Related papers: Approximate polymorphisms
It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…
A near permutation of a set is a bijection between two cofinite subsets, modulo coincidence on smaller cofinite subsets. Near permutations of a set form its near symmetric group. In this monograph, we define near actions as homomorphisms…
The problem of testing low-degree polynomials has received significant attention over the years due to its importance in theoretical computer science, and in particular in complexity theory. The problem is specified by three parameters:…
The number of homomorphisms from a finite graph $F$ to the complete graph $K_n$ is the evaluation of the chromatic polynomial of $F$ at $n$. Suitably scaled, this is the Tutte polynomial evaluation $T(F;1-n,0)$ and an invariant of the cycle…
We consider the problem of testing if a given function f : F_2^n -> F_2 is close to any degree d polynomial in n variables, also known as the Reed-Muller testing problem. The Gowers norm is based on a natural 2^{d+1}-query test for this…
Given a weakly almost additive sequence of continuous functions with bounded variation $\mathcal{F}=\{\log f_n\}_{n=1}^{\infty}$ on a subshift $X$ over finitely many symbols, we study properties of a function $f$ on $X$ such that…
Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$…
The paper considers truncation errors for functions of the form $f(x_1,x_2,\dots)=g(\sum_{j=1}^\infty x_j\,\xi_j)$, i.e., errors of approximating $f$ by $f_k(x_1,\dots,x_k)=g(\sum_{j=1}^k x_j\,\xi_j)$, where the numbers $\xi_j$ converge to…
We show that in finite-dimensional nonlinear approximations, the best $r$-term approximant of a function $f$ almost always exists over $\mathbb{C}$ but that the same is not true over $\mathbb{R}$, i.e., the infimum $\inf_{f_1,\dots,f_r \in…
Let f be a chain mixing continuous onto mapping from the Cantor set onto itself. Let g be a homeomorphism on the Cantor set that is topologically conjugate to a subshift. Then, homeomorphisms that are topologically conjugate to g…
A theorem of Gr\"unbaum, which states that every $m$-polytope is a refinement of an $m$-simplex, implies the following generalization of Tverberg's theorem: if $f$ is a linear function from an $m$-dimensional polytope $P$ to $\mathbb{R}^d$…
We investigate the realizations of a random Gaussian field on a finite domain of ${\mathbb R}^d$ in the limit where a given linear functional of the field is large. We prove that if its variance is bounded, the field converges uniformly and…
Let $\mu_n$ be the standard Gaussian measure on $\mathbb{R}^n$ and $X$ be a random vector on $\mathbb{R}^n$ with the law $\mu_n$. U-conjecture states that if $f$ and $g$ are two polynomials on $\mathbb{R}^n$ such that $f(X)$ and $g(X)$ are…
In this paper, we consider the characterization of norm--parallelism problem in some classical Banach spaces. In particular, for two continuous functions $f, g$ on a compact Hausdorff space $K$, we show that $f$ is norm--parallel to $g$ if…
The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…
Given a piecewise linear (PL) function $p$ defined on an open subset of $\R^n$, one may construct by elementary means a unique polyhedron with multiplicities $\D(p)$ in the cotangent bundle $\R^n\times \R^{n*}$ representing the graph of the…
Suppose that all groups of order $n$ are defined on the same set $G$ of cardinality $n$, and let the \emph{distance} of two groups of order $n$ be the number of pairs $(a,b)\in G\times G$ where the two group operations differ. Given a group…
Given a finite number of samples of a continuous set-valued function F, mapping an interval to non-empty compact subsets of $\mathbb{R}^d$, $F: [a,b] \to K(\mathbb{R}^d)$, we discuss the problem of computing good approximations of F. We…
Lifting theorems are theorems that bound the communication complexity of a composed function $f\circ g^{n}$ in terms of the query complexity of $f$ and the communication complexity of $g$. Such theorems constitute a powerful generalization…
The main purpose of this paper is to give characterization theorems on derivations as well as on linear functions. Among others the following problem will be investigated: Let $n\in\mathbb{Z}$, $f, g\colon\mathbb{R}\to\mathbb{R}$ be…