Related papers: Sidon sets and statistics of the ElGamal function
Let G be a primitive permutation group on a finite set Omega. Let p^2 divide |G|, for a prime p. We show that when G is solvable, there exists a subset of Omega whose stabilizer S has the property that 1<|S|_p<|G|_p. We offer a counting…
Let $G=(V,E)$ be a $d$-regular graph on $n$ vertices and let $\mu_0$ be a probability measure on $V$. The act of moving to a randomly chosen neighbor leads to a sequence of probability measures supported on $V$ given by $\mu_{k+1} = A…
Let $G$ be a $d$-regular graph $G$ on $n$ vertices. Suppose that the adjacency matrix of $G$ is such that the eigenvalue $\lambda$ which is second largest in absolute value satisfies $\lambda=o(d)$. Let $G_p$ with $p=\frac{\alpha}{d}$ be…
We consider the network reliability problem in wireless sensor networks secured by the heterogeneous random key predistribution scheme. This scheme generalizes Eschenauer-Gligor scheme by considering the cases when the network comprises…
In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations…
Let $G$ be a transitive permutation group on a finite set $\Omega$ and recall that a base for $G$ is a subset of $\Omega$ with trivial pointwise stabiliser. The base size of $G$, denoted $b(G)$, is the minimal size of a base. If $b(G)=2$…
Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci-…
An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\le x$. We improve an upper bound for the least such number due to E. Bach, R. Lukes, J. Shallit, and H. C. Williams. The…
Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along…
The prime graph $\Gamma(G)$ of a finite group $G$ (also known as the Gruenberg-Kegel graph) has as its vertices the prime divisors of $|G|$, and $p\text-q$ is an edge in $\Gamma(G)$ if and only if $G$ has an element of order $pq$. Since…
The planar rigidity problem asks, given a set of m pairwise distances among a set P of n unknown points, whether it is possible to reconstruct P, up to a finite set of possibilities (modulo rigid motions of the plane). The celebrated…
Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups:…
We study partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. These permutations are the linear…
The power graph \( \mathcal{G}_G \) of a group \( G \) is a graph whose vertex set is \( G \), and two elements \( x, y \in G \) are adjacent if one is an integral power of the other. In this paper, we determine the adjacency, Laplacian,…
We propose a new method for the calculation of the statistical properties, as e.g. the entropy, of unknown generators of symbolic sequences. The probability distribution $p(k)$ of the elements $k$ of a population can be approximated by the…
There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some…
Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a…
We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the…
In this paper we describe some properties of groups $G$ that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3). We prove that if $G$ is a non-solvable group that contains a solvable subgroup of index…
Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process and edges exist between two points if and only if their distance is less than a fixed given…