Symmetry in the Mathematical Inequalities

This Special Issue brings together original research papers, in all areas of mathematics, that are concerned with inequalities or the role of inequalities. The research results presented in this Special Issue are related to improvements in classical inequalities, highlighting their applications and promoting an exchange of ideas between mathematicians from many parts of the world dedicated to the theory of inequalities. This volume will be of interest to mathematicians specializing in inequality theory and beyond. Many of the studies presented here can be very useful in demonstrating new results. It is our great pleasure to publish this book. All contents were peer-reviewed by multiple referees and published as papers in our Special Issue in the journal Symmetry. These studies give new and interesting results in mathematical inequalities enabling readers to obtain the latest developments in the fields of mathematical inequalities. Finally, we would like to thank all the authors who have published their valuable work in this Special Issue. We would also like to thank the editors of the journal Symmetry for their help in making this volume, especially Mrs. Teresa Yu

Revisiting Relations between Stochastic Ageing and Dependence for Exchangeable Lifetimes with an Extension for the IFRA/DFRA Property

We first review an approach that had been developed in the past years to introduce concepts of "bivariate ageing" for exchangeable lifetimes and to analyze mutual relations among stochastic dependence, univariate ageing, and bivariate ageing. A specific feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. "Archimedean" models have an important role in the present framework. In the second part of the paper, the definitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of "Pseudo-Archimedean" models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD

The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity

We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$F\mapsto W_{_{\rm eH}}(F):=\widehat{W}_{_{\rm eH}}(U):=\left\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\frac{\kappa}{{2\, {\widehat{k}}}}\,e^{\widehat{k}\,[{ \rm tr}(\log U)]^2}&\text{if}& { \rm det} F>0,\\ +\infty &\text{if} &{ \rm det} F\leq 0, \end{array}\right.\quad$$ based on the Hencky-logarithmic (true, natural) strain tensor $\log U$, where $\mu>0$ is the infinitesimal shear modulus, $\kappa=\frac{2\mu+3\lambda}{3}>0$ is the infinitesimal bulk modulus with $\lambda$ the first Lam\'{e} constant, $k,\widehat{k}$ are dimensionless parameters, $F=\nabla \varphi$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n} {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the strain tensor $\log U$. For small elastic strains, $W_{_{\rm eH}}$ approximates the classical quadratic Hencky strain energy $$F\mapsto W_{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U):={\mu}\,\|{\rm dev}_n\log U\|^2+\frac{\kappa}{2}\,[{\rm tr}(\log U)]^2,$$ which is not everywhere rank-one convex. In plane elastostatics, i.e. $n=2$, we prove the everywhere rank-one convexity of the proposed family $W_{_{\rm eH}}$, for $k\geq \frac{1}{4}$ and $\widehat{k}\geq \frac{1}{8}$. Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for $n=2,3$ and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family $W_{_{\rm eH}}$ is not preserved in dimension $n=3$

Logarithmic concavity of Schur and related polynomials

We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients in the special case of Kostka numbers

Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations

Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichm\"uller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using Atiyah-Bott, Forni and M\"oller's works. We obtain several applications to Teichm\"uller dynamics conditional to the conjecture.Comment: 37 pages. We rewrite this paper without changing the mathematics content. arXiv admin note: text overlap with arXiv:1112.5872, arXiv:1204.1707 by other author

Intersection theoretic inequalities via Lorentzian polynomials

We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with respect to $m$-positive classes and Schur classes. We also study its convexity variants -- the geometric inequalities for $m$-convex functions on the sphere and convex bodies. Along the exploration, we prove that any finite subset on the closure of the cone generated by $m$-positive classes can be endowed with a polymatroid structure by a canonical numerical-dimension type function, extending our previous result for nef classes; and we prove Alexandrov-Fenchel inequalities for valuations of Schur type. We also establish various analogs of sumset estimates (Pl\"{u}nnecke-Ruzsa inequalities) from additive combinatorics in our contexts.Comment: 27 pages; comments welcome

A power consensus algorithm for DC microgrids

A novel power consensus algorithm for DC microgrids is proposed and analyzed. DC microgrids are networks composed of DC sources, loads, and interconnecting lines. They are represented by differential-algebraic equations connected over an undirected weighted graph that models the electrical circuit. A second graph represents the communication network over which the source nodes exchange information about the instantaneous powers, which is used to adjust the injected current accordingly. This give rise to a nonlinear consensus-like system of differential-algebraic equations that is analyzed via Lyapunov functions inspired by the physics of the system. We establish convergence to the set of equilibria consisting of weighted consensus power vectors as well as preservation of the weighted geometric mean of the source voltages. The results apply to networks with constant impedance, constant current and constant power loads.Comment: Abridged version submitted to the 20th IFAC World Congress, Toulouse, Franc