Related papers: $\rm{SL}(n)$ covariant function-valued valuations

The spaces of Sp(n)-, Sp(n)U(1)- and Sp(n)Sp(1)- invariant, translation invariant, continuous convex valuations on the quaternionic vector space H^n are studied. Combinatorial dimension formulas involving Young diagrams and Schur…

We prove new kinematic formulas for tensor valuations and simplify previously known Crofton formulas by using the recently developed algebraic theory of translation invariant valuations. The heart of the paper is the computation of the…

This is a semi-expository article concerning Langlands functoriality and Deligne's conjecture on the special values of $L$-functions. The emphasis is on symmetric power $L$-functions associated to a holomorphic cusp form, while appealing to…

In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative…

Regularization plays a key role in a variety of optimization formulations of inverse problems. A recurring theme in regularization approaches is the selection of regularization parameters, and their effect on the solution and on the optimal…

In this paper, we give some properties and remarks of the new fractional Sobolev spaces with variable exponents. We also study the eigenvalue problem involving the new fractional $p(\cdot)$-Laplacian.

We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants…

Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely…

We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincare inequalities for such functions. This leads naturally to the concept of…

Value alignment of Large Language Models (LLMs) requires us to empirically measure these models' actual, acquired representation of value. Among the characteristics of value representation in humans is that they distinguish among value of…

We shown that every continuous local functional on the space of finite convex functions on $\mathbb{R}^n$ is a valuation. This relation is used to establish a homogeneous decomposition for the class of polynomial local functionals as well…

We develop characterizations for Sobolev spaces of potential type on graded Lie groups, by means of Littlewood-Paley square functions, and Strichartz functionals involving second-order differences. A key role is played by some mean value…

A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…

We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. Indeed, in the…

For function spaces equipped with Muckenhoupt weights, the validity of continuous Sobolev embeddings in case $p_0\leq p_1$ is characterized. Extensions to Jawerth-Franke embeddings, vector-valued spaces and examples involving some prominent…

Using explicit constructions of the Weierstrass mock modular form, we offer a closed formula for generating the values of shifted convolution $L$-values for certain elliptic curves that can be computed to arbitrary precision. These…

In this paper, we endow the space of continuous translation invariant valuation on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to…

A more general notion of weight called admissible is introduced and then an investigation is carried out on the a.e. convergence of weighted strong laws of large numbers and their applications to weighted one-sided ergodic Hilbert…

This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of a linear space. In this article we still work…

It is known that for Banach valued functions there are several approaches to define a Sobolev class. We compare the usual definition via weak derivatives with the Reshetnyak-Sobolev space and with the Newtonian space; in particular, we…