Abstract : Mathematical inequalities are of fundamental interest due to their applications in different problems in industrial and applied mathematics. Developing a new inequality is a challenging and crucial task.
On the other hand, entropy plays a key role in theoretical physics and information theory. Investigations in entropy need new mathematical inequalities. In contrast, research in inequality provides novel characteristics of entropy and divergence. For instance, we need matrix inequalities in quantum information theory.
In this mini-symposium, our aim is to provide the recent advances, problems and ideas at the interface of mathematical inequalities and entropy with various applications.
[02362] Refined Hermite-Hadamard inequalities and their applications to some n variable means
Format : Talk at Waseda University
Author(s) :
Kenjiro Yanagi (Josai University)
Abstract : It is well known that the Hermite-Hadamard inequality $($called the HH inequality$)$ refines the definition of convexity of function $f(x)$ defined on $[a,b]$ by using the integral of $f(x)$ from $a$ to $b$. There are many generalizations or refinements of the HH inequality. Futhermore the HH inequality has many applications to several fields of mathematics, including numerical analysis, functional analysis and operator inequality. Recently we gave several types of refined HH inequalities and obtained inequalities which were satisfied by weighted logarithmic means. In this talk, we give $n$ variable HH inequality and apply to some $n$ variable means. Finally we compare these means.
[02747] Generalization of Hermite-Hadamard Mercer Inequalities for Certain Interval Valued Functions
Format : Talk at Waseda University
Author(s) :
Asfand Fahad (Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, Pakistan)
Abstract : Due to its significance in economics, optimization and different fields, convex analysis theory has experienced various advancements and extensions over time. A modern development is the use of cr-convex functions to construct equivalent optimality conditions for constrained and unconstrained nonlinear optimization problems using interval-valued objective functions. By keeping in mind the relationships between convex functions and mathematical inequalities involving the convex functions, we investigated generalizations of well-known Hermite-Hadamard Mercer Inequalities for new types of cr-convex functions. We include several well-known consequences as special cases.
[01760] Generalized spectral radius of operators and related inequalities
Format : Talk at Waseda University
Author(s) :
Kais Feki (University of Monastir)
Abstract : In this talk, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $\mathcal{H}$. We denote this new concept by $r_A(\cdot)$. In this presentation, several basic properties and inequalities involving $r_A(\cdot)$ are investigated. Moreover, we study the connection between the notions of $A$-spectral radius and $A$-spectrum for $A$-bounded operators.
[03970] $q$-deformation of Böttcher-Wenzel inequality
Format : Talk at Waseda University
Author(s) :
Hiromichi Ohno (Shinshu University)
Abstract : The Böttcher-Wenzel inequality states that the 2-norm of the commutator of matrices $A$ and $B$ is less than or equal to $\sqrt{2}$ times the product of the 2-norms of $A$ and $B$. In this talk, we discuss $q$-deformation of the Böttcher-Wenzel inequality in which the commutator is replaced by a q-commutator.
Abstract : We introduce the notion of reduced relative quantum entropy and prove that it is convex. The result is used to give a simplified proof of a theorem of Lieb and Seiringer. We then proceed to describe an interpolation inequality between Golden-Thompson’s trace inequality and Jensen’s trace inequality.
[02970] On the quantum Tsallis relative entropy of real order
Format : Talk at Waseda University
Author(s) :
Yuki Seo (Osaka Kyoiku University)
Abstract : In 2005, Furuichi-Yanagi-Kuriyama showed 1-parameter extension of matrix trace inequalities due to Hiai-Petz, and it revealed relationships between two quantum Tsallis relative entropies. In this talk, we show matrix trace inequalities related to quantum Tsallis relative entropy of real order, and improve on Furuichi-Yanagi-Kuriyama's result by using Furuta inequality.
[02482] On log-sum inequalities
Format : Talk at Waseda University
Author(s) :
Supriyo Dutta (National Institute of Technology Agartala)
Shigeru Furuichi (Nihon University)
Abstract : I shall present our recently published article entitled "On log-sum inequalities" in Linear and Multilinear Algebra. The log-sum inequality is a fundamental tool which indicates the nonnegativity for the relative entropy. We establish a set of inequalities which are similar to the log-sum inequality. We extend these inequalities for the commutative matrices. In addition, utilizing the L ̈owner partial order relation and the Hansen-Pedersen theory for non-commutative positive semi-definite matrices we demonstrate several matrix-inequalities.
[02313] On certain properties of Shannon's Entropy
Format : Talk at Waseda University
Author(s) :
Eleutherius Symeonidis (Faculty of Mathematics and Geography, Catholic University of Eichstaett-Ingolstadt)
Abstract : Let $P:=(p_1,\ldots,p_n)$ be a discrete probability distribution, $$ H(P):=-\sum_{j=1}^n p_j \log p_j $$ its Shannon entropy. Motivated by studies on the permutation entropy of time series of temperatures in combustion experiments we fix an integer $k$, $1\le k\le n$,
and consider the largest set $\Delta\subset {\mathbb R}^k$ such that $$ H(p_1,\ldots,p_{n-k},p_{n-k+1},\ldots,p_n)\ge H\left(0,\ldots,0,\frac{1}{k},\ldots,\frac{1}{k}\right)\;(=\log k) $$ for all $P$ such that $(p_{n-k+1},\ldots,p_n)\in\Delta$. In particular, we are interested in the smallest value of $p$ such that $(p,\ldots,p)\in\Delta$.
[02721] The permutation entropy and its applications on full-scale compartment fire data
Format : Talk at Waseda University
Author(s) :
Flavia-Corina Mitroi-Symeonidis (Department of Applied Mathematics Academy of Economic Studies Calea Dorobanti 15-17, Sector 1 010552 Bucharest)
Abstract : Given the sparse literature on the usefulness of the entropy in characterizing fire data, we investigate the order characteristics of the compartment fire based on experimental data. We compare known algorithms dedicated to the extraction of the underlying probabilities, checking their suitability to point out the abnormal values and structure of the time series. We claim that the permutation entropy is suitable to detect the occurrence of the flashover and unusual data in fire experiments.
[02772] A Spectral Analysis of The Correlated Random Walk
Format : Talk at Waseda University
Author(s) :
Akihiro Narimatsu (The University of Fukuchiyama)
Yusuke Ide (Nihon University)
Abstract : In this talk, we consider a spectral analysis of the Correlated Random Walk with the isospectral coin cases. In the Szegedy's quantum walk, our method gives the arcsine law as the lower bound of the time averaged distribution.
[05367] On a class of k-entanglement witnesses
Format : Talk at Waseda University
Author(s) :
Hiroyuki Osaka (Ritsumeikan University)
Abstract : Recently, Yang et al.showed that each 2-positive map acting from $\mathcal{M}_3(\mathbb{C})$ into itself is decomposable. It is equivalent to the statement that each PPT state on $\mathbb{C}^3\otimes\mathbb{C}^3$ has Schmidt number at most 2. It is a generalization of Perez-Horodecki criterion which states that each PPT state on $\mathbb{C}^2\otimes\mathbb{C}^2$ or $\mathbb{C}^2\otimes\mathbb{C}^3$ has Schmidt rank 1 i.e. is separable. Natural question arises whether the result of Yang at al. stays true for PPT states on $\mathbb{C}^3\otimes\mathbb{C}^4$. This question can be considered also in higher dimensions. We construct a positive maps which is suspected for being a counterexample. More generally, we provide a class of positive maps $\Phi_a$ between matrix algebras whose $k$-positivity properties can be easily controlled.
The estimate bounds on the parameter a are better than those derived from the spectral conditions considered by Chru\'{s}ci\'{n}ski and Kossakowski.
We found that in case where dimensions are differ by one we can give explicit analytic formula for parameter a that guarantee $k$-positivity.
As an apllication we show that $\Phi_a$ detects $k$-entanglement.
This is mainly based on joint work with Tomasz Mlynik and Marcin Marciniak (arXiv:2104.14058v4, 2022).
[03537] Violation of Bell's Inequality by Classical Correlation via Adaptive Dynamics
Format : Talk at Waseda University
Author(s) :
Satoshi Iriyama (Tokyo University of Science)
Abstract : Ohya introduced the adaptive dynamics as subclasses of information dynamics. The notion of adaptive dynamics is helpful to find out characteristic factors in complex systems. In 2001, the chameleon effect proposed by Accardi et al. is the classical dynamics adopting that the local acts of observation may disturb local measurement, and its experimental implementation which can violate Bell's inequality was shown. In this talk, mathematical foundations of the chameleon dynamics and its applications are explained.
