SHIFTED AND SHIFTLESS PARTITION IDENTITIES II

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 $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$.

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.

On a Class of Enumeration Problems in Additive Arithmetics

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

Let IN be the set of positive integers, $${\cal B}$$ = {b1 < ⋅s < bh}⊂IN, N∊IN and N≥ bh. $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sarkozy) such that $${\cal A}$$ {1,..., N} = $${\cal B}$$ and p( $${\cal A}$$ ,n)≡ 0 (mod 2) for n∊IN and n > N, where p( $${\cal A}$$ ,n) denotes the number of partitions of n with parts in $${\cal A}$$ . Let us denote by σ ( $${\cal A}$$ ,n) the sum of the divisors of n belonging to $${\cal A}$$ . In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ( $${\cal A}$$ ,2k n))n≥ 1mod 2k+1 is periodic with an odd period qk. In this paper, we will characterize for any fixed odd positive integer q, the sets $${\cal B}$$ and the integers N such that q0 = q, and those for which qk = q for all k≥ 0. Moreover, a set $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is constructed with the property that its period, i.e. the period of (σ( $${\cal A}$$ ,n))n≥ 1mod 2, is 217, and for which the counting function is asymptotically equal to that of $${\cal A}$$ 0({1,2,3,4,5},5) which is a set of period 31.

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].

The principal result of this paper is the introduction of a readily verifiable sufficient condition for the existence of an invariant measure on transient Markov chains. Specifically, subject to mild additional regularity conditions, it is enough to check that for every compact set $A$ in the state space $X$, there exists a compact $C \supset A$ such that \begin{equation*}\tag{0.1} \lim_{y\in F, y\rightarrow\infty} \frac{P(y, A)}{P(y, C)} = 0\quad\text{if} \bar{F} \text{is not compact and} F = \{y: P(y, A) > 0\},\end{equation*} $P(y, A)$ denote the 1-step transition probabilities of the Markov chain. (0.1) is readily implied by simple conditions on the generating functional of the offspring distribution (Theorems 2.1 and 2.2) for discrete time multi-type Markov processes. These conditions together with some additional regularity-hypotheses are verified for (i) A 1-dimensional neutron branching model of Harris, (ii) Discrete-time age dependent branching processes, (iii) Galton-Watson processes with immigration. A generalization (to discrete-time temporally homogeneous Markov processes with $\sigma$-compact metric state space) of a condition of T. Harris (1957) for existence of invariant measures on transient Markov chains is also given. This condition is, unfortunately, difficult to check in specific examples. The sufficient condition (0.1) involves only 1-step transition probabilities, as opposed to the $n$-step transitions incorporated in the Harris condition, which is also necessary. Throughout this paper, all subsets of a topological space considered are assumed Borel measurable. The closure, interior, boundary and complement of a set $A$ are denoted by $\bar{A}, A^0, \partial A$ and $A^c$ respectively, and $I_A$ denotes the indicator function of $A$. The sets of positive integers, nonnegative integers, are denoted by $I$ and $I_0$ respectively. For any $\sigma$-compact metric space $X$, the following notations are used: $M(X) =$ set of regular measures (Borel measures finite on compact sets) on $X$. $B(X) =$ set of bounded measurable functions on $X$ with sup norm. $B'(X) = \{s \in B(X): \|s\| < 1\}$. $C(X) =$ set of bounded continuous functions on $X$. $C_{00}(X) = \{s\in C(X): s$ has compact support$\}$. $C_0(X) = \{s \in C(X):\lim_{x\rightarrow\infty}s(x) = 0\}.$ $C'(X) = C(X) \cap B'(X)$, similarly for $C_0'(X)$ and $C'_{00}(X)$. For any $\mathscr{L} \subset B(X), \mathscr{L}^+ = \{s \in \mathscr{L}: s \geqq 0\}$. If $\mu \in M(X)$ and $\in B(X)$, then $\mu f = \int_xf(x)\mu(dx)$ when defined.

Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with m parts in A that have r occurrences of 3-letter patterns formed by two (adjacent) instances of levels, rises and drops. We also derive asymptotics for the number of compositions of n that avoid a given pattern. Finally, we obtain the generating function for the number of k-ary words of length m which contain a prescribed number of occurrences of a given pattern as a special case of our results.

We examine the ideals of nowhere dense sets in three topologies on the set of positive integers, namely Furstenberg’s, Rizza’s and the common division topology. We mainly concentrate on inclusions between these ideals, we present a diagram showing these and we explore all possible inclusions between them. We present a formula for the closure of a set in the common division topology. We answer a question posed by Kwela and Nowik (Topol Appl. 248:149–163, 2018) by constructing a set in {{mathcal {I}}}_G {setminus } ({{mathcal {I}}}_K cup {{mathcal {I}}}_F). Therefore, the main diagram of comparison between the ideals of nowhere dense sets in various topologies from the article by M. Kwela and A. Nowik is completed.

A ring or near-ring R is called periodic if for each xCR, there exist distinct positive integers n, m for which x"=x". This paper continues the study of commutativity in such rings and near-rings which was previously undertaken in [1]--[5]. The first section is devoted to new proofs of two basic results Chacron's result that co-algebraic rings are necessarily periodic, and Herstein's result that periodic rings with central nilpotent elements are commutative. The next section deals with commutativity of periodic rings with constraints on certain commutators involving nilpotent elements, and the third is a discussion of structure and commutativity of certain of the 6-rings introduced by PtJWCrIA and YAQUB in [17]. The final section presents related results on commutativity of periodic near-rings. Throughout the paper Z will denote the ring of integers, Z + the set of positive integers, and Z[X] the ring of polynomials in one indeterminate with integer coefficients. The (multiplicative) center of R will be denoted by C, the set of nilpotent elements by N, and the additive group by (R, +). For each subset S of R, the left, right and two-sided annihilators will be denoted respectively by At(S),Ar(S), and A(S); the subring or sub-near-ring generated by S will be denoted by (S). As usual, the symbol [x, y] will stand for the commutator xy -yx ; and the symbol Cg(R) will indicate the commutator ideal of R. Generalized commutators are defined by letting Ix, y]l=[x, y] and extending inductively by the formula [x, y],=[[x, y],_l, y], n~_2. We shall make frequent use of the following properties of periodic rings [4, Lemma 1]: (Px) For each xCR, some power of x is idempotent. (P2) For each xER, there exists an integer n (x )> l for which x-x"(X)EN. (Ps) Each xCR can be expressed in the form y+w, where w~N and y"=y for some n=n(y)>l. (P4) If I is an ideal of R and x + I is a non-zero nilpotent element of R/I, then R contains a nilpotent element u such that u=-x (rood/).

(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

Ramanujan discovered that $$\sum_{n=0}^\infty p(5n+4)q^n=5 \prod_{j=1}^\infty \frac{(1-q^{5j})^5}{(1-q^j)^6}, $$ where p(n) is the number of partitions of n. Recently, H.-C. Chan and S. Cooper, and H.H. Chan and P.C. Toh established several analogues of Ramanujan’s partition identities by employing the theory of modular functions. Very recently, N.D. Baruah and K.K. Ojah studied the partition function \(p_{[c^{l}d^{m}]}(n)\) which is defined by $$\sum_{n=0}^\infty p_{[c^ld^m]}(n)q^n= \frac{1}{\prod_{j=1}^\infty (1-q^{cj})^{l}(1-q^{dj})^m}. $$ They discovered some analogues of Ramanujan’s partition identities and deduced several interesting partition congruences. In this paper, we provide a uniform method to prove some of their results by utilizing an addition formula. In the process, we also establish some new analogues of Ramanujan’s partition identities and congruences for \(p_{[c^{l}d^{m}]}(n)\).