Related papers: The Breadth-one $D$-invariant Polynomial Subspace

The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and…

Let $X$ be a list of vectors that is totally unimodular. In a previous article the author proved that every real-valued function on the set of interior lattice points of the zonotope defined by $X$ can be extended to a function on the whole…

For a fixed symmetric matrix A and symmetric perturbation E we develop purely deterministic bounds on how invariant subspaces of A and A+E can differ when measured by a suitable "row-wise" metric rather than via traditional measures of…

We study D-modules and related invariants on the space of 2 x 2 x n hypermatrices for n >= 3, which has finitely many orbits under the action of G = GL_2 x GL_2 x GL_n. We describe the category of coherent G-equivariant D-modules as the…

In this paper we obtain some statements concerning ideals of polynomials and apply these results in a number of different situations. Among other results, we present new characterizations of $\mathcal{L}_{\infty}$-spaces, Coincidence…

In this paper we study a homological version of the higher-dimensional divergence invariants defined by Brady and Farb. We show that they are quasi-isometry invariants in the class of proper cocompact Hadamard spaces in the sense of…

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…

Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions which…

The aim of this expository article is to present recent developments in the centuries old discussion on the interrelations between continuous and differentiable real valued functions of one real variable. The truly new results include,…

Embedding-based representations in Euclidean space $\mathbb{R}^d$ are a cornerstone of modern machine learning, where a major goal is to use the \emph{smallest dimension} that faithfully captures data relations. In this work, we prove sharp…

Motivated by the need to better understand the properties of sparse cutting-planes used in mixed integer programming solvers, the paper [2] studied the idealized problem of how well a polytope is approximated by the use of sparse valid…

In this paper we construct an injection from the linear space of trigonometric polynomials defined on $\mathbb{T}^d$ with bounded degrees with respect to each variable to a suitable linear subspace $L^1_E\subset L^1(\mathbb{T})$. We give…

An important problem in machine learning theory is to understand the approximation and generalization properties of two-layer neural networks in high dimensions. To this end, researchers have introduced the Barron space…

Let $P_N(R)$ be the space of all real polynomials in $N$ variables with the usual inner product $<, >$ on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form…

In this paper we look at normed spaces of differentiable functions on compact plane sets, including the spaces of infinitely differentiable functions originally considered by Dales and Davie. For many compact plane sets the classical…

In this paper we extend the Shepard-Bernoulli operators introduced in [6] to the bivariate case. These new interpolation operators are realized by using local support basis functions introduced in [23] instead of classical Shepard basis…

The article considers one of the possible generalizations of constraint satisfaction problems where relations are replaced by multivalued membership functions. In this case operations of disjunction and conjunction are replaced by maximum…

We provide sequence space representations for the test function space $\mathcal{D}_{E}$ and the distribution space $\mathcal{D}^{\prime}_{E}$ associated to a Banach space $E$ belonging to a broad class of translation-modulation invariant…

We study monotone neural networks with threshold gates where all the weights (other than the biases) are non-negative. We focus on the expressive power and efficiency of representation of such networks. Our first result establishes that…

We consider approximation problems for a special space of d variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show…