitness benefit when added to a genetic background that already has high fitness: this is known as diminishing returns epistasis. When deleterious mutations also have a smaller fitness effect on high fitness backgrounds, this is known as "synergistic epistasis". However, the effect of deleterious mutations tends on average to be very close to multiplicative, or can even show the opposite pattern, known as "antagonistic epistasis".
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Trevor was the president and largest stockholder in DMSI. In 1990, Dupuy resigned from DMSI, sold his stock and reactivated TNDA. In 1992 TNDA was closed out, and he established the non-profit The Dupuy Institute (TDI). Dupuy's main contribution to military operation analysis is the assessment method Quantified Judgment Method or QJM, where the outcome of a battle is predicted using a fairly complicated multiplicative-additive formula in which various factors relating to the strength and firepower of the fighting parties as well as the circumstances are taken into account.
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If G is cyclic then the transfer takes any element y of G to y[G:H]. A simple case is that seen in the Gauss lemma on quadratic residues, which in effect computes the transfer for the multiplicative group of non-zero residue classes modulo a prime number p, with respect to the subgroup {1, −1}. One advantage of looking at it that way is the ease with which the correct generalisation can be found, for example for cubic residues in the case that p − 1 is divisible by three.
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For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition. Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations. In this sense, an SDE is not a uniquely defined entity when noise is multiplicative and when the SDE is understood as a continuous time limit of a stochastic difference equation. In this case, SDE must be complemented by what is known as "interpretations of SDE" such as Itô or a Stratonovich interpretations of SDEs.
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The problem is NP-hard, and does not admit a polynomial-time approximation scheme unless P = NP. This can be proven using the inapproximability of vertex cover in bounded degree graphs. Although there is no polynomial-time approximation scheme, there is a polynomial-time constant-factor approximation—an algorithm that finds a connector whose Wiener index is within a constant multiplicative factor of the Wiener index of the optimum connector. In terms of complexity classes, the minimum Wiener connector problem is in APX but is not in PTAS unless P = NP.
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In physics, the differential coefficient of a function f(x) is what is now called its derivative df(x)/dx, the (not necessarily constant) multiplicative factor or coefficient of the differential dx in the differential df(x). A coefficient is usually a constant quantity, but the differential coefficient of f is a constant function only if f is a linear function. When f is not linear, its differential coefficient is a function, call it f′, derived by the differentiation of f, hence, the modern term, derivative. The older usage is now rarely seen.
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Tundra Nenets has two verbal aspectual classes, perfective and imperfective. There are several derivational aspectual suffixes which can change the aspectual class of a verb. For example, imperfectivizing suffixes can be used to express durative, frequentative, multiplicative, and iterative meanings, such as in tola-bə 'to keep counting' (from tola- 'to count'). There are also denominal verbs with the meaning 'to use as X, to have as X', which are formed from the accusative plural stem, such as in səb'i-q' 'to use as a hat' (from səwa 'hat').
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In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as in the case of additive identity and multiplicative identity), when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
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Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number- theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory.
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Vol 88, Issue 5, pp. 438-453. The social impact model further explains that social impact is the multiplicative effect of strength (power, status, knowledge), the immediacy (physical proximity and recency), and the number of group members, supporting the view that a minority will be less influential on a larger majority. Clark and Maass (1990) looked at the interaction between minority influence and majorities of varying sizes, and found that, like Latané & Wolf's findings, the minority's influence decreases in a negatively accelerating power function as the majority increases.Clark, R. D. and Maass, A. (1990), The effects of majority size on minority influence.
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The multiplicative case (abbreviated ' or ') is a grammatical case used for marking a number of something ("three times"). The case is found in the Hungarian language,Mentioned in: István Kenesei, Anna Fenyvesi, Robert Michael Vago, Hungarian, page xxviii, 1998 - 472 pages [ Google book search] for example nyolc (eight), nyolcszor (eight times), however it is not considered a real case in modern Hungarian linguistics because of its adverb-forming nature. The case appears also in Finnish as an adverbial (adverb-forming) case. Used with a cardinal number it denotes the number of actions; for example, viisi (five) -> viidesti (five times).
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Kalman filtering can be used to sequentially estimate the attitude, as well as the angular rate. Because attitude dynamics (combination of rigid body dynamics and attitude kinematics) are non-linear, a linear Kalman filter is not sufficient. Because attitude dynamics is not very non-linear, the Extended Kalman filter is usually sufficient (however Crassidis and Markely demonstrated that the Unscented Kalman filter could be used, and can provide benefits in cases where the initial estimate is poor). Multiple methods have been proposed, however the Multiplicative Extended Kalman Filter (MEKF) is by far the most common approach.
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This approach utilizes the multiplicative formulation of the error quaternion, which allows for the unity constraint on the quaternion to be better handled. It is also common to use a technique known as dynamic model replacement, where the angular rate is not estimated directly, but rather the measured angular rate from the gyro is used directly to propagate the rotational dynamics forward in time. This is valid for most applications as gyros are typically far more precise than one's knowledge of disturbance torques acting on the system (which is required for precise estimation of the angular rate).
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The positive (equally non-negative) integers form a cancellative semigroup under addition. The non-negative integers form a cancellative monoid under addition. In fact, any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid embedding into a group (as the above examples clearly do) will obey the cancellative law. In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of a ring that are not zero divisors (which is just the set of all nonzero elements if the ring in question is a domain, like the integers) has the cancellation property.
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Bird covered in oil after oil spill Scope neglect or scope insensitivity is a cognitive bias that occurs when the valuation of a problem is not valued with a multiplicative relationship to its size. Scope neglect is a specific form of extension neglect. In one study, respondents were asked how much they were willing to pay to prevent migrating birds from drowning in uncovered oil ponds by covering the oil ponds with protective nets. Subjects were told that either 2,000, or 20,000, or 200,000 migrating birds were affected annually, for which subjects reported they were willing to pay $80, $78 and $88 respectively.
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Some customary systems of weights and measures had duodecimal ratios, which meant quantities were conveniently divisible by 2, 3, 4, and 6. But it was difficult to do arithmetic with things like pound or foot. There was no system of notation for successive fractions: for example, of of a foot was not an inch or any other unit. But the system of counting in decimal ratios did have notation, and the system had the algebraic property of multiplicative closure: a fraction of a fraction, or a multiple of a fraction was a quantity in the system, like of which is .
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The main reason of the growing interest towards the model of boson sampling is that despite being non-universal it is strongly believed to perform a computational task that is intractable for a classical computer. One of the main reasons behind this is that the probability distribution, which the boson sampling device has to sample from, is related to the permanent of complex matrices. As is known, the computation of the permanent is in the general case an extremely hard task: it falls in the #P-hard complexity class. Moreover, its approximation to within multiplicative error is a #P-hard problem as well.
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The theory of this type of generator is rather complex, and it may not be sufficient simply to choose random values for and . These generators also tend to be very sensitive to initialisation. Generators of this type employ k words of state (they 'remember' the last k values). If the operation used is addition, then the generator is described as an Additive Lagged Fibonacci Generator or ALFG, if multiplication is used, it is a Multiplicative Lagged Fibonacci Generator or MLFG, and if the XOR operation is used, it is called a Two-tap generalised feedback shift register or GFSR.
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Many objects that arise in number theory are naturally Galois representations. For example, if L is a Galois extension of a number field K, the ring of integers OL of L is a Galois module over OK for the Galois group of L/K (see Hilbert–Speiser theorem). If K is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of K and its study leads to local class field theory. For global class field theory, the union of the idele class groups of all finite separable extensions of K is used instead.
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Sepp Hochreiter developed "Factor Analysis for Bicluster Acquisition" (FABIA) for biclustering that is simultaneously clustering rows and columns of a matrix. A bicluster in transcriptomic data is a pair of a gene set and a sample set for which the genes are similar to each other on the samples and vice versa. In drug design, for example, the effects of compounds may be similar only on a subgroup of genes. FABIA is a multiplicative model that assumes realistic non-Gaussian signal distributions with heavy tails and utilizes well understood model selection techniques like a variational approach in the Bayesian framework.
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Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units O× of the ring of integers O. Specifically, it states that O× is isomorphic to G × Zr, where G is the finite cyclic group consisting of all the roots of unity in O, and r = r1 + r2 − 1 (where r1 (respectively, r2) denotes the number of real embeddings (respectively, pairs of conjugate non- real embeddings) of K). In other words, O× is a finitely generated abelian group of rank r1 + r2 − 1 whose torsion consists of the roots of unity in O.
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There is limited work published with regards to the theoretical foundations of this technique, in particular relating to the justification of the five factor methodology. Regardless of the situation, it remains to be assumed that these 5 factors are suffice for an accurate assessment of human performance; as no other factors are considered, this suggests that to solely use these 5 factors to adequately describe the full range of error producing conditions fails to be highly realistic. Further to this, the values of K1-5 are unsubstantiated and the suggested multiplicative relationship has no sufficient theoretical or empirical evidence for justification purposes.
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His article on Markov chain models Markov Chain Models in life insurance, Blätter Deutsch. Gesellsch.Versich.math. has been named one of the four most significant papers in modern actuarial science. Hoem also made contributions to stochastic stable population theory,Stochastic stable population theory with continuous time,(with Niels Keiding). Scand. Act. J. 1976 (3), 150-175 (1976) demographic incidence rates,Demographic incidence rates. Theor. Popul. Biol. 14 (3), 329-337. Bibliographic Note, 18 (2), 195 (1978) and the statistical analysis of multiplicative models. He is best known for his work on event-history analysis—contributions that have helped shape demographic methodology.
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This leads to computable variants of AC and AP, and Universal "Levin" Search (US) solves all inversion problems in optimal time (apart from some unrealistically large multiplicative constant). AC and AP also allow a formal and rigorous definition of randomness of individual strings to not depend on physical or philosophical intuitions about non-determinism or likelihood. Roughly, a string is Algorithmic "Martin-Löf" Random (AR) if it is incompressible in the sense that its algorithmic complexity is equal to its length. AC, AP, and AR are the core sub-disciplines of AIT, but AIT spawns into many other areas.
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Thus it should be clear that maxdiff is a subset of BWS; maxdiff is BWS, but BWS is not necessarily maxdiff. Indeed, maxdiff might not be considered an attractive model on psychological and intuitive grounds: as the number of items increases, the number of possible pairs increases in a multiplicative fashion: n items produces n(n-1) pairs (where best-worst order matters). Assuming respondents do evaluate all possible pairs is a strong assumption. Early work did use the term maxdiff to refer to BWS, but with Marley's return to the field, correct academic terminology has been disseminated throughout Europe and Asia-Pacific.
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The Jacobian on a hyperelliptic curve is an Abelian group and as such it can serve as group for the discrete logarithm problem (DLP). In short, suppose we have an Abelian group G and g an element of G, the DLP on G entails finding the integer a given two elements of G, namely g and g^a. The first type of group used was the multiplicative group of a finite field, later also Jacobians of (hyper)elliptic curves were used. If the hyperelliptic curve is chosen with care, then Pollard's rho method is the most efficient way to solve DLP.
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First, by default it finds only matches that are 512+ byte long, since benchmarking proved that this is optimal setting for overall REP+LZMA compression. Second, it uses a sliding dictionary that's about 1/2 RAM long, so decompression doesn't need to reread data from decompressed file. REP's advantage is its multiplicative rolling hash that is both quick to compute and has near-ideal distribution. Larger minimal match length (512 bytes compared to 32 bytes in rzip) allowed for additional speed optimizations, so that REP provides very fast compression (about 200 MB/s on Intel i3-2100).
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Multiplicative seasonality can be represented as a constant factor, not an absolute amount. Triple exponential smoothing was first suggested by Holt's student, Peter Winters, in 1960 after reading a signal processing book from the 1940s on exponential smoothing. Holt's novel idea was to repeat filtering an odd number of times greater than 1 and less than 5, which was popular with scholars of previous eras. While recursive filtering had been used previously, it was applied twice and four times to coincide with the Hadamard conjecture, while triple application required more than double the operations of singular convolution.
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In 2004, Tao, together with Jean Bourgain and Nets Katz, studied the additive and multiplicative structure of subsets of finite fields of prime order. It is well known that there are no nontrivial subrings of such a field. Bourgain, Katz, and Tao provided a quantitative formulation of this fact, showing that for any subset of such a field, the number of sums and products of elements of the subset must be quantitatively large, as compared to the size of the field and the size of the subset itself. Improvements of their result were later given by Bourgain, Alexey Glibichuk, and Sergei Konyagin.
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Let G be a group and k a field. The group Hopf algebra of G over k, denoted kG (or k[G]), is as a set (and a vector space) the free vector space on G over k. As an algebra, its product is defined by linear extension of the group composition in G, with multiplicative unit the identity in G; this product is also known as convolution. Note that while the group algebra of a finite group can be identified with the space of functions on the group, for an infinite group these are different.
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There are various kinds of proofs that can be given for the Cauchy−Binet formula. The proof below is based on formal manipulations only, and avoids using any particular interpretation of determinants, which may be taken to be defined by the Leibniz formula. Only their multilinearity with respect to rows and columns, and their alternating property (vanishing in the presence of equal rows or columns) are used; in particular the multiplicative property of determinants for square matrices is not used, but is rather established (the case n = m). The proof is valid for arbitrary commutative coefficient rings.
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One must loop over each pair of elements (so we get n2 interactions) and for each pair of elements we loop through Gauss points in the elements producing a multiplicative factor proportional to the number of Gauss-points squared. Also, the function evaluations required are typically quite expensive, involving trigonometric/hyperbolic function calls. Nonetheless, the principal source of the computational cost is this double-loop over elements producing a fully populated matrix. The Green's functions, or fundamental solutions, are often problematic to integrate as they are based on a solution of the system equations subject to a singularity load (e.g.
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Bézout's identity asserts that and are coprime if and only if there exist integers and such that :ns+at=1 Reducing this identity modulo gives :at \equiv 1 \mod n. Thus , or, more exactly, the remainder of the division of by , is the multiplicative inverse of modulo . To adapt the extended Euclidean algorithm to this problem, one should remark that the Bézout coefficient of is not needed, and thus does not need to be computed. Also, for getting a result which is positive and lower than n, one may use the fact that the integer provided by the algorithm satisfies .
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The algorithm is very similar to that provided above for computing the modular multiplicative inverse. There are two main differences: firstly the last but one line is not needed, because the Bézout coefficient that is provided always has a degree less than . Secondly, the greatest common divisor which is provided, when the input polynomials are coprime, may be any non zero elements of ; this Bézout coefficient (a polynomial generally of positive degree) has thus to be multiplied by the inverse of this element of . In the pseudocode which follows, is a polynomial of degree greater than one, and is a polynomial.
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Cycle graphs were investigated by the number theorist Daniel Shanks in the early 1950s as a tool to study multiplicative groups of residue classes. Shanks first published the idea in the 1962 first edition of his book Solved and Unsolved Problems in Number Theory. In the book, Shanks investigates which groups have isomorphic cycle graphs and when a cycle graph is planar. In the 1978 second edition, Shanks reflects on his research on class groups and the development of the baby-step giant-step method: Cycle graphs are used as a pedagogical tool in Nathan Carter's 2009 introductory textbook Visual Group Theory.
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A near- field is a set Q, together with two binary operations, + (addition) and \cdot (multiplication), satisfying the following axioms: :A1: (Q, +) is an abelian group. :A2: (a \cdot b) \cdot c = a \cdot (b \cdot c) for all elements a, b, c of Q (The associative law for multiplication). :A3: (a + b) \cdot c = a \cdot c + b \cdot c for all elements a, b, c of Q (The right distributive law). :A4: Q contains an element 1 such that 1 \cdot a = a \cdot 1 = a for every element a of Q (Multiplicative identity).
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When the abelian group A is the group of roots of unity in a separable closure Ks of K, the p-adic Tate module of A is sometimes referred to as the Tate module (where the choice of p and K are tacitly understood). It is a free rank one module over Zp with a linear action of the absolute Galois group GK of K. Thus, it is a Galois representation also referred to as the p-adic cyclotomic character of K. It can also be considered as the Tate module of the multiplicative group scheme Gm,K over K.
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Being a quadratic polynomial with no multiple root, the defining equation has two distinct solutions, which are equally valid and which happen to be additive and multiplicative inverses of each other. Once a solution of the equation has been fixed, the value , which is distinct from , is also a solution. Since the equation is the only definition of , it appears that the definition is ambiguous (more precisely, not well-defined). However, no ambiguity will result as long as one or other of the solutions is chosen and labelled as "", with the other one then being labelled as .
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In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.
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Note that Z, equipped with these operations of addition and multiplication, is a ring, and is in fact, the prototypical example of a ring. We can also define a total order on Z by writing :(a, b) ≤ (c, d) if and only if a + d ≤ b + c. This will lead to an additive zero of the form (a, a), an additive inverse of (a, b) of the form (b, a), a multiplicative unit of the form (a + 1, a), and a definition of subtraction :(a, b) − (c, d) = (a + d, b + c). This construction is a special case of the Grothendieck construction.
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The PA mechanism, which does not use payments, is analogous to the VCG mechanism, which uses monetary payments. VCG starts by selecting the max-sum allocation, and then for each agent i it calculates the max-sum allocation when i is not present, and pays i the difference (max-sum when i is present)-(max-sum when i is not present). Since the agents are quasilinear, the utility of i is reduced by an additive factor. In contrast, PA does not use monetary payments, and the agents' utilities are reduced by a multiplicative factor, by taking away some of their resources.
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The operator, K, is known as the coagulation kernel and describes the rate at which particles of size x_1 coagulate with particles of size x_2. Analytic solutions to the equation exist when the kernel takes one of three simple forms: : K = 1,\quad K = x_1 + x_2, \quad K = x_1x_2, known as the constant, additive, and multiplicative kernels respectively. For the case K = 1 it could be mathematically proven that the solution of Smoluchowski coagulation equations have asymptotically the dynamic scaling property. This self-similar behaviour is closely related to scale invariance which can be a characteristic feature of a phase transition.
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Explicit lists in the OEIS are J2 in , J3 in , J4 in , J5 in , J6 up to J10 in up to . Multiplicative functions defined by ratios are J2(n)/J1(n) in , J3(n)/J1(n) in , J4(n)/J1(n) in , J5(n)/J1(n) in , J6(n)/J1(n) in , J7(n)/J1(n) in , J8(n)/J1(n) in , J9(n)/J1(n) in , J10(n)/J1(n) in , J11(n)/J1(n) in . Examples of the ratios J2k(n)/Jk(n) are J4(n)/J2(n) in , J6(n)/J3(n) in , and J8(n)/J4(n) in .
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From 1870 onwards Carl Neumann also contributed to this theory. In the 1950s Schwarz's method was generalized in the theory of partial differential equations to an iterative method for finding the solution of an elliptic boundary value problem on a domain which is the union of two overlapping subdomains. It involves solving the boundary value problem on each of the two subdomains in turn, taking always the last values of the approximate solution as the next boundary conditions. It is used in numerical analysis, under the name multiplicative Schwarz method (in opposition to additive Schwarz method) as a domain decomposition method.
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Zenon Borevich completed his master's thesis titled "Regarding the theory of local fields" in 1951 and his doctoral dissertation titled "Regarding the multiplicative groups of normal R-extensions of local fields" in 1967. In 1968, Borevich became Professor of Mathematics at Saint Petersburg State University (then named Leningrad State University) and was put in charge of the Department of Higher Algebra and Number Theory. He became Dean of the university's entire Department of Mathematics and Mechanics in 1973 and remained in that position until 1984. He continued to head up Higher Algebra and Number Theory until his retirement in 1992.
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Kreindler was born on 15 October 1931 in Brăila, Romania into a Jewish family. She obtained in 1951 a fellowship and spent the next four years in the USSR studying mathematics at the Ural State University, located in Sverdlovsk (nowadays Yekaterinburg). In 1955, she completed a master thesis on "Multiplicative Lattices with Additive Basis" under the supervision of Petr Grigor'evich Kontorovich, before returning to Bucharest to join the faculty of Mathematics at the Polytechnic Institute of Bucharest. Next to her duties as assistant professor, she continued with her research in the field of functional analysis under the guidance of Grigore Moisil.
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Aristotle responded to these paradoxes by developing the notion of a potential countable infinity, as well as the infinitely divisible continuum. Unlike the eternal and unchanging cycles of time, he believed that the world is bounded by the celestial spheres and that cumulative stellar magnitude is only finitely multiplicative. The Indian philosopher Kanada, founder of the Vaisheshika school, developed a notion of atomism and proposed that light and heat were varieties of the same substance.Will Durant, Our Oriental Heritage: In the 5th century AD, the Buddhist atomist philosopher Dignāga proposed atoms to be point-sized, durationless, and made of energy.
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The most significant interaction between these two systems (corticotectal interactions) is the connection between the anterior ectosylvian sulcus (AES), which lies at the junction of the parietal, temporal and frontal lobes, and the SC. The AES is divided into three unimodal regions with multisensory neurons at the junctions between these sections. (Jiang & Stein, 2003). Neurons from the unimodal regions project to the deep layers of the SC and influence the multiplicative integration effect. That is, although they can receive inputs from all modalities as normal, the SC can not enhance or depress the effect of multisensory stimulation without input from the AES.
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A mid- squares hash code is produced by squaring the input and extracting an appropriate number of middle digits or bits. For example, if the input is 123,456,789 and the hash table size 10,000, squaring the key produces 1.524157875019e16, so the hash code is taken as the middle 4 digits of the 17-digit number (ignoring the high digit) 8750. The mid-squares method produces a reasonable hash code if there are not a lot of leading or trailing zeros in the key. This is a variant of multiplicative hashing, but not as good, because an arbitrary key is not a good multiplier.
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The third development is a higher adelic study of relations between the arithmetic and analytic ranks of an elliptic curve over a global field, which in conjectural form are stated in the Birch and Swinnerton-Dyer conjecture for the zeta function of elliptic surfaces. This new method uses FIT theory, two adelic structures: the geometric additive adelic structure and the arithmetic multiplicative adelic structure and an interplay between them motivated by higher class field theory. These two adelic structures have some similarity to two symmetries in inter-universal Teichmüller theory of Mochizuki. His contributions include his analysis of class field theories and their main generalizations.
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Anatoli (or Anatoly) Nikolaievich Andrianov (Анатолий Николаевич Андрианов, born 21 July 1936) is а Russian mathematician. Andrianov received in 1962 his Ph.D. under Yuri Linnik at the Leningrad State University with thesis Investigation of quadratic forms by methods of the theory of correspondences and in 1969 his Russian doctorate of sciences (Doctor Nauk). A Community of Scholars, Institute for Advanced Study, Faculty and Members 1930–1980 He is a professor at the Steklov Institute in Saint Petersburg. His research deals with the multiplicative arithmetic of quadratic forms, zeta functions of automorphic forms, modular forms in several variables (such as Siegel modular forms, Hecke operators, spherical functions, and theta functions).
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The axioms of a field imply that it is an abelian group under addition. This group is called the additive group of the field, and is sometimes denoted by when denoting it simply as could be confusing. Similarly, the nonzero elements of form an abelian group under multiplication, called the multiplicative group, and denoted by or just } or . A field may thus be defined as set equipped with two operations denoted as an addition and a multiplication such that is an abelian group under addition, } is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication is distributive over addition.
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Dirac's theory has inspired and continues to inspire a significant body of scientific literature in a variety of disciplines. In the context of geophysics, for instance, Edward Teller seemed to raise a serious objection to LNH in 1948 when he argued that variations in the strength of gravity are not consistent with paleontological data. However, George Gamow demonstrated in 1962 how a simple revision of the parameters (in this case, the age of the Solar System) can invalidate Teller's conclusions. The debate is further complicated by the choice of LNH cosmologies: In 1978, G. Blake argued that paleontological data is consistent with the 'multiplicative' scenario but not the 'additive' scenario.
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Let be a prime power, and be the splitting field of the polynomial :P=X^q-X over the prime field . This means that is a finite field of lowest order, in which has distinct roots (the formal derivative of is , implying that , which in general implies that the splitting field is a separable extension of the original). The above identity shows that the sum and the product of two roots of are roots of , as well as the multiplicative inverse of a root of . In other words, the roots of form a field of order , which is equal to by the minimality of the splitting field.
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The existence of multiplicative inverses in fields is not involved in the axioms defining a vector space. One may thus replace the field of scalars by a ring , and this gives a structure called module over , or -module. The concepts of linear independence, span, basis, and linear maps (also called module homomorphisms) are defined for modules exactly as for vector spaces, with the essential difference that, if is not a field, there are modules that do not have any basis. The modules that have a basis are the free modules, and those that are spanned by a finite set are the finitely generated modules.
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Intuitively, partial function application says "if you fix the first argument of the function, you get a function of the remaining arguments". For example, if function div stands for the division operation x/y, then div with the parameter x fixed at 1 (i.e., div 1) is another function: the same as the function inv that returns the multiplicative inverse of its argument, defined by inv(y) = 1/y. The practical motivation for partial application is that very often the functions obtained by supplying some but not all of the arguments to a function are useful; for example, many languages have a function or operator similar to `plus_one`.
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Multiplicative number theory deals primarily in asymptotic estimates for arithmetic functions. Historically the subject has been dominated by the prime number theorem, first by attempts to prove it and then by improvements in the error term. The Dirichlet divisor problem that estimates the average order of the divisor function d(n) and Gauss's circle problem that estimates the average order of the number of representations of a number as a sum of two squares are also classical problems, and again the focus is on improving the error estimates. The distribution of primes numbers among residue classes modulo an integer is an area of active research.
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In computational number theory and computational algebra, Pollard's kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist J. M. Pollard, in the same paper J. Pollard, Monte Carlo methods for index computation (mod p), Mathematics of Computation, Volume 32, 1978 as his better-known Pollard's rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime p, it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group.
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However, its realism as a description of how humans might actually provide best and worst data can be questioned for the following reason. As the number of items increases, the number of possible pairs increases in a multiplicative fashion: n items produces n(n-1) pairs (where best-worst order matters). To assume that respondents do evaluate all possible pairs is a strong assumption and in 14 years of presentations, the three co-authors have virtually never found a course or conference participant who admitted to using this method to decide their best and worst choices. Virtually all admitted to using sequential models (best then worst or worst then best).
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The short course prevention factor (SCPF) is a multiplicative factor or coefficient used in the sport of athletics, specifically road running, to ensure that the measured length of a course is at least as long as the desired length of the course. World Athletics, the international governing body for athletics, as well as USA Track & Field, the national governing body for the United States, specify the numerical factor of the SCPF to be 1.001. The SCPF has important implications when verifying world record performances. However, just because the SCPF has been added to the measurement calculation, that does not mean that a 10k course is 10,010 meters in length.
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In field theory, a primitive element of a finite field is a generator of the multiplicative group of the field. In other words, is called a primitive element if it is a primitive th root of unity in ; this means that each non- zero element of can be written as for some integer . If is a prime number, the elements of can be identified with the integers modulo . In this case, a primitive element is also called a primitive root modulo For example, 2 is a primitive element of the field and , but not of since it generates the cyclic subgroup of order 3; however, 3 is a primitive element of .
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While approximation algorithms always provide an a priori worst case guarantee (be it additive or multiplicative), in some cases they also provide an a posteriori guarantee that is often much better. This is often the case for algorithms that work by solving a convex relaxation of the optimization problem on the given input. For example, there is a different approximation algorithm for minimum vertex cover that solves a linear programming relaxation to find a vertex cover that is at most twice the value of the relaxation. Since the value of the relaxation is never larger than the size of the optimal vertex cover, this yields another 2-approximation algorithm.
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Intuitively, partial function application says "if you fix the first arguments of the function, you get a function of the remaining arguments". For example, if function div(x,y) = x/y, then div with the parameter x fixed at 1 is another function: div1(y) = div(1,y) = 1/y. This is the same as the function inv that returns the multiplicative inverse of its argument, defined by inv(y) = 1/y. The practical motivation for partial application is that very often the functions obtained by supplying some but not all of the arguments to a function are useful; for example, many languages have a function or operator similar to `plus_one`.
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A primitive normal basis of an extension of finite fields E/F is a normal basis for E/F that is generated by a primitive element of E, that is a generator of the multiplicative group K^\times. (Note that this is a more restrictive definition of primitive element than that mentioned above after the general Normal Basis Theorem: one requires powers of the element to produce every non-zero element of K, not merely a basis.) Lenstra and Schoof (1987) proved that every finite field extension possesses a primitive normal basis, the case when F is a prime field having been settled by Harold Davenport.
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The above chart shows an example application of the DKW inequality in constructing confidence bounds (in purple) around an empirical distribution function (in light blue). In this random draw, the true CDF (orange) is entirely contained within the DKW bounds. In the theory of probability and statistics, the Dvoretzky–Kiefer–Wolfowitz inequality bounds how close an empirically determined distribution function will be to the distribution function from which the empirical samples are drawn. It is named after Aryeh Dvoretzky, Jack Kiefer, and Jacob Wolfowitz, who in 1956 proved the inequality with an unspecified multiplicative constant C in front of the exponent on the right-hand side.
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In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent parts. The multiplicative decomposition expresses an invertible operator as the product of its commuting semisimple and unipotent parts. The decomposition is easy to describe when the Jordan normal form of the operator is given, but it exists under weaker hypotheses than the existence of a Jordan normal form. Analogues of the Jordan-Chevalley decomposition exist for elements of linear algebraic groups, Lie algebras, and Lie groups, and the decomposition is an important tool in the study of these objects.
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The basic algebraic operations for real interval numbers (real closed intervals) can be extended to complex numbers. It is therefore not surprising that complex interval arithmetic is similar to, but not the same as, ordinary complex arithmetic. It can be shown that, as it is the case with real interval arithmetic, there is no distributivity between addition and multiplication of complex interval numbers except for certain special cases, and inverse elements do not always exist for complex interval numbers. Two other useful properties of ordinary complex arithmetic fail to hold in complex interval arithmetic: the additive and multiplicative properties, of ordinary complex conjugates, do not hold for complex interval conjugates.
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Vectorial ptychography needs to be invoked when the multiplicative model of the interaction between the probe and the specimen cannot be described by scalar quantities. This happens typically when polarized light probes an anisotropic specimen, and when this interaction modifies the state of polarization of light. In that case, the interaction needs to be described by the Jones formalism, where field and object are described by a two-component complex vector and a 2×2 complex matrix, respectively. The optical configuration for vectorial ptychography is similar to those of classical (scalar) ptychography, although a control of light polarization (before and after the specimen) needs to be implemented in the setup.
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The powers of ω function is also an exponential function, but does not have the properties desired for an extension of the function on the reals. It will, however, be needed in the development of the base-e exponential, and it is this function that is meant whenever the notation ωx is used in the following. When y is a dyadic fraction, the power function , may be composed from multiplication, multiplicative inverse and square root, all of which can be defined inductively. Its values are completely determined by the basic relation and where defined it necessarily agrees with any other exponentiation that can exist.
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Wraith had become interested in these structures (which he initially dubbed sequentials) while at school. Conway renamed them wracks, partly as a pun on his colleague's name, and partly because they arise as the remnants (or 'wrack and ruin') of a group when one discards the multiplicative structure and considers only the conjugation structure. The spelling 'rack' has now become prevalent. These constructs surfaced again in the 1980s: in a 1982 paper by David Joyce (where the term quandle was coined), in a 1982 paper by Sergei Matveev (under the name distributive groupoids) and in a 1986 conference paper by Egbert Brieskorn (where they were called automorphic sets).
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The supergravity theories that have attracted the most interest contain no spins higher than two. This means, in particular, that they do not contain any fields that transform as symmetric tensors of rank higher than two under Lorentz transformations. The consistency of interacting higher spin field theories is, however, presently a field of very active interest. The supercharges in every super-Poincaré algebra are generated by a multiplicative basis of m fundamental supercharges, and an additive basis of the supercharges (this definition of supercharges is a bit more broad than that given above) is given by a product of any subset of these m fundamental supercharges.
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For example, the result of 5 × 7 = 35 mod 13 = 9\. Such finite fields can be defined for any prime p; using more sophisticated definitions, they can also be defined for any power m of a prime p m. Finite fields are often called Galois fields, and are abbreviated as GF(p) or GF(p m). In such a field with m numbers, every nonzero element a has a unique modular multiplicative inverse, a−1 such that This inverse can be found by solving the congruence equation ax ≡ 1 mod m, or the equivalent linear Diophantine equation : This equation can be solved by the Euclidean algorithm, as described above.
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Suppose that G is a profinite group acting on a module A with a surjective homomorphism π from the G-module A to itself. Suppose also that G acts trivially on the kernel C of π and that the first cohomology group H1(G,A) is trivial. Then the exact sequence of group cohomology shows that there is an isomorphism between AG/π(AG) and Hom(G,C). Kummer theory is the special case of this when A is the multiplicative group of the separable closure of a field k, G is the Galois group, π is the nth power map, and C the group of nth roots of unity.
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The goal is to construct the right ring of fractions R[S−1] with respect to multiplicative subset S. In other words, we want to work with elements of the form as−1 and have a ring structure on the set R[S−1]. The problem is that there is no obvious interpretation of the product (as−1)(bt−1); indeed, we need a method to "move" s−1 past b. This means that we need to be able to rewrite s−1b as a product b1s1−1. Suppose then multiplying on the left by s and on the right by s1, we get .
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For any e, a multiple of and any a relatively prime to p, by Fermat's little theorem we have . Then is likely to produce a factor of n. However, the algorithm fails when has large prime factors, as is the case for numbers containing strong primes, for example. ECM gets around this obstacle by considering the group of a random elliptic curve over the finite field Zp, rather than considering the multiplicative group of Zp which always has order The order of the group of an elliptic curve over Zp varies (quite randomly) between and by Hasse's theorem, and is likely to be smooth for some elliptic curves.
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Hence, the resulting throughput of this distributed implementation is optimal over the class of all routing and scheduling algorithms that use such randomized transmissions. Alternative distributed implementations can roughly be grouped into two classes: The first class of algorithms consider constant multiplicative factor approximations to the max-weight problem, and yield constant-factor throughput results. The second class of algorithms consider additive approximations to the max-weight problem, based on updating solutions to the max-weight problem over time. Algorithms in this second class seem to require static channel conditions and longer (often non-polynomial) convergence times, although they can provably achieve maximum throughput under appropriate assumptions.
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The same function may be an invariant for one group of symmetries and equivariant for a different group of symmetries. For instance, under similarity transformations instead of congruences the area and perimeter are no longer invariant: scaling a triangle also changes its area and perimeter. However, these changes happen in a predictable way: if a triangle is scaled by a factor of , the perimeter also scales by and the area scales by . In this way, the function mapping each triangle to its area or perimeter can be seen as equivariant for a multiplicative group action of the scaling transformations on the positive real numbers.
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In the standard model, the Higgs field is an SU(2) doublet (i.e. the standard representation with two complex components called isospin), which is a scalar under Lorentz transformations. Its electric charge is zero; its weak isospin is and the third component of weak isospin is -(1/2 ) ; its weak hypercharge (the charge for the U(1) gauge group defined up to an arbitrary multiplicative constant) is 1. Under U(1) rotations, it is multiplied by a phase, which thus mixes the real and imaginary parts of the complex spinor into each other, combining to the standard two-component complex representation of the group U(2).
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Any element g of a linear algebraic group over a perfect field can be written uniquely as the product g = gugs of commuting unipotent and semisimple elements gu and gs. In the case of the group GLn(C), this essentially says that any invertible complex matrix is conjugate to the product of a diagonal matrix and an upper triangular one, which is (more or less) the multiplicative version of the Jordan–Chevalley decomposition. There is also a version of the Jordan decomposition for groups: any commutative linear algebraic group over a perfect field is the product of a unipotent group and a semisimple group.
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Modules are to rings what vector spaces are to fields: the same axioms, applied to a ring R instead of a field F, yield modules. The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases, as the Z-module (that is, abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field, with the elements being called vectors.
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Raman Tool Set is a free software package for processing and analysis of Raman spectroscopy datasets. It has been developed mainly aiming to Raman spectra analysis, but since it works with 2-columns datafiles (Intensity vs Frequency) it can deal with the results of many spectroscopy techniques. Beyond the spectra preprocessing steps, such as baseline subtraction, normalization of spectra, smoothing and scaling, Raman Tool Set allows the user for chemometric analysis by means of principal component analysis (PCA), extended multiplicative signal correction (EMSC) and cluster analysis. Chemometric and multivariate data analysis can also be applied to hyperspectral maps, using PCA, independent component analysis (ICA) and cluster analysis.
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The Reichardt model is a more complex form of the simplest Hassenstein–Reichardt detector model, which is considered to be a pairwise model with a common quadratic nonlinearity. As Fourier method is considered to be linear method, Reichardt Model introduces multiplicative nonlinearity when our visual responses to luminance changes at different element locations are combined. In this model, one photoreceptor input would be delayed by a filter to be compared by the multiplication with the other input from a neighboring location. The input would be filtered two times in a mirror-symmetrical manner, one before the multiplication and one after the multiplication, which gives a second-order motion estimation.
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Donald Knuth, who uses decimal notation like 1 MB = 1000 kB,The Art of Computer Programming Volume 1, Donald Knuth, pp. 24 and 94 expressed "astonishment" that the IEC proposal was adopted, calling them "funny-sounding" and opining that proponents were assuming "that standards are automatically adopted just because they are there." Knuth proposed that the powers of 1024 be designated as "large kilobytes" and "large megabytes" (abbreviated KKB and MMB, as "doubling the letter connotes both binary-ness and large-ness"). Double prefixes were already abolished from SI, however, having a multiplicative meaning ("MMB" would be equivalent to "TB"), and this proposed usage never gained any traction.
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He investigated the completeness of systems of own elements of one class of not self-interfaced operators, polynomially – and rationally – depending on parameter. His theorems of completeness and basis were proven, is given definitions of the best approach of linear operators finite-dimensional operators, its exact expression is found, given definitions of repeated completeness of systems of elements in linear spaces and many thorough theorems in this direction are proved. Theorems of new type of completeness and basis systems of own and attached elements in Banach and Hilbert spaces are proved. Necessary and sufficient conditions closure, limitations and quite continuity of operators in multiplicative terms are found.
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Given a group G and a subgroup H, and an element a ∈ G, one can consider the corresponding left coset: aH := { ah : h ∈ H }. Cosets are a natural class of subsets of a group; for example consider the abelian group G of integers, with operation defined by the usual addition, and the subgroup H of even integers. Then there are exactly two cosets: 0 + H, which are the even integers, and 1 + H, which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation). For a general subgroup H, it is desirable to define a compatible group operation on the set of all possible cosets, { aH : a ∈ G }.
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The cosets of the fourth roots of unity N in the twelfth roots of unity G. The twelfth roots of unity, which are points on the complex unit circle, form a multiplicative abelian group G, shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup N made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue. One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.).
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The underlying set F may not be required to be a field but instead allowed to simply be a ring, R, and concurrently the exponential function is relaxed to be a homomorphism from the additive group in R to the multiplicative group of units in R. The resulting object is called an exponential ring. An example of an exponential ring with a nontrivial exponential function is the ring of integers Z equipped with the function E which takes the value +1 at even integers and −1 at odd integers, i.e., the function n \mapsto (-1)^n. This exponential function, and the trivial one, are the only two functions on Z that satisfy the conditions.
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One of the axioms defining a group is the identity m(x, i(x)) = e; another is m(x,e) = x. The axioms can be represented as trees. These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group. Some structures do not form varieties, because either: # It is necessary that 0 ≠ 1, 0 being the additive identity element and 1 being a multiplicative identity element, but this is a nonidentity; # Structures such as fields have some axioms that hold only for nonzero members of S. For an algebraic structure to be a variety, its operations must be defined for all members of S; there can be no partial operations.
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In contrast to the more structured approach of most open-world games, Breath of the Wild features a large and fully interactive world that is generally unstructured and rewards the exploration and manipulation of its world. Inspired by the original 1986 Legend of Zelda, the open world of Breath of the Wild integrates multiplicative gameplay, where "objects react to the player's actions and the objects themselves also influence each other." Along with a physics engine, the game's open-world also integrates a chemistry engine, "which governs the physical properties of certain objects and how they relate to each other," rewarding experimentation. Nintendo has described the game's approach to open- world design as "open-air".
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Using so-called multifractal formalism, it can be shown that, under some well- suited assumptions, there exists a correspondence between the singularity spectrum D(h) and the multi-scaling exponents \zeta(q) through a Legendre transform. While the determination of D(h) calls for some exhaustive local analysis of the data, which would result in difficult and numerically unstable calculations, the estimation of the \zeta(q) relies on the use of statistical averages and linear regressions in log-log diagrams. Once the \zeta(q) are known, one can deduce an estimate of D(h), thanks to a simple Legendre transform. Multifractal systems are often modeled by stochastic processes such as multiplicative cascades.
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In the literature, an approximation ratio for a maximization (minimization) problem of c - ϵ (min: c + ϵ) means that the algorithm has an approximation ratio of c ∓ ϵ for arbitrary ϵ > 0 but that the ratio has not (or cannot) be shown for ϵ = 0. An example of this is the optimal inapproximability — inexistence of approximation — ratio of 7 / 8 + ϵ for satisfiable MAX-3SAT instances due to Johan Håstad. As mentioned previously, when c = 1, the problem is said to have a polynomial-time approximation scheme. An ϵ-term may appear when an approximation algorithm introduces a multiplicative error and a constant error while the minimum optimum of instances of size n goes to infinity as n does.
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Quasi- likelihood models can be fitted using a straightforward extension of the algorithms used to fit generalized linear models. Instead of specifying a probability distribution for the data, only a relationship between the mean and the variance is specified in the form of a variance function giving the variance as a function of the mean. Generally, this function is allowed to include a multiplicative factor known as the overdispersion parameter or scale parameter that is estimated from the data. Most commonly, the variance function is of a form such that fixing the overdispersion parameter at unity results in the variance-mean relationship of an actual probability distribution such as the binomial or Poisson.
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Some researchers have preferred to partition gamma diversity into additive rather than multiplicative components. Then the beta component of diversity becomes βA = γ - α This quantifies how much more species diversity the entire dataset contains than an average subunit within the dataset. This can also be interpreted as the total amount of species turnover among the subunits in the dataset. When there are two subunits, and presence-absence data are used, this can be calculated with the following equation: \beta_A=(S_1-c)+(S_2-c) where, S1= the total number of species recorded in the first community, S2= the total number of species recorded in the second community, and c= the number of species common to both communities.
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Multiplying a number by −1 is equivalent to changing the sign on the number. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity: for x real, we have :x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0 \cdot x=0 where we used the fact that any real x times 0 equals 0, implied by cancellation from the equation :0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x \, i, and −i in the complex or cartesian plane In other words, :x+(-1)\cdot x=0 \, so (−1) · x, or −x, is the arithmetic inverse of x.
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The conjecture is studied in the more general context of graph homomorphisms, especially because of interesting relations to the category of graphs (with graphs as objects and homomorphisms as arrows). For any fixed graph K, one considers graphs G that admit a homomorphism to K, written G → K. These are also called K-colorable graphs. This generalizes the usual notion of graph coloring, since it follows from definitions that a k-coloring is the same as a Kk-coloring (a homomorphism into the complete graph on k vertices). A graph K is called multiplicative if for any graphs G, H, the fact that G × H → K holds implies that G → K or H → K holds.
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In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor. For example, a homogeneous real- valued function of two variables x and y is a real-valued function that satisfies the condition f(\alpha x,\alpha y)=\alpha^k f(x,y) for some constant k and all real numbers α. The constant k is called the degree of homogeneity. More generally, if is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if for all nonzero and .
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It is a multiplicative model that estimates an excess risk per exposure unit. It takes into account age, elapsed time since exposure, and duration and length of exposure, and its parameters allow for taking smoking habits into account. In the absence of other causes of death, the absolute risks of lung cancer by age 75 at usual radon concentrations of 0, 100, and 400 Bq/m3 would be about 0.4%, 0.5%, and 0.7%, respectively, for lifelong nonsmokers, and about 25 times greater (10%, 12%, and 16%) for cigarette smokers. There is great uncertainty in applying risk estimates derived from studies in miners to the effects of residential radon, and direct estimates of the risks of residential radon are needed.
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The above result can be restated in a different way. For a semisimple algebra A = A1 ×...× An expressed in terms of its simple factors, consider the units ei ∈ Ai. The elements Ei = (0,...,ei,...,0) are idempotent elements in A and they lie in the center of A. Furthermore, Ei A = Ai, EiEj = 0 for i ≠ j, and Σ Ei = 1, the multiplicative identity in A. Therefore, for every semisimple algebra A, there exists idempotents {Ei} in the center of A, such that #EiEj = 0 for i ≠ j (such a set of idempotents is called central orthogonal), #Σ Ei = 1, #A is isomorphic to the Cartesian product of simple algebras E1 A ×...× En A.
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Gauss, BQ § 8 In the vocabulary of group theory, the first set is a subgroup of index 4 (of the multiplicative group Z/pZ×), and the other three are its cosets. The first set is the biquadratic residues, the third set is the quadratic residues that are not quartic residues, and the second and fourth sets are the quadratic nonresidues. Gauss proved that −1 is a biquadratic residue if p ≡ 1 (mod 8) and a quadratic, but not biquadratic, residue, when p ≡ 5 (mod 8).Gauss, BQ § 10 2 is a quadratic residue mod p if and only if p ≡ ±1 (mod 8). Since p is also ≡ 1 (mod 4), this means p ≡ 1 (mod 8).
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In the book Arithmetices principia, nova methodo exposita, Giuseppe Peano proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication: :x \times 0 = 0 :x \times S(y) = (x \times y) + x Here S(y) represents the successor of y, or the natural number that follows y. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic including induction. For instance S(0), denoted by 1, is a multiplicative identity because :x \times 1 = x \times S(0) = (x \times 0) + x = 0 + x = x The axioms for integers typically define them as equivalence classes of ordered pairs of natural numbers.
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The extended Euclidean algorithm is also the main tool for computing multiplicative inverses in simple algebraic field extensions. An important case, widely used in cryptography and coding theory, is that of finite fields of non-prime order. In fact, if is a prime number, and , the field of order is a simple algebraic extension of the prime field of elements, generated by a root of an irreducible polynomial of degree . A simple algebraic extension of a field , generated by the root of an irreducible polynomial of degree may be identified to the quotient ring K[X]/\langle p\rangle,, and its elements are in bijective correspondence with the polynomials of degree less than .
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However, in 2002 Sapir and Olshanskii found finitely presented counterexamples: non-amenable finitely presented groups that have a periodic normal subgroup with quotient the integers. For finitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative: every subgroup of GL(n,k) with k a field either has a normal solvable subgroup of finite index (and therefore is amenable) or contains the free group on two generators. Although Tits' proof used algebraic geometry, Guivarc'h later found an analytic proof based on V. Oseledets' multiplicative ergodic theorem. Analogues of the Tits alternative have been proved for many other classes of groups, such as fundamental groups of 2-dimensional simplicial complexes of non-positive curvature.
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The setup, especially the Nisnevich topology, is chosen as to make algebraic K-theory representable by a spectrum, and in some aspects to make a proof of the Bloch-Kato conjecture possible. After the Morel-Voevodsky construction there have been several different approaches to homotopy theory by using other model category structures or by using other sheaves than Nisnevich sheaves (for example, Zariski sheaves or just all presheaves). Each of these constructions yields the same homotopy category. There are two kinds of spheres in the theory: those coming from the multiplicative group playing the role of the -sphere in topology, and those coming from the simplicial sphere (considered as constant simplicial sheaf).
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For every positive integer n, the set of the integers modulo n that are relatively prime to n is written as (Z/nZ)×; it forms a group under the operation of multiplication. This group is not always cyclic, but is so whenever n is 1, 2, 4, a power of an odd prime, or twice a power of an odd prime ... This is the multiplicative group of units of the ring Z/nZ; there are φ(n) of them, where again φ is the Euler totient function. For example, (Z/6Z)× = {1,5}, and since 6 is twice an odd prime this is a cyclic group. In contrast, (Z/8Z)× = {1,3,5,7} is a Klein 4-group and is not cyclic.
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An alternative arrangement is to keep generations distributed across all points in time is by deleting (or overwriting) past generations (except the oldest and the most-recent-n generations) when necessary in a weighted-random fashion. For each deletion, the weight assigned to each deletable generation corresponds to the probability of it being deleted. One acceptable weight is a constant exponent (possibly the square) of the multiplicative inverse of the duration (possibly expressed in the number of days) between the dates of the generation and the generation preceding it. Using a larger exponent leads to a more uniform distribution of generations, whereas a smaller exponent leads to a distribution with more recent and fewer older generations.
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Countless results in number theory invoke the fundamental theorem of arithmetic and the algebraic properties of even numbers, so the above choices have far-reaching consequences. For example, the fact that positive numbers have unique factorizations means that one can determine whether a number has an even or odd number of distinct prime factors. Since 1 is not prime, nor does it have prime factors, it is a product of 0 distinct primes; since 0 is an even number, 1 has an even number of distinct prime factors. This implies that the Möbius function takes the value , which is necessary for it to be a multiplicative function and for the Möbius inversion formula to work.
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The Diffie–Hellman problem is stated informally as follows: : Given an element g and the values of gx and gy, what is the value of gxy? Formally, g is a generator of some group (typically the multiplicative group of a finite field or an elliptic curve group) and x and y are randomly chosen integers. For example, in the Diffie–Hellman key exchange, an eavesdropper observes gx and gy exchanged as part of the protocol, and the two parties both compute the shared key gxy. A fast means of solving the DHP would allow an eavesdropper to violate the privacy of the Diffie–Hellman key exchange and many of its variants, including ElGamal encryption.
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There were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry. Leonhard Euler considered algebraic operations on numbers modulo an integer—modular arithmetic—in his generalization of Fermat's little theorem. These investigations were taken much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law for the composition of forms, but like Euler before him, he seems to have been more interested in concrete results than in general theory.
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When experiment participants were verbally presented with certain simple concepts, the processing of the information causes electrical stimulation in the region. When the same participants were verbally presented with a single complex concept formed from the combination of the aforementioned simple concepts, the stimulation recorded was equivalent to the sum of the stimulation that resulted from each individual component simple concept. In other words, the stimulation caused by a complex concept is equivalent to the total stimulation caused by its component concepts. More recent data contradicts those results by indicating a multiplicative effect in which the activation caused by a complex concept is the product of the activation levels caused by its component concepts, rather than the sum.
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One form of TFR (or TFD) can be formulated by the multiplicative comparison of a signal with itself, expanded in different directions about each point in time. Such representations and formulations are known as quadratic or "bilinear" TFRs or TFDs (QTFRs or QTFDs) because the representation is quadratic in the signal (see Bilinear time–frequency distribution). This formulation was first described by Eugene Wigner in 1932 in the context of quantum mechanics and, later, reformulated as a general TFR by Ville in 1948 to form what is now known as the Wigner–Ville distribution, as it was shown in B. Boashash, "Note on the use of the Wigner distribution for time frequency signal analysis", IEEE Trans. on Acoust. Speech.
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Building Emissions Study accessed at California Integrated Waste Management web site Volatile organic compounds (VOC) can be found in any indoor environment coming from a variety of different sources. VOCs have a high vapor pressure and low water solubility, and are suspected of causing sick building syndrome type symptoms. This is because many VOCs have been known to cause sensory irritation and central nervous system symptoms characteristic to sick building syndrome, indoor concentrations of VOCs are higher than in the outdoor atmosphere, and when there are many VOCs present, they can cause additive and multiplicative effects. Green products are usually considered to contain fewer VOCs and be better for human and environmental health.
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The multiplicative weights update method is an algorithmic technique most commonly used for decision making and prediction, and also widely deployed in game theory and algorithm design. The simplest use case is the problem of prediction from expert advice, in which a decision maker needs to iteratively decide on an expert whose advice to follow. The method assigns initial weights to the experts (usually identical initial weights), and updates these weights multiplicatively and iteratively according to the feedback of how well an expert performed: reducing it in case of poor performance, and increasing it otherwise. It was discovered repeatedly in very diverse fields such as machine learning (AdaBoost, Winnow, Hedge), optimization (solving linear programs), theoretical computer science (devising fast algorithm for LPs and SDPs), and game theory.
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A given set of parameters describes a single distribution within the family sharing the functional form of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables, are part of the normalization factor of a distribution (the multiplicative factor that ensures that the area under the density—the probability of something in the domain occurring— equals 1). This normalization factor is outside the kernel of the distribution. Since the parameters are constants, reparametrizing a density in terms of different parameters, to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones.
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In most people, the substance increases self-confidence, concentration, and willingness to take risks while at the same time reducing sensitivity to pain, hunger, and the need for sleep. In September 1939, Ranke tested the drug on 90 university students and concluded that Pervitin could help the Wehrmacht win the war. Cocaine, whose effects substantially overlap with those of amphetamine but feature greater euphoria, was later added to the formulation to increase its potency through the multiplicative effects of drug interaction and to reinforce its use by individuals. Medical authorities described this plan, under which the distributed pills numbered in the millions, as having the negative consequence that many soldiers became addicted to drugs and useless in any military capacity, whether combat or supporting.
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It is common to draw only the black points of a clean dessin and to leave the white points unmarked; one can recover the full dessin by adding a white point at the midpoint of each edge of the map. Thus, any embedding of a graph in a surface in which each face is a disk (that is, a topological map) gives rise to a dessin by treating the graph vertices as black points of a dessin, and placing white points at the midpoint of each embedded graph edge. If a map corresponds to a Belyi function f, its dual map (the dessin formed from the preimages of the line segment [1, ∞]) corresponds to the multiplicative inverse ., pp. 120–121.
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Under the recent unique games conjecture, this factor is even the best possible one. NP- hard problems vary greatly in their approximability; some, such as the knapsack problem, can be approximated within a multiplicative factor 1 + \epsilon, for any fixed \epsilon > 0, and therefore produce solutions arbitrarily close to the optimum (such a family of approximation algorithms is called a polynomial time approximation scheme or PTAS). Others are impossible to approximate within any constant, or even polynomial, factor unless P = NP, as in the case of the maximum clique problem. Therefore, an important benefit of studying approximation algorithms is a fine-grained classification of the difficulty of various NP-hard problems beyond the one afforded by the theory of NP-completeness.
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A Hall divisor(also called a unitary divisor) of an integer n is a divisor d of n such that d and n/d are coprime. The easiest way to find the Hall divisors is to write the prime factorization for the number in question and take any product of the multiplicative terms (the full power of any of the prime factors), including 0 of them for a product of 1 or all of them for a product equal to the original number. For example, to find the Hall divisors of 60, show the prime factorization is 22·3·5 and take any product of {3,4,5}. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60.
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More recently, a series of papers by Getty shows that Grafen's analysis, like that of Spence, is based on the critical simplifying assumption that signalers trade off costs for benefits in an additive fashion, the way humans invest money to increase income in the same currency. This assumption that costs and benefits trade off in an additive fashion might be valid for some biological signaling systems, but is not valid for multiplicative tradeoffs, such as the survival cost – reproduction benefit tradeoff that is assumed to mediate the evolution of sexually selected signals. Charles Godfray (1991) modeled the begging behavior of nestling birds as a signaling game. The nestlings begging not only informs the parents that the nestling is hungry, but also attracts predators to the nest.
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The orthogonal group of all orthogonal real matrices (intuitively the set of all rotations and reflections of -dimensional space that keep the origin fixed) is isomorphic to a semidirect product of the group (consisting of all orthogonal matrices with determinant , intuitively the rotations of -dimensional space) and . If we represent as the multiplicative group of matrices }, where is a reflection of -dimensional space that keeps the origin fixed (i.e., an orthogonal matrix with determinant representing an involution), then is given by for all H in and in . In the non-trivial case ( is not the identity) this means that is conjugation of operations by the reflection (a rotation axis and the direction of rotation are replaced by their "mirror image").
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Every Boolean algebra (A, ∧, ∨) gives rise to a ring (A, +, ·) by defining a + b := (a ∧ ¬b) ∨ (b ∧ ¬a) = (a ∨ b) ∧ ¬(a ∧ b) (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a · a = a for all a in A; rings with this property are called Boolean rings. Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + (x · y) and x ∧ y := x · y.
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The Tsirelson space is reflexive () and finitely universal, which means that for some constant , the space contains -isomorphic copies of every finite-dimensional normed space, namely, for every finite-dimensional normed space , there exists a subspace of the Tsirelson space with multiplicative Banach-Mazur distance to less than . Actually, every finitely universal Banach space contains almost-isometric copies of every finite- dimensional normed space,this is because for every , and ε, there exists such that every -isomorph of ℓ∞ contains a -isomorph of ℓ∞n, by James' blocking technique (see Lemma 2.2 in Robert C. James "Uniformly Non-Square Banach Spaces", Annals of Mathematics, Vol. 80, 1964, pp. 542-550), and because every finite-dimensional normed space -embeds in ℓ∞ when is large enough.
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In most people, the substance increases self-confidence, concentration, and willingness to take risks while at the same time reducing sensitivity to pain, hunger, thirst, and the need for sleep. In September 1939, Ranke tested the drug on 90 university students and concluded that Pervitin could help the Wehrmacht win the war. Cocaine, whose effects substantially overlap with those of amphetamine but feature greater euphoria, was later added to the formulation to increase potency through the multiplicative effects of drug interaction and to reinforce use by individuals. Some medical authorities criticized this drug regimen, with distributed pills numbered in the millions, as having the negative consequence that many soldiers became addicted to drugs and useless in any military capacity, whether combat or supporting.
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The Chinese City Creativity Index is based on the theoretical methods related to Michael Porter's diamond Model, Systems Theory, etc. A Chinese City Creativity Index model is established, which consists of 4 modules including factor driving force, demand pull force, relevant support force and industrial influence force, 9 secondary indexes and 18 tertiary indexes. The model takes into account not only the promoting effect of various resources such as talents, funds, technologies and culture, but also the pulling function of cultural needs and consumption potentials as well as the supporting role of relevant industries such as communication and network. Moreover, all the indexes adopt the form of both absolute value and relative value, and the index summarization method uses a multiplicative model rather than an additive model.
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The nimber multiplicative inverse of the nonzero ordinal is given by , where is the smallest set of ordinals (nimbers) such that # 0 is an element of ; # if and is an element of , then is also an element of . For all natural numbers , the set of nimbers less than form the Galois field of order . In particular, this implies that the set of finite nimbers is isomorphic to the direct limit as of the fields . This subfield is not algebraically closed, since no other field (so with not a power of 2) is contained in any of those fields, and therefore not in their direct limit; for instance the polynomial , which has a root in , does not have a root in the set of finite nimbers.
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The p-adic cyclotomic character is the p-adic Tate module of the multiplicative group scheme Gm,Q over Q. As such, its representation space can be viewed as the inverse limit of the groups of pnth roots of unity in . In terms of cohomology, the p-adic cyclotomic character is the dual of the first p-adic étale cohomology group of Gm. It can also be found in the étale cohomology of a projective variety, namely the projective line: it is the dual of H2ét( P1 ). In terms of motives, the p-adic cyclotomic character is the p-adic realization of the Tate motive Z(1). As a Grothendieck motive, the Tate motive is the dual of H2( P1 ).
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Early work centered on the discovery and elucidation of cortical gain fields, a general rule of multiplicative computation used by many areas of the cortex.Andersen RA, Mountcastle VB (1983) The influence of the angle of gaze upon the excitability of the light-sensitive neurons of the posterior parietal cortex. J Neurosci 3:532–548Andersen RA, Essick GK, Siegel RM (1985) The encoding of spatial location by posterior parietal neurons. Science 230:456–458 Andersen and Zipser of UCSD developed one of the first neural network models of cortical function, which generated a mathematical basis for testing hypotheses based on laboratory findings.Zipser D, Andersen RA (1988) A back-propagation programmed network that simulates response properties of a subset of posterior parietal neurons.
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It has been argued that research endeavours working within the conventional linear paradigm necessarily end up in replication difficulties. Problems arise if the causal processes in the system under study are "interaction-dominant" instead of "component dominant", multiplicative instead of additive, and with many small non-linear interactions producing macro-level phenomena, that are not reducible to their micro-level components. In the context of such complex systems, conventional linear models produce answers that are not reasonable, because it is not in principle possible to decompose the variance as suggested by the General Linear Model (GLM) framework – aiming to reproduce such a result is hence evidently problematic. The same questions are currently being asked in many fields of science, where researchers are starting to question assumptions underlying classical statistical methods.
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A more general framework where the term exponential polynomial may be found is that of exponential functions on abelian groups. Similarly to how exponential functions on exponential fields are defined, given a topological abelian group G a homomorphism from G to the additive group of the complex numbers is called an additive function, and a homomorphism to the multiplicative group of nonzero complex numbers is called an exponential function, or simply an exponential. A product of additive functions and exponentials is called an exponential monomial, and a linear combination of these is then an exponential polynomial on G.László Székelyhidi, On the extension of exponential polynomials, Mathematica Bohemica 125 (2000), pp.365-370.P. G. Laird, On characterizations of exponential polynomials, Pacific Journal of Mathematics 80 (1979), pp.503-507.
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Consider the simple set of simultaneous congruences : x ≡ 3 (mod 4) : x ≡ 5 (mod 6) Now, for x ≡ 3 (mod 4) to be true, x = 3 + 4j for some integer j. Substitute this in the second equation : 3+4j ≡ 5 (mod 6) since we are looking for a solution to both equations. Subtract 3 from both sides (this is permitted in modular arithmetic) : 4j ≡ 2 (mod 6) We simplify by dividing by the greatest common divisor of 4,2 and 6. Division by 2 yields: : 2j ≡ 1 (mod 3) The Euclidean modular multiplicative inverse of 2 mod 3 is 2. After multiplying both sides with the inverse, we obtain: : j ≡ 2 × 1 (mod 3) or : j ≡ 2 (mod 3) For the above to be true: j = 2 + 3k for some integer k.
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Fuller took an intuitive approach to his studies, often going into exhaustive empirical detail while at the same time seeking to cast his findings in their most general philosophical context. For example, his sphere packing studies led him to generalize a formula for polyhedral numbers: 2 P F2 \+ 2, where F stands for "frequency" (the number of intervals between balls along an edge) and P for a product of low order primes (some integer). He then related the "multiplicative 2" and "additive 2" in this formula to the convex versus concave aspects of shapes, and to their polar spinnability respectively. These same polyhedra, developed through sphere packing and related by tetrahedral mensuration, he then spun around their various poles to form great circle networks and corresponding triangular tiles on the surface of a sphere.
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For odd prime p, the p-adic Hamilton quaternions are isomorphic to the 2×2 matrices over the p-adics. To see the p-adic Hamilton quaternions are not a division algebra for odd prime p, observe that the congruence x2 \+ y2 = −1 mod p is solvable and therefore by Hensel's lemma — here is where p being odd is needed — the equation :x2 \+ y2 = −1 is solvable in the p-adic numbers. Therefore the quaternion :xi + yj + k has norm 0 and hence doesn't have a multiplicative inverse. One way to classify the F-algebra isomorphism classes of all quaternion algebras for a given field, F is to use the one-to-one correspondence between isomorphism classes of quaternion algebras over F and isomorphism classes of their norm forms.
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In abstract algebra, a Valya algebra (or Valentina algebra) is a nonassociative algebra M over a field F whose multiplicative binary operation g satisfies the following axioms: 1\. The skew-symmetry condition :g (A, B) =-g (B, A) for all A,B \in M. 2\. The Valya identity : J (g (A_1, A_2), g (A_3, A_4), g (A_5, A_6)) =0 for all A_k \in M, where k=1,2,...,6, and J (A, B, C):= g (g (A, B), C)+g (g (B, C), A)+g (g (C, A), B). 3\. The bilinear condition : g(aA+bB,C)=ag(A,C)+bg(B,C) for all A,B,C \in M and a,b \in F. We say that M is a Valya algebra if the commutant of this algebra is a Lie subalgebra.
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A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a semigroup rather than a monoid. Such structures are called hemiringsJonathan S. Golan, Semirings and their applications, Chapter 1, p1 or pre-semirings.Michel Gondran, Michel Minoux, Graphs, Dioids, and Semirings: New Models and Algorithms, Chapter 1, Section 4.2, p22 A further generalization are left-pre-semirings,Michel Gondran, Michel Minoux, Graphs, Dioids, and Semirings: New Models and Algorithms, Chapter 1, Section 4.1, p20 which additionally do not require right-distributivity (or right-pre- semirings, which do not require left-distributivity). Yet a further generalization are near-semirings: in addition to not requiring a neutral element for product, or right-distributivity (or left-distributivity), they do not require addition to be commutative.
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The negative value means that heat is produced and the system is exothermic. Endothermic: A + B + Heat → C, ΔH > 0 Exothermic: A + B → C + Heat, ΔH < 0 Since enthalpy is a state function, the ΔH given for a particular reaction is only true for that exact reaction. Physical states (of reactants or products) matter, as do molar concentrations. This matter of ΔH being dependent on physical state and molar concentration means that thermochemical equations must be stoichiometrically correct. If one agent of the equation is changed through multiplication, then all agents must be proportionally changed, including ΔH. (See Manipulating Thermochemical Equations, below.) Thermochemical equation’s multiplicative property is largely due to the First Law of Thermodynamics, which says that energy can be neither created nor destroyed, a concept commonly known as the conservation of energy.
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For m=0 the generalized Jacobian Jm is just the usual Jacobian J, an abelian variety of dimension g, the genus of C. For m a nonzero effective divisor the generalized Jacobian is an extension of J by a connected commutative affine algebraic group Lm of dimension deg(m)−1. So we have an exact sequence :0 → Lm → Jm → J → 0 The group Lm is a quotient :0 → Gm → ΠRi → Lm → 0 of a product of groups Ri by the multiplicative group Gm of the underlying field. The product runs over the points Pi in the support of m, and the group Ri is the group of invertible elements of the local ring modulo those that are 1 mod m. The group Ri has dimension ni, the number of times Pi occurs in m.
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Let E be an elliptic curve defined over a local field K and p the prime ideal of the ring of integers of K. We consider a minimal equation for E: a generalised Weierstrass equation whose coefficients are p-integral and with the valuation of the discriminant νp(Δ) as small as possible. If the discriminant is a p-unit then E has good reduction at p and the exponent of the conductor is zero. We can write the exponent f of the conductor as a sum ε + δ of two terms, corresponding to the tame and wild ramification. The tame ramification part ε is defined in terms of the reduction type: ε=0 for good reduction, ε=1 for multiplicative reduction and ε=2 for additive reduction.
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If we write F× for the multiplicative group of F (excluding 0), then the determinant is a group homomorphism :det: GL(n, F) → F×. that is surjective and its kernel is the special linear group. Therefore, by the first isomorphism theorem, is isomorphic to F×. In fact, can be written as a semidirect product: :GL(n, F) = SL(n, F) ⋊ F× The special linear group is also the derived group (also known as commutator subgroup) of the GL(n, F) (for a field or a division ring F) provided that n e 2 or k is not the field with two elements., Theorem II.9.4 When F is R or C, is a Lie subgroup of of dimension . The Lie algebra of consists of all matrices over F with vanishing trace.
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A tight lower bound is not known on the number of required additions, although lower bounds have been proved under some restrictive assumptions on the algorithms. In 1973, Morgenstern proved an Ω(N log N) lower bound on the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true for most but not all FFT algorithms). This result, however, applies only to the unnormalized Fourier transform (which is a scaling of a unitary matrix by a factor of \sqrt N), and does not explain why the Fourier matrix is harder to compute than any other unitary matrix (including the identity matrix) under the same scaling. Pan (1986) proved an Ω(N log N) lower bound assuming a bound on a measure of the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear.
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Whenever a formal series :f(X)=\sum_k f_k X^k \in RX has f0 = 0 and f1 being an invertible element of R, there exists a series :g(X)=\sum_k g_k X^k that is the composition inverse of f, meaning that composing f with g gives the series representing the identity function x = 0 + 1x + 0x^2+ 0x^3+\cdots. The coefficients of g may be found recursively by using the above formula for the coefficients of a composition, equating them with those of the composition identity X (that is 1 at degree 1 and 0 at every degree greater than 1). In the case when the coefficient ring is a field of characteristic 0, the Lagrange inversion formula (discussed below) provides a powerful tool to compute the coefficients of g, as well as the coefficients of the (multiplicative) powers of g.
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Each digit of the product depends only on a neighborhood of two digits in the given number: the digit in the same position and the digit one position to the right. More generally, multiplication or division of doubly infinite digit sequences in any radix , by a multiplier or divisor all of whose prime factors are also prime factors of , is an operation that forms a cellular automaton because it depends only on a bounded number of nearby digits, and is reversible because of the existence of multiplicative inverses., p. 1093. Multiplication by other values (for instance, multiplication of decimal numbers by three) remains reversible, but does not define a cellular automaton, because there is no fixed bound on the number of digits in the initial value that are needed to determine a single digit in the result.
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Hantzsch–Widman nomenclature, also called the extended Hantzsch–Widman system, is a type of systematic chemical nomenclature used for naming heterocyclic parent hydrides having no more than ten ring members. Some common heterocyclic compounds have retained names that do not follow the Hantzsch–Widman pattern.. A Hantzsch–Widman name will always contain a prefix, which indicates the type of heteroatom present in the ring, and a stem, which indicates both the total number of atoms and the presence or absence of double bonds. The name may include more than one prefix, if more than one type of heteroatom is present; a multiplicative prefix if there are several heteroatoms of the same type; and locants to indicate the relative positions of the different atoms. Hantzsch–Widman names may be combined with other aspects of organic nomenclature, to indicate substitution or fused-ring systems.
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The importance of the computational complexity of matrix multiplication relies on the facts that many algorithmic problems may be solved by means of matrix computation, and most problems on matrices have a complexity which is either the same as that of matrix multiplication (up to a multiplicative constant), or may be expressed in term of the complexity of matrix multiplication or its exponent \omega. There are several advantages of expressing complexities in terms of the exponent \omega of matrix multiplication. Firstly, if \omega is improved, this will automatically improve the known upper bound of complexity of many algorithms. Secondly, in practical implementations, one never uses the matrix multiplication algorithm that has the best asymptotical complexity, because the constant hidden behind the big O notation is too large for making the algorithm competitive for sizes of matrices that can be manipulated in a computer.
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This constraint limits the maximum amount of information a PURB's total length can leak to O(\log \log M) bits, a significant asymptotic reduction and the best achievable in general for variable-length encrypted formats whose multiplicative overhead is limited to a constant factor of the unpadded payload size. This asymptotic leakage is the same as one would obtain by padding encrypted objects to a power of some base, such as to a power of two. Allowing some significant mantissa bits in the length's representation rather than just an exponent, however, significantly reduces the overhead of padding. For example, padding to the next power of two can impose up to 100% overhead by nearly doubling the object's size, while a PURB's padding imposes overhead of at most 12% for small strings and decreasing gradually (to 6%, 3%, etc.) as objects get larger.
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Northern Song Dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule. Yang Hui triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD Four outstanding mathematicians arose during the Song Dynasty and Yuan Dynasty, particularly in the twelfth and thirteenth centuries: Yang Hui, Qin Jiushao, Li Zhi (Li Ye), and Zhu Shijie. Yang Hui, Qin Jiushao, Zhu Shijie all used the Horner-Ruffini method six hundred years earlier to solve certain types of simultaneous equations, roots, quadratic, cubic, and quartic equations. Yang Hui was also the first person in history to discover and prove "Pascal's Triangle", along with its binomial proof (although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD).
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For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or the fact that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero with a regular element of R as value on some pair of arguments implies that R is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant.
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The order of operations, which is used throughout mathematics, science, technology and many computer programming languages, is expressed here: # exponentiation and root extraction # multiplication and division # addition and subtraction This means that if, in a mathematical expression, a subexpression appears between two operators, the operator that is higher in the above list should be applied first. The commutative and associative laws of addition and multiplication allow adding terms in any order, and multiplying factors in any order—but mixed operations must obey the standard order of operations. In some contexts, it is helpful to replace a division by multiplication by the reciprocal (multiplicative inverse) and a subtraction by addition of the opposite (additive inverse). For example, in computer algebra, this allows one to handle fewer binary operations, and makes it easier to use commutativity and associativity when simplifying large expressions (for more, see ).
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Thus, the creation of the renal angina index was done by a multiplicative index (instead of sum). The RAI score is a composite of risk strata and clinical signs. Risk strata were given point values that were essentially the epidemiologic risk compared to general pediatric risk divided by 10: 5 (very high risk), 3 (high risk), and 1 (moderate risk). Clinical signs of injury are based on changes in estimated creatinine clearance (eCCl) or % fluid overload (% FO). The assigned point values are: 1 (ICU status and no decrease in eCCl or <5% FO), 2 (> 5% FO or eCCl decrease of 0-25%), 4 (>10% FO or eCCl decrease of 25-50%), or 8 (>15% FO or eCCl decrease of > 50%). The composite range of the RAI is therefore: 1, 2, 3, 4, 5, 6, 8, 10, 12, 20, 24, and 40.
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Representation of the lunar gravity well, illustrating how resources needed only for the trip home don't have to be carried down and back up the "well" The main advantage of LOR is the spacecraft payload saving, due to the fact that the propellant necessary to return from lunar orbit back to Earth need not be carried as dead weight down to the Moon and back into lunar orbit. This has a multiplicative effect, because each pound of "dead weight" propellant used later has to be propelled by more propellant sooner, and also because increased propellant requires increased tankage weight. The resultant weight increase would also require more thrust for lunar landing, which means larger and heavier engines. Another advantage is that the lunar lander can be designed for just that purpose, rather than requiring the main spacecraft to also be made suitable for a lunar landing.
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Given a subset S in Rn, a vector field is represented by a vector-valued function V: S → Rn in standard Cartesian coordinates (x1, ..., xn). If each component of V is continuous, then V is a continuous vector field, and more generally V is a Ck vector field if each component of V is k times continuously differentiable. A vector field can be visualized as assigning a vector to individual points within an n-dimensional space. Given two Ck-vector fields V, W defined on S and a real valued Ck- function f defined on S, the two operations scalar multiplication and vector addition : (fV)(p) := f(p)V(p)\, : (V+W)(p) := V(p) + W(p)\, define the module of Ck-vector fields over the ring of Ck-functions where the multiplication of the functions is defined pointwise (therefore, it is commutative with the multiplicative identity being fid(p) := 1).
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This construction achieves many of the desired properties of F1-geometry: consists of a single point, so behaves similarly to the spectrum of a field in conventional geometry, and the category of affine monoid schemes is dual to the category of multiplicative monoids, mirroring the duality of affine schemes and commutative rings. Furthermore, this theory satisfies the combinatorial properties expected of F1 mentioned in previous sections; for instance, projective space over of dimension as a monoid scheme is identical to an apartment of projective space over of dimension when described as a building. However, monoid schemes do not fulfill all of the expected properties of a theory of F1-geometry, as the only varieties which have monoid scheme analogues are toric varieties. More precisely, if is a monoid scheme whose base extension is a flat, separated, connected scheme of finite type, then the base extension of is a toric variety.
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Going one step ahead of universal algebra, most algebraic structures are F-algebras. For example, abelian groups are F-algebras for the same functor F(G) = 1 + G + G×G as for groups, with an additional axiom for commutativity: m∘t = m, where t(x,y) = (y,x) is the transpose on GxG. Monoids are F-algebras of signature F(M) = 1 + M×M. In the same vein, semigroups are F-algebras of signature F(S) = S×S Rings, domains and fields are also F-algebras with a signature involving two laws +,•: R×R -> R, an additive identity 0: 1 -> R, a multiplicative identity 1: 1 -> R, and an additive inverse for each element -: R -> R. As all these functions share the same codomain R they can be glued into a single signature function 1 + 1 + R + R×R + R×R -> R, with axioms to express associativity, distributivity, and so on.
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In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification.
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NMF decomposes a non-negative matrix to the product of two non-negative ones, which has been a promising tool in fields where only non-negative signals exist, such as astronomy. NMF is well known since the multiplicative update rule by Lee & Seung, which has been continuously developed: the inclusion of uncertainties , the consideration of missing data and parallel computation , sequential construction which leads to the stability and linearity of NMF, as well as other updates including handling missing data in digital image processing. is able to preserve the flux in direct imaging of circumstellar structures in astromony, as one of the methods of detecting exoplanets, especially for the direct imaging of circumstellar disks. In comparison with PCA, NMF does not remove the mean of the matrices which leads to unphysical non-negative fluxes, therefore NMF is able to preserve more information than PCA as demonstrated by Ren et al.
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This work led in turn, in the winter of 1966–67, to the now well known conjectures making up what is often called the Langlands program. Very roughly speaking, they propose a huge generalization of previously known examples of reciprocity, including (a) classical class field theory, in which characters of local and arithmetic abelian Galois groups are identified with characters of local multiplicative groups and the idele quotient group, respectively; (b) earlier results of Martin Eichler and Goro Shimura in which the Hasse–Weil zeta functions of arithmetic quotients of the upper half plane are identified with -functions occurring in Hecke's theory of holomorphic automorphic forms. These conjectures were first posed in relatively complete form in a famous letter to Weil, written in January 1967. It was in this letter that he introduced what has since become known as the -group and along with it, the notion of functoriality.
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Illustration of Brünnhilde by Odilon Redon, 1885 The plot revolves around a magic ring that grants the power to rule the world, forged by the Nibelung dwarf Alberich from gold he stole from the Rhine maidens in the river Rhine. Wagner described the Ring itself as a Rune-magic taufr ("tine", or "talisman") intended to rule the feminine multiplicative power by a fearful magical act termed "denial of love" (Liebesverzicht). With the assistance of the god Loge, Wotan – the chief of the gods – steals the ring from Alberich, but is forced to hand it over to the giants, Fafner and Fasolt in payment for building the home of the gods, Valhalla, or they will take Freia, who provides the gods with the golden apples that keep them young. Wotan's schemes to regain the ring, spanning generations, drive much of the action in the story.
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Arnold said that the creative process "is an intellectual process ... whereby you combine and re-combine all your past experience or selected aspects of it ... [and] end up with a new combination, a new pattern, a new configuration that somehow satisfies some basic expressed or implied need of a man."Arnold, 1959b, p. 35. Arnold stated that the creative process is a particular kind of problem-solving, distinguished from decision-making by four requirements that make the result creative: #a better combination, not just something different #tangible, something you can see, or feel or react to in some fashion, not just an idea #forward-looking in time, relating to society's needs, not merely "recreative" #a "synergetic" quality—the value achieved in the combination is much greater than the sum of the parts (a multiplicative effect). The value of a creative result was measured by increased function, improved performance, and lowered cost.
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The Burnside ring of a finite group G is constructed from the category of finite G-sets as a Grothendieck group. More precisely, let M(G) be the commutative monoid of isomorphism classes of finite G-sets, with addition the disjoint union of G-sets and identity element the empty set (which is a G-set in a unique way). Then A(G), the Grothendieck group of M(G), is an abelian group. It is in fact a free abelian group with basis elements represented by the G-sets G/H, where H varies over the subgroups of G. (Note that H is not assumed here to be a normal subgroup of G, for while G/H is not a group in this case, it is still a G-set.) The ring structure on A(G) is induced by the direct product of G-sets; the multiplicative identity is the (isomorphism class of any) one-point set, which becomes a G-set in a unique way.
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In Alfred North Whitehead's book A Treatise on Universal Algebra, published in 1898, the term universal algebra had essentially the same meaning that it has today. Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself. At the time structures such as Lie algebras and hyperbolic quaternions drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review Alexander Macfarlane wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures."Alexander Macfarlane (1899) Review:A Treatise on Universal Algebra (pdf), Science 9: 324–8 via Internet Archive At the time George Boole's algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities.
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Preferred metric sizes are a set of international standards and de facto standards that are designed to make using the metric system easier and simpler, especially in engineering and construction practices. One of the methods used to arrive at these preferred sizes is the use of preferred numbers and convenient numbers such as the Renard series, the 1-2-5 series to limit the number of different sizes of components needed. One of the largest benefits of such limits is an ensuing multiplicative or exponential reduction in the number of parts, tools and other items needed to support the installation and maintenance of the items built using these techniques. This occurs because eliminating one diameter fastener will typically allow the elimination of a large number of variations on that diameter (multiple thread pitches, multiple lengths, multiple tip types, multiple head types, multiple drive types, and the tools needed for installing each, including multiple drill bits (one for each different thread pitch, material, and fit combination).
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This approach of taking higher-order similarities takes the latent semantic structure of the whole corpus into consideration with the result of generating a better clustering of the documents and words. In text databases, for a document collection defined by a document by term D matrix (of size m by n, m: number of documents, n: number of terms) the cover-coefficient based clustering methodology yields the same number of clusters both for documents and terms (words) using a double-stage probability experiment. According to the cover coefficient concept number of clusters can also be roughly estimated by the following formula (m \times n) / t where t is the number of non-zero entries in D. Note that in D each row and each column must contain at least one non-zero element. In contrast to other approaches, FABIA is a multiplicative model that assumes realistic non-Gaussian signal distributions with heavy tails.
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The 13 possible strict weak orderings on a set of three elements {a, b, c} In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n elements (orderings of the elements into a sequence allowing ties, such as might arise as the outcome of a horse race).. Because of this application, de Koninck calls these numbers "horse numbers", but this name does not appear to be in widespread use. Starting from n = 0, these numbers are :1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... . The ordered Bell numbers may be computed via a summation formula involving binomial coefficients, or by using a recurrence relation. Along with the weak orderings, they count several other types of combinatorial objects that have a bijective correspondence to the weak orderings, such as the ordered multiplicative partitions of a squarefree number or the faces of all dimensions of a permutohedron. (e.g.
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Naturally, CSV got the inspiration to become a mathematician from his great professor, Ananda Rau, and also, he got an opportunity to get introduced to another mathematical genius R. Vaidyanathaswamy, who had already established himself at the University of Madras in the 1930s and had set up a tradition and an academic atmosphere that gave Madras an international recognition in the field of mathematics. CSV was selected as a research scholar in the Department of Mathematics of Madras University to do research in Theory of Arithmetic function under the guidance of R Vaidyanathaswamy. In fact, there he was in the eminent company of senior research scholars like P Kesava Menon and K G Ramanathan and the theory of multiplicative function formed the material for the dissertations of P Kesava Menon and CSV. The theory of arithmetic functions was initiated in the 1930s by E. T. Bell of the California Institute of Technology and independently by R Vaidyanathaswamy.
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The Chebotarev density theorem may be viewed as a generalisation of Dirichlet's theorem on arithmetic progressions. A quantitative form of Dirichlet's theorem states that if N≥2 is an integer and a is coprime to N, then the proportion of the primes p congruent to a mod N is asymptotic to 1/n, where n=φ(N) is the Euler totient function. This is a special case of the Chebotarev density theorem for the Nth cyclotomic field K. Indeed, the Galois group of K/Q is abelian and can be canonically identified with the group of invertible residue classes mod N. The splitting invariant of a prime p not dividing N is simply its residue class because the number of distinct primes into which p splits is φ(N)/m, where m is multiplicative order of p modulo N; hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to N.
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One says that a is a two-sided divisor of b if it is both a left divisor and a right divisor of b; in this case, it is not necessarily true that (using the previous notation) x=y, only that both some x and some y which each individually satisfy the previous equations in R exist in R. When R is commutative, a left divisor, a right divisor and a two-sided divisor coincide, so in this context one says that a is a divisor of b, or that b is a multiple of a, and one writes a \mid b . Elements a and b of an integral domain are associates if both a \mid b and b \mid a . The associate relationship is an equivalence relation on R, and hence divides R into disjoint equivalence classes. Notes: These definitions make sense in any magma R, but they are used primarily when this magma is the multiplicative monoid of a ring.
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Let R be a fixed commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is an additive abelian group A which has the structure of both a ring and an R-module in such a way that the scalar multiplication satisfies :r\cdot(xy) = (r\cdot x)y = x(r\cdot y) for all r ∈ R and x, y ∈ A. Furthermore, A is assumed to be unital, which is to say it contains an element 1 such that :1 x = x = x 1 for all x ∈ A. Note that such an element 1 is necessarily unique. In other words, A is an R-module together with a R-bilinear binary operation A × A → A that is associative, and has an identity. Technical note: the multiplicative identity is a datum (there is the forgetful functor from the category of unital associative algebras to the category of possibly non-unital associative algebras) while associativity is a property.
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An associative algebra over K is given by a K-vector space A endowed with a bilinear map A × A → A having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K → A identifying the scalar multiples of the multiplicative identity. If the bilinear map A × A → A is reinterpreted as a linear map (i. e., morphism in the category of K-vector spaces) A ⊗ A → A (by the universal property of the tensor product), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form A ⊗ A → A and one of the form K → A) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams that describe the algebra axioms; this defines the structure of a coalgebra.
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The Ore condition can be generalized to other multiplicative subsets, and is presented in textbook form in and . A subset S of a ring R is called a right denominator set if it satisfies the following three conditions for every a, b in R, and s, t in S: # st in S; (The set S is multiplicatively closed.) # aS ∩ sR is not empty; (The set S is right permutable.) # If , then there is some u in S with ; (The set S is right reversible.) If S is a right denominator set, then one can construct the ring of right fractions RS−1 similarly to the commutative case. If S is taken to be the set of regular elements (those elements a in R such that if b in R is nonzero, then ab and ba are nonzero), then the right Ore condition is simply the requirement that S be a right denominator set. Many properties of commutative localization hold in this more general setting.
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Many real-world examples of Benford's law arise from multiplicative fluctuations. For example, if a stock price starts at $100, and then each day it gets multiplied by a randomly chosen factor between 0.99 and 1.01, then over an extended period the probability distribution of its price satisfies Benford's law with higher and higher accuracy. The reason is that the logarithm of the stock price is undergoing a random walk, so over time its probability distribution will get more and more broad and smooth (see above). (More technically, the central limit theorem says that multiplying more and more random variables will create a log-normal distribution with larger and larger variance, so eventually it covers many orders of magnitude almost uniformly.) To be sure of approximate agreement with Benford's law, the distribution has to be approximately invariant when scaled up by any factor up to 10; a lognormally distributed data set with wide dispersion would have this approximate property.
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A generalization of Newton's method as used for a multiplicative inverse algorithm may be convenient, if it is convenient to find a suitable starting seed: :X_{k+1} = 2X_k - X_k A X_k. Victor Pan and John Reif have done work that includes ways of generating a starting seed. Byte magazine summarised one of their approaches. Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough.
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The handicap principle has proven hard to test empirically, partly because of inconsistent interpretations of Zahavi's metaphor and Grafen's marginal fitness model, and partly because of conflicting empirical results: in some studies individuals with bigger signals seem to pay higher costs, in other studies they seem to be paying lower costs. A possible explanation for the inconsistent empirical results is given in a series of papers by Getty, who shows that Grafen's proof of the handicap principle is based on the critical simplifying assumption that signallers trade off costs for benefits in an additive fashion, the way humans invest money to increase income in the same currency. But the assumption that costs and benefits trade off in an additive fashion is true only on a logarithmic scale; for the survival cost – reproduction benefit tradeoff is assumed to mediate the evolution of sexually selected signals. Fitness depends on producing offspring, which is a multiplicative function of reproductive success given an individual is still alive times the probability of still being alive, given investment in signals.
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A linear transformation f: V → V is an endomorphism of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is the identity map id: V → V. An endomorphism of V that is also an isomorphism is called an automorphism of V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a group, the automorphism group of V which is denoted by Aut(V) or GL(V). Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V). If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n × n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n × n invertible matrices with entries in K.
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Given a base scheme S, an algebraic torus over S is defined to be a group scheme over S that is fpqc locally isomorphic to a finite product of copies of the multiplicative group scheme Gm/S over S. In other words, there exists a faithfully flat map X → S such that any point in X has a quasi- compact open neighborhood U whose image is an open affine subscheme of S, such that base change to U yields a finite product of copies of GL1,U = Gm/U. One particularly important case is when S is the spectrum of a field K, making a torus over S an algebraic group whose extension to some finite separable extension L is a finite product of copies of Gm/L. In general, the multiplicity of this product (i.e., the dimension of the scheme) is called the rank of the torus, and it is a locally constant function on S. Most notions defined for tori over fields carry to this more general setting.
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Suppose the game of Nim is played as usual with heaps of objects, but that at the start of play, every heap is restricted to have either one or two objects in it. In the normal-play convention, players take turns to remove any number of objects from a heap, and the last player to take an object from a heap is declared the winner of the game; in Misere play, that player is the loser of the game. Regardless of whether the normal or misere play convention is in effect, the outcome of such a position is necessarily of one of two types: ; N : The Next player to move has a forced win in best play; or ; P : The Previous player to move has a forced win. We can write down a commutative monoid presentation for the misere quotient of this 1- and 2-pile Nim game by first recasting its conventional nimber-based solution into a multiplicative form, and then modifying that slightly for misere play.
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This proof builds on Lagrange's result that if p=4n+1 is a prime number, then there must be an integer m such that m^2 + 1 is divisible by p (we can also see this by Euler's criterion); it also uses the fact that the Gaussian integers are a unique factorization domain (because they are a Euclidean domain). Since does not divide either of the Gaussian integers m + i and m-i (as it does not divide their imaginary parts), but it does divide their product m^2 + 1, it follows that p cannot be a prime element in the Gaussian integers. We must therefore have a nontrivial factorization of p in the Gaussian integers, which in view of the norm can have only two factors (since the norm is multiplicative, and p^2 = N(p), there can only be up to two factors of p), so it must be of the form p = (x+yi)(x-yi) for some integers x and y. This immediately yields that p = x^2 + y^2.
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The period of a lag-r MWC generator is the order of b in the multiplicative group of numbers modulo p = abr − 1\. While it is theoretically possible to choose a non-prime modulus, a prime modulus eliminates the possibility of the initial seed sharing a common divisor with the modulus, which would reduce the generator's period. Because 2 is a quadratic residue of numbers of the form 8k±1, b = 2k cannot be a primitive root of p = abr − 1\. Therefore, MWC generators with base 2k have their parameters chosen so their period is (abr−1)/2. This is one of the difficulties that use of b = 2k − 1 overcomes. The basic form of an MWC generator has parameters a, b and r, and r+1 words of state. The state consists of r residues modulo b : 0 ≤ x0, x1, x2,..., xr−1 < b, and a carry cr−1 < a. The initial state ("seed") values are arbitrary, except that they must not be all zero, nor all at the maximum permitted values (xi = b−1 and cr−1 = a−1).
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1\. Let , the cyclic group of order 3, with generator a and identity element 1G. An element r of C[G] can be written as :r = z_0 1_G + z_1 a + z_2 a^2\, where z0, z1 and z2 are in C, the complex numbers. This is the same thing as a polynomial ring in variable a such that a^3=a^0=1 i.e. C[G] is isomorphic to the ring C[a]/(a^3-1). Writing a different element s as s=w_0 1_G +w_1 a +w_2 a^2\, their sum is :r + s = (z_0+w_0) 1_G + (z_1+w_1) a + (z_2+w_2) a^2\, and their product is :rs = (z_0w_0 + z_1w_2 + z_2w_1) 1_G +(z_0w_1 + z_1w_0 + z_2w_2)a +(z_0w_2 + z_2w_0 + z_1w_1)a^2. Notice that the identity element 1G of G induces a canonical embedding of the coefficient ring (in this case C) into C[G]; however strictly speaking the multiplicative identity element of C[G] is 1⋅1G where the first 1 comes from C and the second from G. The additive identity element is zero.
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In modern mathematical language class field theory can be formulated as follows. Consider the maximal abelian extension A of a local or global field K. It is of infinite degree over K; the Galois group G of A over K is an infinite pro- finite group, so a compact topological group, and it is abelian. The central aims of class field theory are: to describe G in terms of certain appropriate topological objects associated to K, to describe finite abelian extensions of K in terms of open subgroups of finite index in the topological object associated to K. In particular, one wishes to establish a one-to-one correspondence between finite abelian extensions of K and their norm groups in this topological object for K. This topological object is the multiplicative group in the case of local fields with finite residue field and the idele class group in the case of global fields. The finite abelian extension corresponding to an open subgroup of finite index is called the class field for that subgroup, which gave the name to the theory.
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1–17 Other techniques supported include principal component analysis (PCA),S. De Vries, Cajo J.F. Ter Braak (1995) Prediction error in partial least squares regression: a critique on the deviation used in The Unscrambler Chemometrics and Intelligent Laboratory Systems 30:239-245 PDF 3-way PLS, multivariate curve resolution, design of experiments, supervised classification, unsupervised classification and cluster analysis.Kristian Helland (1991) UNSCRAMBLER 11, version 3.10: A program for multivariate analysis with PLS and PCA/PCR Journal of Chemometrics 5(4):413-415 The software is used in spectroscopy (IR, NIR, Raman, etc.), chromatography, and process applications in research and non-destructive quality control systems in pharmaceutical manufacturing,M.R. Maleki, A.M. Mouazen, H. Ramon and J. De Baerdemaeker (2006) Multiplicative Scatter Correction during On-line Measurement with Near Infrared Spectroscopy Biosystems Engineering 96(3):427-433 Tatavarti AS, Fahmy R, Wu H, Hussain AS, Marnane W, Bensley D, Hollenbeck G, Hoag SW. Assessment of NIR Spectroscopy for Nondestructive Analysis of Physical and Chemical Attributes of Sulfamethazine Bolus Dosage Forms AAPS PharmSciTech. 2005; 06(01): E91-E99.
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By contrast, multiple jeopardy is founded on the idea that each mode of discrimination is multiplicative, and thus the relationship between racism, sexism and classism would instead be represented as "racism multiplied by sexism multiplied by classism". King uses this equation to argue that the institutional context behind the ways that race, sex, and class are treated in society can create unique types of discrimination that differ vastly from the discrimination associated with each of these factors, such that the discrimination experienced by a black woman is much more than just the sum of the discrimination that a black man and a white woman would experience. King illustrates this concept by recounting the ill treatment of black women during the era of slavery in the United States. At that time, black workers were subjected to demanding physical labor and brutal punishments. Black men and black women were both victims of this practice, but black women also endured subjugation exclusive to women; as Angela Davis explained in Women, Race, and Class, “If the most violent punishments of men consisted in floggings and mutilations, women were flogged and mutilated, as well as raped.
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The original construction of tmf uses the obstruction theory of Hopkins, Miller, and Paul Goerss, and is based on ideas of Dwyer, Kan, and Stover. In this approach, one defines a presheaf Otop ("top" stands for topological) of multiplicative cohomology theories on the etale site of the moduli stack of elliptic curves and shows that this can be lifted in an essentially unique way to a sheaf of E-infinity ring spectra. This sheaf has the following property: to any etale elliptic curve over a ring R, it assigns an E-infinity ring spectrum (a classical elliptic cohomology theory) whose associated formal group is the formal group of that elliptic curve. A second construction, due to Jacob Lurie, constructs tmf rather by describing the moduli problem it represents and applying general representability theory to then show existence: just as the moduli stack of elliptic curves represents the functor that assigns to a ring the category of elliptic curves over it, the stack together with the sheaf of E-infinity ring spectra represents the functor that assigns to an E-infinity ring its category of oriented derived elliptic curves, appropriately interpreted.
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In statistics, fixed-effect Poisson models are used for static panel data when the outcome variable is count data. Hausman, Hall, and Griliches pioneered the method in the mid 1980s. Their outcome of interest was the number of patents filed by firms, where they wanted to develop methods to control for the firm fixed effects.Hausman, J. A., B. H. Hall, and Z. Griliches (1984): "Econometric Models for Count Data with an Application to the Patents-R&D; Relationship." Econometrica (46), pp. 909–938 Linear panel data models use the linear additivity of the fixed effects to difference them out and circumvent the incidental parameter problem. Even though Poisson models are inherently nonlinear, the use of the linear index and the exponential link function lead to multiplicative separability, more specifically Cameron, C. A. and P. K. Trivedi (2015) "Count Panel Data," Oxford Handbook of Panel Data, ed. by B. Baltagi, Oxford University Press, pp. 233–256 : E[yit ∨ xi1... xiT, ci ] = m(xit, ci, b0 ) = exp(ci \+ xit b0 ) = ai exp(xit b0 ) = μti (1) This formula looks very similar to the standard Poisson premultiplied by the term ai.
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HOS identify such multi-input multi-output (MIMO) systems by removing the rotational (unitary matrix) ambiguity present with second-order statistics – a basic result that led to the renown tool of independent component analysis and further enabled blind separation of sources received by sensor arrays. Highly regarded are also Giannakis’ identification of linear time-varying systems using basis expansion models including Fourier bases, and optimally chosen wavelet bases and multiresolution depths; HOS-based Gaussianity and linearity tests, detection, estimation, pattern recognition, noise cancellation, object registration, image motion estimation, and the first proof that HOS can estimate directions of arrival of more sources with less antenna elements. Besides non-Gaussian stationary signals, he contributed influential results on consistency and asymptotic normality of HOS for a class of non-stationary and cyclostationary processes. For those, he developed widely applied statistical tests for the presence of cyclostationarity, as well as algorithms for retrieval of harmonics in the presence of multiplicative and additive noise; time series analysis with random and periodic misses; delay-Doppler estimators based on the high-order ambiguity function; multi-component polynomial phase signals for synthetic-aperture radar, and their impact to time-varying image motion estimation.
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