Subsets Of Positive Integers Research Articles

Signs behaviour of sums of weighted numbers of compositions

Let A be a subset of positive integers. For a given positive integer n and 0≤i≤n, let cA(i,n) denote the number of A-compositions of n with exactly i parts. In this note, we investigate the sign behaviour of the sequence (SA,k(n))n∈N, where SA,k(n)=∑i=0n(-1)kikcA(i,n). We prove that for a broad class of subsets A, the number (-1)nSA,k(n) is non-negative for all sufficiently large n. Moreover, we show that there exists A⊂N+ such that the sign behaviour of SA,k(n) is not periodic.

On the non-existence of linear perfect Lee codes: The Zhang–Ge condition and a new polynomial criterion

The Golomb–Welch conjecture (1968) states that there are no e-perfect Lee codes in Zn for n≥3 and e≥2. This conjecture remains open even for linear codes. A recent result of Zhang and Ge establishes the non-existence of linear e-perfect Lee codes in Zn for infinitely many dimensions n, for e=3 and 4. In this paper we extend this result in two ways. First, using the non-existence criterion of Zhang and Ge together with a generalized version of Lucas’ theorem we extend the above result for almost all e (i.e. a subset of positive integers with density 1). Namely, if e contains a digit 1 in its base-3 representation which is not in the unit place (e.g. e=3,4) there are no linear e-perfect Lee codes in Zn for infinitely many dimensions n. Next, based on a family of polynomials (the Q-polynomials), we present a new criterion for the non-existence of certain lattice tilings. This criterion depends on a prime p and a tile B. For p=3 and B being a Lee ball we recover the criterion of Zhang and Ge.

Lacunary ideal summability and its applications to approximation theorem

An ideal I is a family of subsets of positive integers $$\mathbf {N}$$ which is closed under taking finite unions and subsets of its elements. In this paper, we define and study the notion of $$I_{\theta }$$ -convergence as a variant of the notion of ideal convergence, where $$\theta = (h_{r})$$ is a nondecreasing sequence of positive real numbers. We further apply this notion of summability to prove a Korovkin type approximation theorem.

On the Size of the Quotient of Two Subsets of Positive Integers

We obtain a nontrivial lower bound for the size of the set A/B, where A and B are subsets of the interval [1,Q].

Cycle frames and the Oberwolfach problem

For an integer λ,G(λ) denotes a graph G with uniform edge-multiplicity λ. Let J be a subset of positive integers. A 2-regular subgraph of m-partite graph Km⊗In containing vertices of all but one partite set is called partial 2-factor, where ⊗ denotes wreath product and In is an independent set on n vertices. If (Km⊗In)(λ) can be partitioned into edge-disjoint partial 2-factors such that each partial 2-factor consists of cycles of lengths from J, then we say that (Km⊗In)(λ)has a(J,λ)-cycle frame. The Oberwolfach problem OP(m1α1,m2α2,…,mtαt), raised by Ringel, asks the existence of a 2-factorization of Kn (when n is odd) or Kn−I (when n is even), in which each 2-factor consists of exactly αi cycles of length mi, i=1,2,…,t. In this paper, we show that there exists a ({4,6},λ)-cycle frame of (Km⊗In)(λ) if and only if λn≡0(mod2), m≥3, (n,m)∉{(1,3),(2t+1,2s)∣t≥0,s≥2}. Further we show that there exists a ({3,4},1)-cycle frame of Km⊗In if and only if m≥3 and n≡0(mod2). As a consequence, we solve OP(3a,4b), OP(3a,6b) and OP(5a,10b) with some restrictions on a,b∈N.

On ideal convergent interval valued generalized difference classes defined by Orlicz function

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. A sequence (xk) of real numbers is said to be I - convergent to a real number ℓ, if for each ε>0 the set {k : |xk - ℓ|≥ε} belongs to I . In this paper, using ideal convergence, the difference operator Δm and Orlicz functions, we introduce and examine some generalized difference sequences of interval numbers. We prove completeness properties of these spaces. Further, we investigate some inclusion relations related to these spaces.

Ideally slowly oscillating sequences

An ideal $I$ is a family of subsets of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. In this paper, we introduce the notion of ideally slowly oscillating sequences, which is lying between ideal convergent and ideal quasi-Cauchy sequences, and study on ideally slowly oscillating continuous functions, and ideally slowly oscillating compactness.

ON THE INTEGERS OF THE FORM $p+b$

Let $B$ be a subset of positive integers, and $\mathcal{P}$ the set of all positive primes. For a subset $A$ of positive integers, $A(x)$ denotes the number of integers in $A$ not exceeding $x$. Let $\mathcal{S}$ denote the set of integers of the form $p+b$ with $p\in \mathcal{P}$ and $b\in B$. In this paper, we prove that if $B(x)\gg \log x/\log\log x$ and $B(cx)\gg B(x)$ for some positive constant $c<1$, then $\mathcal{S}(x)\gg x/\log \log x$. This result is best possible in a sense: For any positive integer $m$, we construct an explicit subset $B$ of positive integers with $B(x)\gg (\log x)^m$ and $B(cx)\gg B(x)$ for any positive constant $c<1$ such that $\mathcal{S}(x)\ll x/\log\log x$. We also give an application to the integers of the form $p+2^{a^2}+2^{b^2}$, where $p\in \mathcal{P}$ and $a,b$ are integers. Two open problems are posed for further research.

ON DIFFERENCE IDEAL CONVERGENCE OF DOUBLE SEQUENCES IN RANDOM 2-NORMED SPACES

An ideal I is a family of subsets of positive integers ℕ which is closed under taking finite unions and subsets of its elements. We define and study the notions of Δn-ideal convergence and Δn-ideal Cauchy double sequences in random 2-normed spaces and prove some interesting theorems.

On λ-ideal convergent interval valued difference classes defined by Musielak–Orlicz function

An ideal I is a family of subsets of positive integers \(\mathbb{N}\) which is closed under taking finite unions and subsets of its elements. In this paper, using λ-ideal convergence as a variant of the notion of ideal convergence, the difference operator Δn and Musielak–Orlicz functions, we introduce and examine some generalized difference sequences of interval numbers, where λ=(λm) is a nondecreasing sequence of positive real numbers such that λm+1≤λm+1,λ1=1,λm→∞(m→∞). We prove completeness properties of these spaces. Further, we investigate some inclusion relations related to these spaces.

Approximation complexity of tensor product-type random fields with heavy spectrum

We consider a sequence of Gaussian tensor product-type random fields Open image in new window , where Open image in new window and Open image in new window are all positive eigenvalues and eigenfunctions of the covariance operator of the process X1, Open image in new window are standard Gaussian random variables, and Open image in new window is a subset of positive integers. For each d ∈ ℕ, the sample paths of Xd almost surely belong to L2([0, 1]d) with norm ∥·∥2,d. The tuples Open image in new window, are the eigenpairs of the covariance operator of Xd. We approximate the random fields Xd, d ∈ Open image in new window, by the finite sums Xd(n) corresponding to the n maximal eigenvalues λk, Open image in new window.

Minimal sets of periods for Morse–Smale diffeomorphisms on non-orientable compact surfaces without boundary

We study the minimal set of (Lefschetz) periods of the C 1 Morse–Smale diffeomorphisms on a non-orientable compact surface without boundary inside its class of homology. In fact our study extends to the C 1 diffeomorphisms on these surfaces having finitely many periodic orbits, all of them hyperbolic and with the same action on the homology as the Morse–Smale diffeomorphisms. We mainly have two kinds of results. First, we completely characterize the possible minimal sets of periods for the C 1 Morse–Smale diffeomorphisms on non-orientable compact surface without boundary of genus g with . But the proof of these results provides an algorithm for characterizing the possible minimal sets of periods for the C 1 Morse–Smale diffeomorphisms on non-orientable compact surfaces without boundary of arbitrary genus. Second, we study what kind of subsets of positive integers can be minimal sets of periods of the C 1 Morse–Smale diffeomorphisms on a non-orientable compact surface without boundary.

Wijsman Orlicz Asymptotically Ideal -Statistical Equivalent Sequences

An ideal is a family of subsets of positive integers which is closed under taking finite unions and subsets of its elements. In this paper, we introduce a new definition of asymptotically ideal -statistical equivalent sequence in Wijsman sense and present some definitions which are the natural combination of the definition of asymptotic equivalence, statistical equivalent, -statistical equivalent sequences in Wijsman sense. Finally, we introduce the notion of Cesaro Orlicz asymptotically -equivalent sequences in Wijsman sense and establish their relationship with other classes.

BIPARTITE DIVISOR GRAPH FOR THE PRODUCT OF SUBSETS OF INTEGERS

AbstractThe bipartite divisor graph B(X), for a set Xof positive integers, and some of its properties have recently been studied. We construct the bipartite divisor graph for the product of subsets of positive integers and investigate some of its properties. We also give some applications in group theory.

Romanoff theorem in a sparse set

Let A be any subset of positive integers, and P the set of all positive primes. Two of our results are: (a) the number of positive integers which are less than x and can be represented as 2 k + p (resp. p − 2 k ) with k ∈ A and p ∈ P is more than 0.03A(log x/ log 2)π(x) for all sufficiently large x; (b) the number of positive integers which are less than x and can be represented as 2 q + p with p, q ∈ P is (1 + o(1))π(log x/ log 2)π(x). Four related open problems and one conjecture are posed.

On new spectral multiplicities for ergodic maps

It is shown that each subset of positive integers that contains 2 is re- alizable as the set of essential values of the multiplicity function for the Koopman operator of some weakly mixing transformation.

A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If \(\mathcal F\) is an initially hereditary family of finite subsets of positive integers (i.e., if \(F \in \mathcal F\) and G is initial segment of F then \(G \in \mathcal F\)) and M an infinite subset of positive integers then we define an ordinal index \(\alpha_{M}( \mathcal F )\) . We prove that if \(\mathcal F\) is a family of finite subsets of positive integers such that for every \(F \in \mathcal F\) the characteristic function χF is isolated point of the subspace $$X_{\mathcal F}= \{ \chi_{G}: G \mbox{ is initial segment of $F$ for some } F \in \mathcal F \}$$ of { 0,1 }N with the product topology then \(\alpha_{M}( \overline{\mathcal F} )< \omega_{1}\) for every \(M \subseteq {\rm N}\) infinite, where \(\overline{\mathcal F}\) is the set of all initial segments of the members of \(\mathcal F\) and ω1 is the first uncountable ordinal. As a consequence of this result we prove that \(\mathcal F\) is Ramsey, i.e., if \(\{ {\mathcal P}_{1}, {\mathcal P}_{2} \}\) is a partition of \(\mathcal F\) then there exists an infinite subset M of positive integers such that $$\mathcal F \cap [M]^{< \omega} \subseteq {\mathcal P}_{1} \quad \mbox{or} \quad \mathcal F \cap [M]^{< \omega} \subseteq {\mathcal P}_{2},$$ where [M]< ω is the family of all finite subsets of M.

The characterization of PGL(2,p) for some p by their element orders

For any group G, πe(G) denotes the set of orders of its elements. If Ω is a subset of positive integers, h(Ω) stands for the number of distinct isomorphism classes of finite groups G such that πe(G) = Ω. Let Ω = πe(PGL(2 ,p )), where p is a prime number and 5 ≤ p< 100. In this paper, we prove that h(Ω) ∈{ 1, ∞}.

The Characterization of Almost Simple Groups PGL(2, p ) by Their Element Orders#

For any group G, π e (G) denotes the set of orders of its elements. If Ω is a subset of positive integers, h(Ω) stands for the number of isomorphism classes of finite groups G such that π e (G) = Ω. Let Ω = π e (PGL(2, p)), where p ≥ 5 is a prime number. In this paper, we prove that h(Ω) ∈ {1, ∞ }.

Almost Combinatorial Selector Sets

Classes of subsets of positive integers based on the notion of selector functions similar to the Jockusch selector of a semirecursive set are studied.