On S-Magic Graphs

The Cayley graph generated by a set A is the graph ΓA(V) on a set of positive integers V such that a pair (u, v) ∈ V×V is an edge of the graph if and only if |u−v| ∈ A or u+v ∈ A. We denote by (n,m) the class of graphs G = (V, E) such that G is a union of chains and cycles and |V| = n, |E| = m. In this paper, we present an upper bound for the number of independent sets in Cayley graphs ΓA(V) such that A = {⌈n/2⌉ − t, ⌈n/2⌉ − ƒ}, V ⊆ [⌊n/2⌋ + 1, ⌊n/2⌋+t] ∪ [n − t + 1, n], where n, t, ƒ ∈ N and ƒ < t < n/4. We also describe the graph with the maximum number of independent sets in the family (n, m).

In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

In this paper we study the semigroup $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $\tau$ on $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N}),\tau)$ is a semitopological semigroup is discrete, describe the closure of $(\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$.

(1940). A Theorem on Partitions of the Set of Positive Integers. The American Mathematical Monthly: Vol. 47, No. 3, pp. 152-154.

On a System of Axioms Which Has no Recursively Enumerable Arithmetic Model

On a Class of Enumeration Problems in Additive Arithmetics

Let [Formula: see text] be the set of integers and [Formula: see text] the set of positive integers. For a nonempty set [Formula: see text] of integers and any integers [Formula: see text], [Formula: see text] with [Formula: see text], define [Formula: see text] as the number of solutions of [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] A set [Formula: see text] of integers is defined as a basis of order [Formula: see text] for [Formula: see text] if [Formula: see text] for every integer [Formula: see text]. In 2004, Nešetřil and Serra considered the Erdős–Turán conjecture for a class of bounded bases. In this paper, we generalize the above result and obtain that: for any function [Formula: see text], there exists a bounded basis of order [Formula: see text] for [Formula: see text] such that [Formula: see text] for every integer [Formula: see text].

We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence- 4 internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations, we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space \mathcal M_{0,4} . We also conjecture that the corresponding polynomials are log-concave , on the basis of hundreds of explicit computations.

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form [Formula: see text] Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M = 32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form [Formula: see text] In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

For h ge 2 and an infinite set of positive integers A, let R_{A,h}(n) denote the number of representations of the positive integer n as the sum of h distinct terms from A. A set of positive integers A is called a B_h[g] set if every positive integer can be written as the sum of h not necessarily distinct terms from A at most g different ways. We say a set A is a basis of order h if every positive integer can be represented as the sum of h terms from A. Recently, Vu [17] proved the existence of a thin basis of order h formed by perfect powers. In this paper, we study weak B_{h}[g] sets formed by perfect powers. In particular, we prove the existence of a set A formed by perfect powers with almost possible maximal density such that R_{A,h}(n) is bounded by using probabilistic methods.

S-functions are mappings from the class of finite graphs into the set of integers, such that certain formal conditions are fulfilled which are shared by the chromatic number, the vertex-connectivity, and the homomorphism-degree. The S-functions form a complete lattice (with respect to their natural partial order). The classes of graphs with values <n under some S-function are studied from a general point of view, and uncountably many S-functions are constructed. Further for every n≥5 a non-trivial base-element of (see K. WAGNER [7]) is constructed.

Let $G=(V,E)$ be an undirected graph on $n$ vertices and $\lambda:E\to 2^{\mathbb{N}}$ a mapping that assigns to every edge a non-empty set of positive integer labels. These labels can be seen as discrete times when the edge is present. Such a labeled graph ${\cal G}=(G,\lambda)$ is said to be temporally connected if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar (STOC 2000) asked whether, given such a temporal graph, a sparse subset of edges can always be found whose labels suffice to preserve temporal connectivity---a temporal spanner. Axiotis and Fotakis (ICALP 2016) answered negatively by exhibiting a family of $\Theta(n^2)$-dense temporal graphs which admit no temporal spanner of density $o(n^2)$. The natural question is then whether sparse temporal spanners can always be found in some classes of dense graphs. In this paper, we answer this question affirmatively, by showing that if the underlying graph $G$ is a complete graph, then one can always find temporal spanners of density $O(n \log n)$. The best known result for complete graphs so far was that spanners of density ${n \choose 2}- \lfloor n/4 \rfloor = O(n^2)$ always exist. Our result is the first positive answer as to the existence of $o(n^2)$ sparse spanners in adversarial instances of temporal graphs since the original question by Kempe et al., focusing here on complete graphs. The proofs are constructive and directly adaptable as an algorithm.

Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are infinitely many elements x in X with d_A(e,x)=h. It is proved that if P is a nonempty finite set of prime numbers and A is the set of positive integers whose prime factors all belong to P, then the diameter of the metric space (\Z,d_A) is infinite. Let \lambda_A(h) denote the smallest positive integer x with d_A(e,x)=h. It is an open problem to compute \lambda_A(h) and estimate its growth rate.

On generalized perfect difference sets constructed from Sidon sets

In this paper, we examine properties of the division topology on the set of positive integers introduced by Rizza in 1993. The division topology on [Formula: see text] with the division order is an example of [Formula: see text]-Alexandroff topology. We mainly concentrate on closures of arithmetic progressions and connected and compact sets. Moreover, we show that in the division topology on [Formula: see text], the continuity is equivalent to the Darboux property.