Related papers: Switch Functions

Let $d$ and $n$ be positive integers such that $d|n$. Let $[n]=\{1,2,\ldots,n\}$ and $T$ be an endofunction on $[n]$. A subset $W$ of $[n]$ of cardinality $n/d$ is said to be $d$-splitting if $W \cup TW \cup \cdots \cup T^{d-1}W =[n]$. Let…

We study whether a uniformly random Boolean function $f : \{-1,1\}^p \to \{-1,1\}$ is determined by its Walsh--Fourier coefficients of degree at most $d$. We show that the threshold lies at $p/2$ up to an $O(\sqrt{p \log p})$ window: if \[…

To a pair of subspaces wandering with respect to a row isometry we associate a transfer function which in general is multi-Toeplitz and in interesting special cases is multi-analytic. Then we describe in an expository way how characteristic…

In courses on integration theory, Chasles property is usually considered as elementary and so "natural" that this is sometimes left to the reader. When the functions take their values in finite dimensional spaces, the property is always…

The Wigner functions on the one dimensional lattice are studied. Contrary to the previous claim in literature, Wigner functions exist on the lattice with any number of sites, whether it is even or odd. There are infinitely many solutions…

We show that if the complexity difference function p(n+1)-p(n) of a infinite minimal shift is bounded, then the the automorphism group of the one-sided shift is finite, and the automorphism group of the corresponding two-sided shift "modulo…

For all natural numbers a,b and d > 0, we consider the function f_{a,b,d} which associates n/d to any integer n when it is a multiple of d, and an + b otherwise; in particular f_{3,1,2} is the Collatz function. Coding in base a > 1 with b <…

We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,...,n}. These classes include the following: (1) both w and w^{-1} are alternating, (2) w has certain special shapes, such as…

Transient algebra is a multi-valued algebra for hazard detection in gate circuits. Sequences of alternating 0's and 1's, called transients, represent signal values, and gates are modeled by extensions of boolean functions to transients.…

Let $C({\mathbb R}^n)$ denote the set of real valued continuous functions defined on ${\mathbb R}^n$. We prove that for every $n\ge 2$ there are positive numbers $\lambda _1 , \ldots , \lambda _n$ and continuous functions $\phi_1 ,\ldots ,…

Given a subset $\Lambda$ of $\mathbb Z_+:=\{0,1,2,\dots\}$, let $H^\infty(\Lambda)$ denote the space of bounded analytic functions $f$ on the unit disk whose coefficients $\widehat f(k)$ vanish for $k\notin\Lambda$. Assuming that either…

We say that a plane set $A$ is {\it graph-null,} if there is a function $g\colon [0,1] \to \mathbb{R}$ such that $\lambda_2 (A+{\rm graph}\, g)=0$. A plane set $A$ has the {\it translational Kakeya property} if, for every translated copy…

The asymmetric switch process is a binary stochastic process that alternates between the values one and minus one, where the distributions of the time in these states may differ. Two versions of the process are considered: a non-stationary…

For each of the functions $f \in \{\phi, \sigma, \omega, \tau\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for…

A permutation \tau contains another permutation \sigma as a pattern if \tau has a subsequence whose elements are in the same order with respect to size as the elements in \sigma. This defines a partial order on the set of all permutations,…

We study $\mathbb{R}_{\textrm{an},\exp}$-definable functions $f:\mathbb{R}\to \mathbb{R}$ that take integer values at all sufficiently large positive integers. If $|f(x)|= O\big(2^{(1+10^{-5})x}\big)$, then we find polynomials $P_1, P_2$…

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly…

This text gives a rough, but linear summary covering some key definitions, notations, and propositions from Lambda Calculus: Its Syntax and Semantics, the classical monograph by Barendregt. First, we define a theory of untyped extensional…

There are no two identical leaves in the world, so how to find effective markers or features to distinguish them is an important issue. Function transformation, such as f(x,y) and f(x,y,z), can transform two, three, or multiple…

We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function mapping from R^n to R^(n-1); in combination with the work of others, this completes the investigation of…