We consider a 2-player permutation game inspired by the celebrated Erdős-Szekeres Theorem. The game depends on two positive integer parameters $a$ and $b$ and we determine the winner and give a winning strategy when $a \geq b$ and $b \in \left\{2,3,4,5\right\}$. 18 pages

Partitions of natural numbers and their representation function

A ∗-ring [Formula: see text] is a ring with an involution ∗. Let [Formula: see text] denote the set of all nonzero zero-divisors of [Formula: see text]. We associate a simple (undirected) graph [Formula: see text] with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] or [Formula: see text], for some positive integer [Formula: see text]. We find the diameter and girth of [Formula: see text]. The characterizations are obtained for ∗-rings having [Formula: see text] a connected graph, a complete graph, and a star graph. Further, we have shown that for a ring [Formula: see text], there is an involution on [Formula: see text] such that [Formula: see text] is disconnected if and only if [Formula: see text] is an integral domain.

Abstract For all sufficiently large , in any arithmetic progression in which and are relatively prime there exists a positive integer with at most two prime factors (counted with multiplicity) which is asymptotically less than . The proof uses the weighted sieve of Greaves–Halberstam–Richert with bilinear remainder terms and Selberg's sieve.

This paper presents all Padovan numbers that can be written as the concatenation of three Padovan or Perrin numbers under a certain constraint. Namely, we consider the Diophantine equations \[ P_{k}=10^{d+l}P_{m}+10^{l}P_{n}+P_{r} \] and \[ P_{k}=10^{d+l}R_{m}+10^{l}R_{n}+R_{r} \] where k, m, n, r, d and l are positive integers satisfying n ≤ m. The parameters $d$ and $l$ denote the numbers of digits in the integers $P_{n}$ (or $R_{n}$) and $P_{r}$ (or $R_{r}$), respectively. The solutions to these equations can be written in the form $P_{18}=\overline{P_{2}P_{2}P_{6}}=114$ for all m, n, r ≥ 2 and, similarly, $P_{30}=\overline{R_{3}R_{3}R_{12}}=3329$, $P_{33}=\overline{R_{7}R_{7}R_{13}}=7739$ for all m ≥ 3 and n, r ≥ 1.

Generalizing the notion of a multiplicative inequality among minors of a totally positive matrix, we describe, over full rank cluster algebras of finite type, the cone of Laurent monomials in cluster variables that are bounded as real-valued functions on the positive locus of the cluster variety. We prove that the extreme rays of this cone are the u -variables of the cluster algebra. Using this description, we prove that all bounded ratios are bounded by 1 and give a sufficient condition for all such ratios to be subtraction free. This allows us to show in Gr ( 2 , n ) , Gr ( 3 , 6 ) , Gr ( 3 , 7 ) , and Gr ( 3 , 8 ) that every bounded Laurent monomial in Plücker coordinates factors into a positive integer combination of so-called primitive ratios. In Gr ( 4 , 8 ) this factorization does not exists, but we provide the full list of extreme rays of the cone of bounded Laurent monomials in Plücker coordinates.

The Riemann Zeta function, usually denoted by the Greek letter ζ , was defined in 1737 by a Swiss mathematician Leonhard Euler. This function is an infinite converging sum of powers of natural numbers, and it has explicit expressions in terms of π at positive even integers. In this paper we will discuss various irrationality proofs, focusing on irrationality of certain values of the Zeta function.

A general approach to multistability analysis for fuzzy multidimensional-valued NNs with memristor.

Abstract Let and . A Diophantine tuple with property is a set of positive integers such that is a th power for all with . Such generalizations of classical Diophantine tuples have been studied extensively. In this paper, we prove several results related to robust versions of such Diophantine tuples and discuss their applications to product sets contained in a nontrivial shift of the set of all perfect powers or some of its special subsets. In particular, we substantially improve several results by Bérczes–Dujella–Hajdu–Luca, and Yip. We also prove several interesting conditional results. Our proofs are based on a novel combination of ideas from sieve methods, Diophantine approximation, and extremal graph theory.

Abstract We study the derivative of the characteristic polynomial of Haar‐distributed unitary matrices. We obtain new explicit formulae for complex‐valued moments when the spectral variable is inside the unit disc, in the limit . These formulae are expressed in terms of the confluent hypergeometric function of the first kind. We explore the connection between these moments and those of the derivative of the Riemann zeta function away from the critical line. Under the Lindelöf hypothesis, we prove that all positive integer moments agree with our random matrix results up to a well‐known arithmetic factor. Inspired by this finding, we propose a conjecture on the asymptotics of noninteger moments of the derivative of the Riemann zeta function off the critical line. Within random matrix theory, we also investigate the microscopic regime where the spectral variable satisfies for a fixed constant . We obtain an asymptotic formula for the moments in this regime as a determinant involving the finite temperature Bessel kernel, which reduces to the Bessel kernel when .

We study approximation algorithms for the Min-Max Rural Postmen Cover Problem (MMRPCP). Given an undirected graph [Formula: see text], and a required subset [Formula: see text] of edges, where each edge in [Formula: see text] has a nonnegative weight, the objective is to find at most [Formula: see text] closed walks covering all the edges in [Formula: see text] such that the maximum weight of the closed walks is minimum. We propose a bicriteria [Formula: see text]-approximation algorithm for the MMRPCP. More exactly, given any instance [Formula: see text] of the MMRPCP consisting of a positive integer [Formula: see text], a graph [Formula: see text] and a required edge set [Formula: see text], the algorithm can produce at most [Formula: see text] closed walks covering all the edges in [Formula: see text] such that the maximum weight of the closed walks is no more than [Formula: see text] times the optimal value of [Formula: see text]. Previously, the best-known approximation ratio for the MMRPCP is [Formula: see text]. Our result demonstrates that a moderate relaxation of the constraint on the number of closed walks is helpful to reduce the approximation ratio.

Let ( u ( n ) ) n ∈ N be an arithmetic progression of natural integers in base b ∈ N ∖ { 0 , 1 } . We consider the following sequences: s ( n ) = u ( 0 ) u ( 1 ) ⋯ u ( n ) ¯ b formed by concatenating the first n + 1 terms of ( u ( n ) ) n ∈ N in base b from the right; s g ( n ) = u ( n ) u ( n − 1 ) ⋯ u ( 0 ) ¯ b ; and ( s ∗ ( n ) ) n ∈ N , given by s ∗ ( 0 ) = u ( 0 ) , s ∗ ( n ) = s ( n ) s g ( n − 1 ) ¯ b , n ≥ 1 . We construct explicit formulae for these sequences and use basic concepts of linear difference operators to prove they are not P-recursive (holonomic). We also present an alternative proof that follows directly from their definitions. We implemented ( s ( n ) ) n ∈ N and ( s g ( n ) ) n ∈ N in the decimal base when ( u ( n ) ) n ∈ N = N ∖ { 0 } .

A radio labeling of a graph 𝐺 is a function 𝑓 from the vertex set 𝑉(𝐺) to the set of non-negative integers such that |𝑓(𝑢) − 𝑓(𝑣)| ≥ diam(𝐺)+1−𝑑𝐺(𝑢,𝑣), where diam(𝐺) and 𝑑𝐺(𝑢,𝑣) are the diameter of 𝐺 and the distance between 𝑢 and v in 𝐺, respectively. The radio number rn(𝐺) of 𝐺 is the smallest number 𝑘 such that 𝐺 has radio labeling with max{𝑓(𝑣):𝑣 ∈ 𝑉(𝐺)} = 𝑘. A tree 𝑇 is called a caterpillar if it has a path 𝑃 of maximum length such that all the vertices other than the path 𝑃 are at most distance 1 from the path 𝑃. In this paper, we determine the radio number of some special types of caterpillars.

Abstract The Padovan ( P n ) n ≥0 and Perrin ( R n ) n ≥0 sequences are third-order linear recurrences, both defined by the relation u n = u n− 2 + u n− 3 for n ≥ 3. They differ in their initial conditions resulting in different sequences. The Padovan sequence begins with P 0 = P 1 = P 2 = 1, whereas the Perrin sequence starts with R 0 = 3, R 1 = 0 , and R 2 = 2. Motivated by the work of Gómez and Luca [Tribonacci Diophantine quadruples, Glas. Mat., Ser. III , 50(1):17–24, 2015], we investigate whether there exist quadruples of positive integers a 1 < a 2 < a 3 < a 4 such that all pairwise products a i a j + 1 (for i ≠ j ) belong to the Padovan or Perrin sequence, and we prove that the answer is negative.

In this paper we are concerned with a family of sums involving the floor function. With $r$ a nonnegative integer and $n$ and $m$ positive integers we consider the sums$$\mathbf{S}_{r}(n,m):=\sum_{k=1}^{n-1}{\left\lfloor \frac{km}{n}}\right\rfloor ^r.$$ While a formula for $\mathbf{S}_1$ is well known, we provide closed-form formulas for $\mathbf{S}_2$ and $\mathbf{S}_3$ as well as the reciprocity laws they satisfy. Additionally, one can find a closed-form formula for the classical Dedekind sum using the Euclidean algorithm. Finally, we provide a general formula for $\mathbf{S}_r$ showing its dependency on generalized Dedekind sums.

This paper gives a clear account of how prime numbers form the basic structure of arithmetic. Using the Fundamental Theorem of Arithmetic, I show that every natural number can be written as a product of primes and that this makes it possible to picture numbers as points in a lattice, each one defined by its prime factors. In this way, arithmetic is not built from isolated numbers but from the network of relations among primes. What is real, on this view, is not the numbers themselves but the structure that connects them.

A subset $S$ of the vertex set $V(G)$ of a graph $G$ is called an equitable fair dominating set of $G$ if $S$ is an equitable dominating set of $G$ and for any $v,$ $w \in V(G) \backslash S$, $|N_G(v) \cap S| = |N_G(w) \cap S| \geq 1$. The equitable fair domination number of $G$ denoted by $\gamma_{efd}(G)$ is the minimum cardinality of an EFD-set of $G$. $S$ is called an equitable $k$-fair dominating set (abbreviated E$k$FD-set) of $G$ if $|N_G(v) \cap S| = k$ for any $v \in V(G) \backslash S$ where $k$ is a positive integer. The equitable $k$-fair domination number of $G$ denoted by $\gamma_{_{kfd}}^e (G)$ is the minimum cardinality of an E$k$FD-set. An equitable $k$-fair dominating set of cardinality $\gamma_{_{kfd}}^e (G)$ is called a $\gamma_{_{kfd}}^e$-set of $G$. In this paper, we characterize the notions of equitable k-fair domination in graphs, study the EkFD-sets under some binary operations of graphs, and determine exact values or bounds for this domination variant.

In this paper, for a positive integer n, we compute the number of all n degree representations for a dihedral group G of order 2m, m ≥ 3. We evaluate dimensions of the spaces of invariant bilinear forms corresponding to each of the representation over the field of complex numbers ℂ (in fact over a number field consisting of a primitive 4mth root of unity). We also, assure that the same results hold equally good when considered over a field of characteristic ≡ 1 (mod 4m}. We explicitly discuss the existence of a non-degenerate invariant bilinear form.

Rooted binary perfect phylogenies provide a generalization of rooted binary unlabeled trees. In a rooted binary perfect phylogeny, each leaf is assigned a positive integer value that corresponds in a biological setting to the count of the number of indistinguishable lineages associated with the leaf. For the rooted binary unlabeled trees, these integers equal 1. We enumerate rooted binary perfect phylogenies with leaves and sample size , : the rooted binary unlabeled trees with leaves in which a sample of size lineages is distributed across the leaves. (1) First, we recursively enumerate rooted binary perfect phylogenies with sample size , summing over all possible , . We obtain an equation for the generating function, showing that asymptotically, the number of rooted binary perfect phylogenies with sample size grows with , faster than the rooted binary unlabeled trees, which grow with ≈ . (2) Next, we recursively enumerate rooted binary perfect phylogenies with a specific number of leaves and sample size . We report closed-form counts of the rooted binary perfect phylogenies with sample size and leaves. We provide a recurrence for the generating function describing, for each number of leaves , the number of rooted binary perfect phylogenies with leaves as the sample size increases. We also obtain an equation satisfied by the bivariate generating function counting rooted binary perfect phylogenies with leaves and sample size , as well as an asymptotic normal distribution for the number of leaves in a randomly chosen perfect phylogeny with sample size . (3) We find a generating function for the number of rooted binary perfect phylogenies with the -leaf caterpillar shape, growing with . We also find a generating function and exact count for the number of rooted binary perfect phylogenies with sample size and any caterpillar tree shape. A bivariate generating function counting rooted binary perfect phylogenies with leaves, sample size , and a caterpillar shape produces an asymptotic normal distribution for the number of leaves in a randomly chosen caterpillar perfect phylogeny with sample size . (4) Finally, we provide initial results recursively enumerating rooted binary perfect phylogenies with any specific unlabeled tree shape and sample size . The enumerations further characterize the rooted binary perfect phylogenies, which include the rooted binary unlabeled trees, and which can provide a set of structures useful for various biological contexts.

We present a fully elementary method for evaluating the infinite series $S_k = \sum_{n=1}^{\infty} \frac{n^k}{2^n}$, where k is a fixed natural number. The method relies only on repeated scaling, term-by-term subtraction, and the systematic use of finite differences. No tools from calculus, generating functions, or special functions are required. Starting from explicit computations for $k=1,2,3,4$, we show how a stable pattern emerges and how this pattern can be described and proved using a difference matrix. Finally, we present an interesting combinatorial identity.