Convex Cone Research Articles

Anomalous linear and quadratic nodeless surface Dirac cones in three-dimensional Dirac semimetals

Anomalous linear and quadratic nodeless surface Dirac cones in three-dimensional Dirac semimetals

Coneability of Anti-Fubini Functions and Other Lineability Properties

It is proved that the family of measurable real functions on the product measure space satisfying (or not) the conclusion of Fubini’s Theorem (along with a number of related families) is algebraically large, in the sense that it contains large convex cones or even large vector subspaces (except for zero) under rather general assumptions. Several earlier related results are improved, mainly regarding the cardinality of the generator set of the corresponding cones or subspaces.

Lorentzian polynomials, Segre classes, and adjoint polynomials of convex polyhedral cones

We consider polynomials expressing the cohomology classes of subvarieties of products of projective spaces, and limits of positive real multiples of such polynomials. We study the relation between these covolume polynomials and Lorentzian polynomials. While these are distinct notions, we prove that, like Lorentzian polynomials, covolume polynomials have M-convex support and generalize the notion of log-concave sequences. In fact, we prove that covolume polynomials are ‘sectional log-concave’, that is, the coefficients of suitable restrictions of these polynomials form log-concave sequences.We observe that Chern classes of globally generated bundles give rise to covolume polynomials, and use this fact to prove that certain polynomials associated with Segre classes of subschemes of products of projective spaces are covolume polynomials. We conjecture that the same polynomials may be Lorentzian after a standard normalization operation.Finally, we obtain a combinatorial application of a particular case of our Segre class result. We prove that the adjoint polynomial of a convex polyhedral cone contained in the nonnegative orthant, and sharing a face with it, is a covolume polynomials. This implies that these adjoint polynomials are M-convex and sectional log-concave, and in fact dually Lorentzian, that is, Lorentzian after a certain change of variables.

Exposedness of elementary positive maps between matrix algebras

ABSTRACT The positive linear maps Ad s which send matrices x to s ∗ x s play important roles in quantum information theory as well as matrix theory. It was proved by [Marciniak M. Rank properties of exposed positive maps. Linear Multilinear Alg. 2013;61:970–975] that the map Ad s generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.

Polyhedral Approximation of Spectrahedral Shadows via Homogenization

This article is concerned with the problem of approximating a not necessarily bounded spectrahedral shadow, a certain convex set, by polyhedra. By identifying the set with its homogenization, the problem is reduced to the approximation of a closed convex cone. We introduce the notion of homogeneous δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-approximation of a convex set and show that it defines a meaningful concept in the sense that approximations converge to the original set if the approximation error δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} diminishes. Moreover, we show that a homogeneous δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-approximation of the polar of a convex set is immediately available from an approximation of the set itself under mild conditions. Finally, we present an algorithm for the computation of homogeneous δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-approximations of spectrahedral shadows and demonstrate it on examples.

Dirac cones in three-layered plates

The proper Dirac-like cones are observed in dispersion portraits of guided waves propagating in isotropic three-layered traction-free plates. It is revealed that if both layers have equal Poisson’s ratios and satisfy Wiechert condition, the Dirac-like cones appear regardless of the Poisson’s ratio value. This phenomenon opens a new approach for the creation of the layered plates possessing Dirac-like cones. The analysis is based on the Cauchy sextic formalism combined with the exponential fundamental matrix method.

Monotonicity and Contraction on Polyhedral Cones

Monotonicity and Contraction on Polyhedral Cones

О структуре пространства весов голосующих процедур

The structure of the self-dual threshold functions space of weights used in voting decision-making procedures has been studied for dimensions 2-6. Extremal vectors of polyhedral cones representing these threshold functions in the space of weights have been found.

Dual Brunn-Minkowski inequality for $ C $-star bodies

<abstract><p>In this paper, we introduced the concept of $ C $-star bodies in a fixed pointed closed convex cone $ C $ and studied the dual mixed volume for $ C $-star bodies. For $ C $-star bodies, we established the corresponding dual Brunn-Minkowski inequality, dual Minkowski inequality, and dual Aleksandrov-Fenchel inequality. Moveover, we found that the dual Brunn-Minkowski inequality for $ C $-star bodies can strengthen the Brunn-Minkowski inequality for $ C $-coconvex sets.</p></abstract>

Constrained mean-variance investment-reinsurance under the Cramér–Lundberg model with random coefficients

In this paper, we study an optimal mean-variance investment-reinsurance problem for an insurer (she) under the Cramér–Lundberg model with random coefficients. At any time, the insurer can purchase reinsurance or acquire new business and invest her surplus in a security market consisting of a risk-free asset and multiple risky assets, subject to a general convex cone investment constraint. We reduce the problem to a constrained stochastic linear-quadratic control problem with jumps whose solution is related to a system of partially coupled stochastic Riccati equations (SREs). Then we devote ourselves to establishing the existence and uniqueness of solutions to the SREs by pure backward stochastic differential equation (BSDE) techniques. We achieve this with the help of approximation procedure, comparison theorems for BSDEs with jumps, log transformation and BMO martingales. The efficient investment-reinsurance strategy and efficient mean-variance frontier are explicitly given through the solutions of the SREs, which are shown to be a linear feedback form of the wealth process and a half-line, respectively.

Attractor-repeller construction of Shintani domains for totally complex quartic fields

The units of a number field k act naturally on the real vector space k⊗QR, and so on open subsets of (k⊗QR)⁎ that are stable under the units. A Shintani domain for this action consists of a finite number of polyhedral cones, all having generators in k, whose union is a fundamental domain. Aside from the trivial case of imaginary quadratic fields, no practical method for computing Shintani domains for totally complex number fields has been published. Here we give a quick way to compute Shintani domains for totally complex quartic number fields.To construct these domains we exploit the existence of a forward attractor and backward repeller set for this action. We prove that the number of polyhedral cones needed for the Shintani domain is bounded by an absolute constant, a fact previously known only for cubic or quadratic fields. We also show that our algorithm for finding a Shintani domain runs in polynomial time and we give a table describing the results of running the algorithm on more than 168,000 fields.

Stability of linear operators in locally convex cones

The theory of locally convex cones generalizes locally convex topological vector spaces and provides many different examples and applications than locally convex vector spaces. The stability problem in the sense of Ulam has not yet received any attention in the theory of locally convex cones. Therefore, in this paper, we introduce the stability problem in locally convex cones and provide the stability of linear operators in locally convex cones.

Decomposition of an integrally convex set into a Minkowski sum of bounded and conic integrally convex sets

Every polyhedron can be decomposed into a Minkowski sum (or vector sum) of a bounded polyhedron and a polyhedral cone. This paper establishes similar statements for some classes of discrete sets in discrete convex analysis, such as integrally convex sets, L♮\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {L}^{\ atural }$$\\end{document}-convex sets, and M♮\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {M}^{\ atural }$$\\end{document}-convex sets.

Tip of the Quantum Entropy Cone.

Relations among von Neumann entropies of different parts of an N-partite quantum system have direct impact on our understanding of diverse situations ranging from spin systems to quantum coding theory and black holes. Best formulated in terms of the set Σ_{N}^{*} of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure Σ[over ¯]_{N}^{*}, which is a convex cone. Further homogeneous constrained inequalities are also known. In this Letter we provide (nonhomogeneous) inequalities that constrain Σ_{N}^{*} near the apex (the vector of zero entropies) of Σ[over ¯]_{N}^{*}, in particular showing that Σ_{N}^{*} is not a cone for N≥3. Our inequalities apply to vectors with certain entropy constraints saturated and, in particular, they show that while it is always possible to upscale an entropy vector to arbitrary integer multiples it is not always possible to downscale it to arbitrarily small size, thus answering a question posed by Winter.

Lagrangian Duality in Convex Conic Programming with Simple Proofs

In this paper, we study Lagrangian duality aspects in convex conic programming over general convex cones. It is known that the duality in convex optimization is linked with specific theorems of alternatives. We formulate and prove the strong alternative theorems to the strict feasibility and analyze the relation between the boundedness of the optimal solution sets and the existence of the relative interior points in the feasible set. We also provide sufficient conditions under which the duality gap is zero and the optimal solution sets are unbounded. As a consequence, we obtain several new sufficient conditions that guarantee the strong duality between primal and dual convex conic programs. Our proofs are based only on fundamental convex analysis and linear algebra results.

Existence of Nash equilibria for generalized multiobjective games through the vector extension of Weierstrass and Berge maximum theorems

In this work, we deal with a vector extension of the generalized the Weierstrass and Berge maximum theorems, notably in mathematical economics. This is carried out by introducing the notions of transfer continuity and pseudo-continuity for those functions. Here the preference relation is given via a closed convex cone, having possibly empty interior. As a consequence, we present an existence of strong-Nash equilibria for generalized multiobjective games, and the Rosen model is revisited.

Mixed lattice structures and cone projections

Problems related to projections on closed convex cones are frequently encountered in optimization theory and related fields. To study these problems, various unifying ideas have been introduced, including asymmetric vector-valued norms and certain generalized lattice-like operations. We propose a new perspective on these studies by describing how the problem of cone projection can be formulated using an order-theoretic formalism developed in this paper. The underlying mathematical structure is a partially ordered vector space that generalizes the notion of a vector lattice by using two partial orderings and having certain lattice-type properties with respect to these orderings. In this note we introduce a generalization of these so-called mixed lattice spaces, and we show how such structures arise quite naturally in some of the applications mentioned above.

A Numerical Study on the Capacity Region of a Three-Layer Wiretap Network.

In this paper, we study a three-layer wiretap network including the source node in the top layer, N nodes in the middle layer and L sink nodes in the bottom layer. Each sink node recovers the message generated from the source node correctly via the middle layer nodes that it has access to. Furthermore, it is required that an eavesdropper eavesdropping a subset of the channels between the top layer and the middle layer learns absolutely nothing about the message. For each pair of decoding and eavesdropping patterns, we are interested in finding the capacity region consisting of (N+1)-tuples, with the first element being the size of the message successfully transmitted and the remaining elements being the capacity of the N channels from the source node to the middle layer nodes. This problem can be seen as a generalization of the secret sharing problem. We show that when the number of middle layer nodes is no larger than four, the capacity region is fully characterized as a polyhedral cone. When such a number is 5, we find the capacity regions for 74,222 decoding and eavesdropping patterns. For the remaining 274 cases, linear capacity regions are found. The proving steps are: (1) Characterizing the Shannon region, an outer bound of the capacity region; (2) Characterizing the common information region, an outer bound of the linear capacity region; (3) Finding linear schemes that achieve the Shannon region or the common information region.

Ordering stability of Nash equilibria for a class of differential games

Abstract This study is concerned with the stability of Nash equilibria for a class of n n -person noncooperative differential games. More precisely, due to a preorder induced by a convex cone on a real linear normed space, we define a new concept called ordering stability of equilibria against the perturbation of the right-hand side functions of state equations for the differential game. Moreover, using the set-valued analysis theory, we present the sufficient conditions of the ordering stability for such differential games.

Further remarks on local K-boundedness of K-subadditive set-valued maps

AbstractLet X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map $$F:X\rightarrow \mathcal{C}\mathcal{C}(Y)$$ F : X → C C ( Y ) , for which the functionals $$f_{y^*}(x)=\inf y^*(F(x))$$ f y ∗ ( x ) = inf y ∗ ( F ( x ) ) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional $$y^*$$ y ∗ , has to be locally K-bounded on X. Our results refer to the papers Banakh and Jabłońska (Israel J Math 230:361–386, 2019), Jabłońska and Nikodem (Math Inequal Appl 22:1081–1089, 2019) and Nikodem (Aequationes Math 62:175–183, 2001).