Related papers: Multidimensional exact classes, smooth approximati…

We present two classes of exact solutions to a geometric model which describes the kinetics of fragmentation of $d$-dimensional hypercuboid-shaped objects. The first class of exact solutions is described by a fragmentation rate…

This is a preliminary version of a book on infinite-dimensional Lie groups. It covers the basics of calculus and manifolds in the context of locally convex spaces, based on Bastiani's notion of a smooth map. Starting from this concept, we…

Sequential multi-class diagnosis, also known as multi-hypothesis testing, is a classical sequential decision problem with broad applications. However, the optimal solution remains, in general, unknown as the dynamic program suffers from the…

We prove that, for every polyhedral or $C^1$ norm on $\mathbb{R}^d$ and every set $E \subseteq \mathbb{R}^d$ of packing dimension $s$, the packing dimension of the distance set of $E$ with respect to that norm is at least $\tfrac{s}{d}$.…

The notion of an $\mathcal{M}$-coextensive object is introduced in an arbitrary category $\mathbb{C}$, where $\mathcal{M}$ is a distinguished class of morphisms from $\mathbb{C}$. This notion allows for a categorical treatment of the strict…

This paper focuses on the relation between computational learning theory and resource-bounded dimension. We intend to establish close connections between the learnability/nonlearnability of a concept class and its corresponding size in…

We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out…

Multi-dimensional classification (MDC) is the supervised learning problem where an instance is associated with multiple classes, rather than with a single class, as in traditional classification problems. Since these classes are often…

In the paper we define three new complexity classes for Turing Machine undecidable problems inspired by the famous Cook/Levin's NP-complete complexity class for intractable problems. These are U-complete (Universal complete), D-complete…

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R^4 can be described as certain 2-morphisms in the 2-category of `2-tangles in 4…

Let $\Omega$ be a smooth real analytic submanifold of a complex manifold $X$. We establish and study the link between the following 3 subjects: 1) topological properties of smooth families of attached analytic discs, the manifold $\Omega$…

We complete and precise the results of [B.13] and we prove a strong version of the semi-proper direct image theorem with values in the space C f n (M) of finite type closed n--cycles in a complex space M. We describe the strongly…

Let $n$ be a positive integer, and let $R$ be a (possibly infinite dimensional) finitely presented algebra over a computable field of characteristic zero. We describe an algorithm for deciding (in principle) whether $R$ has at most finitely…

We show the existence of a deep neural network capable of approximating a wide class of high-dimensional approximations. The construction of the proposed neural network is based on a quasi-optimal polynomial approximation. We show that this…

A class of singular integral operators, encompassing two physically relevant cases arising in perturbative QCD and in classical fluid dynamics, is presented and analyzed. It is shown that three special values of the parameters allow for an…

We demonstrate a compactness result holding broadly across supervised learning with a general class of loss functions: Any hypothesis class $H$ is learnable with transductive sample complexity $m$ precisely when all of its finite…

We show that the group ${\Cal D}(M)$ of pseudoisotopy classes of diffeomorphisms of a manifold of dimension $\geq 5$ and of finite fundamental group is commensurable to an arithmetic group. As a result $\pi_0(\text{{\it Diff\,M}})$ is a…

We consider the regression problem of estimating functions on $\mathbb{R}^D$ but supported on a $d$-dimensional manifold $ \mathcal{M} \subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear…

The article [14] gives a list of 51 symplectic hypergeometric monodromy groups corresponding to primitive pairs of degree four polynomials, which are products of cyclotomic polynomials, and for which, the absolute value of the leading…

We present higher order polynomial algebras which are the dynamical symmetry algebras of a wide class of multi-mode boson systems in non-linear optics. We construct their unitary representations and the corresponding single-variable…