Related papers: On Nash approximation of complex analytic sets in …

We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $X\subset \mathbb{P}^n(\mathbb{R})$. The main tool is a refinement of the simplex lemma, which…

An approximation scheme is a family of homogeneous subsets $(A_n)$ of a quasi-Banach space $X$, such that $A_1 \subsetneq A_2 \subsetneq ... \subsetneq X$, $A_n + A_n \subset A_{K(n)}$, and $\bar{\cup_n A_n} = X$. Continuing the line of…

We prove that the classical Oka property of a complex manifold Y, concerning the existence and homotopy classification of holomorphic mappings from Stein manifolds to Y, is equivalent to a Runge approximation property for holomorphic maps…

We consider the range of random analytic functions with finite radius of convergence. We show that any unbounded random Taylor series with rotationally invariant coefficients has dense image in the plane. We moreover show that if in…

We prove an analogue of the classical Bernstein theorem concerning the rate of polynomial approximation of piecewise analytic functions on a compact subset of the real line.

We prove a version of the Khinchine--Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This…

We consider a game in which the action set of each player is uncountable, and show that, from weak assumptions on the common prior, any mixed strategy has an approximately equivalent pure strategy. The assumption of this result can be…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

In this paper, a sample-based procedure for obtaining simple and computable approximations of chance-constrained sets is proposed. The procedure allows to control the complexity of the approximating set, by defining families of…

Let $E$ be a real Banach space with dimension at least 2. In this paper, we prove that if $D\subset E$ is a John domain which is homeomorphic to an inner uniform domain via a CQH map, then each neargeodesic in $D$ is a cone arc.

We explore representing the compact subsets of a given represented space by infinite sequences over Plotkin's $\mathbb{T}$. We show that computably compact computable metric spaces admit representations of their compact subsets in such a…

For a germ $(X,0) \subset (\mathbb{C}^n,0)$ of reduced, equidimensional complex analytic singularity its Nash modification can be constructed as an analytic subvariety $ Z \subset \mathbb{C}^n \times G(k,n)$. We give a characterization of…

The theory of universal Taylor series can be extended to the case of Pad\'e approximants where the universal approximation is not realized by polynomials any more, but by rational functions, namely the Pad\'e approximants of some power…

The general theory of greedy approximation with respect to arbitrary dictionaries is well developed in the case of real Banach spaces. Recently, some of results proved for the Weak Chebyshev Greedy Algorithm (WCGA) in the case of real…

For a compact subset $K$ of the complex plane $\mathbb C,$ let $C(K)$ denote the algebra of continuous functions on $K$. For an open subset $U \subset K,$ let $A(K,U) \subset C(K)$ be the algebra of functions that are analytic in $U.$ We…

Using a recently developed $\mathcal H$-calculus we propose a unified approach to the study of rational approximations of holomorphic semigroups on Banach spaces. We provide unified and simple proofs to a number of basic results on…

In this paper we show that a holomorphic function, defined on an open subset $D$ of $\mathbb{C}^n$, is a complex Nash function if and only if its real part (or equivalently its imaginary part) is a real Nash function.

We show that any subset of the natural numbers with positive logarithmic Banach density contains a set that is within a factor of two of a geometric progression, improving the bound on a previous result of the authors. Density conditions on…

We study the computational complexity of (deterministic or randomized) algorithms based on point samples for approximating or integrating functions that can be well approximated by neural networks. Such algorithms (most prominently…

Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…