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We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the inverse…
To every log-concave function $f$ one may associate a pair of measures $(\mu_{f},\nu_{f})$ which are the surface area measures of $f$. These are a functional extension of the classical surface area measure of a convex body, and measure how…
We construct a singular minimizing map ${\bf u}$ from $\mathbb{R}^3$ to $\mathbb{R}^2$ of a smooth uniformly convex functional of the form $\int_{B_1} F(D{\bf u})\,dx$.
This is the continuation of previous article. For subspaces $M^n(t)$ and $M^{n-m}(t)$ which are invariant manifolds of the differential equation under consideration we build a change of variables which splits this equation into a system of…
The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with…
We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and…
On fractals, different measures (mutually singular in general) are involved to measure volumes of sets and energies of functions. Singularity of measures brings difficulties in (especially non-linear) analysis on fractals. In this paper, we…
A generalization of Mercer inequality for h-convex function is presented. As application, a weighted generalization of triangle inequality is given.
We investigate polyconvexity of the double well function $f(X)\,:= |X-X\_1|^2|X-X\_2|^2$ for given matrices $X\_1, X\_2 \in \R^{n \times n}$. Such functions are fundamental in the modeling of phase transitions in materials, but their…
In what follows we improve an inequality related to matrix theory. T. Laffey proved (2013) a weaker form of this inequality [2].
The study of solving inverse singular value problems for nonnegative matrices has been around for decades. It is clear that an inverse singular problem is trivial if the desirable matrix is not restricted to a certain structure. Provided…
In this paper we are concerned with the solutions of the differential equation $f'''+ff''+g(f')=0$ on $[0,\infty)$, satisfying the boundary conditions $f(0)=\alpha$, $f'(0)=\beta\geq 0$, $f'(\infty)=\l$, and where $g$ is some given…
The main objective of this paper is to study the existence of solutions to some basic fractional difference equations. The tools employed are Krasnosel'skii fixed point theorem which guarantee at least two positive solutions.
We prove inequalities on symmetric tensor sums of positive definite operators. In particular, we prove multivariable operator inequalities inspired by generalizations to the well-known Hlawka and Popoviciu inequalities. As corollaries, we…
A function $\mathfrak{F}$ with simple and nice algebraic properties is defined on a subset of the space of complex sequences. Some special functions are expressible in terms of $\mathfrak{F}$, first of all the Bessel functions of first…
Let f be a non-negative concave function on the positive half-line. Let A and B be two positive matrices. Then, for all symmetric norms, || f(A+B) || is less than || f(A)+f(B) ||. When f is operator concave, this was proved by Ando and…
Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions and non-commutative $\mathbb{L}_p$ spaces, and have proven useful in bounding the time evolution of classical/quantum Markov processes, among…
We prove existence of positive solutions to a boundary value problem depending on discrete fractional operators. Then, corresponding discrete fractional Lyapunov-type inequalities are obtained.
The classification of continuous, translation invariant Minkowski valuations which are contravariant (or covariant) with respect to the complex special linear group is established in a 2-dimensional complex vector space. Every such…
We consider the inequality $f \geqslant f\star f$ for real integrable functions on $d$ dimensional Euclidean space where $f\star f$ denotes the convolution of $f$ with itself. We show that all such functions $f$ are non-negative, which is…