The risk of this twofold loss is known as negative convexity.
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Technology, like finance, relies on scale and convexity of returns to thrive.
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The big picture: The previous record was set by Convexity at $6.3 billion in 2006.
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Carmignac's approach is to "play convexity" — trading the currency through options rather than direct exposure, Saint-Georges said.
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DRW meanwhile has invested in real estate since 2009, eventually creating its own real estate investment arm known as Convexity Properties.
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Convexity buying might occur when homeowners refinance their mortgages, eliminating securities that fund managers had expected to hold for several more years.
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You pay a premium just to bring convexity in your portfolio because it could really go a big way in both directions.
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They started with the skeleton: Gold is flexible and nonreactive, and they designed the spine and ribs to have a natural convexity.
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His father is the chief executive of Convexity Capital Management, a hedge fund in Boston, and is the chairman of the Boston Ballet.
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While convexity is to blame for the distortion of our vehicle's mirror, complexity is the culprit in the distortion of our initial releases of economic data.
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"The demand for 50-year is there for good reason, there's a lot of convexity value for pension funds, and for asset managers," said Rieger of Commerzbank.
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I've regretted my sun-spurning pallor, I've felt bad about my neck (for reasons other than Nora Ephron's — mine is just really short), I've registered dismay at my stomach's persistent convexity.
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These convexity flow continue to hammer rates lower and only now are risk assets actually catching up, which is again threatening to create yet another leg lower in rates, Goldberg said.
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Bond pros said some of the week's market moves have been perplexing, and many blamed them on quarter end portfolio maneuvers and on other technical factors, such as convexity buying or hedging.
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Tanguy Le Saout, head of European fixed income at Pioneer Investments, said the attraction lies in another bond market calculation called "convexity" - or how the duration of the bond changes with movements in rates.
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It plans to discuss systematic flows and liquidity dynamics in equities, to what extent is high-frequency trading to blame for drops in market depth and convexity hedging in interest rate markets, the invitation shows.
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His other hand ran experimentally up her shirt, and Greer stood in shocked suspension for a moment as he found the convexity of her breast and encircled it, all the while looking her in the eye, not blinking, just looking .
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In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.
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In mathematics -- specifically, in Riemannian geometry -- geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds. It is common to drop the prefix "geodesic" and refer simply to "convexity" of a set or function.
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Hotelling made pioneering studies of non-convexity in economics. In economics, non-convexity refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences and convex budget sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood. When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failures, where supply and demand differ or where market equilibria can be inefficient.
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However, strict-convexity of preferences implies both of them. It is obvious that strict-convexity implies weak-convexity (theorem 1). To see that it implies the condition of theorem 2, suppose there are two different allocations x,y with the same utility profile u. Define z = x/2+y/2.
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Convexity also gives an idea of the spread of future cashflows. (Just as the duration gives the discounted mean term, so convexity can be used to calculate the discounted standard deviation, say, of return.) Note that convexity can be positive or negative. A bond with positive convexity will not have any call features - i.e. the issuer must redeem the bond at maturity - which means that as rates fall, both its duration and price will rise.
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The first published proof to the existence of super- proportional division was as a corollary to the Dubins–Spanier convexity theorem. This was a purely existential proof based on convexity arguments.
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In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalizations in the setting of symplectic geometry. A few important generalizations are Kostant's convexity theorem, Atiyah–Guillemin–Sternberg convexity theorem, Kirwan convexity theorem.
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The convexity of all the sublevel sets characterizes quasiconvex functions.
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The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms. Given a set , a convexity over is a collection of subsets of satisfying the following axioms: #The empty set and are in #The intersection of any collection from is in . #The union of a chain (with respect to the inclusion relation) of elements of is in . The elements of are called convex sets and the pair is called a convexity space.
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For the ordinary convexity, the first two axioms hold, and the third one is trivial. For an alternative definition of abstract convexity, more suited to discrete geometry, see the convex geometries associated with antimatroids.
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Opisthocoelous vertebrae are the opposite, possessing anterior convexity and posterior concavity.
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The notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. The common name "generalized convexity" is used, because the resulting objects retain certain properties of convex sets.
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In finance, bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates, the second derivative of the price of the bond with respect to interest rates (duration is the first derivative). In general, the higher the duration, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance. Convexity was based on the work of Hon-Fei Lai and popularized by Stanley Diller.
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The peak of this convexity is at the distal end of the humeral head.
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The convexity can be used to interpret derivative pricing: mathematically, convexity is optionality – the price of an option (the value of optionality) corresponds to the convexity of the underlying payout. In Black–Scholes pricing of options, omitting interest rates and the first derivative, the Black–Scholes equation reduces to \Theta = -\Gamma, "(infinitesimally) the time value is the convexity". That is, the value of an option is due to the convexity of the ultimate payout: one has the option to buy an asset or not (in a call; for a put it is an option to sell), and the ultimate payout function (a hockey stick shape) is convex – "optionality" corresponds to convexity in the payout. Thus, if one purchases a call option, the expected value of the option is higher than simply taking the expected future value of the underlying and inputting it into the option payout function: the expected value of a convex function is higher than the function of the expected value (Jensen inequality).
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Refining a model to account for non- linearities is referred to as a convexity correction.
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In economics, non-convexity refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences (that do not prefer extremes to in-between values) and convex budget sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood. When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failures, where supply and demand differ or where market equilibria can be inefficient. Non-convex economies are studied with nonsmooth analysis, which is a generalization of convex analysis.
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Specifically, duration can be formulated as the first derivative of the price with respect to the interest rate, and convexity as the second derivative (see: Bond duration closed-form formula; Bond convexity closed-form formula; Taylor series). Continuing the above example, for a more accurate estimate of sensitivity, the convexity score would be multiplied by the square of the change in interest rate, and the result added to the value derived by the above linear formula.
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Strictly speaking, convexity refers to the second derivative of output price with respect to an input price. In derivative pricing, this is referred to as Gamma (Γ), one of the Greeks. In practice the most significant of these is bond convexity, the second derivative of bond price with respect to interest rates. As the second derivative is the first non-linear term, and thus often the most significant, "convexity" is also used loosely to refer to non-linearities generally, including higher-order terms.
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Journal of Algebra, vol. 270 (2003), no. 1, pp. 133–149.M. Elder and S. Hermiller, Minimal almost convexity.
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An example of generalized convexity is orthogonal convexity.Rawlins G.J.E. and Wood D, "Ortho-convexity and its generalizations", in: Computational Morphology, 137-152. Elsevier, 1988. A set in the Euclidean space is called orthogonally convex or ortho-convex, if any segment parallel to any of the coordinate axes connecting two points of lies totally within .
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If the shell is kept horizontal, its convexity towards the observer, the mucronation of the septum appears on the left side.
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Convexity and Concavity Much like ascent and descent, these mirrored aspects of the same element heavily rely on one another. Working with the depth of built form, convexity and concavity act as connector and divider of urban space. They inform the volume of urban form and can be taken advantage of to make urban form more dramatic.
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Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers. It can be applied under differentiability and convexity.
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An anatomical variation is that the left vertebral artery can arise from the aortic arch instead of the left subclavian artery. The arch of the aorta forms two curvatures: one with its convexity upward, the other with its convexity forward and to the left. Its upper border is usually about 2.5 cm. below the superior border to the manubrium sterni.
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The Swedish functional analyst Edgar Asplund, then Professor of Mathematics at Aarhus University in Denmark, assisted Ribe as supervisor of his 1972 thesis,Acknowledgement in before dying of cancer in 1974. Ribe's results concerned topological vector spaces without assuming local convexity; Ribe constructed a counter-example to naive extensions of the Hahn–Banach theorem to topological vector spaces that lack local convexity.
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In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative.
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Under convexity, these conditions are also sufficient. If some of the functions are non-differentiable, subdifferential versions of Karush–Kuhn–Tucker (KKT) conditions are available.
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In mathematics, a function f is logarithmically convex or superconvexKingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.
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Forward rate agreements (FRAs) are interconnected with short term interest rate futures (STIR futures). Because STIR futures settle against the same index as a subset of FRAs, IMM FRAs, their pricing is related. The nature of each product has a distinctive gamma (convexity) profile resulting in rational, no arbitrage, pricing adjustments. This adjustment is called futures convexity adjustment (FCA) and is usually expressed in basis points.
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The neighbourhood swept out has similar properties to balls in Euclidean space, namely any two points in it are joined by a unique geodesic. This property is called "geodesic convexity" and the coordinates are called "normal coordinates". The explicit calculation of normal coordinates can be accomplished by considering the differential equation satisfied by geodesics. The convexity properties are consequences of Gauss's lemma and its generalisations.
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In mathematical finance, convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative (or, loosely speaking, higher-order terms) of the modeling function. Geometrically, the model is no longer flat but curved, and the degree of curvature is called the convexity.
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Bond convexity is one of the most basic and widely used forms of convexity in finance. For a bond with an embedded option, the standard yield to maturity based calculations here do not consider how changes in interest rates will alter the cash flows due to option exercise. To address this, effective duration and effective convexity are introduced. These values are typically calculated using a tree-based model, built for the entire yield curve (as opposed to a single yield to maturity), and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see Lattice model (finance)#Interest rate derivatives.
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When reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of "-space" and "Fréchet space" requires local convexity.
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The jaw is straight, with a slight convexity on the cutting-edge and no median projection. The radula is broad, with about 100 rows of teeth: 145 .
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This can be explained by the argument that high levels of homophily may significantly decrease the viability of networks, hence making convexity less frequent in complex networks.
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Even though every equilibrium is efficient, neither of the above two theorems say anything about the equilibrium existing in the first place. To guarantee that an equilibrium exists, it suffices that consumer preferences be strictly convex. With enough consumers, the convexity assumption can be relaxed both for existence and the second welfare theorem. Similarly, but less plausibly, convex feasible production sets suffice for existence; convexity excludes economies of scale.
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In this more general setting, a few properties are lost: for example, the Legendre transform is no longer its own inverse (unless there are extra assumptions, like convexity).
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Formally, the convexity adjustment arises from the Jensen inequality in probability theory: the expected value of a convex function is greater than or equal to the function of the expected value: :E[f(X)] \geq f(E[X]). Geometrically, if the model price curves up on both sides of the present value (the payoff function is convex up, and is above a tangent line at that point), then if the price of the underlying changes, the price of the output is greater than is modeled using only the first derivative. Conversely, if the model price curves down (the convexity is negative, the payoff function is below the tangent line), the price of the output is lower than is modeled using only the first derivative. The precise convexity adjustment depends on the model of future price movements of the underlying (the probability distribution) and on the model of the price, though it is linear in the convexity (second derivative of the price function).
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However, orthogonal hulls and tight spans differ for point sets with disconnected orthogonal hulls, or in higher-dimensional Lp spaces. describes several other results about orthogonal convexity and orthogonal visibility.
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In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936.
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The price of the option – the value of the optionality – thus reflects the convexity of the payoff function. This value is isolated via a straddle – purchasing an at-the-money straddle (whose value increases if the price of the underlying increases or decreases) has (initially) no delta: one is simply purchasing convexity (optionality), without taking a position on the underlying asset – one benefits from the degree of movement, not the direction. From the point of view of risk management, being long convexity (having positive Gamma and hence (ignoring interest rates and Delta) negative Theta) means that one benefits from volatility (positive Gamma), but loses money over time (negative Theta) – one net profits if prices move more than expected, and net loses if prices move less than expected.
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Norics were characterized by tall stature, brachycephaly, nasal convexity, long face and broad forehead. Their complexion was said to be light, and blondness combined with light eyes to be their anthropologic characteristic.
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Segal-Halevi et al use simulations from similar distributions to show that, in many cases, there exist allocations that are necessarily fair based on a certain convexity assumption on the agents' preferences.
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The tibial facet is divided into two basins by a low convexity, although this flexion is somewhat indistinct compared to that of suchians and avemetatarsalians. The calcaneal facet comes in the form of a flat surface overlying a convex 'peg', a feature characteristic of crurotarsal joints. The peg is poorly developed, more similar to that of phytosaurs rather than other suchians. The front edge of the astragalus has a large concave surface (astragalar hollow) overlying a small convexity (astragalar ball).
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Valuation of constant maturity swaps depend on volatilities of different forward rates and therefore requires a stochastic yield curve model or some approximated methodology like a convexity adjustment, see for example Brigo and Mercurio (2006).
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It is natural to generalize orthogonal convexity to restricted-orientation convexity, in which a set is defined to be convex if all lines having one of a finite set of slopes must intersect in connected subsets; see e.g. , , or . In addition, the tight span of a finite metric space is closely related to the orthogonal convex hull. If a finite point set in the plane has a connected orthogonal convex hull, that hull is the tight span for the Manhattan distance on the point set.
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Duration is a linear measure of how the price of a bond changes in response to interest rate changes. As interest rates change, the price does not change linearly, but rather is a convex function of interest rates. Convexity is a measure of the curvature of how the price of a bond changes as the interest rate changes. Specifically, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question, and the convexity as the second derivative.
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If the Gaussian curvature of a surface is everywhere positive, then the Euler characteristic is positive so is homeomorphic (and therefore diffeomorphic) to . If in addition the surface is isometrically embedded in , the Gauss map provides an explicit diffeomorphism. As Hadamard observed, in this case the surface is convex; this criterion for convexity can be viewed as a 2-dimensional generalisation of the well-known second derivative criterion for convexity of plane curves. Hilbert proved that every isometrically embedded closed surface must have a point of positive curvature.
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In the Arrow–Debreu model of general economic equilibrium, agents are assumed to have convex budget sets and convex preferences. These assumptions of convexity in economics can be used to prove the existence of an equilibrium. When actual economic data is non-convex, it can be made convex by taking convex hulls. The Shapley–Folkman theorem can be used to show that, for large markets, this approximation is accurate, and leads to a "quasi- equilibrium" for the original non-convex market.. See in particular Section 16.9, Non Convexity and Approximate Equilibrium, pp. 209–210.
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It is desirable to limit the class of possible solutions to only those that are typical of the class of the images which contains the image being reconstructed by using a priori information, such as convexity or connectedness.
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A significant part of Tian's research has been focused on characterization of the existence of equilibrium in discontinuous games. Tian and his co-authors systematically studied the existence of Nash equilibrium in discontinuous games, and characterized the existence of Nash equilibrium by weakening the traditional continuity, convexity and transitivity assumptions. By using very weak properties of continuity, convexity and transitivity: transfer continuity, transfer convexity, and transfer transitivity, he, along with Michael R. Baye and Jianxin Zhou, was the first to characterize the existence of Nash equilibria for discontinuous and non-convex games. In a paper published in 2015 entitled 'On The Existence of Equilibria in Games with Arbitrary Strategy Spaces and Preferences', Tian provided a full characterization on the existence of Nash equilibria in games with arbitrary strategy spaces and preferences that may be non-total, non-transitive, discontinuous, non-convex, or non-monotonic.
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This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on is demanded additionally. The case must be treated differently because is not normalizable at infinity (the sum of the reciprocals doesn't converge).
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It has sometimes been found that results can be derived under their definition without assuming convexity in the proof (the first fundamental theorem of welfare economics being an example). It is not apodeictic that such results even have any meaning.
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In measure and probability theory in mathematics, a convex measure is a probability measure that -- loosely put -- does not assign more mass to any intermediate set "between" two measurable sets A and B than it does to A or B individually. There are multiple ways in which the comparison between the probabilities of A and B and the intermediate set can be made, leading to multiple definitions of convexity, such as log-concavity, harmonic convexity, and so on. The mathematician Christer Borell was a pioneer of the detailed study of convex measures on locally convex spaces in the 1970s.
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Press, Cambridge, MA, 1995. By modifying the integral methods he developed in 1984, Huisken and Carlo Sinestrari carried out an elaborate inductive argument on the elementary symmetric polynomials of the second fundamental form to show that any singularity model resulting from such rescalings must be a mean curvature flow which moves by translating a single convex hypersurface in some direction. This passage from mean-convexity to full convexity is comparable with the much easier Hamilton- Ivey estimate for Ricci flow, which says that any singularity model of a Ricci flow on a closed 3-manifold must have nonnegative sectional curvature.
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If the concept of equilibrium includes local optima such as x, then equilibrium may be attainable but sub-optimal; if such points are excluded, then equilibrium may be optimal but unattainable. The differences caused by non-convexity become more deep-rooted when we look at the second fundamental theorem. Not every Pareto optimum is a competitive equilibrium (though it may still be a resting place for the economy). Consequently the theorem needs either to be given convexity of preferences as a premise, or else to be stated in such a way that ‘equilibrium’ is not understood as ‘competitive equilibrium’ as defined above.
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On the other hand, a bond with call features - i.e. where the issuer can redeem the bond early - is deemed to have negative convexity as rates approach the option strike, which is to say its duration will fall as rates fall, and hence its price will rise less quickly. This is because the issuer can redeem the old bond at a high coupon and re-issue a new bond at a lower rate, thus providing the issuer with valuable optionality. Similar to the above, in these cases it may be more correct to calculate an effective convexity.
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Fat pads should be carefully examined for convexity, which implies joint effusion (e.g., in the hip and elbow). However, the radiographic technique (positioning in particular) must be optimal for this evaluation to be valid. Osseous lines should be checked for integrity (e.g.
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The white umbilicus is funnel-shaped. It is margined by a slight convexity terminating below the columellar tooth. This is a peculiar little species, of globose form, with truncated columella, lirate interior, and finely decussated surface. The color pattern is very variable.
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For complex bilevel problems, classical methods fail due to difficulties like non-linearity, discreteness, non- differentiability, non-convexity etc. In such situations, evolutionary methods, though computationally demanding, could be an alternative tool to offset some of these difficulties and lead to an approximate optimal solution.
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Other evidence from single-cell electrophysiology in monkeys implicates ventrolateral PFC (inferior prefrontal convexity) in the control of motor responses. For example, cells that increase their firing rate to NoGo signals as well as a signal that says "don't look there!" have been identified.
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Convexity can be extended for a totally ordered set endowed with the order topology.Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). . Let . The subspace is a convex set if for each pair of points in such that , the interval is contained in .
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The surface shows strong, elevated, radiating wrinkles or lamellae, but no spiral markings when adult. The 6 to 11 perforations are small, subcircular, and separated by spaces greater than their own diameter. The two sides are about equally curved. The convexity varies with age.
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Extensions of convex optimization include the optimization of biconvex, pseudo-convex, and quasiconvex functions. Extensions of the theory of convex analysis and iterative methods for approximately solving non-convex minimization problems occur in the field of generalized convexity, also known as abstract convex analysis.
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These definitions are made by analogy with the classical theory of convexity, in which is convex if, for every line , the intersection of with is empty, a point, or a single segment. Orthogonal convexity restricts the lines for which this property is required to hold, so every convex set is orthogonally convex but not vice versa. For the same reason, the orthogonal convex hull itself is a subset of the convex hull of the same point set. A point belongs to the orthogonal convex hull of if and only if each of the closed axis-aligned orthants having as apex has a nonempty intersection with .
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Recent research in economics has recognized non-convexity in new areas of economics. In these areas, non- convexity is associated with market failures, where equilibria need not be efficient or where no competitive equilibrium exists because supply and demand differ. Non-convex sets arise also with environmental goods (and other externalities),Pages 106, 110–137, 172, and 248: and with market failures, and public economics.Pages 63–65: Starrett discusses non-convexities in his textbook on public economics (pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236): Non-convexities occur also with information economics, and with stock markets (and other incomplete markets).
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A more practical alternative immunisation method is duration matching. Here, the duration of the assets is matched with the duration of the liabilities. To make the match actually profitable under changing interest rates, the assets and liabilities are arranged so that the total convexity of the assets exceed the convexity of the liabilities. In other words, one can match the first derivatives (with respect to interest rate) of the price functions of the assets and liabilities and make sure that the second derivative of the asset price function is set to be greater than or equal to the second derivative of the liability price function.
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In microeconomic theory, cost-minimization by consumers and by firms implies the existence of supply and demand correspondences for which market clearing equilibrium prices exist, if there are large numbers of consumers and producers. Under convexity assumptions or under some marginal-cost pricing rules, each equilibrium will be Pareto efficient: In large economies, non- convexity also leads to quasi-equilibria that are nearly efficient. However, the concept of market equilibrium has been criticized by Austrians, post- Keynesians and others, who object to applications of microeconomic theory to real-world markets, when such markets are not usefully approximated by microeconomic models. Heterodox economists assert that micro-economic models rarely capture reality.
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London: The Economist (Profile Books). pp. 145–6. . of a long tailed investment (i.e. mortality risk). It is similar in nature to bond convexity or gamma that are exhibited in financial products such as bonds or options but is specific to portfolios replicating indices of shorter maturities.
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Kim and Altmann (2017) find that homophily may affect the evolution of the degree distribution of scale- free networks. More specifically, homophily may cause a bias towards convexity instead of the often hypothesised concave shape of networks.Kim, K., & Altmann, J. (2017). "Effect of Homophily on Network Formation".
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In mathematics, Hanner's inequalities are results in the theory of Lp spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of Lp spaces for p ∈ (1, +∞) than the approach proposed by James A. Clarkson in 1936.
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The dorsal exoskeleton consists of a cephalon, a pygidium and two or three thoracic somites with articulating half-rings, all non-calcified, supposedly of medium convexity. The axis is poorly defined. The cephalon is transversely oval, widest at midlength. The cephalon is wider than the pygidium.
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While it is possible to define a cost function ad hoc, frequently the choice is determined by the function's desirable properties (such as convexity) or because it arises from the model (e.g. in a probabilistic model the model's posterior probability can be used as an inverse cost).
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A cingulum is a convexity mesiodistally resembling a girdle, encircling the lingual surface at the cervical third, found on the lingual surface of anterior teeth. It is frequently identifiable as an inverted V-shaped ridge,Gray, Henry. XI. Splanchnology. 2a. The Mouth, from "Gray's Anatomy of the Human Body".
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Given that their definition is not in fact sound when indifference curves may be non-convex, it is reasonable to suppose that they meant the assumption of convexity in the former sense. Whether or not this is so, the definition has been widely adopted without any restriction of domain.
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The palmar metacarpal arteries (volar metacarpal arteries, palmar interosseous arteries), three or four in number, arise from the convexity of the deep volar arch. They run distally upon the Interossei, and anastomose at the clefts of the fingers with the common digital branches of the superficial volar arch.
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Starr (1969) applied the Shapley–Folkman–Starr theorem to prove that even without convex preferences there exists an approximate equilibrium. The Shapley–Folkman–Starr results bound the distance from an "approximate" economic equilibrium to an equilibrium of a "convexified" economy, when the number of agents exceeds the dimension of the goods. Following Starr's paper, the Shapley–Folkman–Starr results were "much exploited in the theoretical literature", according to Guesnerie, who wrote the following: > some key results obtained under the convexity assumption remain > (approximately) relevant in circumstances where convexity fails. For > example, in economies with a large consumption side, nonconvexities in > preferences do not destroy the standard results of, say Debreu's theory of > value.
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To address this, "effective" duration and -convexity are introduced. Here, similar to rho and vega above, the interest rate tree is rebuilt for an upward and then downward parallel shift in the yield curve and these measures are calculated numerically given the corresponding changes in bond value.See Fabozzi under Bibliography.
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Equivalently, it is a surjective TVS embedding. Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms. ;A necessary condition for a vector topology All of the above conditions are consequently a necessity for a topology to form a vector topology.
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Early examinations of the properties of equilibrium were based on an implicit definition as tangency, and convexity seems to have been implicitly assumed.Oscar Lange, ‘The Foundations of Welfare Economics’ (1942). There was no doubt that equilibrium would be reached: gradient ascent would lead to it. But the results lacked generality.
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ACM Names Fellows for Computing Advances that Are Driving Innovation , Association for Computing Machinery, December 8, 2011. In 2014 he was elected as a member of Academia Europaea,. and in 2015 as a fellow of the American Mathematical Society "for contributions to discrete and combinatorial geometry and to convexity and combinatorics.".
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The Plough and Fleece, occupying a 16th-century building, opened as a pub in the 19th century. The Millennium Green contains a sculpture called Convexity created by local artist Matthew Sanderson.Sanderson Sculpture The Millennium pavilion is also situated here. These were created as part of the Millennium celebrations at the start of 2000.
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More central members of a category are "between" the peripheral members. He postulates that most natural categories exhibit a convexity in conceptual space, in that if x and y are elements of a category, and if z is between x and y, then z is also likely to belong to the category.
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The conditions for the second theorem are stronger than those for the first, as consumers' preferences and production sets now need to be convex (convexity roughly corresponds to the idea of diminishing marginal rates of substitution i.e. "the average of two equally good bundles is better than either of the two bundles").
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A source for the report said that some of the positions in the endowment which had to be liquidated were in hedge funds run by Meyer's Convexity Capital and Seth Klarman's Baupost Group.Fabrikant, Geraldine, "Endowment Director Is on Harvard’s Hot Seat", The New York Times, Feb. 20, 2009. Retrieved 2-22-09.
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This idea was adapted by Marks et al. to the crystallographic phase problem. With a feasible set approach, constraints can be considered convex (highly convergent) or non- convex (weakly convergent). Imposing these constraints with the algorithm detailed earlier can converge towards unique or non-unique solutions, depending on the convexity of the constraints.
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Vector spaces (particularly, n-spaces) over an ordered field exhibit some special properties and have some specific structures, namely: orientation, convexity, and positively-definite inner product. See Real coordinate space#Geometric properties and uses for discussion of those properties of Rn, which can be generalized to vector spaces over other ordered fields.
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However, when the change in the value of the underlier is not small, the second-order term, \Gamma\,, cannot be ignored: see Convexity (finance). In practice, maintaining a delta neutral portfolio requires continuous recalculation of the position's Greeks and rebalancing of the underlier's position. Typically, this rebalancing is performed daily or weekly.
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Closing out a contract almost always involves reaching out to the counterparty. Compared to their futures counterparts, forwards (especially Forward Rate Agreements) need convexity adjustments, that is a drift term that accounts for future rate changes. In futures contracts, this risk remains constant whereas a forward contract's risk changes when rates change.
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In the mid-1960s with Michael Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms. They discovered the grand antiprism in the process, the only non-Wythoffian uniform polychoron.J. H. Conway, "Four-dimensional Archimedean polytopes", Proc. Colloquium on Convexity, Copenhagen 1965, Kobenhavns Univ. Mat.
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However, a key difference is that the dual subgradient algorithm is typically analyzed under restrictive strict convexity assumptions that are needed for the primal variables x(t) to converge. There are many important cases where these variables do not converge to the optimal solution, and never even get near the optimal solution (this is the case for most linear programs, as shown below). On the other hand, the drift-plus- penalty algorithm does not require strict convexity assumptions. It ensures that the time averages of the primals converge to a solution that is within O(1/V) of optimality, with O(V) bounds on queue sizes (it can be shown that this translates into an O(V2) bound on convergence time).
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Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics. It has close connections to convex analysis, optimization and functional analysis and important applications in number theory. Convex geometry dates back to antiquity. Archimedes gave the first known precise definition of convexity.
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By the nineteenth century, the design and usage began to be specific within each region. In Western Europe the custom became to use the quotation mark pairs with the convexity pointing outward. In Britain those marks were elevated to the same height as the top of capital letters: . Clearly distinguishable apostrophe and angular quotation marks.
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589 and Indic scripts. On the other hand, Greek, Cyrillic, Arabic and Ethiopic adopted the French "angular" quotation marks, . The Far East angle bracket quotation marks, , are also a development of the in-line angular quotation marks. In Central Europe, however, the practice was to use the quotation mark pairs with the convexity pointing inward.
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The surface is smooth, except for the base which contains two or three very faint concentric striae. The circular aperture has a continuous peristome. The columella has a thick callus that connects to the convexity of the penultimate whorl. it covers for the greater part the umbilicus, that is reduced to a narrow, arched chink.
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Convexity is an important topic in economics. In the Arrow–Debreu model of general economic equilibrium, agents have convex budget sets and convex preferences: At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. The profit function is the convex conjugate of the cost function. Convex analysis is the standard tool for analyzing textbook economics.
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As noted by Lindblad and Uhlmann , if, in equation (), one takes K=1 and r=1-p, A=\rho and B=\sigma and differentiates in p at p=0 one obtains the Joint convexity of relative entropy : i.e., if \rho=\sum_k\lambda_k\rho_k, and \sigma=\sum_k\lambda_k\sigma_k, then where \lambda_k\geq 0 with \sum_k\lambda_k=1.
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By strict convexity, all agents strictly prefer z to x and to y. Hence, x and y cannot be weakly-PE. Theorem 3 (Svensson): If all agents' preferences are strongly monotone, and for every PE utility-profile u, the set A(u) is convex, then PEEF allocations exist. The proof uses the Kakutani fixed-point theorem.
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The height of the shell attains 39 mm. The umbilicated shell is large, polished, solid, and contains 8 whorls. It is straw-yellow lineated with red-brown, and has a broad rose-colored peripheral band. The walls of the umbilicus are marked with incremental lines, slightly excavated near the carina, above convex, the convexity revolving with the whorl.
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Now standard proofs of the fact that the set of geodesic words in a word-hyperbolic group is a regular language also use finiteness of the number of cone types. Cannon's work also introduced an important notion of almost convexity for Cayley graphs of finitely generated groups,James W. Cannon. Almost convex groups. Geometriae Dedicata, vol.
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The assumption of local convexity for the ambient space is necessary, because constructed a counter-example for the non-locally convex space where . Linearity is also needed, because the statement fails for weakly compact convex sets in CAT(0) spaces, as proved by . However, proved that the Krein–Milman theorem does hold for metrically compact CAT(0) spaces.
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Bregman divergences can be interpreted as limit cases of skewed Jensen divergences (see Nielsen and Boltz, 2011). Jensen divergences can be generalized using comparative convexity, and limit cases of these skewed Jensen divergences generalizations yields generalized Bregman divergence (see Nielsen and Nock, 2017). The Bregman chord divergence is obtained by taking a chord instead of a tangent line.
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Today's church in Peterswald was built in 1765 and 1766. The church, built in the middle of the village, is an aisleless church with an offset quire. The former west gable got its current façade through tower construction in 1923. The bell tower's features include sound slats on all sides, a convexity in the upswept roof and a cross.
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Lanelater fuscipes can reach a length of . These large click beetles have a dark brown body. Usually they show a puncturation of the pronotum and an evident striation of the elytra, but the species is quite variable, especially in the length and in degree of convexity of the prothorax. Its larvae live in the coconut palms.
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Starting from the cardiac orifice at the incisura cardiaca, it forms an arch backward, upward, and to the left; the highest point of the convexity is on a level with the sixth left costal cartilage. From this level it may be followed downward and forward, with a slight convexity to the left as low as the cartilage of the ninth rib; it then turns to the right, to the end of the pylorus. Directly opposite the incisura angularis of the lesser curvature the greater curvature presents a dilatation, which is the left extremity of the pyloric part; this dilatation is limited on the right by a slight groove, the sulcus intermedius, which is about 2.5 cm, from the duodenopyloric constriction. The portion between the sulcus intermedius and the duodenopyloric constriction is termed the pyloric antrum.
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The concave > portions of the indifference curves and their many-dimensional > generalizations, if they exist, must forever remain in unmeasurable > obscurity.: : Following Hotelling's pioneering research on non-convexities in economics, research in economics has recognized non-convexity in new areas of economics. In these areas, non-convexity is associated with market failures, where any equilibrium need not be efficient or where no equilibrium exists because supply and demand differ. Non-convex sets arise also with environmental goods and other externalities,Pages 106, 110–137, 172, and 248: and with market failures, and public economics.Pages 63–65: Starrett discusses non-convexities in his textbook on public economics (pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236): Non-convexities occur also with information economics, and with stock markets (and other incomplete markets).
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They are slightly fretted by the longitudinals . Between them are little rounded furrows of about twice their breadth. Colour: the spiral threads are porcellaneous, the furrows translucent white, and the surface is a little glossy. The spire is rather short, conical, but slightly concave, with hardly any interruption in its profile-lines by the very slightly impressed suture and convexity of the whorls.
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However, during sleep access to the motor system is blocked (by inactivation of the dorsolateral frontal convexity). As a result, activation moves backwards toward the perceptual areas. This is why the dreamer doesn't engage in motivated behaviours but imagines them. Furthermore, there is inactivation of the reflective system in the limbic brain which leads the dreamer to mistake the dream for reality.
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The 3 to 5 subcircular holes are somewhat tubular. The right side is decidedly straighter than the left, the convexity being variable. The color of the shell is brown, variously marked with white and green. The spiral lirae are deeply cut and number 24-30 (counting along the lip edge) exclusive of 5 or 6 below the row of holes.
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James Andrew Clarkson was an American mathematician and professor of mathematics who specialized in number theory. He is known for proving inequalities in Hölder spaces, and derived from them, the uniform convexity of . His proofs are known in mathematics as Clarkson's inequalities. He was an operations' analyst during World War II, and was awarded the Medal of Freedom for his achievements.
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American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. xlii+521 pp. In 2005, Max-K. von Renesse and Karl-Theodor Sturm showed that the a lower bound of the Ricci curvature on a Riemannian manifold could be characterized by optimal transportation, in particular by the convexity of a certain "entropy" functional along geodesics of the associated Wasserstein metric space.
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The plantar metatarsal arteries (digital branches) are four in number, arising from the convexity of the plantar arch. They run forward between the metatarsal bones and in contact with the Interossei. They are located in the fourth layer of the foot. Each divides into a pair of plantar digital arteries which supply the four webs and the adjacent sides of the toes.
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In economics, convex hulls can be used to apply methods of convexity in economics to non-convex markets. In geometric modeling, the convex hull property Bézier curves helps find their crossings, and convex hulls are part of the measurement of boat hulls. And in the study of animal behavior, convex hulls are used in a standard definition of the home range.
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The complete bow is (5 ft) long. Bows of Holmegaard-type were in use until the Bronze Age; the convexity of the midsection has decreased with time. Mesolithic pointed shafts have been found in England, Germany, Denmark, and Sweden. They were often rather long, up to (4 ft) and made of European hazel (Corylus avellana), wayfaring tree (Viburnum lantana) and other small woody shoots.
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In mathematics, Tonelli's theorem in functional analysis is a fundamental result on the weak lower semicontinuity of nonlinear functionals on Lp spaces. As such, it has major implications for functional analysis and the calculus of variations. Roughly, it shows that weak lower semicontinuity for integral functionals is equivalent to convexity of the integral kernel. The result is attributed to the Italian mathematician Leonida Tonelli.
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Four of those are arranged on the dorso- lateral part of the elytron. The two anterior spots form an roughly half-moon shaped oval with the convexity directed towards the suture of the elytron. The two posterior ones make a more irregular shape, formed by the intersection of two circular spots. Finally, the fifth spot covers the length of the elytron's suture, enlarging towards the posterior stretch.
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Much of the ground structure is undecorated, above intricately decorated. The overwhelmingly vertical decoration of the facade is granted liveliness by horizontal convexity. In his will, this bachelor called this church his beloved daughter. He also renovated the exterior renewal of the ancient Santa Maria della Pace (1656–1667), and the façade (with an unusual loggia) of Santa Maria in Via Lata (appr. 1660).
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Noticeably, on the contrary of the usual Arrhenius case, the barrier or activation energy is temperature dependent and k_{d}(T) has different concavities depending on the value of the d parameter (see Figs.1 and 1a). Thus, a positive convexity means that E_{a} decreases with increasing temperature. This general result is explained by a new Tolman-like interpretation of the activation energy through Eq.(8).
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The convexity is straw-color, a deep brown band revolving just within the carina. The spiral sculpture outside the carina, which is not very sharp, consists of two strong beaded spirals alternating with two fine simple brown elevated lines. Then follow nine subequal, finer spirals, less coarsely beaded, the upper angle of the aperture being at the ninth. All these are straw-colored with brown interspaces.
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The lateral dorsal nucleus is a nucleus of the thalamus. It is the most anterior of the dorsal lateral nuclei. It acts in concert with the anterior nuclei of thalamus. It receives significant input from several subdivisions of visual cortex, and has a primary output to parietal cortex on the dorsolateral cortical convexity, giving it access to limbic forebrain nuclei important for emotion and behavior functions.
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This formula can be used to calculate the VaR of the portfolio by ignoring higher order terms. Typically cubic or higher terms are truncated. Quadratic terms, when included, can be expressed in terms of (multi-variate) bond convexity. One can make assumptions about the joint distribution of the interest rates and then calculate VaR by Monte Carlo simulation or, in some special cases (e.g.
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Not every planar graph corresponds to a convex polyhedron in this way: the trees do not, for example. Steinitz's theorem says that the polyhedral graphs formed from convex polyhedra are precisely the finite 3-connected simple planar graphs. More generally, Euler's formula applies to any polyhedron whose faces are simple polygons that form a surface topologically equivalent to a sphere, regardless of its convexity.
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Arches and barrel roofs are completely absent. Gable and eave curves are gentler than in China and columnar entasis (convexity at the center) limited. The roof is the most visually impressive component, often constituting half the size of the whole edifice. The slightly curved eaves extend far beyond the walls, covering verandas, and their weight must therefore be supported by complex bracket systems called tokyō.
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Opisthocoelous vertebrae are the opposite, possessing anterior convexity and posterior concavity. They are found in salamanders, and in some non-avian dinosaurs. Heterocoelous vertebrae have saddle-shaped articular surfaces. This type of configuration is seen in turtles that retract their necks, and birds, because it permits extensive lateral and vertical flexion motion without stretching the nerve cord too extensively or wringing it about its long axis.
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Adjacent areas of the body are represented by adjacent areas in the cortex. When body parts are drawn in proportion to the density of their innervation, the result is a "little man": the cortical homunculus. Many textbooks have reproduced the outdated Penfield-Rasmussen diagram [ref?], with the toes and genitals on the mesial surface of the cortex when they are actually represented on the convexity.
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The crossing of these two sets of lines produces a finely cancellated sculpture over the whole surface, but the transverse lines are usually more evident on the convexity of the whorls, while the spiral lines are more conspicuous anteriorly, and on the siphon. The aperture is relatively large, oblong-elliptical, slightly obtusely angled posteriorly. The sinus is shallow, but distinct, evenly concave. The outer lip is elsewhere evenly convex.
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This means that the affine combination belongs to , for all and in , and in the interval . This implies that convexity (the property of being convex) is invariant under affine transformations. This implies also that a convex set in a real or complex topological vector space is path-connected, thus connected. A set is ' if every point on the line segment connecting and other than the endpoints is inside the interior of .
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The collection of convex subsets of a vector space, an affine space, or a Euclidean space has the following properties:Soltan, Valeriu, Introduction to the Axiomatic Theory of Convexity, Ştiinţa, Chişinău, 1984 (in Russian). #The empty set and the whole space are convex. #The intersection of any collection of convex sets is convex. #The union of a sequence of convex sets is convex, if they form a non-decreasing chain for inclusion.
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In the Indian River Lagoon in Florida, two other scallops occur: the bay scallop, (Argopecten irradians), which generally has a uniform gray to gray-brown coloration with distinct convexity of the right (lower) valve. The other is the rough scallop, (Aequipecten muscosus), which has unequal 'ears' and has sharp scales on the lower surface of the ribs. The color of the rough scallop is yellow, orange, red or brownish.
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The tooth sockets are rectangular in upper view. The jaw carries at least ten teeth. These are relatively slender, but the front teeth have a D-shaped cross section with the convexity facing outward; the rear teeth are dagger-shaped and more flattened. The cutting edges are convex and show up to thirteen denticles per five millimetres at the crown base, and up to eight denticles near the apex.
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Further research was largely grounded in Gestalt psychology and Perception, which largely corroborated and expanded upon Dent's model. In summarizing the work to date, MacEachren added Orientation and Convexity to Dent's list, with the acknowledgment that these are relatively minor influences compared to the others.MacEachren, Alan M., How Maps Work, Guilford Press, 1995 MacEachren discussed the concept of visual levels as "related," but not equal, to figure- ground contrast.
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In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin. This theorem bounds the norms of linear maps acting between spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others.
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The shell contains about 12 whorls. These are angulate in the middle and show obliquely nodular plicae (these are more attenuate in the lower portions). The whorls are concave above and subconvex below. In the body whorl there is a slight convexity or rounded ridge just below the suture and above the excavation, below which occur the oblique nodose plications which gradually diminish in strength as the aperture is approached.
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The original tooth shows, as far as can be deduced from the surviving illustrations, the rare combination of being spatulate and having a convex inner side, though the convexity is slight. Its crown is short and wide, slightly curving to the inside. The outer side is strongly convexly curved from the front to the rear. On this side a shallow groove is present, running parallel to the rear edge.
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Previous work includes the invention of the microcanonical ensemble approach to lattice gauge theory with Aneesur Rahman, work on the convexity of the effective potential of quantum field theory, work on Langevin dynamics in quantum field theory with John R. Klauder, a monograph on quantum triviality, constraints on the Higgs boson and papers on black holes and superconductors. His work in these areas is highly cited and notable.
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A voting system is consistent if, whenever the electorate is divided (arbitrarily) into several parts and elections in those parts garner the same result, then an election of the entire electorate also garners that result. Smith calls this property separability and Woodall calls it convexity. It has been proven a ranked voting system is "consistent if and only if it is a scoring function", i.e. a positional voting system.
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Moshe Shaked was a Fellow of the Institute of Mathematical Statistics. Shaked was a leading figure in stochastic order and distribution theory. He published widely in applied probability and statistics. He became most celebrated internationally for his collection of influential papers on stochastic order and multivariate dependence. Shaked’s contribution also includes pioneering studies on stochastic convexity and on multivariate phase-type distributions, with important applications in reliability modelling and queueing analysis.
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However, polynomially convex sets do not behave as nicely as convex sets. Kallin studied conditions under which unions of convex balls are polynomially convex, and found an example of three disjoint cubical cylinders whose union is not polynomially convex.. As part of her work on polynomial convexity, she proved a result now known as Kallin's lemma, giving conditions under which the union of two polynomially convex sets remains itself polynomially convex...
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The abdominal aorta begins at the level of the diaphragm, crossing it via the aortic hiatus, technically behind the diaphragm, at the vertebral level of T12. It travels down the posterior wall of the abdomen, anterior to the vertebral column. It thus follows the curvature of the lumbar vertebrae, that is, convex anteriorly. The peak of this convexity is at the level of the third lumbar vertebra (L3).
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The envy-free cake-cutting problem is to partition the cake to n disjoint pieces, one piece per agent, such for each agent, the value of his piece is weakly larger than the values of all other pieces (so no agent envies another agent's share). A corollary of the Dubins–Spanier convexity theorem (1961) is that there always exists a "consensus partition" - a partition of the cake to n pieces such that every agent values every piece as exactly 1/n. A consensus partition is of course EF, but it is not PE. Moreover, another corollary of the Dubins–Spanier convexity theorem is that, when at least two agents have different value measures, there exists a division that gives each agent strictly more than 1/n. This means that the consensus partition is not even weakly PE. Envy- freeness, as a criterion for fair allocation, were introduced into economics in the 1960s and studied intensively during the 1970s.
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The infraspinatous fossa (infraspinatus fossa, infraspinous fossa) of the scapula is much larger than the supraspinatous fossa; toward its vertebral margin a shallow concavity is seen at its upper part; its center presents a prominent convexity, while near the axillary border is a deep groove which runs from the upper toward the lower part. The medial two-thirds of the fossa give origin to the Infraspinatus; the lateral third is covered by this muscle.
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Radius of metric space is the infimum of radii of metric balls which contain the space completely. Radius of convexity at a point p of a Riemannian manifold is the largest radius of a ball which is a convex subset. Ray is a one side infinite geodesic which is minimizing on each interval Riemann curvature tensor Riemannian manifold Riemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the same time.
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Gable and eave curves are gentler than in China and columnar entasis (convexity at the center) limited. The roof is the most visually impressive component, often constituting half the size of the whole edifice. The slightly curved eaves extend far beyond the walls, covering verandas, and their weight must therefore be supported by complex bracket systems called tokyō. These oversize eaves give the interior a characteristic dimness, which contributes to the temple's atmosphere.
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Countershading acts as a form of camouflage by 'painting out' the self-shadowing of the body or object. The result is a 'flat' appearance, instead of the 'solid' appearance (with visual convexity) of the body before countershading. Hannah Rowland, reviewing countershading 100 years after Abbott Thayer, observed that countershading, which she defines as "darker pigmentation on those surfaces exposed to the most lighting" is a common but poorly understood aspect of animal coloration.Rowland, 2009.
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The external surface form an intritacalx (= the chalky outer layer) which gives it a chalky appearance, even on fresh specimens, as is characteristic of the genus. The body whorl has maximum convexity just below the deep suture. Then it is rather parallel sided at the periphery and markedly constricted around the siphonal canal. The aperture is strongly thickened externally by the last varix, with an indistinct inner rim, bearing 5 faint denticles on aged specimens.
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Reverse convertibles are a less common variation, mostly issued synthetically. They would be opposite of the vanilla structure: the conversion price would act as a knock-in short put option: as the stock price drops below the conversion price the investor would start to be exposed the underlying stock performance and no longer able to redeem at par its bond. This negative convexity would be compensated by a usually high regular coupon payment.
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The eyes themselves are small with not much convexity and the ommatidia are separated from each other by hairs. The antennae are not as robust as other species, with scapes that protrude past the rear borders of the eyes. The mesosoma is slightly shorter than the head length, with a distinct suture line at the pronotum. The petiole node is scale shaped and nearly four times wider than long, double that of other Bradoponera species.
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Second-order approximation is the term scientists use for a decent-quality answer. Few simplifying assumptions are made, and when a number is needed, an answer with two or more significant figures ("the town has or thirty-nine hundred residents") is generally given. In mathematical finance, second-order approximations are known as convexity corrections. As in the examples above, the term "2nd order" refers to the number of exact numerals given for the imprecise quantity.
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The shell is very small, measuring 2 mm. It is semitranslucent, bluish-white. The nuclear whorls are quite large, forming a moderately elevated, helicoid spire, whose axis is at right angles to that of the succeeding turns, in the first of which it is about one-fourth immersed. The five post-nuclear whorls are decidedly rounded, with the greatest convexity falling on the anterior third of the whorls, between the sutures, appressed at the summit.
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Composer Sergei Lyapunov, mathematician Aleksandr Lyapunov, and philologist Boris Lyapunov were close relatives of Alexey Lyapunov. In 1928, Lyapunov enrolled at Moscow State University to study mathematics, and in 1932 he became a student of Nikolai Luzin. Under his mentorship, Lyapunov began his research in descriptive set theory. He became world-wide known for his theorem on the range of an atomless vector-measure in finite dimensions, now called the Lyapunov Convexity Theorem.
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For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. Quasiconvexity and quasiconcavity extend to functions with multiple arguments the notion of unimodality of functions with a single real argument.
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An affine transformation preserves: # collinearity between points: three or more points which lie on the same line (called collinear points) continue to be collinear after the transformation. # parallelism: two or more lines which are parallel, continue to be parallel after the transformation. # convexity of sets: a convex set continues to be convex after the transformation. Moreover, the extreme points of the original set are mapped to the extreme points of the transformed set.
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Of the two borders the superior is concave, the inferior convex; they afford attachment to the intercostales interni: the upper border of the sixth gives attachment also to the pectoralis major. The inferior borders of the sixth, seventh, eighth, and ninth cartilages present heel-like projections at the points of greatest convexity. These projections carry smooth oblong facets which articulate with facets on slight projections from the upper borders of the seventh, eighth, ninth, and tenth cartilages, respectively.
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The length of the shell varies between 15 mm and 50 mm. The rather thin, smooth shell is ovate and conical. Its ground color is whitish, with spots of a more or less dark red, and upon the lower whorl, a very large spot of the same tint, but deeper. Upon the convexity of this whorl, may be counted nine or ten distant, parallel and transverse lines of a bright chestnut color, sometimes brown, at other times blackish.
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The 1-norm is not strictly convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. Correspondingly, the median (in this sense of minimizing) is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation. The 0-"norm" is not convex (hence not a norm). Correspondingly, the mode is not unique – for example, in a uniform distribution any point is the mode.
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Further back, immediately in front of the nasoantorbital fenestra, the palatal ridge became a strong, blunt, convex keel. This convexity fit into the symphyseal shelf at the front end of the lower jaw, and they would have tightly interlocked when the jaws were closed. The palatal ridge ended in a strongly concave area unique to this species. The postpalatine fenestrae (openings behind the palatine bone) were oval and very small, differing from those of related species.
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Coxeter developed the theory further. The conceptual issues raised by complex polytopes, non-convexity, duality and other phenomena led Grünbaum and others to the more general study of abstract combinatorial properties relating vertices, edges, faces and so on. A related idea was that of incidence complexes, which studied the incidence or connection of the various elements with one another. These developments led eventually to the theory of abstract polytopes as partially ordered sets, or posets, of such elements.
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In humans, the total optical power of the relaxed eye is approximately 60 dioptres. The cornea accounts for approximately two-thirds of this refractive power (about 40 dioptres) and the crystalline lens contributes the remaining one-third (about 20 dioptres). In focusing, the ciliary muscle contracts to reduce the tension or stress transferred to the lens by the suspensory ligaments. This results in increased convexity of the lens which in turn increases the optical power of the eye.
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The ten whorls of the teleoconch are moderately rounded. They are somewhat overhanging, the greatest convexity being on the lower third of the exposed portion of the whorls, traversed by 14 broad, coarse and strong, oblique, and somewhat flexuous axial ribs on the fourth and seventh whorl and 18 on the eighth. These ribs extend over the angulated periphery to the umbilical region, appearing fainter on the base. The deep intercostal grooves terminate at the periphery, i. e.
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Heinrich Walter Guggenheimer (born 21 July 1924) is a German-born Swiss- American mathematician who has contributed to knowledge in differential geometry, topology, algebraic geometry, and convexity. He has also contributed volumes on Jewish sacred literature. Heinrich Guggenheimer was born in Nuremberg, Germany. He is the son of Marguerite Bloch and the Physicist Dr. Siegfried Guggenheimer. He studied in Zurich, Switzerland at the Eidgenössiche Technische Hochschule, receiving his diploma in 1947 and a D.Sc. in 1951.
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The palatal view of this tooth considers the portion of the tooth visible from the side where the tongue would be. The palatal side of the maxillary central incisor has a small convexity, called a cingulum near the cervical line and has a large concavity, called the lingual fossa. Along the mesial and distal sides are slightly raised portions called marginal ridges. The lingual incisal edge is also raised slightly to the level of the marginal ridges.
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In actual markets, the assumption of constant interest rates and even changes is not correct, and more complex models are needed to actually price bonds. However, these simplifying assumptions allow one to quickly and easily calculate factors which describe the sensitivity of the bond prices to interest rate changes. Convexity does not assume the relationship between Bond value and interest rates to be linear. For large fluctuations in interest rates, it is a better measure than duration.
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The FPP is also preserved by any retraction. According to Brouwer fixed point theorem every compact and convex subset of a Euclidean space has the FPP. More generally, according to the Schauder-Tychonoff fixed point theorem every compact and convex subset of a locally convex topological vector space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP.
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Unlike both Western and some Chinese architecture, the use of stone is avoided except for certain specific uses, for example temple podia and pagoda foundations. The general structure is almost always the same: posts and lintels support a large and gently curved roof, while the walls are paper-thin, often movable and never load-bearing. Arches and barrel roofs are completely absent. Gable and eave curves are gentler than in China and columnar entasis (convexity at the center) limited.
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Williston explains that the humerus and femur of Dissorophus are solidly built and stouter. The humerus has "deep lateral curvatures and wide supracondicular ridges" while the femur is a lot stronger built compared to the humerus. He also mentions that the articular surface of Dissorophus femur is "flattened with sharp rims on the antero-posterior convexity". He adds that both femur and humerus are both "expanded on the inner and outer side and narrow in the middle".
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Sculpture—spirals con sist of numerous fine lines on the keel and above and below it, indistinct in some specimens, on the body whorl usually more pronounced. The axials consist of growthlines only, variable in strength, arcuate between the keel and suture above. The aperture is small, ovate and produced into a long siphonal canal. The outer lip is angled at the keel, thence concave followed by a convexity, and beneath this rapidly narrowing to the siphonal canal.
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Convexity is not strictly necessary for BFPT. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, BFPT is equivalent to forms in which the domain is required to be a closed unit ball D^n. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also closed, bounded, connected, without holes, etc.). The following example shows that BFPT doesn't work for domains with holes.
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The kopis sword was a one-handed weapon. Early examples had a blade length of up to 65 cm (25.6 inches), making it almost equal in size to the spatha. Later Macedonian examples tended to be shorter with a blade length of about 48 cm (18.9 inches). The kopis had a single-edged blade that pitched forward towards the point, the edge being concave on the part of the sword nearest the hilt, but swelling to convexity towards the tip.
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The length of the shell attains 22 mm, its diameter 11 mm. (Original description) The stout-fusiform shell has a rather short, regularly tapered spire, a broad and deep posterior sinus, and a very short and wide siphonal canal. It contains seven, moderately convex whorls with a wide, concave subsntural band, which is covered with regular, strongly receding, raised lines, but destitute of spiral sculpture. The shoulder is rather prominent where the concave band joins the convexity of the whorl.
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An exact division (aka consensus division) is a partition of the cake into n pieces such that each agent values each piece at exactly 1/n. The existence of such a division is a corollary of the Dubins–Spanier convexity theorem. Moreover, there exists such a division with at most n(n-1)^2 cuts; this is a corollary of the Stromquist–Woodall theorem and the necklace splitting theorem. In general, an exact division cannot be found by a finite algorithm.
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In K. guimarotae, the two halves of the dentary diverge from each other at an angle of 20° near the front, then 40° near the back. It also bears two convexities on the bottom of the jaw, one at the third and fourth teeth and another at the eighth to tenth teeth. The latter convexity is replaced by a concavity in K. langenbergensis. In both species, the top margin of the jaw behind the tooth row slopes upwards in a straight line.
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The closures come after assets plummeted from a high of over $3 billion in 2014 in its flagship credit strategy. Tricadia will return around $800 million to investors in its Credit Strategy Fund and the company has already given customers over $150 million to its Convexity fund. In August 2019, Tricadia buys First Midwest Bancorp Inc, iShares Core S&P; 500 ETF. As of the 2nd Quarter of 2019, Tricadia Capital Management, LLC owns 3 stocks with value totalling $43 million.
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The larvae feed on Dichanthelium dichotomum. They mine the leaves of their host plant. They mine a small basal leaf in the spring, eating out almost the entire substance of the leaf. Just before pupation, it enters one of the lower stem leaves, in which it makes a small inconspicuous mine, scarcely larger than the larva, but broadening at its anterior end towards the tip of the leaf, slightly inflated, and showing us a convexity on the upper surface of the leaf.
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In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph, (U ∪ V, E), is said to be convex over the vertex set U if U can be enumerated such that for all v ∈ V the vertices adjacent to v are consecutive. Convexity over V is defined analogously. A bipartite graph (U ∪ V, E) that is convex over both U and V is said to be biconvex or doubly convex.
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Parallel wavy lines (rivers) were spaced apart with the purple contours on the inside and orange on the outside. Opposite of the principle, the rivers were not perceived as filed in, but the interspaces between the rivers were perceived to be filled in, or as figure in this case. The fifth experiment was watercolor illusion compared to convexity. According to the “law of the inside” the concave regions of the stimulus should be perceived as ground and the convex ones perceived as figure.
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The interventricular septum (IVS, or ventricular septum, or during development septum inferius) is the stout wall separating the ventricles, the lower chambers of the heart, from one another. The ventricular septum is directed obliquely backward to the right and curved with the convexity toward the right ventricle; its margins correspond with the anterior and posterior interventricular sulci. The lower part of the septum, which is the major part, is thick and muscular, and its much smaller upper part is thin and membraneous.
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He was an invited speaker at the International Congress of Mathematicians in Seoul in 2014. He was elected a Fellow of the American Mathematical Society in 2012. He invented the displacement interpolation between probability measures and studied the convexity of various entropies and energies along it, later linking these to Ricci curvature and eventually to the Einstein equations of general relativity. He has pioneered applications of optimal transport to economic problems such as hedonic matching, investment to match, and multidimensional screening.
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PEEF allocations might fail to exist even when all preferences are convex, if there is production and the technology has increasing-marginal-returns. Proposition 6 (Vohra): There exist economies in which all preferences are continuous strongly-monotone and convex, the only source of non-convexity in the technology is due to fixed costs, and there exists no PEEF allocation. Thus, the presence of increasing returns introduces a fundamental conflict between efficiency and fairness. However, envy-freeness can be weakened in the following way.
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Rådström's Ph.D. students included Per Enflo and Martin Ribe, both of whom wrote Ph.D. theses in functional analysis. In the uniform and Lipschitz categories of topological vector spaces, Enflo's results concerned spaces with local convexity, especially Banach spaces. In 1970,: Hans Rådström died of a heart attack. Enflo supervised one of Rådström's Linköping students, Lars-Erik Andersson, from 1970–1971, helping him with his 1972 thesis, On connected subgroups of Banach spaces, on Hilbert's fifth problem for complete, normed spaces.
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A. plautini, enrolledThe headshield (or cephalon) and tailshield (or pygidium) are semicircular and without a border (defined by a furrow or a change in convexity parallel to its margin). The cephalon is of approximately equal size as the pygidium (or isopygous). The central raised area of the cephalon (or glabella) is long, reaching the frontal margin. It may have faint lateral glabellar furrows or be smooth, and sometimes an inconspicuous tubercle is present just in front of the hardly discernible occipital ring.
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This convexity is less prominent than more derived titanosaurs, but is still diagnostic of the clade as a whole. The first caudal has a flat anterior and slightly convex posterior face, different from the subsequent vertebrae similar to in Epachthosaurus. A prominent depression is present on the bottom surface of some anterior caudals, a feature present in diplodocids and multiple titanosaurs. Anterior caudals are shorter proportionally, the bones becoming almost double the proportional length towards the end of the tail.
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Its most prominent point behind corresponds to the spinous process of the seventh thoracic vertebra. This curve is known as a kyphotic curve. Lateral lumbar X-ray of a 34-year-old male The lumbar curve is more marked in the female than in the male; it begins at the middle of the last thoracic vertebra, and ends at the sacrovertebral angle. It is convex anteriorly, the convexity of the lower three vertebrae being much greater than that of the upper two.
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Between them are little rounded furrows of about twice their breadth. The spiral threads are porcellanous, the furrows translucent white, and the surface is a little glossy. The spire is rather short, conical, but slightly concave, with hardly any interruption in its profile-lines by the very slightly impressed suture and the convexity of the whorls. The protoconch consists of 2½ rounded subcylindrical whorls rising to a small rounded point, where the extreme tip hardly projects and is bent down on one side.
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Eva Marianne Kallin Pohlmann is a professor emerita of mathematics at Brown University. Her research concerns function algebras, polynomial convexity, and Tarski's axioms for Euclidean geometry. Kallin attended the University of California, Berkeley as an undergraduate, and graduated with an A.B. in mathematics in 1953 and an M.S. in 1956.Commencement program from Berkeley in 1950 showing Kallin as the freshman recipient of a scholarship; Commencement program from 1956 showing her with an A.B. in 1953 and an M.S. in 1956.
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These are a little more prominent upon the convexity of the longitudinal ribs than in their interstices, and it is their prolongation in this part, which causes them to resemble small spines. Its color is reddish, varied with fawn-colored or clear chestnut-brown spots. Oftentimes the lower whorl presents, towards its middle, a transverse brown band, the half only of which can be seen upon the upper whorls, the whole length of the sutures. The aperture is whitish, ovate, elongated, and narrowed towards its base.
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Historically, ideas from linear programming have inspired many of the central concepts of optimization theory, such as duality, decomposition, and the importance of convexity and its generalizations. Likewise, linear programming was heavily used in the early formation of microeconomics and it is currently utilized in company management, such as planning, production, transportation, technology and other issues. Although the modern management issues are ever-changing, most companies would like to maximize profits and minimize costs with limited resources. Therefore, many issues can be characterized as linear programming problems.
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L. salmonicolor can be found throughout most of Europe (see map) from September to November, the species' growing season. The fungus grows primarily near to the roots of fir trees, where it receives nutrients for its growth, participating in ectomycorrhiza with the roots of its host plant. The top of the pileus is an orange-reddish color, with rare spots of green in older decaying specimens. The cap is also slightly depressed in the center after an initial convexity, and irregularly shaped, sometimes with lobes.
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In the "local theory of Banach spaces", Pisier and Bernard Maurey developed the theory of Rademacher type, following its use in probability theory by J. Hoffman–Jorgensen and in the characterization of Hilbert spaces among Banach spaces by S. Kwapień. Using probability in vector spaces, Pisier proved that super-reflexive Banach spaces can be renormed with the modulus of uniform convexity having "power type". His work (with Per Enflo and Joram Lindenstrauss) on the "three–space problem" influenced the work on quasi–normed spaces by Nigel Kalton.
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23 in the 2nd edition.) about decreasing marginal rates of substitutionHicks, Sir John Richard; Value and Capital, Chapter I. "Utility and Preference" §7–8. would then have to be introduced to have convexity of indifference curves. For those who accepted that indifference curve analysis superseded earlier marginal utility analysis, the latter became at best perhaps pedagogically useful, but "old fashioned" and observationally unnecessary.Samuelson, Paul Anthony; "Complementarity: An Essay on the 40th Anniversary of the Hicks-Allen Revolution in Demand Theory", Journal of Economic Literature vol 12 (1974).
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The Kakeya needle problem asks whether there is a minimum area of a region D in the plane, in which a needle of unit length can be turned through 360°. This question was first posed, for convex regions, by . The minimum area for convex sets is achieved by an equilateral triangle of height 1 and area 1/, as Pál showed. Kakeya seems to have suggested that the Kakeya set D of minimum area, without the convexity restriction, would be a three-pointed deltoid shape.
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Lydekker also used the improved spelling "Hoplosaurus" but the original Oplosaurus has priority. The tooth is large, 85 mm (3.35 in) tall in total, with a spatulate crown 52 mm (2.05 in) tall, comparable to Brachiosaurus; it has a pointed tip, a slightly compressed form "cheek" to tongue, a slight convexity to the base of the tongue-facing side, and wear facets. It is vaguely like a Brachiosaurus tooth, which is why the genus has for a time been referred to the Brachiosauridae.McIntosh, J.S. (1990). Sauropoda.
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They are marked by two strong, spiral keels between the sutures, a third at the periphery, and a fourth on the middle of the base, the last two somewhat less strong than the rest. The posterior keel forms the strong tabulation at the summit of the whorls and is strongly tuberculated, 14 tubercles appearing upon the second and 20 upon the remaining whorls. The space between the keels is marked by rather strong lines of growth. The greatest convexity coincides at the superperipheral keel.
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In 1970 Figiel graduated in mathematics at the University of Warsaw. He received his doctorate in 1972 under the supervision of Aleksander Pełczyński and then habilitated in 1975 with habilitation thesis O modułach wypukłości i gładkości (On modules of convexity and smoothness) at the (Instytut Matematyczny PAN). There Figiel was appointed in 1983 an associate professor and in 1990 a full professor. He is the head of the Gdańsk Branch of the Polish Academy of Sciences and the editor-in-chief of the journal Studia Mathematica.
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Kelly is known for posing the reconstruction conjecture with his advisor Ulam, which states that every graph is uniquely determined by the ensemble of subgraphs formed by deleting one vertex in each possible way.. He also proved a special case of this conjecture, for trees.. He is the coauthor of three textbooks: Projective geometry and projective metrics (1953, with Herbert Busemann), Geometry and convexity: A study in mathematical methods (1979, with Max L. Weiss), and The non-Euclidean, hyperbolic plane: Its structure and consistency (1981, with Gordon Matthews).
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It forms a convex varix at the base, and terminates near this point by a straight and somewhat pointed projection. The general color is whitish or reddish, marked upon the convexity of the lowest whorl, with a large red or fawn-colored spot, the rest of the spire sometimes sprinkled with other smaller spots of the same color.Kiener (1840). General species and iconography of recent shells : comprising the Massena Museum, the collection of Lamarck, the collection of the Museum of Natural History, and the recent discoveries of travellers; Boston :W.
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The reason for this exemption from glaciation is the converse of that for the southward convexity of the morainic loops. For while they mark the paths of greatest glacial advance along lowland troughs (lake basins), the driftless zone is a district protected from ice invasion by reason of the obstruction which the highlands of northern Wisconsin and Michigan (part of the Superior upland) offered to glacial advance. The course of the upper Mississippi River is largely consequent upon glacial deposits. Its sources are in the morainic lakes in northern Minnesota.
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The Bohr–Mollerup theorem is useful because it is relatively easy to prove logarithmic convexity for any of the different formulas used to define the gamma function. Taking things further, instead of defining the gamma function by any particular formula, we can choose the conditions of the Bohr–Mollerup theorem as the definition, and then pick any formula we like that satisfies the conditions as a starting point for studying the gamma function. This approach was used by the Bourbaki group. Borwein & Corless review three centuries of work on the gamma function.
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The opponens digiti minimi (opponens digiti quinti in older texts) is a muscle in the hand. It is of a triangular form, and placed immediately beneath the palmaris brevis, abductor digiti minimi and flexor digiti minimi brevis. It is one of the three hypothenar muscles that control the little finger. It arises from the convexity of the hamulus of the hamate bone and the contiguous portion of the transverse carpal ligament; it is inserted into the whole length of the metacarpal bone of the little finger, along its ulnar margin.
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Relatedly, as regards corporate debt pricing, the relationship between equity holders' limited liability and potential Chapter 11 proceedings has also been modelled via lattice. The calculation of "Greeks" for interest rate derivatives proceeds as for equity. There is however an additional requirement, particularly for hybrid securities: that is, to estimate sensitivities related to overall changes in interest rates. For a bond with an embedded option, the standard yield to maturity based calculations of duration and convexity do not consider how changes in interest rates will alter the cash flows due to option exercise.
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Bond indices can be categorized based on their broad characteristics, such as whether they are composed of government bonds, municipal bonds, corporate bonds, high-yield bonds, mortgage-backed securities, syndicated or leveraged loans, etc. They can also be classified based on their credit rating or maturity. Bond indices tend to be total rate-of-return indices and are used mostly as such: to look at performance of a market over time. In addition to returns, bond indices generally also have yield, duration, and convexity, which is aggregated up from individual bonds.
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In physical chemistry terms, the AFPs adsorbed onto the exposed ice crystal force the growth of the ice crystal in a convex fashion as the temperature drops, which elevates the ice vapour pressure at the nucleation sites. Ice vapour pressure continues to increase until it reaches equilibrium with the surrounding solution (water), at which point the growth of the ice crystal stops. The aforementioned effect of AFPs on ice crystal nucleation is lost at the thermal hysteresis point. At a certain low temperature, the maximum convexity of the ice nucleation site is reached.
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Three common palmar digital arteries arise from the convexity of the superficial palmar arch and proceed distally on the second, third, and fourth lumbricales muscles. Alternative names for these arteries are: common volar digital arteries,Palmar and volar may be used synonymously, but volar is less common. ulnar metacarpal arteries, arteriae digitales palmares communes,This is the official and international Latin term as defined by the Terminologia Anatomica (TA), but in English speaking countries and especially the US, common palmar digital arteries is more commonly used. or aa.
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The cryptic genus Buluniella and species B. borealis was described in 1986 by V. Jermak from three fossils found in Northern Siberia. The two right and one left disarticulated valves known show a slightly convexity of the hinge, central umbo and lack of a row of muscle scars were used to the genus from Fordilla. The less distinct umbones were suggested as reason to separate Buluniella from Pojetaia. Due to the high variation in characters of Cambrian bivalve species the validity of Buluniella as a separate genus and species has been questioned several times.
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The Department of Mathematics conducts a master's degree programme in Mathematics and Operations Research and Computer Applications as well as M.Phil and PhD programmes. Research scholars in the department receive funding from Government of India agencies such as the University Grants Commission, the Council of Scientific and Industrial Research, and the National Board for Higher Mathematics (NBHM). Research areas include analysis, algebra, operator theory, functional analysis, general topology, fuzzy mathematics, graph theory, combinatorics, convexity theory, fluid dynamics, non-linear waves, stability, stochastic processes and random graphs, operations research and the history of mathematics.
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This category includes findings that are minor and not suggestive of TB disease. These findings require no follow-up evaluation.. Chest x-ray of pleural thickening post-primary tuberculosis #Pleural thickening - Irregularity or abnormal prominence of the pleural margin, including apical capping (thickening of the pleura in the apical region). Pleural thickening can be calcified. #Diaphragmatic tenting - A localized accentuation of the normal convexity of the hemidiaphragm as if “pulled upwards by a string.” #Blunting of costophrenic angle (in adults)—Loss of sharpness of one or both costophrenic angles.
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According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.
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Indian-specification vehicle's side-view mirror with the eponymous legend Wing mirror on a South Korean-specification vehicle. Legend in Korean reads "Objects in mirror are closer than they appear". The phrase "objects in (the) mirror are closer than they appear" is a safety warning that is required to be engraved on passenger side mirrors of motor vehicles in many places including United States, Canada, Nepal, India, and Saudi Arabia. It is present because while these mirrors' convexity gives them a useful field of view, it also makes objects appear smaller.
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Dorsal valve more convex than ventral one, with greatest convexity at anterior part, but recurved anteriorly; fold occurring at anterior 1/2 to 1/3 of valve, generally narrow and lower, with rounded top, moderately raised above slopes, giving valve trilobate appearance. Dorsal umbones rarely slightly sulcate or depressed. Numerous fine subangular costae separated by deep intervals, on each valve numbering 20-26, with 4-7 on fold and 3-6 in sinus; shell also bearing innumerable fine, conspicuous growth lines, becoming feebly lamellose or imbricated toward anterior margin.
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14 the point x is a Pareto optimum which does not satisfy the definition of competitive equilibrium. The question of whether the economy would settle at such a point is quite separate from whether it satisfies an arbitrary definition of equilibrium; evidently in this case it would do so. Arrow and Debreu always included the convexity of indifference curves amongst their ‘assumptions’. Unfortunately the term ‘assumptions’ is a vague one which might refer to a presupposition underlying definitions as well as theorems, or to a premise which is needed only for the latter.
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In complex geometry and analysis, the notion of convexity and its generalizations play an important role in understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and the plurisubharmonic functions. Geometrically, these classes of functions correspond to convex domains and pseudoconvex domains, but there are also other types of domains, for instance lineally convex domains which can be generalized using convex analysis. A great deal is already known about these domains, but there remain some fascinating, unsolved problems.
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It has a lozenge-shaped disc and a maximum length of . The margin of the disc is almost straight from the tip of the snout to the tip of the disc, with a slight convexity from the level of the nostrils to the level of the gills. The straight disc margin can be used to distinguish B. peruana from B. aguja which has undulating margins. Coloration on most of the surface is brownish-grey, but ranges from brown to yellow on the margins of the nostrils, mouth, pelvic fins, and base of the tail.
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Choice under uncertainty is often characterized as the maximization of expected utility. Utility is often assumed to be a function of profit or final portfolio wealth, with a positive first derivative. The utility function whose expected value is maximized is convex for a risk- seeker, concave for a risk-averse agent, and linear for a risk-neutral agent. Its convexity in the risk-seeking case has the effect of causing a mean- preserving spread of any probability distribution of wealth outcomes to be preferred over the unspread distribution.
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Its middle third is broad, slightly concave, and gives origin to the Abductor pollicis longus above, and the extensor pollicis brevis muscle below. Its lower third is broad, convex, and covered by the tendons of the muscles which subsequently run in the grooves on the lower end of the bone. The lateral surface (facies lateralis; external surface) is convex throughout its entire extent and is known as the convexity of the radius, curving outwards to be convex at the side. Its upper third gives insertion to the supinator muscle.
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Duration is a linear measure or 1st derivative of how the price of a bond changes in response to interest rate changes. As interest rates change, the price is not likely to change linearly, but instead it would change over some curved function of interest rates. The more curved the price function of the bond is, the more inaccurate duration is as a measure of the interest rate sensitivity. Convexity is a measure of the curvature or 2nd derivative of how the price of a bond varies with interest rate, i.e.
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M. Grötschel, L. Lovász, A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer, 1988.L. Lovász: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986.V. Chandru and M.R.Rao, Linear Programming, Chapter 31 in Algorithms and Theory of Computation Handbook, edited by M. J. Atallah, CRC Press 1999, 31-1 to 31-37.V. Chandru and M.R.Rao, Integer Programming, Chapter 32 in Algorithms and Theory of Computation Handbook, edited by M.J.Atallah, CRC Press 1999, 32-1 to 32-45.
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Pg. 101 But instead of being flat it needed to be convex to deemphasize the edge. The disc was painted where the disc bulges out (point closest to the viewer) the same color as the wall (point furthest away from the viewer) to give it a floating effect. But the combination of convexity and color made it so the viewer had a difficult time determining whether the disc was convex, concave, or flat. (Originally he painted the discs with dots, but the way they turned out was unsatisfactory.
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Because of the distance from the driver's eye to the passenger side mirror, a useful field of view can be achieved only with a convex or aspheric mirror. However, the convexity also minifies the objects shown. Since such objects seem farther away than they actually are, a driver might make a maneuver such as a lane change assuming an adjacent vehicle is a safe distance behind, when in fact it is quite a bit closer. In the United States, Canada, India, Korea and Australia, non-planar mirrors are etched or printed with the warning legend .
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Under his guidance and supervision, studies were conducted on various fuzzy mathematical structures like, fuzzy topology, fuzzy topological semigroups, fuzzy convexity, fuzzy inner-product spaces, fuzzy measures, fuzzy topological games, fuzzy commutative algebra, fuzzy matroids, etc. During the 79th Annual Conference of the Indian Mathematical Society held at Rajagiri School of Engineering & Technology, Kochi, during 28–31 December 2013, Dr. Thrivikraman, along with two other mathematicians from Kerala Prof. K. S. S. Nambooripad and Prof. R. Sivaramakrishnan, were honoured in a special event called Guruvandanam for their lifetime contributions to mathematics.
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The transverse colon is the longest and most movable part of the colon. It crosses the abdomen from the ascending colon at the hepatic or right colic flexure with a downward convexity to the descending colon where it curves sharply on itself beneath the lower end of the spleen forming the splenic or left colic flexure. In its course, it describes an arch, the concavity of which is directed backward and a little upward. Toward its splenic end there is often an abrupt U-shaped curve which may descend lower than the main curve.
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It is somewhat crescent-shaped, with its convexity directed forward: Medially, it is in relation with the internal carotid artery and the posterior part of the cavernous sinus. The motor root runs in front of and medial to the sensory root, and passes beneath the ganglion; it leaves the skull through the foramen ovale, and, immediately below this foramen, joins the mandibular nerve. The greater superficial petrosal nerve lies also underneath the ganglion. The ganglion receives, on its medial side, filaments from the carotid plexus of the sympathetic.
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The head is globular and forms rather more than a hemisphere, is directed upward, medialward, and a little forward, the greater part of its convexity being above and in front. The femoral head's surface is smooth. It is coated with cartilage in the fresh state, except over an ovoid depression, the fovea capitis, which is situated a little below and behind the center of the femoral head, and gives attachment to the ligament of head of femur. The diameter of the femoral head is generally larger in men than in women.
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In contrast with the classical convexity where there exist several equivalent definitions of the convex hull, definitions of the orthogonal convex hull made by analogy to those of the convex hull result in different geometric objects. So far, researchers have explored the following four definitions of the orthogonal convex hull of a set K \subset \R^d: #Maximal definition: The definition described in the introduction of this article. It is based on the Maxima of a point set. #Classical definition: The orthogonal convex hull of K is the intersection of all orthogonally convex supersets of K; .
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Econometrica, Vol. 62 Issue 1, pp. 157-180. pointed out in 1994 that the assumptions conventionally used to justify the use of comparative statics on optimization problems are not actually necessary—specifically, the assumptions of convexity of preferred sets or constraint sets, smoothness of their boundaries, first and second derivative conditions, and linearity of budget sets or objective functions. In fact, sometimes a problem meeting these conditions can be monotonically transformed to give a problem with identical comparative statics but violating some or all of these conditions; hence these conditions are not necessary to justify the comparative statics.
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In 1984 he started to work in the field of computer vision at the University of Edinburgh. Together with Andrew Blake they wrote the book Visual reconstruction published in 1987, which is considered one of the seminal works in the field of computer vision. According to Fitzgibbon (2008) this publication was "one of the first treatments of the energy minimisation approach to include an algorithm (called "graduated non-convexity") designed to directly address the problem of local minima, and furthermore to include a theoretical analysis of its convergence."Andrew Fitzgibbon (2008) "Andrew Zisserman, BMVA Distinguished Fellow 2008 " Bmva.
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The prostatic urethra, the widest and most dilatable part of the urethra canal, is about 3 cm long. It runs almost vertically through the prostate from its base to its apex, lying nearer its anterior than its posterior surface; the form of the canal is spindle-shaped, being wider in the middle than at either extremity, and narrowest below, where it joins the membranous portion. A transverse section of the canal as it lies in the prostate is horse-shoe- shaped, with the convexity directed forward. The keyhole sign, in ultrasound, is associated with a dilated bladder and prostatic urethra.
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The mean (L2 center) and midrange (L∞ center) are unique (when they exist), while the median (L1 center) and mode (L0 center) are not in general unique. This can be understood in terms of convexity of the associated functions (coercive functions). The 2-norm and ∞-norm are strictly convex, and thus (by convex optimization) the minimizer is unique (if it exists), and exists for bounded distributions. Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point.
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For example, for small interest rate changes, the duration is the approximate percentage by which the value of the bond will fall for a 1% per annum increase in market interest rate. So the market price of a 17-year bond with a duration of 7 would fall about 7% if the market interest rate (or more precisely the corresponding force of interest) increased by 1% per annum. Convexity is a measure of the "curvature" of price changes. It is needed because the price is not a linear function of the discount rate, but rather a convex function of the discount rate.
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This common structure may be represented in an underlying abstract polytope, a purely algebraic partially-ordered set which captures the pattern of connections or incidences between the various structural elements. The measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope. What is true for traditional polytopes (also called classical or geometric polytopes) may not be so for abstract ones, and vice versa. For example, a traditional polytope is regular if all its facets and vertex figures are regular, but this is not necessarily so for an abstract polytope.
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Eberlein studied from 1936 to 1942 at the University of Wisconsin and at Harvard University, where he received in 1942 a PhD for the thesis Closure, Convexity, and Linearity in Banach Spaces under the direction of Marshall Stone. He was married twice—to Mary Bernarda Barry and Patricia Ramsay James. He had four children with Mary Barry, including Patrick Barry Eberlein, another renowned mathematician. Patricia Ramsay James was a mathematician who moved into computer science as the field opened up; their one child is Kristen James Eberlein, the chair of the OASIS Darwin Information Typing Architecture Technical Committee.
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The Macedonian phalangite shield, also termed the 'Telamon shield', was circular and displayed a slight convexity; its outer surface was faced by a thin bronze sheet. The inner face of the shield was of wood or a multilayered leather construction, with a band for the forearm fixed to the centre of the shield. Plutarch noted that the phalangites (phalanx soldiers) carried a small shield on their shoulder. This probably meant that, as both hands were needed to hold the sarissa, the shield was worn suspended by a shoulder strap and steadied by the left forearm passing through the armband.
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Although the size of clusters might affect the magnitude of relative homophily. A higher level of homophily can be associated to a more convex cumulative degree distribution instead of a concave one. Although not as salient, the link density of the network might also lead to short-term, localized deviations in the shape of the distribution. In the development of the shape of the cumulative degree distribution curve the effects of the link structure of existing nodes (among themselves and with new nodes) and homophily work against each other, with the former leading to concavity and homophily causing convexity.
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NAS bonds are designed to protect investors from volatility and negative convexity resulting from prepayments. NAS tranches of bonds are fully protected from prepayments for a specified period, after which time prepayments are allocated to the tranche using a specified step down formula. For example, an NAS bond might be protected from prepayments for five years, and then would receive 10% of the prepayments for the first month, then 20%, and so on. Recently, issuers have added features to accelerate the proportion of prepayments flowing to the NAS class of bond in order to create shorter bonds and reduce extension risk.
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Exact formulas are known for enumerating polyominoes of special classes, such as the class of convex polyominoes and the class of directed polyominoes. The definition of a convex polyomino is different from the usual definition of convexity, but is similar to the definition used for the orthogonal convex hull. A polyomino is said to be vertically or column convex if its intersection with any vertical line is convex (in other words, each column has no holes). Similarly, a polyomino is said to be horizontally or row convex if its intersection with any horizontal line is convex.
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Variance risk premium is a phenomenon on the variance swap market, of the variance swap strike being greater than the realized variance on average. For most trades, the buyer of variance ends up with a loss on the trade, while the seller profits. The amount that the buyer of variance typically loses in entering into the variance swap, is known as the variance risk premium. The variance risk premium can be naively justified by taking into account the large negative convexity of a short variance position; variance during rare times of crisis can be 50-100 times that of normal market conditions.
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He visited the Institute for Advanced Study (IAS) in Princeton from 1948 to 1950, where he co-organized a seminar on convexity. Together with Olof Hanner, who, like Rådström, would earn his Ph.D. from Stockholm University in 1952, he improved Werner Fenchel's version of Carathéodory's lemma. In the 1950s, he obtained important results on convex sets. He proved the Rådström embedding theorem, which implies that the collection of all nonempty compact convex subsets of a normed real vector- space (endowed with the Hausdorff distance) can be isometrically embedded as a convex cone in a normed real vector-space.
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Other factors that could contribute to higher risks of ACL tears in women include patient weight and height, the size and depth of the intercondylar notch, the diameter of the ACL, the magnitude of the tibial slope, the volume of the tibial spines, the convexity of the lateral tibiofemoral articular surfaces, and the concavity of the medial tibial plateau. While anatomical factors are most talked about, extrinsic factors, including dynamic movement patterns, might be the most important risk factor when it comes to ACL injury. Environmental factors also play a big role. Extrinsic factors are controlled by the individual.
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The large sculpture comprises five quadrilateral slab-like metal elements: a horizontal base, on which stand two thick parallel square slabs on edge, on which stand two similar parallel square slabs at right angles to the lower two. Each of the standing slabs is pierced off-centre by a round hole. Although the five slabs are similar in appearance, each around square, so approximately the height of a typical person, each differs slightly in its linear dimensions, convexity, and surface detail. One of each pair of the standing elements is slightly larger than the other, so they are arranged in echelon.
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The EPH must be approximated in the IDM technique before the decision maps are displayed. Methods for approximating the EPH depend on the convexity properties of the EPH. Approximation methods are typically based either on approximation of the EPH by a convex polyhedral set or on approximation of the EPH by a large but finite number of domination cones in objective space with vertices that are close to the Pareto front. The first form can be applied only in the convex problems, while the second form is universal and can be used in general nonlinear problems.
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K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality of the (s,S) policy in inventory control theory. The policy is characterized by two numbers and , S \geq s, such that when the inventory level falls below level , an order is issued for a quantity that brings the inventory up to level , and nothing is ordered otherwise. Gallego and Sethi Gallego, G. and Sethi, S. P. (2005). K-convexity in ℜn. _Journal of Optimization Theory & Applications,_ 127(1):71-88.
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700 BC), the echinus moulding has become a more definite form: this in the Parthenon reaches its culmination, where the convexity is at the top and bottom with a delicate uniting curve. The sloping side of the echinus becomes flatter in the later examples, and in the Colosseum at Rome forms a quarter round (see Doric order). In versions where the frieze and other elements are simpler the same form of capital is described as being in the Tuscan order. Doric reached its peak in the mid-5th century BC, and was one of the orders accepted by the Romans.
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The 3D model of the element to be inspected can be projected in the image plane for more complex shapes. The evaluation is based on indices such as the uniformity of segmented regions, convexity of their forms, or periodicity of the image pixels' intensity. The feature extraction using speeded up robust features (SURF) is also able to perform the inspection of certain elements having two possible states, such as pitot probes or static ports being covered or not covered. A pairing is performed between images of the element to be inspected in different states and that present on the scene.
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Whorls are very slightly convex beneath, strongly spirally ribbed and grooved. The ribs are six in number on the upper whorls and rounded; the two above are much more slender than the four beneath; the uppermost borders the suture; the next lies in the concavity at the top of the whorls; and the rest surround the slight convexity, and are three times as broad as the sulci separating them. All the whorls, with the exception of the last four, are coronated at the slight angle below the excavation with very short, hollow, oblique spinules. Some of the spiral grooves exhibit rows of fine granules.
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Manfredo Perdigão do Carmo (15 August 1928 – 30 April 2018) was a Brazilian mathematician, doyen of Brazilian differential geometry, and former president of the Brazilian Mathematical Society.Biography from the Guggenheim Foundation He was at the time of his death an emeritus researcher at the IMPA. He is known for his research on Riemannian manifolds, topology of manifolds, rigidity and convexity of isometric immersions, minimal surfaces, stability of hypersurfaces, isoperimetric problems, minimal submanifolds of a sphere, and manifolds of constant mean curvature and vanishing scalar curvature. He earned his Ph.D. from the University of California, Berkeley in 1963 under the supervision of Shiing-Shen Chern.
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Bandini coupé in profile The body sports coupé, produced by Bandini completely in alloy, allegedly closely follows that of the Barchetta of 1966. The front is very similar to the predecessor, but to make the organic roof and windscreen, the bonnet is more inclined and acquires a convexity shared with the rest of the body. Consequently, the doors have a greater height, but remain unchanged in their shape, as does the rear engine cover, which maintains the air side but not higher. The upper part of the rear engine cover is entirely dominated by a long and convex back window that leaves nice view of the engine and mechanics.
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Beginning of the interventricular septum shown at 28 days The interventricular septum is the stout wall separating the ventricles, the lower chambers of the heart, from one another. The ventricular septum is directed obliquely backward to the right, and curved with the convexity toward the right ventricle; its margins correspond with the anterior and posterior longitudinal sulci. The greater portion of it is thick and muscular and constitutes the muscular interventricular septum. Its upper and posterior part, which separates the aortic vestibule from the lower part of the right atrium and upper part of the right ventricle, is thin and fibrous, and is termed the membranous ventricular septum.
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In mathematics, Kostant's convexity theorem, introduced by , states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a generalization of a result of , and for hermitian matrices. They proved that the projection onto the diagonal matrices of the space of all n by n complex self-adjoint matrices with given eigenvalues Λ = (λ1, ..., λn) is the convex polytope with vertices all permutations of the coordinates of Λ. Kostant used this to generalize the Golden–Thompson inequality to all compact groups.
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Scarf's fixed-point method was a break-through in the mathematics of computation generally, and specifically in optimization and computational economics. Later researchers continued to develop iterative methods for computing fixed-points, both for topological models like Scarf's and for models described by functions with continuous second derivatives or convexity or both. Of course, "global Newton methods"Stephen Smale, Global analysis and economics, Handbook of Mathematical Economics, K. J. Arrow and M. D. Intrilligator, North-Holland, Amsterdam, 1 (1981), pp. 331--370. for essentially convex and smooth functions and path-following methods for diffeomorphisms converged faster than did robust algorithms for continuous functions, when the smooth methods are applicable.
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In the late 1950s, Jerome Lettvin and his colleagues began to expand the feature detection hypothesis and clarify the relationship between single neurons and sensory perception. In their paper "What the Frog's Eye Tells the Frog's Brain", Lettvin et al. (1959) looked beyond the mechanisms for signal- noise discrimination in the frog's retina and were able to identify four classes of ganglion cells in the frog retina: sustained contrast detectors, net convexity detectors (or bug detectors), moving edge detectors, and net dimming detectors. In the same year, David Hubel and Torsten Wiesel began investigating properties of neurons in the visual cortex of cats, processing in the mammalian visual system.
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As phase values can only vary from zero to 2π, then repeat in either direction (termed phase wrapping), changing the piston coefficient changes the zero phase value contour locations across the wavefront. This property is critical to the operation of phase-measuring interferometers, which give not only the magnitude but also the sign (convexity or concavity) of a wavefront under test. Piston is physically created in the interferometer by piezoelectric actuators that translate the Fizeau interferometer reference surface along the optical axis by precise fractions of the test wavelength, usually by one quarter of a wavelength. This changes the interferometric fringe patterns and allows direct calculation of the exact wavefront error.
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Surely aware of the sea's convexity, he may have designed his map on a slightly rounded metal surface. The centre or “navel” of the world ( omphalós gẽs) could have been Delphi, but is more likely in Anaximander's time to have been located near Miletus. The Aegean Sea was near the map's centre and enclosed by three continents, themselves located in the middle of the ocean and isolated like islands by sea and rivers. Europe was bordered on the south by the Mediterranean Sea and was separated from Asia by the Black Sea, the Lake Maeotis, and, further east, either by the Phasis River (now called the Rioni) or the Tanais.
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The protoconch is smooth white, pointed, drawn out. The sculpture consists of longitudinal ribs thirteen or fourteen on the body whorl, obsolete on the lower third of the whorl and not extending to the suture, below which is a smooth band only marked by oblique lines of growth. The ribs are slightly nodulous at their posterior terminations (where they are united by a slight carina) strong on the upper whorls, slightly flexuous on the convexity of the whorl. The whorl below the carina is marked by very faint grooves close together and passing over the ribs, stronger at the anterior end of the body whorl.
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The resulting tilt in the scan lines is very small, and is dwarfed in effect by screen convexity and other modest geometrical imperfections. There is a misconception that once a scan line is complete, a CRT display in effect suddenly jumps internally, by analogy with a typewriter or printer's paper advance or line feed, before creating the next scan line. As discussed above, this does not exactly happen: the vertical sweep continues at a steady rate over a scan line, creating a small tilt. Steady-rate sweep is done, instead of a stairstep of advancing every row, because steps are hard to implement technically, while steady-rate is much easier.
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Observe that a set is absorbing in if and only if } is absorbing in every 1-dimensional vector subspace , where . Thus, it is necessary and sufficient to show that contains an open -ball around the origin in . The condition implies that every "open ray" in starting at the origin (i.e. a set of the form for some ) contains an element of so that in particular, and (where so that ) so that now the convexity of makes it clear that for every , the convex set is a line segment (possibly open, closed, or half-closed, and possibly bounded or unbounded) containing an open sub-interval that contains the origin.
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It was pointed out very early in the controlled fusion program that such a device has a natural instability in the magnetic field arrangement. In any area where there is convexity in the field, there is a natural tendency for the ions to want to move to the outside of their original trajectory when they undergo collision. As a result of this motion, they wander outwards through the confinement area. When enough ions do this in any particular area, their electric charge modifies the magnetic field in such a way to further increase the curvature, causing a runaway effect that results in the plasma pouring out of the confinement area.
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The earliest written record of what was probably a meningioma is from the 1600s, when Felix Plater (1536–1614) of the University of Basel performed an autopsy on Sir Caspar Bonecurtius. Surgery for removal of meningiomas was first attempted in the sixteenth century, but the first known successful surgery for removal of a meningioma of the convexity (parasagittal) was performed in 1770 by Anoine Luis. The first documented successful removal of a skull base meningioma was performed in 1835 by Zanobi Pecchioli, Professor of Surgery at the University of Siena. Other notable meningioma researchers have been William Macewen (1848–1924), and William W. Keen (1837–1932).
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Bregman divergences correspond to convex functions on convex sets. Given a strictly convex, continuously-differentiable function on a convex set, known as the Bregman generator, the Bregman divergence measures the convexity of: the error of the linear approximation of from as an approximation of the value at : :D_F(p, q) = F(p)-F(q)-\langle abla F(q), p-q\rangle. The dual divergence to a Bregman divergence is the divergence generated by the convex conjugate of the Bregman generator of the original divergence. For example, for the squared Euclidean distance, the generator is , while for the relative entropy the generator is the negative entropy .
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Economists have increasingly studied non-convex sets with nonsmooth analysis, which generalizes convex analysis. Convex analysis centers on convex sets and convex functions, for which it provides powerful ideas and clear results, but it is not adequate for the analysis of non-convexities, such as increasing returns to scale.: "Non-convexities in [both] production and consumption ... required mathematical tools that went beyond convexity, and further development had to await the invention of non-smooth calculus": For example, Clarke's differential calculus for Lipschitz continuous functions, which uses Rademacher's theorem and which is described by and ,Chapter 8 "Applications to economics", especially Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495: according to .
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In finance, interest rate immunisation, as developed by Frank Redington is a strategy that ensures that a change in interest rates will not affect the value of a portfolio. Similarly, immunisation can be used to ensure that the value of a pension fund's or a firm's assets will increase or decrease in exactly the opposite amount of their liabilities, thus leaving the value of the pension fund's surplus or firm's equity unchanged, regardless of changes in the interest rate. Interest rate immunisation can be accomplished by several methods, including cash flow matching, duration matching, and volatility and convexity matching. It can also be accomplished by trading in bond forwards, futures, or options.
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Has a crush on Bakumaru. Being a rabbit, she has good hearing, and can even hear if her teammates are in trouble from far away. ; (Dragon) : :Has a white cloud for personal transportation which he can expand for others to travel on as well and can turn into a giant dragon by counting to three in Mandarin and activating his special crystal ball somewhat like a ninja smoke bomb. He is also very intelligent, can command weather in normal or giant dragon form and while a giant dragon he can use his wings to conjure hurricanes and has various breath powers like fire and a purple breath similar to Spyro's convexity breath.
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Pricing quanto derivatives involves modeling financial variables (stocks, interest rates etc.) in a currency which is different from their actual currency. In order to write the dynamics of the modeled financial variables under foreign currency pricing measure one has to apply Girsanov theorem leading to a drift term which depends on its volatility, the FX rate volatility (FX rate between the pricing currency and the modeled variable currency) and correlation between both.D. Papaioannou (2011): "Applied Multidimensional Girsanov Theorem", SSRN. This drift term leads to an adjustment in the pricing that is referred to as "quanto adjustment" and falls into the more general category of what is called in mathematical finance convexity adjustments.
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Consequently, there is a level of homophily such that the two effects cancel each other out and the cumulative degree distribution reaches a linear shape in a log-log scale. Large variety of shapes observed in empirical studies of real complex networks may be explained by these phenomena. A low level of homophily could then be linked to a convex shape of cumulative degree distributions which have been observed in networks of Facebook wall posts, Flickr users, and message boards; while linear shapes have been noted in the networks of and software class dependency, Yahoo adverts, and YouTube users. Compared to these two shapes, convexity seems to be much less prevalent with examples in Google Plus and Filmtipset networks.
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Although generally (assuming convexity) an equilibrium will exist and will be efficient, the conditions under which it will be unique are much stronger. The Sonnenschein–Mantel–Debreu theorem, proven in the 1970s, states that the aggregate excess demand function inherits only certain properties of individual's demand functions, and that these (Continuity, Homogeneity of degree zero, Walras' law and boundary behavior when prices are near zero) are the only real restriction one can expect from an aggregate excess demand function. Any such function can represent the excess demand of an economy populated with rational utility-maximizing individuals. There has been much research on conditions when the equilibrium will be unique, or which at least will limit the number of equilibria.
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As stated above, for univariate parameters, median-unbiased estimators remain median-unbiased under transformations that preserve order (or reverse order). Note that, when a transformation is applied to a mean-unbiased estimator, the result need not be a mean-unbiased estimator of its corresponding population statistic. By Jensen's inequality, a convex function as transformation will introduce positive bias, while a concave function will introduce negative bias, and a function of mixed convexity may introduce bias in either direction, depending on the specific function and distribution. That is, for a non-linear function f and a mean-unbiased estimator U of a parameter p, the composite estimator f(U) need not be a mean-unbiased estimator of f(p).
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Jacques Salbert indicates that it had an original place in the Laval altarpiece by its double convexity of the wings framing the large recessed central painting. In 1672, he had the following transported from Nantes to and at the hâvre of Pontrieux marble and tufa.three cents of tufa, 6 columns of 7 feet marble, 4 columns of 6 feet, 2 of 5, 18 half balusters Jacques Salbert thinks that an altarpiece was built in the chapel of the Château du Taureau, and that the rest of the material was used to build the altarpiece of Guingamp.Attributed to Olivier Martinet by Hervé du Halgouët, Les retables de chevet au XVIIe et au XVIIIe, .
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Vanilla convertible bonds are the most plain convertible structures. They grant the holder the right to convert into a certain number of shares determined according to a conversion price determined in advance. They may offer coupon regular payments during the life of the security and have a fixed maturity date where the nominal value of the bond is redeemable by the holder. This type is the most common convertible type and is typically providing the asymmetric returns profile and positive convexity often wrongly associated to the entire asset class: at maturity the holder would indeed either convert into shares or request the redemption at par depending on whether or not the stock price is above the conversion price.
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Artstein won the Haim Nessyahu Prize in Mathematics, an annual dissertation award of the Israel Mathematical Union, in 2006. In 2008 she won the Krill Prize for Excellence in Scientific Research, from the Wolf Foundation. In 2015 she won the Anna and Lajos Erdős Prize in Mathematics. The award cited her "solution of Shannon's long standing problem on monotonicity of entropy (with K. Ball, F. Barthe and A. Naor), profound and unexpected development of the concept of duality, Legendre and Fourier transform from axiomatic viewpoint (with V. Milman) and discovery of an astonishing link between Mahler's conjecture in convexity theory and an isoperimetric-type inequality involving symplectic capacities (with R. Karasev and Y. Ostrover)".
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"In the dum-dum bullet the jacket ends by leaving a small piece of the core uncovered. The effect of this modification is to produce a certain extension or convexity of the point, and to give a force more pronounced than that of a bullet which is completely jacketed, at the same time, however, less effective than that of the Enfield, Snider, or Martini bullets, all of which have greater calibre." The German protests were effective, however, resulting in the ban of the use of expanding bullets in warfare. The British replaced the hollow-point bullets with new full metal jacket bullets, and used the remaining stocks of expanding bullets for practice.
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The Laguna Copperplate Inscription shows the use of mathematics in precolonial Philippine societies. A standard system of weights and measures is demonstrated by the use of precise measurement for gold, and familiarity with rudimentary astronomy is shown by fixing the precise day within the month in relation to the phases of the moon. Shipbuilding showed geometric thinking and mastery of convexity, concavity, and the proper proportion between ship breadth and length to ensure sailing efficiency. The practice of constructing as much as twelve ships and boats to fit inside each other, not unlike matryoshka dolls containing each other, can be interpreted as large three- dimensional wooden demonstration of sets, subsets, volumes, and ordinality.
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An anglers catch of brownback trevallyThe brownback trevally is a relatively small species of carangid, growing to a maximum recorded length of 25 cm, but is much more common at lengths less than 16 cm. The species has a similar body profile to other trevallies in the same genus, having an elongate, compressed form with the dorsal and ventral profiles approximately equal in convexity. The dorsal fin is in two parts, the first consisting of 8 spines while the second has 1 spine and 21 to 24 soft rays, with both dorsal fins of approximately equal height. The anal fin has two anteriorly detached spines followed by 1 spine and 18 to 20 soft rays.
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For two bonds with the same par value, coupon, and maturity, convexity may differ depending on what point on the price yield curve they are located. Suppose both of them have at present the same price yield (p-y) combination; also you have to take into consideration the profile, rating, etc. of the issuers: let us suppose they are issued by different entities. Though both bonds have the same p-y combination, bond A may be located on a more elastic segment of the p-y curve compared to bond B. This means if yield increases further, the price of bond A may fall drastically while the price of bond B won’t change; i.e.
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The Yudin-Nemirovsky analysis implies that no method can be fast on high- dimensional problems that lack convexity: > "The catastrophic growth [in the number of iterations needed to reach an > approximate solution of a given accuracy] as [the number of dimensions > increases to infinity] shows that it is meaningless to pose the question of > constructing universal methods of solving ... problems of any appreciable > dimensionality 'generally'. It is interesting to note that the same > [conclusion] holds for ... problems generated by uni-extremal [that is, > unimodal] (but not convex) functions." > > Page 7 summarizes the later discussion of : When applied to twice continuously differentiable problems, the LJ heuristic's rate of convergence decreases as the number of dimensions increases.
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In 1986, Hamilton and Michael Gage applied Hamilton's Nash-Moser theorem and well-posedness result for parabolic equations to prove the well-posedness for mean curvature flow; they considered the general case of a one-parameter family of immersions of a closed manifold into a smooth Riemannian manifold. Then, they specialized to the case of immersions of the circle into the two-dimensional Euclidean space , which is the simplest context for curve shortening flow. Using the maximum principle as applied to the distance between two points on a curve, they proved that if the initial immersion is an embedding, then all future immersions in the mean curvature flow are embeddings as well. Furthermore, convexity of the curves is preserved into the future.
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In a 1956 paper, Hanner introduced the Hanner polytopes and the Hanner spaces having these polytopes as their metric balls. Hanner was interested in a Helly property of these shapes, later used to characterize them by : unlike other convex polytopes, it is not possible to find three translated copies of a Hanner polytope that intersect pairwise but do not have a point of common intersection.. Subsequently, the Hanner polytopes formed a class of important examples for the Mahler conjecture. and for Kalai's 3d conjecture.. In another paper from the same year, Hanner proved a set of inequalities related to the uniform convexity of Lp spaces, now known as Hanner's inequalities. Other contributions of Hanner include (with Hans Rådström) improving Werner Fenchel's version of Carathéodory's lemma,.
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By Hopf–Rinow theorem there always exist length minimizing curves (at least in small enough neighborhoods) on (M, F). Length minimizing curves can always be positively reparametrized to be geodesics, and any geodesic must satisfy the Euler–Lagrange equation for E[γ]. Assuming the strong convexity of F2 there exists a unique maximal geodesic γ with γ(0) = x and γ'(0) = v for any (x, v) ∈ TM \ 0 by the uniqueness of integral curves. If F2 is strongly convex, geodesics γ : [0, b] → M are length-minimizing among nearby curves until the first point γ(s) conjugate to γ(0) along γ, and for t > s there always exist shorter curves from γ(0) to γ(t) near γ, as in the Riemannian case.
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Jack Meyer managed HMC from 1990 to September 30, 2005, beginning with an endowment worth $4.8 billion and ending with a value of $25.9 billion (including new contributions). During the last decade of his tenure, the endowment earned an annualized return of 15.9%. In part after compensation disagreements, a number of HMC managers including Meyer himself left to form their own investment management firms. Bloomberg in 2011 reviewed a group of the resultant firms—Adage Capital Management LP, Charlesbank Capital Partners LLC, Convexity Capital Management LP (Meyer's), Highfields Capital Management LP and Regiment Capital Advisors LLC—which at that time between them managed $43 billion in assets.Wee, Gillian, "Harvard's Crimson Cubs With $43 Billion Dwarf Their Former Endowment Home", Bloomberg, March 2, 2011. Retrieved 2012-11-07.
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His set consists of the numbers whose digits are restricted to the range from 0 to d-1 (so that addition of these numbers has no carries), with the extra constraint that the sum of the squares of the digits is some chosen value k. If the digits of each number are thought of as coordinates of a vector, this constraint describes a sphere in the resulting vector space, and by convexity the average of two distinct values on this sphere will be interior to the sphere rather than on it. Therefore, if two elements of Behrend's set are the endpoints of an arithmetic progression, the middle value of the progression (their average) will not be in the set. Thus, the resulting set is progression-free.
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Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that it is necessary for any optimal control along with the optimal state trajectory to solve the so-called Hamiltonian system, which is a two- point boundary value problem, plus a maximum condition of the Hamiltonian. These necessary conditions become sufficient under certain convexity conditions on the objective and constraint functions. The maximum principle was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students, Reprinted in and its initial application was to the maximization of the terminal speed of a rocket.
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A small blue runner in profile The blue runner is moderately large in size, growing to a maximum confirmed length of 70 cm and 5.05 kg in mass, but is more common at lengths less than 35 cm. The blue runner is morphologically similar to a number of other carangids, having an elongated, moderately compressed body with dorsal and ventral profiles of approximately equal convexity and a slightly pointed snout. The posterior section of the eye is covered by a moderately well developed adipose eyelid, and the posterior extremity of the jaw is vertically under the center of the eye. The dorsal fin is in two parts, the first consisting of 8 spines and the second of 1 spine followed by 22 to 25 soft rays.
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The black jack has characteristic black fins and scutesThe black jack is a large fish, and is confidently known to grow to a length of 1 m and a weight of 17.9 kg, although is more common at lengths under 70 cm. At least one source asserts a fish of 2.21 m has been reported, which if true would make the black jack the second largest species of carangid behind the yellowtail amberjack (2.5 m). The black jack has a similar overall body shape to the other members of Caranx, having an oblong, compressed form, with the dorsal profile more convex than the ventral profile. This convexity is most pronounced at the head, which slopes steeply downwards, giving the head profile a very angular appearance.
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So, except for tiny effects of convexity bias (due to earning or paying interest on margin), futures and forwards with equal delivery prices result in the same total loss or gain, but holders of futures experience that loss/gain in daily increments which track the forward's daily price changes, while the forward's spot price converges to the settlement price. Thus, while under mark to market accounting, for both assets the gain or loss accrues over the holding period; for a futures this gain or loss is realized daily, while for a forward contract the gain or loss remains unrealized until expiry. With an exchange-traded future, the clearing house interposes itself on every trade. Thus there is no risk of counterparty default.
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The right colic artery arises from about the middle of the concavity of the superior mesenteric artery, or from a stem common to it and the ileocolic. It passes to the right behind the peritoneum, and in front of the right internal spermatic or ovarian vessels, the right ureter and the Psoas major, toward the middle of the ascending colon; sometimes the vessel lies at a higher level, and crosses the descending part of the duodenum and the lower end of the right kidney. At the colon it divides into a descending branch, which anastomoses with the ileocolic, and an ascending branch, which anastomoses with the middle colic. These branches form arches, from the convexity of which vessels are distributed to the ascending colon.
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Olenellus chiefensis, suborder Olenellina, showing the visual surface has broken away, the lack of dorsal sutures, and the enlarged pleurae of the 3rd thorax segment from the frontMost redlichiids are rather flat (or have low dorso-ventral convexity) and their exoskeleton typically has an oval outline, about 1½× longer than wide. Each back edge of the headshield (or cephalon) very often carries a spine, termed a genal spine. The eye lobes are sickle-shaped, long and extend from the frontal lobe of the central raised area of the cephalon (or glabella) curving outward and increasingly backwards and sometimes eventually inwards again. The visual surface, that contains the calcite lenses is surrounded by fracture lines (or circumocular sutures), so that it has most often broken away from the rest of the cephalon.
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A school of bar jacks over a shallow reef Caranx ruber closeup The bar jack is a moderately large species, growing to a recorded maximum length of 69 cm, and a weight of 6.8 kg, but is commonly encountered at lengths of less than 40 cm. The bar jack displays the typical body shape of most of the jacks, having an elongate, moderately deep and compressed form, with dorsal and ventral profiles of approximately equal convexity. The dorsal fin is divided into two sections, the first consisting of 8 spines while the second has 1 spine followed by 26 to 30 soft rays. The anal fin is composed of 2 anteriorly detached spines followed by 1 spine and 23 to 26 soft rays, with both the anal and soft dorsal fin lobes being slightly elongated.
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In 1919, Weed and McKibben, biomedical researchers at Johns Hopkins Medical School, were the first ones to document the use and effect of osmotically active substances on brain mass. While studying transfer of salt solutions from blood to Cerebrospinal Fluid (CSF), they first noted that concentrated sodium chloride intravenous (IV) injection led to collapse of the thecal sac which prevented them from withdrawing CSF from the lumbar cistern. In order to further study the effect, they conducted lab experiments on anesthetized cats which underwent craniotomy. They observed changes to the convexity of cat's brain upon IV injection, specifically, they noted that Hypertonic Saline IV injection resulted in maximum shrinkage of the brain in 15-30 mins, while administration of hypotonic solutions resulted in protrusion and rupture of the brain tissue.
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The holder of any debt is subject to interest rate risk and credit risk, inflationary risk, currency risk, duration risk, convexity risk, repayment of principal risk, streaming income risk, liquidity risk, default risk, maturity risk, reinvestment risk, market risk, political risk, and taxation adjustment risk. Interest rate risk refers to the risk of the market value of a bond changing due to changes in the structure or level of interest rates or credit spreads or risk premiums. The credit risk of a high- yield bond refers to the probability and probable loss upon a credit event (i.e., the obligor defaults on scheduled payments or files for bankruptcy, or the bond is restructured), or a credit quality change is issued by a rating agency including Fitch, Moody's, or Standard & Poors.
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The bluefin trevally is a large fish, growing to a maximum known length of 117 cm and a weight of 43.5 kg, however it is rare at lengths greater than 80 cm. It is similar in shape to a number of other large jacks and trevallies, having an oblong, compressed body with the dorsal profile slightly more convex than the ventral profile, particularly anteriorly. This slight convexity leads to the species having a much more pointed snout than most other members of Caranx. The dorsal fin is in two parts, the first consisting of 8 spines and the second of 1 spine followed by 21 to 24 soft rays. The anal fin consists of 2 anteriorly detached spines followed by 1 spine and 17 to 20 soft rays.
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Lawrence E. Blume is a Goldwin Smith Professor of Economics and professor of information science at Cornell University, US. He is a visiting research professor at IHS Vienna and a member of the external faculty at the Santa Fe Institute, where he has served as co-director of the economics program and on the institute's steering committee. He teaches and conducts research in general equilibrium theory and game theory, and also has research projects on natural resource management and network design. A Fellow of the Econometric Society, he received a BA in economics from Washington University and a PhD in economics from the University of California, Berkeley. Blume was one of the general editors of The New Palgrave Dictionary of Economics, to which he contributed several articles on mathematical economics: Convexity, convex programming, and duality.
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More specifically, the existence of competitive equilibrium implies both price-taking behaviour and complete markets, but the only additional assumption is the local non-satiation of agents' preferences – that consumers would like, at the margin, to have slightly more of any given good. The first fundamental theorem is said to capture the logic of Adam Smith's invisible hand, though in general there is no reason to suppose that the "best" Pareto efficient point (of which there are a set) will be selected by the market without intervention, only that some such point will be. The second fundamental theorem states that given further restrictions, any Pareto efficient outcome can be supported as a competitive market equilibrium. These restrictions are stronger than for the first fundamental theorem, with convexity of preferences and production functions a sufficient but not necessary condition.
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Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming when George Dantzig described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming. Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Paul Gordan (1873), which was later popularized by Karmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex).
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Rockafellar was a Fulbright Scholar at the University of Bonn in 1957–58 and completed a Master of Science degree at Marquette University in 1959. Formally under the guidance of Professor Garrett Birkhoff, Rockafellar completed his Doctor of Philosophy degree in mathematics from Harvard University in 1963 with the dissertation “Convex Functions and Dual Extremum Problems.” However, at the time there was little interest in convexity and optimization at Harvard and Birkhoff was neither involved with the research nor familiar with the subject. The dissertation was inspired by the duality theory of linear programming developed by John von Neumann, which Rockafellar learned about through volumes of recent papers compiled by Albert W. Tucker at Princeton University. Rockafellar’s dissertation together with the contemporary work by Jean-Jacques Moreau in France are regarded as the birth of convex analysis.
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The green jack is moderately large in size, growing to a maximum confirmed length of 55 cm and 2.81 kg in weight. Unconfirmed reports indicate the species may grow much larger; up to 1 m in size, but it is most commonly seen at lengths below 40 cm. The green jack is morphologically similar to a number of other carangids, having an elongated, moderately compressed fusiform body with dorsal and ventral profiles of approximately equal convexity and a slightly pointed snout. The posterior section of the eye is covered by a moderately well developed adipose eyelid, and the posterior extremity of the jaw is vertically under the center of the eye. The dorsal fin is in two parts, the first consisting of 8 spines and the second of 1 spine followed by 22 to 25 soft rays.
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What can we infer about the relation of the second property to the road? The spatial configuration can be formalized in RCC8 as the following constraint network: house1 DC house2 house1 {TPP, NTPP} property1 house1 {DC, EC} property2 house1 EC road house2 { DC, EC } property1 house2 NTPP property2 house2 EC road property1 { DC, EC } property2 road { DC, EC, TPP, TPPi, PO, EQ, NTPP, NTPPi } property1 road { DC, EC, TPP, TPPi, PO, EQ, NTPP, NTPPi } property2 Using the RCC8 composition table and the path-consistency algorithm, we can refine the network in the following way: road { PO, EC } property1 road { PO, TPP } property2 That is, the road either overlaps with the second property, or is even (tangential) part of it. Other versions of the region connection calculus include RCC5 (with only five basic relations - the distinction whether two regions touch each other are ignored) and RCC23 (which allows reasoning about convexity).
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It leaves the body at an acute angle and retreats towards the right to form the sinus, which is open and near but not immediately at the body: from the sinus the lip-edge advances with a strong forward convexity to the point of the siphonal canal. Laterally it is also rather convex, but is contracted into the aperture, along the edge of which it is pretty straight with a somewhat oblique direction towards the left, and here it is patulous. The inner lip is porcellanous, smooth, narrow, cut off, and slightly twisted in front, and running out at the point to a sharp edge along the siphonal canal, the point of which is then rounded and patulous. Report on the scientific results of the voyage of H.M.S. Challenger during the years 1873-76 under the command of Captain George S. Nares ... and the late Captain Frank Tourle Thomson, R.N.; Zoology vol.
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The WFEs were developed out of the Regen exercise (also called “squat exercise”), advocated in the 1930s by Eugene M. Regen (1900-1983), a Tennessee orthopedic surgeon, and which consist in squatting and emphasizing the convexity of the lumbar area. (The Regen exercise was originally publicized in a film by the Veterans Administration.) Williams first published his own modified exercise program in 1937 for patients with chronic low back pain in response to his clinical observation that the majority of patients who experienced low back pain had degenerative vertebrae secondary to degenerative disk disease.Williams, Paul C. (1965), The Lumbosacral Spine: Emphasizing Conservative Management; 202 pp, 87 illus, New York: Blakiston Division, McGraw-Hill Book Co. These exercises were initially developed for men under 50 and women under 40 who had exaggerated lumbar lordosis, whose x-ray films showed decreased disc space between lumbar spine segments (L1-S1), and whose symptoms were chronic, but low grade.
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For relative convex hulls of simple polygons, an alternative but equivalent definition of convexity can be used. A simple polygon P within another simple polygon Q is relatively convex or Q-convex if every line segment contained in Q that connects two points of P lies within P. The relative convex hull of a simple polygon P within Q can be defined as the intersection of all Q-convex polygons that contain P, as the smallest Q-convex polygon that contains P, or as the minimum-perimeter simple polygon that contains P and is contained by Q. generalizes linear time algorithms for the convex hull of a simple polygon to the relative convex hull of one simple polygon within another. The resulting generalized algorithm is not linear time, however: its time complexity depends on the depth of nesting of certain features of one polygon within another. In this case, the relative convex hull is itself a simple polygon.
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Traditionally, the "crocodile-normal" and "advanced mesotarsal" ankle arrangements have been considered as a dichotomy among archosaurs: early archosaurs and pseudosuchians possess the more mobile "crocodile-normal" configuration, while pterosaurs and dinosauromorphs (including birds) possess the stiffer "advanced mesotarsal" configuration. The presence of the "crocodile-normal" ankle in Teleocrater (convex joint with the astragalus, presence of a tuber, and the convexity of the fibular facet on the calcaneum) indicates that this configuration was probably plesiomorphic for archosaurs, including avemetatarsalians, supported by reconstructions of character state evolution using the two datasets. At the same time, features associated with the "advanced mesotarsal" ankle (lack of a tuber and the concavity of the fibular facet on the calcaneum) were reconstructed as having appeared at least two different times among ornithodirans, with basal dinosaurs also possessing a mixture of "crocodile-normal" and "advanced mesotarsal" characteristics. This demonstrates that the evolution of ankle morphology in avemetatarsalians is more complex than previously thought, and led Nesbitt et al.
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The fibers of the oculomotor nerve arise from a nucleus in the midbrain, which lies in the gray substance of the floor of the cerebral aqueduct and extends in front of the aqueduct for a short distance into the floor of the third ventricle. From this nucleus the fibers pass forward through the tegmentum, the red nucleus, and the medial part of the substantia nigra, forming a series of curves with a lateral convexity, and emerge from the oculomotor sulcus on the medial side of the cerebral peduncle. The nucleus of the oculomotor nerve does not consist of a continuous column of cells, but is broken up into a number of smaller nuclei, which are arranged in two groups, anterior and posterior. Those of the posterior group are six in number, five of which are symmetrical on the two sides of the middle line, while the sixth is centrally placed and is common to the nerves of both sides.
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In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, that the liquid after stirring is contained within the space originally taken up by it, and that the glass (and stirred surface shape) maintain a convex volume. Ordering a cocktail shaken, not stirred defeats the convexity condition ("shaking" being defined as a dynamic series of non-convex inertial containment states in the vacant headspace under a lid). In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state.
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For example, the famous Buck 110 folding hunting knife is described as having a "hollow grind" - meaning the faces of the blade are ground into a concave – but the blade also contains a second, less acute, conventional bevel that makes up the cutting edge. A classic Opinel folding knife has a "flat grind" blade, meaning that the faces of the blade are flat, without convexity or concavity, tapering towards the cutting edge: but the actual cutting edge is again formed of another, less acute bevel ground on the narrow edge. A classic Morakniv has a saber or "Scandi" grind, with flat, perpendicular sides on the body, with a secondary bevel formed below to create a tapered edge, but again, the actual cutting edge comprises a third, less-acute bevel. Thus the "grind" of the blade most often refers to the overall cross-section of the blade and should not be confused with the actual style of cutting edge put in the blade, even though this cutting edge is created by grinding as well.
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Among Krantz's research interests include: several complex variables, harmonic analysis, partial differential equations, differential geometry, interpolation of operators, Lie theory, smoothness of functions, convexity theory, the corona problem, the inner functions problem, Fourier analysis, singular integrals, Lusin area integrals, Lipschitz spaces, finite difference operators, Hardy spaces, functions of bounded mean oscillation, geometric measure theory, sets of positive reach, the implicit function theorem, approximation theory, real analytic functions, analysis on the Heisenberg group, complex function theory, and real analysis.Washington University News and Information He applied wavelet analysis to plastic surgery, creating software for facial recognition. Krantz has also written software for the pharmaceutical industry. Krantz has worked on the inhomogeneous Cauchy–Riemann equations (he obtained the first sharp estimates in a variety of nonisotropic norms), on separate smoothness of functions (most notably with hypotheses about smoothness along integral curves of vector fields), on analysis on the Heisenberg group and other nilpotent Lie groups, on harmonic analysis in several complex variables, on the function theory of several complex variables, on the harmonic analysis of several real variables, on partial differential equations, on complex geometry, on the automorphism groups of domains in complex space, and on the geometry of complex domains.
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The transverse sinuses are of large size and begin at the internal occipital protuberance; one, generally the right, being the direct continuation of the superior sagittal sinus, the other of the straight sinus. Each transverse sinus passes lateral and forward, describing a slight curve with its convexity upward, to the base of the petrous portion of the temporal bone, and lies, in this part of its course, in the attached margin of the tentorium cerebelli; it then leaves the tentorium and curves downward and medialward (an area sometimes referred to as the sigmoid sinus) to reach the jugular foramen, where it ends in the internal jugular vein. In its course it rests upon the squama of the occipital, the mastoid angle of the parietal, the mastoid part of the temporal, and, just before its termination, the jugular process of the occipital; the portion which occupies the groove on the mastoid part of the temporal is sometimes termed the sigmoid sinus. The transverse sinuses are frequently of unequal size, with the one formed by the superior sagittal sinus being the larger; they increase in size as they proceed, from back to center.
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If a polynomial is SOS-convex, then it is also convex. Since establishing whether a polynomial is SOS-convex amounts to solving a semidefinite programming problem, SOS-convexity can be used as a proxy to establishing if a polynomial is convex. In contrast, deciding if a generic polynomial of degree large than four is convex is a NP-hard problem. The first counterexample of a polynomial which is convex but not SOS-convex was constructed by Amir Ali Ahmadi and Pablo Parrilo in 2009. The polynomial is a homogeneous polynomial that is sum-of-squares and given by: > p(x)= 32 x_{1}^{8}+118 x_{1}^{6} x_{2}^{2}+40 x_{1}^{6} x_{3}^{2}+25 > x_{1}^{4} x_{2}^{4} -43 x_{1}^{4} x_{2}^{2} x_{3}^{2}-35 x_{1}^{4} > x_{3}^{4}+3 x_{1}^{2} x_{2}^{4} x_{3}^{2} -16 x_{1}^{2} x_{2}^{2} > x_{3}^{4}+24 x_{1}^{2} x_{3}^{6}+16 x_{2}^{8} +44 x_{2}^{6} x_{3}^{2}+70 > x_{2}^{4} x_{3}^{4}+60 x_{2}^{2} x_{3}^{6}+30 x_{3}^{8} In the same year, Grigoriy Blekherman proved in a non-constructive manner that there exist convex forms that is not representable as sum of squares.
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