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While the reversibility of multidimensional cellular automata is undecidable and there exists a criterion for determining if a multidimensional linear cellular automaton is reversible, there are only a few results about the reversibility…

A partial Steiner triple system is is $sequenceable$ if the points can be sequenced so that no proper segment can be partitioned into blocks. We show that, if $0 \leq a \leq (n-1)/3$, then there exists a nonsequenceable PSTS$(n)$ of size…

In this paper, two approximation algorithms are given. Let N be an odd composite number. The algorithms give new directions regarding primality test of given N. The first algorithm is given using a new method called digital coding method.…

Abelian string matching problems are becoming an object of considerable interest in last years. Very recently, Alatabbi et al. \cite{AILR2015} presented the first solution for the longest common Abelian factor problem for a pair of strings,…

The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC'15]. In recent work, Larsen and…

Andrews and Merca [J. Combin. Theory Ser. A 203 (2024), Art. 105849] recently obtained two interesting results on the sum of the parts with the same parity in the partitions of $n$ (the modulo $2$ case), the proof of which relies on…

The bloom clock is a space-efficient, probabilistic data structure designed to determine the partial order of events in highly distributed systems. The bloom clock, like the vector clock, can autonomously detect causality violations by…

We consider the uniform distribution of solutions $(x,y)$ to $xy=N \mod a$, and obtain a bound on the second moment of the number of solutions in squares of length approximately $a^{1/2}$. We use this to study a new factoring algorithm that…

In this paper we propose a sequence of tests which gives a definitive test for checking $2\times M$ separability. The test is definitive in the sense that each test corresponds to checking membership in a cone, and that the closure of the…

We consider string matching with variable length gaps. Given a string $T$ and a pattern $P$ consisting of strings separated by variable length gaps (arbitrary strings of length in a specified range), the problem is to find all ending…

Modular composition is the problem of computing the composition of two univariate polynomials modulo a third one. For a long time, the fastest algebraic algorithm for this problem was that of Brent and Kung (1978). Recently, we improved…

Model checking is an automatic formal verification technique that is widely used in hardware verification. The state-of-the-art complete model-checking techniques, based on IC3/PDR and its general variant CAR, are based on computing…

In the parity problem, a given cellular automaton has to classify any initial configuration into two classes according to its parity. Elementary cellular automaton rule 60 can solve the parity problem in periodic boundary conditions with…

An ordered triple $(s,p,n)$ is called admissible if there exist two different multisets $X=\{x_1,x_2,\dotsc,x_n\}$ and $Y=\{y_1,y_2,\dotsc,y_n\}$ such that $X$ and $Y$ share the same sum $s$, the same product $p$, and the same size $n$. We…

The online square detection problem is to detect the first occurrence of a square in a string whose characters are provided as input one at a time. Recall that a square is a string that is a concatenation of two identical strings. In this…

PARITY is the problem of determining the parity of a string $f$ of $n$ bits given access to an oracle that responds to a query $x\in\{0,1,...,n-1\}$ with the $x^{\rm th}$ bit of the string, $f(x)$. Classically, $n$ queries are required to…

We assume the permutation $\pi$ is given by an $n$-element array in which the $i$-th element denotes the value $\pi(i)$. Constructing its inverse in-place (i.e. using $O(\log{n})$ bits of additional memory) can be achieved in linear time…

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of…

This paper studies the number conservation property of 1-dimensional non-uniform cellular automata (CAs). In a non-uniform cellular automaton (CA), different cells may follow different rules. The present work considers that the cells follow…

We study the problem of finding positive integers $n$ such that all the decimal digits of $2^n$ are even, i.e., belong to $\{0, 2, 4, 6, 8\}$. Computational checks up to $n = 10^{15}$ reveal the known cases $n = 1, 2, 3, 6, 11$ and no…