Related papers: Synchronizable functions on integers

We express some general type of infinite series such as $$ \sum^\infty_{n=1}\frac{F(H_n^{(m)}(z),H_n^{(2m)}(z),\ldots,H_n^{(\ell m)}(z))} {(n+z)^{s_1}(n+1+z)^{s_2}\cdots (n+k-1+z)^{s_k}}, $$ where $F(x_1,\ldots,x_\ell)\in\mathbb…

Fast matching of regular expressions with bounded repetition, aka counting, such as (ab){50,100}, i.e., matching linear in the length of the text and independent of the repetition bounds, has been an open problem for at least two decades.…

The Collatz conjecture implies that an iterated function sequence under a certain linear operator, beginning with a certain complex valued function, must converge to a certain complex function.

The function f:X -> Y is called k-monotonically increasing if there is a partition X = X_1 U ... U X_k such that f|X_i : X_i -> Y is monotonically increasing for i=1,...,k. It is proved that a one-to-one function f:N -> N is k-monotonically…

Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum…

We analyze the stopping-time and cycle structure of the normalized Collatz iteration. Using a recursive description of admissible binary sequences, we show that every integer $m \equiv 3 \pmod{4}$ arises uniquely and derive new bounds for…

We define Collatz representations for a subset of rational numbers and prove that each real number \( x \notin (-1,1) \) can be approximated arbitrarily well by rational numbers which have only \( 2 \)'s and \( 1 \)'s in their Collatz…

We construct a H\"older continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We say that a function with…

It is shown that for $A_\R(\D)$ functions $f_1$ and $f_2$ with $$ \inf_{z\in\bar{\D}}(\abs{f_1(z)}+\abs{f_2(z)})\geq\delta>0 $$ and $f_1$ being positive on real zeros of $f_2$ then there exists $A_\R(\D)$ functions $g_2$ and $g_1,g_1^{-1}$…

We analyze certain compositions of rational inner functions in the unit polydisk $\mathbb{D}^{d}$ with polydegree $(n,1)$, $n\in \mathbb{N}^{d-1}$, and isolated singularities in $\mathbb{T}^d$. Provided an irreducibility condition is met,…

The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms…

We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. The…

In many regular cases, there exists a (properly defined) limit of iterations of a function in several real variables, and this limit satisfies the functional equation (1-z)f(x)=f(f(xz)(1-z)/z); here z is a scalar and x is a vector. This is…

In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$. Let $a$ and $b$ be…

We explore the cycles and convergence of Generalized Collatz Sequence, where $3n+1$ in original collatz function is replaced with $3n+k$. We present a generating function for cycles of GCS and show a particular inheritance structure of…

We introduce simple group-theoretic techniques for classifying conformally-invariant tensor-structures. With them, we classify tensor structures of general n-point functions of non-conserved operators, and $n\geq 4$-point functions of…

For any real sequence {c(n)} tending to infinity as n tends to infinity, this constructs a function f which is continuous and integrable, and such that for every nonzero x, limsup c(n) f(n x) is infinite.

We construct a real sequence $\{\lambda_n\}_{n=1}^{\infty}$ satisfying $\lambda_n = n + o(1)$, and a Schwartz function $f$ on $\mathbb{R}$, such that for any $N$ the system of translates $\{f(x - \lambda_n)\}$, $n > N$, is complete in the…

Let $\Bbb Z$ be the set of integers. For positive integers $a,b,c$ and $n$ let $N(a,b,c;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2$, and let $t(a,b,c;n)$ be the number of representations of $n$ by…

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$…