Related papers: Sharp geometric inequalities for the general $p$-a…
In this paper we investigate the shape invariance property of a potential in one dimension. We show that a simple ansatz allows us to reconstruct all the known shape invariant potentials in one dimension. This ansatz can be easily extended…
We discuss new models of an `affine' theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein's proposal to specify the space - time…
We introduce the notion of amenability for affine algebras. We characterize amenability by Folner-sequences, paradoxicality and the existence of finitely invariant dimension-measures. Then we extend the results of Rowen on ranks, from…
In this paper we consider topological mappings defined by $p$-capacity inequalities in domains of $\mathbb R^n$. In the case $p=n-1$ we prove the Lipschitz continuity of such mappings, that extends the result by F.~W.~Gehring.
In this paper we introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previous known results, and also extend the range of such…
It is well known that general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, we present a number…
We present a purely affine gravitational model in four dimensions built up entirely on the bases of full diffeomorphism invariance, and power- counting renormalizability. We show that its non-relativistic limit around a homogeneous and…
In the paper, by establishing the monotonicity of some functions involving the sine and cosine functions, the authors provide concise proofs of some known inequalities and find some new sharp inequalities involving the Seiffert,…
In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional…
We derive various sharp upper bounds for the $p$-capacity of a smooth compact set $K$ in the hyperbolic space $\mathbb{H}^n$ and the Euclidean space $\mathbb{R}^n$. Firstly, using the inverse mean curvature flow, for the mean convex and…
This work demonstrates that a complete description of the interaction of matter and all forces, gravitational and non-gravitational, can in fact be realized within a quantum affine algebraic framework. Using the affine group formalism, we…
A sharp lower bound for the first Dirichlet eigenvalue of the $p$-laplacian in Gaussian space is derived for sets with prescribed generalized torsional rigidity. The result provides an extension of the classical spectral inequality due to…
Let $K$ be a convex body in $\mathbb R^n$. We introduce a new affine invariant, which we call $\Omega_K$, that can be found in three different ways: as a limit of normalized $L_p$-affine surface areas, as the relative entropy of the cone…
Let G be a reductive algebraic group and H a closed subgroup of G. An affine embedding of the homogeneous space G/H is an affine G-variety with an open G-orbit isomorphic to G/H. We start with some basic properties of affine embeddings and…
Self-similar potentials generalize the concept of shape-invariance which was originally introduced to explore exactly-solvable potentials in quantum mechanics. In this article it is shown that previously introduced algebraic approach to the…
In this paper, we study some properties such as the monotonicity, logarithmically complete monotonicity, logarithmic convexity, and geometric convexity, of the combinations of gamma function and power function. The results we obtain…
This paper is devoted to a systematic study of certain geometric integral inequalities which arise in continuum combinatorial approaches to $L^p$-improving inequalities for Radon-like transforms over polynomial submanifolds of intermediate…
This paper is devoted to exploring the relationship between the $[1,n)\ni p$-capacity and the surface-area in $\mathbb R^{n\ge 2}$ which especially shows: if $\Omega\subset\mathbb R^n$ is a convex, compact, smooth set with its interior…
We study here systems of symmetries on $|1|$--graded parabolic geometries. We are interested in smooth systems of symmetries and we discuss non--flat homogeneous $|1|$--graded geometries. We show the existence of an invariant admissible…
Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of…