5,046 research outputs found
Discrete concavity and the half-plane property
Murota et al. have recently developed a theory of discrete convex analysis
which concerns M-convex functions on jump systems. We introduce here a family
of M-concave functions arising naturally from polynomials (over a field of
generalized Puiseux series) with prescribed non-vanishing properties. This
family contains several of the most studied M-concave functions in the
literature. In the language of tropical geometry we study the tropicalization
of the space of polynomials with the half-plane property, and show that it is
strictly contained in the space of M-concave functions. We also provide a short
proof of Speyer's hive theorem which he used to give a new proof of Horn's
conjecture on eigenvalues of sums of Hermitian matrices.Comment: 14 pages. The proof of Theorem 4 is corrected
On discrete integrable equations with convex variational principles
We investigate the variational structure of discrete Laplace-type equations
that are motivated by discrete integrable quad-equations. In particular, we
explain why the reality conditions we consider should be all that are
reasonable, and we derive sufficient conditions (that are often necessary) on
the labeling of the edges under which the corresponding generalized discrete
action functional is convex. Convexity is an essential tool to discuss
existence and uniqueness of solutions to Dirichlet boundary value problems.
Furthermore, we study which combinatorial data allow convex action functionals
of discrete Laplace-type equations that are actually induced by discrete
integrable quad-equations, and we present how the equations and functionals
corresponding to (Q3) are related to circle patterns.Comment: 39 pages, 8 figures. Revision of the whole manuscript, reorder of
sections. Major changes due to additional reality conditions for (Q3) and
(Q4): new Section 2.3; Theorem 1 and Sections 3.5, 3.6, 3.7 update
On Statistical Properties of Jizba-Arimitsu Hybrid Entropy
Jizba-Arimitsu entropy (also called hybrid entropy) combines axiomatics of
R\'enyi and Tsallis entropy. It has many common properties with them, on the
other hand, some aspects as e.g., MaxEnt distributions, are completely
different from the former two entropies. In this paper, we demonstrate the
statistical properties of hybrid entropy, including the definition of hybrid
entropy for continuous distributions, its relation to discrete entropy and
calculation of hybrid entropy for some well-known distributions. Additionally,
definition of hybrid divergence and its connection to Fisher metric is also
discussed. Interestingly, the main properties of continuous hybrid entropy and
hybrid divergence are completely different from measures based on R\'enyi and
Tsallis entropy. This motivates us to introduce average hybrid entropy, which
can be understood as an average between Tsallis and R\'enyi entropy
A Combinatorial Formula for Principal Minors of a Matrix with Tree-metric Exponents and Its Applications
Let be a tree with a vertex set . Denote by
the distance between vertices and . In this paper, we present an
explicit combinatorial formula of principal minors of the matrix
, and its applications to tropical geometry, study of
multivariate stable polynomials, and representation of valuated matroids. We
also give an analogous formula for a skew-symmetric matrix associated with .Comment: 16 page
Stability of Three- and Four-Body Coulomb Systems
We discuss the stability of three- and four-particle system interacting by
pure Coulomb interactions, as a function of the masses and charges of the
particles. We present a certain number of general properties which allow to
answer a certain number of questions without or with less numerical
calculations.Comment: latex file, 15 pages, 8 figure
Approximation of the critical buckling factor for composite panels
This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented
Concavity analysis of the tangent method
The tangent method has recently been devised by Colomo and Sportiello
(arXiv:1605.01388 [math-ph]) as an efficient way to determine the shape of
arctic curves. Largely conjectural, it has been tested successfully in a
variety of models. However no proof and no general geometric insight have been
given so far, either to show its validity or to allow for an understanding of
why the method actually works. In this paper, we propose a universal framework
which accounts for the tangency part of the tangent method, whenever a
formulation in terms of directed lattice paths is available. Our analysis shows
that the key factor responsible for the tangency property is the concavity of
the entropy (also called the Lagrangean function) of long random lattice paths.
We extend the proof of the tangency to -deformed paths.Comment: published version, 22 page
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