Related papers: A functional view of upper bounds on codes
The paper is pertaining to the spectral theory of operators and boundary value problems for differential equations on manifolds. Eigenvalues of such problems are studied as functionals on the space of domains. Resolvent continuity of the…
The paper considers the minimization of a separable convex function subject to linear ascending constraints. The problem arises as the core optimization in several resource allocation scenarios, and is a special case of an optimization of a…
It is well known that there is a deep relationship between codes and lattices. Concepts from coding theory are related to concepts of lattice theory as, for example, weight enumerators to theta series, MacWilliams identity to Jacobi…
In this paper we consider a nonlinear equation $-\mathcal{L} u(x) = f(x, u(x))$ with a super-quadratic nonlinearity, $f$, and a nonlocal operator, $\mathcal{L}$, generated by a special class of radially symmetric $L^1$ convolution kernels…
In this paper, we give a geometric interpretation of optimal functionals in the context of intersection of symmetry planes and cyclic polytopes. For 1D CFTs, we demonstrate that at given derivative order, the functional is given by a…
The abstract theory of boundary triples is applied to the classical Jacobi differential operator and its powers in order to obtain the Weyl $m$-function for several self-adjoint extensions with interesting boundary conditions: separated,…
In a recent work, Coronel et al. initiated the study of the relation between the diversity-multiplexing tradeoff (DMT) performance of a multiuser multiple-input multiple-output (MU-MIMO) lattice code and the rate of the decay of the…
A unified approach is given to kernel functions which intertwine Ruijsenaars difference operators of type A and of type BC. As an application of the trigonometric cases, new explicit formulas for Koornwinder polynomials attached to single…
We give a new construction of linear codes over finite fields on higher dimensional varieties using Grothendieck's theory of residues. This generalizes the construction of differential codes over curves to varieties of higher dimensions.
Controlling the spectral norm of the Jacobian matrix, which is related to the convolution operation, has been shown to improve generalization, training stability and robustness in CNNs. Existing methods for computing the norm either tend to…
We show that a problem on minimal periods of solutions of Lipschitz functional differential equations is closely related to the unique solvability of the periodic problem for linear functional differential equations. Sharp bounds for…
We study a variant of the Alt, Caffarelli, and Friedman free boundary problem with many phases and a slightly different volume term, which we originally designed to guess the localization of eigenfunctions of a Schr\"odinger operator in a…
In the present paper, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on 2-parameters weight functions. Such classes englobe the well…
We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps.…
Linear fixed point equations in Hilbert spaces arise in a variety of settings, including reinforcement learning, and computational methods for solving differential and integral equations. We study methods that use a collection of random…
Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function $W({\bf…
Using linear functional-based duality of modules, we generalize the syndrome decoding algorithm of linear codes over finite fields to those over finite commutative rings. Moreover, If the ring is local the algorithm is simplified by…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…
We develop a rigorous framework for extending neural operators to handle out-of-distribution input functions. We leverage kernel approximation techniques and provide theory for characterizing the input-output function spaces in terms of…
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…