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Journal articles on the topic 'Convex Algebraic Geometry'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 February 2022

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1

Ewald, Günter. "CONVEX BODIES AND ALGEBRAIC GEOMETRY." Annals of the New York Academy of Sciences 440, no. 1 Discrete Geom (1985): 196–204. http://dx.doi.org/10.1111/j.1749-6632.1985.tb14554.x.

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2

Escobar, Laura, and Kiumars Kaveh. "Convex Polytopes, Algebraic Geometry, and Combinatorics." Notices of the American Mathematical Society 67, no. 08 (2020): 1. http://dx.doi.org/10.1090/noti2137.

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3

Bettiol, Renato G., Mario Kummer, and Ricardo A. E. Mendes. "Convex Algebraic Geometry of Curvature Operators." SIAM Journal on Applied Algebra and Geometry 5, no. 2 (2021): 200–228. http://dx.doi.org/10.1137/20m1350777.

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4

Sturmfels, Bernd, and Caroline Uhler. "Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry." Annals of the Institute of Statistical Mathematics 62, no. 4 (2010): 603–38. http://dx.doi.org/10.1007/s10463-010-0295-4.

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5

CORBALAN, A. G., M. MAZON, and T. RECIO. "GEOMETRY OF BISECTORS FOR STRICTLY CONVEX DISTANCES." International Journal of Computational Geometry & Applications 06, no. 01 (1996): 45–58. http://dx.doi.org/10.1142/s0218195996000046.

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In this paper we study some unexpected geometric properties of the family of bisector lines for a convex distance d, showing that bisectors do not always have an asymptotic line (Section 2). Moreover, although bisectors are homeomorphic to lines, pairs of them can exist intersecting infinitely many times (Section 3). This leads to the conclusion that convex distances are not always nice in the sense of Klein and Wood.7 On the other hand, we prove that distances d, having d-balls whose boundary is given by finitely many algebraic conditions, produce nice distances (Section 3).

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6

Zhang, Kewei. "Quasi-convex functions on subspaces and boundaries of quasi-convex sets." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 4 (2004): 783–99. http://dx.doi.org/10.1017/s0308210500003486.

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We embed truncations of the epi-graph of quasi-convex functions defined on linear subspaces E ⊂ MN × n of real matrices into MN × n to bound quasi-convex sets by the graph of the functions. We also characterize subspaces E on which all quasi-convex functions are convex and show, by using the Tarski–Seidenberg theorem in real algebraic geometry, that if dim (E) > N + n − 1, then there exist non-trivial quasi-convex functions on E.

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7

TANG, KAI, CHARLIE C. L. WANG, and DANNY Z. CHEN. "MINIMUM AREA CONVEX PACKING OF TWO CONVEX POLYGONS." International Journal of Computational Geometry & Applications 16, no. 01 (2006): 41–74. http://dx.doi.org/10.1142/s0218195906001926.

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Given two convex polygons P and Q in the plane that are free to translate and rotate, a convex packing of them is the convex hull of a placement of P and a placement of Q whose interiors do not intersect. A minimum area convex packing of P and Q is one whose area is minimized. The problem of designing a deterministic algorithm for finding a minimum area convex packing of two convex polygons has remained open. We address this problem by first studying the contact configurations between P and Q and their algebraic structures. Crucial geometric and algebraic properties on the area function are then derived and analyzed which enable us to successfully discretize the search space. This discretization, together with a delicate algorithmic design and careful complexity analysis, allows us to develop an efficient O((n + m)nm) time deterministic algorithm for finding a true minimum area convex packing of P and Q, where n and m are the numbers of vertices of P and Q, respectively.

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Jiang, Yuhan, and Bernd Sturmfels. "Bad projections of the PSD cone." Collectanea Mathematica 72, no. 2 (2021): 261–80. http://dx.doi.org/10.1007/s13348-021-00319-4.

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AbstractThe image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.

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9

Reid, Miles. "Book Review: Convex bodies and algebraic geometry: An introduction to toric varieties." Bulletin of the American Mathematical Society 21, no. 2 (1989): 360–65. http://dx.doi.org/10.1090/s0273-0979-1989-15864-3.

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10

Looijenga, Eduard. "Discrete automorphism groups of convex cones of finite type." Compositio Mathematica 150, no. 11 (2014): 1939–62. http://dx.doi.org/10.1112/s0010437x14007404.

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AbstractWe investigate subgroups of $\text{SL}(n,\mathbb{Z})$ which preserve an open nondegenerate convex cone in $\mathbb{R}^{n}$ and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on self-dual cones, Weyl groups of certain Kac–Moody algebras, and they do occur in algebraic geometry as the automorphism groups of projective manifolds acting on their ample cones.

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11

Ghosh, Arpita, and Stephen Boyd. "Upper bounds on algebraic connectivity via convex optimization." Linear Algebra and its Applications 418, no. 2-3 (2006): 693–707. http://dx.doi.org/10.1016/j.laa.2006.03.006.

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12

JIANG, D., and N. F. STEWART. "FLOATING-POINT ARITHMETIC FOR COMPUTATIONAL GEOMETRY PROBLEMS WITH UNCERTAIN DATA." International Journal of Computational Geometry & Applications 19, no. 04 (2009): 371–85. http://dx.doi.org/10.1142/s0218195909003015.

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It has been suggested in the literature that ordinary finite-precision floating-point arithmetic is inadequate for geometric computation, and that researchers in numerical analysis may believe that the difficulties of error in geometric computation can be overcome by simple approaches. It is the purpose of this paper to show that these suggestions, based on an example showing failure of a certain algorithm for computing planar convex hulls, are misleading, and why this is so. It is first shown how the now-classical backward error analysis can be applied in the area of computational geometry. This analysis is relevant in the context of uncertain data, which may well be the practical context for computational-geometry algorithms such as, say, those for computing convex hulls. The exposition will illustrate the fact that the backward error analysis does not pretend to overcome the problem of finite precision: it merely provides a way to distinguish those algorithms that overcome the problem to whatever extent it is possible to do so. It is then shown that often the situation in computational geometry is exactly parallel to other areas, such as the numerical solution of linear equations, or the algebraic eigenvalue problem. Indeed, the example mentioned can be viewed simply as an example of the use of an unstable algorithm, for a problem for which computational geometry has already discovered provably stable algorithms. Finally, the paper discusses the implications of these analyses for applications in three-dimensional solid modeling. This is done by considering a problem defined in terms of a simple extension of the planar convex-hull algorithm, namely, the verification of the well-formedness of extruded objects. A brief discussion concerning more difficult problems in solid modeling is also included.

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13

Tran, Ngoc Mai, and Josephine Yu. "Product-Mix Auctions and Tropical Geometry." Mathematics of Operations Research 44, no. 4 (2019): 1396–411. http://dx.doi.org/10.1287/moor.2018.0975.

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In a recent and ongoing work, Baldwin and Klemperer explore a connection between tropical geometry and economics. They give a sufficient condition for the existence of competitive equilibrium in product-mix auctions of indivisible goods. This result, which we call the unimodularity theorem, can also be traced back to the work of Danilov, Koshevoy, and Murota in discrete convex analysis. We give a new proof of the unimodularity theorem via the classical unimodularity theorem in integer programming. We give a unified treatment of these results via tropical geometry and formulate a new sufficient condition for competitive equilibrium when there are only two types of products. Generalizations of our theorem in higher dimensions are equivalent to various forms of the Oda conjecture in algebraic geometry.

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14

Kohn, Kathlén, and Kristian Ranestad. "Projective Geometry of Wachspress Coordinates." Foundations of Computational Mathematics 20, no. 5 (2019): 1135–73. http://dx.doi.org/10.1007/s10208-019-09441-z.

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Abstract We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.

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15

Emiris, Ioannis Z. "A Complete Implementation for Computing General Dimensional Convex Hulls." International Journal of Computational Geometry & Applications 08, no. 02 (1998): 223–53. http://dx.doi.org/10.1142/s0218195998000126.

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We present a robust implementation of the Beneath-Beyond algorithm for computing convex hulls in arbitrary dimension. Certain techniques used are of independent interest in the implementation of geometric algorithms. In particular, two important, and often complementary, issues are studied, namely exact arithmetic and degeneracy. We focus on integer arithmetic and propose a general and efficient method for its implementation based on modular arithmetic. We suggest that probabilistic modular arithmetic may be of wide interest, as it combines the advantages of modular arithmetic with the speed of randomization. The use of perturbations as a method to cope with input degeneracy is also illustrated. A computationally efficient scheme is implemented which, moreover, greatly simplifies the task of programming. We concentrate on postprocessing, often perceived as the Achilles' heel of perturbations. Experimental results illustrate the dependence of running time on the various input parameters and attempt a comparison with existing programs. Lastly, we discuss the visualization capabilities of our software and illustrate them for problems in computational algebraic geometry. All code is publicly available.

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16

HARRINGTON, PAUL, COLM Ó. DÚNLAING, and CHEE K. YAP. "OPTIMAL VORONOI DIAGRAM CONSTRUCTION WITH n CONVEX SITES IN THREE DIMENSIONS." International Journal of Computational Geometry & Applications 17, no. 06 (2007): 555–93. http://dx.doi.org/10.1142/s0218195907002483.

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This paper presents a worst-case optimal algorithm for constructing the Voronoi diagram for n disjoint convex and rounded semi-algebraic sites in 3 dimensions. Rather than extending optimal 2-dimensional methods,32,16,20,2 we base our method on a suboptimal 2-dimensional algorithm, outlined by Lee and Drysdale and modified by Sharir25,30 for computing the diagram of circular sites. For complexity considerations, we assume the sites have bounded complexity, i.e., the algebraic degree is bounded as is the number of algebraic patches making up the site. For the sake of simplicity we assume that the sites are what we call rounded. This assumption simplifies the analysis, though it is probably unnecessary. Our algorithm runs in time O(C(n)) where C(n) is the worst-case complexity of an n-site diagram. For spherical sites C(n) is θ(n2), but sharp estimates do not seem to be available for other classes of site.

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17

Katz, Eric, and Stefano Urbinati. "Newton–Okounkov Bodies over Discrete Valuation Rings and Linear Systems on Graphs." International Mathematics Research Notices 2019, no. 14 (2018): 4516–48. http://dx.doi.org/10.1093/imrn/rnx248.

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Abstract The theory of Newton–Okounkov bodies attaches a convex body to a line bundle on a variety equipped with a flag of subvarieties. This convex body encodes the asymptotic properties of sections of powers of the line bundle. In this article, we study Newton–Okounkov bodies for schemes defined over discrete valuation rings. We give the basic properties and then focus on the case of toric schemes and semistable curves. We provide a description of the Newton–Okounkov bodies for semistable curves in terms of the Baker–Norine theory of linear systems on graphs, finding a connection with tropical geometry. We do this by introducing an intermediate object, the Newton–Okounkov linear system of a divisor on a curve. We prove that it is equal to the set of effective elements of the real Baker–Norine linear system of the specialization of that divisor on the dual graph of the curve. As a bonus, we obtain an asymptotic algebraic geometric description of the Baker–Norine linear system.

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18

Mehmood, Faisal, Fu-Gui Shi, Khizar Hayat, and Xiao-Peng Yang. "The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities." Mathematics 8, no. 3 (2020): 411. http://dx.doi.org/10.3390/math8030411.

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In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings.

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19

Ahmad, Shamsatun Nahar, Nor’ Aini Aris, and Azlina Jumadi. "The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials." Scientific Research Journal 16, no. 2 (2019): 1. http://dx.doi.org/10.24191/srj.v16i2.5507.

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Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

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20

Ahmad, Shamsatun Nahar, Nor’Aini Aris, and Azlina Jumadi. "The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials." Scientific Research Journal 16, no. 2 (2019): 1. http://dx.doi.org/10.24191/srj.v16i2.9346.

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Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

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21

Gubler, Walter, and Klaus Künnemann. "Positivity properties of metrics and delta-forms." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 752 (2019): 141–77. http://dx.doi.org/10.1515/crelle-2016-0060.

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Abstract In previous work, we have introduced δ-forms on the Berkovich analytification of an algebraic variety in order to study smooth or formal metrics via their associated Chern δ-forms. In this paper, we investigate positivity properties of δ-forms and δ-currents. This leads to various plurisubharmonicity notions for continuous metrics on line bundles. In the case of a formal metric, we show that many of these positivity notions are equivalent to Zhang’s semipositivity. For piecewise smooth metrics, we prove that plurisubharmonicity can be tested on tropical charts in terms of convex geometry. We apply this to smooth metrics, to canonical metrics on abelian varieties and to toric metrics on toric varieties.

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22

MORETTI, VALTER. "ASPECTS OF NONCOMMUTATIVE LORENTZIAN GEOMETRY FOR GLOBALLY HYPERBOLIC SPACETIMES." Reviews in Mathematical Physics 15, no. 10 (2003): 1171–217. http://dx.doi.org/10.1142/s0129055x03001886.

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Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally-hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of C*-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C*-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the role of a Lorentzian metric. Specializing back the formalism to the usual globally-hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events.

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23

Perri, Tal, and Louis H. Rowen. "Kernels in tropical geometry and a Jordan–Hölder theorem." Journal of Algebra and Its Applications 17, no. 04 (2018): 1850066. http://dx.doi.org/10.1142/s0219498818500664.

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When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, [Formula: see text]-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield[Formula: see text][Formula: see text], we pass to the semifield[Formula: see text][Formula: see text] of fractions of the polynomial semiring[Formula: see text], for which there already exists a well developed theory of kernels, which are normal convex subgroups of [Formula: see text]; the parallel of the zero set now is the [Formula: see text]-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to [Formula: see text]-kernels (Definition 4.1.4) and [Formula: see text]-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The [Formula: see text]-kernels corresponding to tropical hypersurfaces are the [Formula: see text]-sets of what we call “corner internal rational functions,” and we describe [Formula: see text]-kernels corresponding to “usual” tropical geometry as [Formula: see text]-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of [Formula: see text]-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between [Formula: see text]-sets and a class of [Formula: see text]-kernels of the rational [Formula: see text]-semifield[Formula: see text] called polars, originating from the theory of lattice-ordered groups. When [Formula: see text] is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal [Formula: see text]-kernels, intersected with the [Formula: see text]-kernel generated by [Formula: see text]. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of [Formula: see text]-kernels.

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24

Wilson, P. M. H. "CONVEX BODIES AND ALGEBRAIC GEOMETRY An Introduction to the Theory of Toric Varieties (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 15)." Bulletin of the London Mathematical Society 21, no. 6 (1989): 604–5. http://dx.doi.org/10.1112/blms/21.6.604.

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25

Yaliraki, Sophia N., and Mauricio Barahona. "Chemistry across scales: from molecules to cells." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365, no. 1861 (2007): 2921–34. http://dx.doi.org/10.1098/rsta.2007.0015.

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Many important biological functions are strongly dependent on specific chemical interactions. Modelling how the physicochemical molecular details emerge at much larger scales is an active area of research, currently pursued with a variety of methods. We describe a series of theoretical and computational approaches that aim to derive bottom-up descriptions that capture the specificity that ensues from atomistic detail by extracting relevant features at the different scales. The multiscale models integrate the descriptions at different length and time scales by exploiting the idea of mechanical responses. The methodologies bring together concepts and tools developed in seemingly unrelated areas of mathematics such as algebraic geometry, model reduction, structural graph theory and non-convex optimization. We showcase the applicability of the framework with examples from protein engineering and enzyme catalysis, protein assembly, and with the description of lipid bilayers at different scales. Many challenges remain as it is clear that no single methodology will answer all questions in such multidimensional complex problems.

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26

Acharyya, Amrita, Sudip Kumar Acharyya, Sagarmoy Bag, and Joshua Sack. "Intermediate rings of complex-valued continuous functions." Applied General Topology 22, no. 1 (2021): 47. http://dx.doi.org/10.4995/agt.2021.13165.

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<p>For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on X, let C ∗ (X, C) be its subring of bounded functions, and let Σ(X, C) denote the collection of all the rings that lie between C ∗ (X, C) and C(X, C). We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z ◦ -ideals in the rings P(X, C) in Σ(X, C) and in their real-valued counterparts P(X, C) ∩ C(X). These correlations culminate to the fact that the structure space of any such P(X, C) is βX. For any ideal I in C(X, C), we observe that C ∗ (X, C)+I is a member of Σ(X, C), which is further isomorphic to a ring of the type C(Y, C). Incidentally these are the only C-type intermediate rings in Σ(X, C) if and only if X is pseudocompact. We show that for any maximal ideal M in C(X, C), C(X, C)/M is an algebraically closed field, which is furthermore the algebraic closure of C(X)/M ∩C(X). We give a necessary and sufficient condition for the ideal CP (X, C) of C(X, C), which consists of all those functions whose support lie on an ideal P of closed sets in X, to be a prime ideal, and we examine a few special cases thereafter. At the end of the article, we find estimates for a few standard parameters concerning the zero-divisor graphs of a P(X, C) in Σ(X, C).</p>

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Selby, John H., and Ciarán M. Lee. "Compositional resource theories of coherence." Quantum 4 (September 11, 2020): 319. http://dx.doi.org/10.22331/q-2020-09-11-319.

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Quantum coherence is one of the most important resources in quantum information theory. Indeed, preventing the loss of coherence is one of the most important technical challenges obstructing the development of large-scale quantum computers. Recently, there has been substantial progress in developing mathematical resource theories of coherence, paving the way towards its quantification and control. To date however, these resource theories have only been mathematically formalised within the realms of convex-geometry, information theory, and linear algebra. This approach is limited in scope, and makes it difficult to generalise beyond resource theories of coherence for single system quantum states. In this paper we take a complementary perspective, showing that resource theories of coherence can instead be defined purely compositionally, that is, working with the mathematics of process theories, string diagrams and category theory. This new perspective offers several advantages: i) it unifies various existing approaches to the study of coherence, for example, subsuming both speakable and unspeakable coherence; ii) it provides a general treatment of the compositional multi-system setting; iii) it generalises immediately to the case of quantum channels, measurements, instruments, and beyond rather than just states; iv) it can easily be generalised to the setting where there are multiple distinct sources of decoherence; and, iv) it directly extends to arbitrary process theories, for example, generalised probabilistic theories and Spekkens toy model---providing the ability to operationally characterise coherence rather than relying on specific mathematical features of quantum theory for its description. More importantly, by providing a new, complementary, perspective on the resource of coherence, this work opens the door to the development of novel tools which would not be accessible from the linear algebraic mind set.

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Kaplan, Haim, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. "Dynamic Planar Voronoi Diagrams for General Distance Functions and Their Algorithmic Applications." Discrete & Computational Geometry 64, no. 3 (2020): 838–904. http://dx.doi.org/10.1007/s00454-020-00243-7.

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Abstract We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include $$L_p$$ L p -norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses $$O(n \log ^3 n)$$ O ( n log 3 n ) storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat, and Sharir which required $$O(n^{\varepsilon })$$ O ( n ε ) time for an update and $$O(\log n)$$ O ( log n ) time for a query [SICOMP 1999]. Our data structure has numerous applications. In all of them, it gives faster algorithms, typically reducing an $$O(n^{\varepsilon })$$ O ( n ε ) factor in the previous bounds to polylogarithmic. In addition, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph. To obtain this data structure, we combine and extend various techniques from the literature. Along the way, we obtain several side results that are of independent interest. Our data structure depends on the existence and an efficient construction of “vertical” shallow cuttings in arrangements of bivariate algebraic functions. We prove that an appropriate level in an arrangement of a random sample of a suitable size provides such a cutting. To compute it efficiently, we develop a randomized incremental construction algorithm for computing the lowest k levels in an arrangement of bivariate algebraic functions (we mostly consider here collections of functions whose lower envelope has linear complexity, as is the case in the dynamic nearest-neighbor context, under both types of norm). To analyze this algorithm, we also improve a longstanding bound on the combinatorial complexity of the vertical decomposition of these levels. Finally, to obtain our structure, we combine our vertical shallow cutting construction with Chan’s algorithm for efficiently maintaining the lower envelope of a dynamic set of planes in $${{\mathbb {R}}}^3$$ R 3 . Along the way, we also revisit Chan’s technique and present a variant that uses a single binary counter, with a simpler analysis and improved amortized deletion time (by a logarithmic factor; the insertion and query costs remain asymptotically the same).

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Berg, Mark de, Dan Halperin, Mark Overmars, and Marc van Kreveld. "Sparse Arrangements and the Number of Views of Polyhedral Scenes." International Journal of Computational Geometry & Applications 07, no. 03 (1997): 175–95. http://dx.doi.org/10.1142/s0218195997000120.

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In this paper we study several instances of the problem of determining the maximum number of topologically distinct two-dimensional images that three-dimensional scenes can induce. To bound this number, we investigate arrangements of curves and of surfaces that have a certain sparseness property. Given a collection of n algebraic surface patches of constant maximum degree in 3-space with the property that any vertical line stabs at most k of them, we show that the maximum combinatorial complexity of the entire arrangement that they induce is Θ(n2 k). We extend this result to collections of hypersurfaces in 4-space and to collections of (d > 1)-simplices in d-space, for any fixed d. We show that this type of arrangements (sparse arrangements) is relevant to the study of the maximum number of topologically different views of a polyhedral terrain. For polyhedral terrains with n edges and vertices, we introduce a lower bound construction inducing Ω(n5 α(n)) distinct views, and we present an almost matching upper bound. We then analyze the case of perspective views, point to the potential role of sparse arrangements in obtaining a sharp bound for this case, and present a lower bound construction inducing Ω(n8α(n)) distinct views. For the number of views of a collection of k convex polyhedra with a total of n faces, we show a bound of O(n4 k2) for views from infinity and O(n6 k3) for perspective views. We also present lower bound constructions for such scenes, with Ω(n4 + n2 k4) distinct views from infinity and Ω(n6 + n3 k6) views when the viewpoint can be anywhere in 3-space.

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CZÉDLI, GÁBOR, and ANNA B. ROMANOWSKA. "GENERALIZED CONVEXITY AND CLOSURE CONDITIONS." International Journal of Algebra and Computation 23, no. 08 (2013): 1805–35. http://dx.doi.org/10.1142/s0218196713500458.

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Convex subsets of affine spaces over the field of real numbers are described by so-called barycentric algebras. In this paper, we discuss extensions of the geometric and algebraic definitions of a convex set to the case of more general coefficient rings. In particular, we show that the principal ideal subdomains of the reals provide a good framework for such a generalization. Since the closed intervals of these subdomains play an essential role, we provide a detailed analysis of certain cases, and discuss differences from the "classical" intervals of the reals. We introduce a new concept of an algebraic closure of "geometric" convex subsets of affine spaces over the subdomains in question, and investigate their properties. We show that this closure provides a purely algebraic description of topological closures of geometric generalized convex sets. Our closure corresponds to one instance of the very general closure introduced in an earlier paper of the authors. The approach used in this paper allows to extend some results from that paper. Moreover, it provides a very simple description of the closure, with concise proofs of existence and uniqueness.

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31

Lavenda, B. H. "Geometric Entropies of Mixing (EOM)." Open Systems & Information Dynamics 13, no. 01 (2006): 91–101. http://dx.doi.org/10.1007/s11080-006-7270-9.

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Trigonometric and trigonometric-algebraic entropies are introduced and are given an axiomatic characterization. Regularity increases the entropy and the maximal entropy is shown to result when a regular n-gon is inscribed in a circle. A regular n-gon circumscribing a circle gives the largest entropy reduction, or the smallest change in entropy from the state of maximum entropy, which occurs in the asymptotic infinite n-limit. The EOM are shown to correspond to minimum perimeter and maximum area in the theory of convex bodies, and can be used in the prediction of new inequalities for convex sets. These expressions are shown to be related to the phase functions obtained from the WKB approximation for Bessel and Hermite functions.

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32

Milman, V., and L. Rotem. "“Irrational” constructions in Convex Geometry." St. Petersburg Mathematical Journal 29, no. 1 (2017): 165–75. http://dx.doi.org/10.1090/spmj/1487.

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33

Rabiei, Nima, and Elias G. Saleeby. "On the sample-mean method for computing hyper-volumes." Monte Carlo Methods and Applications 25, no. 2 (2019): 163–76. http://dx.doi.org/10.1515/mcma-2019-2034.

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Abstract Estimating hyper-volumes of convex and non-convex sets are of interest in a number of areas. In this article we develop further a simple geometric Monte Carlo method, known also as the sample-mean method, which transforms the domain to an equivalent hyper-sphere with the same volume. We first examine the performance of the method to compute the volumes of star-convex unit balls and show that it gives accurate estimates of their volumes. We then examine the use of this method for computing the volumes of nonstar-shaped domains. In particular, we develop two algorithms, which couple the sample-mean method with algebraic and geometric techniques, to generate and compute the volumes of low-dimensional stability domains in parameter space.

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34

Adaricheva, K. V., and J. B. Nation. "Largest extension of a finite convex geometry." algebra universalis 52, no. 2-3 (2005): 185–95. http://dx.doi.org/10.1007/s00012-004-1844-6.

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35

Rotem, Liran. "Algebraically inspired results on convex functions and bodies." Communications in Contemporary Mathematics 18, no. 06 (2016): 1650027. http://dx.doi.org/10.1142/s0219199716500279.

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We show how algebraic identities, inequalities and constructions, which hold for numbers or matrices, often have analogs in the geometric classes of convex bodies or convex functions. By letting the polar body [Formula: see text] or the dual function [Formula: see text] play the role of the inverses “[Formula: see text]” and “[Formula: see text]”, we are able to conjecture many new results, which often turn out to be correct. As one example, we prove that for every convex function [Formula: see text] one has [Formula: see text] where [Formula: see text]. We also prove several corollaries of this identity, including a Santal type inequality and a contribution to the theory of summands. We proceed to discuss the analogous identity for convex bodies, where an unexpected distinction appears between the classical Minkowski addition and the more modern 2-addition. In the final section of the paper we consider the harmonic and geometric means of convex bodies and convex functions, and discuss their concavity properties. Once again, we find that in some problems the 2-addition of convex bodies behaves even better than the Minkowski addition.

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36

Harutyunyan, Davit. "Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 2 (2018): 479–94. http://dx.doi.org/10.1051/cocv/2017004.

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In this paper we revisit the anisotropic isoperimetric and the Brunn−Minkowski inequalities for convex sets. The best known constant C(n) = Cn7 depending on the space dimension n in both inequalities is due to Segal [A. Segal, Lect. Notes Math., Springer, Heidelberg 2050 (2012) 381–391]. We improve that constant to Cn6 for convex sets and to Cn5 for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form Cn2, i.e., quadratic in n. The tools are the Brenier’s mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.

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37

Yost, David. "Strictly Convex Banach Algebras." Axioms 10, no. 3 (2021): 221. http://dx.doi.org/10.3390/axioms10030221.

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We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In C∗-algebras, we exhibit one striking example of the tighter relationship that exists between algebra and geometry there.

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38

Nitica, V., and I. Singer. "Contributions to max–min convex geometry. II: Semispaces and convex sets." Linear Algebra and its Applications 428, no. 8-9 (2008): 2085–115. http://dx.doi.org/10.1016/j.laa.2007.11.015.

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39

Romaguera, S., E. A. Sánchez Pérez, and O. Valero. "Dominated extensions of functionals and V-convex functions of cancellative cones." Bulletin of the Australian Mathematical Society 67, no. 1 (2003): 87–94. http://dx.doi.org/10.1017/s0004972700033542.

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Let C be a cancellative cone and consider a subcone C0 of C. We study the natural problem of obtaining conditions on a non negative homogeneous function φ: C → R+ so that for each linear functional f defined in C0 which is bounded by φ, there exists a linear extension to C. In order to do this we assume several geometric conditions for cones related to the existence of special algebraic basis of the linear span of these cones.

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40

Nitica, V., and I. Singer. "Contributions to max–min convex geometry. I: Segments." Linear Algebra and its Applications 428, no. 7 (2008): 1439–59. http://dx.doi.org/10.1016/j.laa.2007.09.032.

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41

Ahmad, Bakhtiar, Muhammad Ghaffar Khan, Mohamed Kamal Aouf, Wali Khan Mashwani, Zabidin Salleh, and Huo Tang. "Applications of a New q -Difference Operator in Janowski-Type Meromorphic Convex Functions." Journal of Function Spaces 2021 (April 15, 2021): 1–10. http://dx.doi.org/10.1155/2021/5534357.

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The main aim of the present article is the introduction of a new differential operator in q -analogue for meromorphic multivalent functions which are analytic in punctured open unit disc. A subclass of meromorphic multivalent convex functions is defined using this new differential operator in q -analogue. Furthermore, we discuss a number of useful geometric properties for the functions belonging to this class such as sufficiency criteria, coefficient estimates, distortion theorem, growth theorem, radius of starlikeness, and radius of convexity. Also, algebraic property of closure is discussed of functions belonging to this class. Integral representation problem is also proved for these functions.

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42

Makeev, V. V. "Some geometric properties of convex bodies. II." St. Petersburg Mathematical Journal 15, no. 06 (2004): 867–75. http://dx.doi.org/10.1090/s1061-0022-04-00836-2.

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43

Lin, Si-Da, Fu-Min Xiao, Zun-Quan Xia, and Li-Ping Pang. "Research on Adjoint Kernelled Quasidifferential." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/131482.

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The quasidifferential of a quasidifferentiable function in the sense of Demyanov and Rubinov is not uniquely defined. Xia proposed the notion of the kernelled quasidifferential, which is expected to be a representative for the equivalence class of quasidifferentials. Although the kernelled quasidifferential is known to have good algebraic properties and geometric structure, it is still not very convenient for calculating the kernelled quasidifferentials of−fandminfi∣i∈a finite index set I, wherefandfiare kernelled quasidifferentiable functions. In this paper, the notion of adjoint kernelled quasidifferential, which is well-defined for−fandminfi∣i∈I, is employed as a representative of the equivalence class of quasidifferentials. Some algebraic properties of the adjoint kernelled quasidifferential are given and the existence of the adjoint kernelled quasidifferential is explored by means of the minimal quasidifferential and the Demyanov difference of convex sets. Under some condition, a formula of the adjoint kernelled quasidifferential is presented.

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44

Ivanov, A. O., and A. A. Tuzhilin. "Geometry of convex polygons and locally minimal binary trees spanning these polygons." Sbornik: Mathematics 190, no. 1 (1999): 71–110. http://dx.doi.org/10.1070/sm1999v190n01abeh000378.

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45

Väisälä, Jussi. "Triangles in convex distance planes." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 59, no. 4 (2018): 797–804. http://dx.doi.org/10.1007/s13366-018-0389-3.

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46

Zarichnyi, M. "FUNCTORS AND SPACES IN IDEMPOTENT MATHEMATICS." Bukovinian Mathematical Journal 9, no. 1 (2021): 171–79. http://dx.doi.org/10.31861/bmj2021.01.14.

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Idempotent mathematics is a branch of mathematics in which idempotent operations (for example, max) on the set of reals play a central role. In recent decades, we have seen intensive research in this direction. The principle of correspondence (this is an informal principle analogous to the Bohr correspondence principle in the quantum mechanics) asserts that each meaningful concept or result of traditional mathematics corresponds to a meaningful concept or result of idempotent mathematics. In particular, to the notion of probability measure there corresponds that if Maslov measure (also called idempotent measure) as well as more recent notion of max-min measure. Also, there are idempotent counterparts of the convex sets; these include the so-called max-plus and max min convex sets. Methods of idempotent mathematics are used in optimization problems, dynamic programming, mathematical economics, game theory, mathematical biology and other disciplines. In this paper we provide a survey of results that concern algebraic and geometric properties of the functors of idempotent and max-min measures.

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47

Inassaridze, Hvedri. "K-regularity of locally convex algebras." Journal of Homotopy and Related Structures 11, no. 4 (2016): 869–84. http://dx.doi.org/10.1007/s40062-016-0155-x.

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Richter, Christian. "Self-affine convex discs are polygons." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 53, no. 1 (2011): 219–24. http://dx.doi.org/10.1007/s13366-011-0044-8.

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49

David, Liana. "The Bochner-flat cone of a CR manifold." Compositio Mathematica 144, no. 3 (2008): 747–73. http://dx.doi.org/10.1112/s0010437x07003363.

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AbstractWe construct a Kähler structure (which we call a generalised Kähler cone) on an open subset of the cone of a strongly pseudo-convex CR manifold endowed with a one-parameter family of compatible Sasaki structures. We determine those generalised Kähler cones which are Bochner-flat and we study their local geometry. We prove that any Bochner-flat Kähler manifold of complex dimension bigger than two is locally isomorphic to a generalised Kähler cone.

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50

Aujla, Jaspal Singh, and Jean-Christophe Bourin. "Eigenvalue inequalities for convex and log-convex functions." Linear Algebra and its Applications 424, no. 1 (2007): 25–35. http://dx.doi.org/10.1016/j.laa.2006.02.027.

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