Related papers: Universal Functions

We prove that in random matrix theory there exists a universal relation between the one-point Green's function $G$ and the connected two- point Green's function $G_c$ given by \vfill $ N^2 G_c(z,w) = {\part^2 \over \part z \part w} \log…

We present a new result on the nonexistence of generalized bent functions (GBFs)from (Z/tZ)^n to Z/tZ (called type [n, t]) for a large class. Assume p is an odd prime number. By showing certain quadratic norm form equations having no…

A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this paper, we prove in $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice)…

Let $\mathcal S$ denote the class of all functions of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$ which are analytic and univalent in the open unit disk $\ID$ and, for $\lambda >0$, let $\Phi_\lambda (n,f)=\lambda a_n^2-a_{2n-1}$ denote the…

Swierczkowski's Lemma - as it is usually formulated - asserts that if f is an at least quaternary operation on a finite set A and every operation obtained from f by identifying a pair of variables is a projection, then f is a…

By universal formulas we understand parameterized analytic expressions that have a fixed complexity, but nevertheless can approximate any continuous function on a compact set. There exist various examples of such formulas, including some in…

Let $X$ and $Y$ be the Hausdorff topological spaces and let $A$ be both an $\fs$- and $\gd$- subset of $X$. Let also $f\cn A\to Y$ be a function for which the inverse image of every open subset $U\subset Y$ is $\fs$ in $X$. We show that $f$…

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is…

We prove an inversion theorem for the Fourier transform defined for normal functions, in the case when such functions are of moderate decrease, and in dimensions 2 and 3. This improves on Carleson's general almost everywhere convergence…

For any pair of bounded observables $A$ and $B$ with pure point spectra, we construct an associated "joint observable" which gives rise to a notion of a joint (projective) measurement of $A$ and $B$, and which conforms to the intuition that…

Given sets $X,Y$ and a regular cardinal $\mu$, let $\Phi(X,Y,\mu)$ be the statement that for any function $f : X \times Y \to \mu$, there are functions $g_1 : X \to \mu$ and $g_2 : Y \to \mu$ such that or all $(x,y) \in X \times Y$,…

A recent refinement of Ker\'ekj\'art\'o's Theorem has shown that in $\mathbb R$ and $\mathbb R^2$ all $\mathcal C^l$-solutions of the functional equation $f^n =\textrm{Id}$ are $\mathcal C^l$-linearizable, where $l\in \{0,1,\dots \infty\}$.…

We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…

Let A be a set of integers. For every integer n, let r_{A,h}(n) denote the number of representations of n in the form n = a_1 + a_2 + ... + a_h, where a_1, a_2,...,a_h are in A and a_1 \leq a_2 \leq ... \leq a_h. The function r_{A,h}: Z \to…

In this article, by comparing the characteristic functions, we prove that for any $\nu$-valued algebroid function $w(z)$ defined in the unit disk with $\limsup_{r\to1-}T(r,w)/\log\frac{1}{1-r}=\infty$ and the hyper order $\rho_2(w)=0$, the…

This paper concerns the universality of the two-layer neural network with the $k$-rectified linear unit activation function with $k=1,2,\ldots$ with a suitable norm without any restriction on the shape of the domain. This type of result is…

We suggest the necessary/sufficient criteria for the existence of a (order-by-order) solution y(x) of a functional equation F(x,y)=0 over a ring. In full generality, the criteria hold in the category of filtered groups, this includes the…

In a recent paper [5] a smooth function f : [0; 1] --> R with all derivatives vanishing at 0 has been considered and a global condition, showing that f is indeed identically 0, has been presented. The purpose of this note is to replace the…

We consider classes $ \mathcal{A}_M(S) $ of functions holomorphic in an open plane sector $ S $ and belonging to a strongly non-quasianalytic class on the closure of $ S $. In $ \mathcal{A}_M(S) $, we construct functions which are flat at…

We prove the following two results. 1. If $X$ is a completely regular space such that for every topological space $Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the first Baire class, then every Lindel\"of subspace…