Odd Positive Integer Research Articles

A Reverse Converter and Sign Detectors for an Extended RNS Five-Moduli Set

This paper deals with the extended five moduli set $ \left({{2^{2n+p}, 2^{n}-1, 2^{n}+1, 2^{n}-2^{\frac {n+1}{2}}+1, 2^{n}+2^{\frac {n+1}{2}}+1}}\right)$ where $n$ is a positive odd integer and $p$ is nonnegative integer such that $p\leq \frac {n-5}{2}$ . The paper proposes an efficient residue-to-binary converter along with a converter-based sign detector for this extended set. The paper also presents a residue-to-residue transformer that transforms the same five-moduli set to the three-moduli set $(2^{2n+p}, 2^{2n}-1, 2^{2n}+1)$ . Such a transformer enables the five-moduli set to utilize components that are (or will be) designed for the three-moduli set such as sign detectors.

Unimodality of the independence polynomials of some composite graphs

Let I(G;x) denote the independence polynomial of a graph G. In this paper we study the unimodality properties of I(G; x) for some composite graphs G. Given two graphs G1 and G2, let G1[G2] denote the lexicographic product of G1 and G2. Assume I(G1; x) = P i_0 aixi and I(G2; x) = P i_0 bixi, where I(G2; x) is log-concave. Then we prove (i) if I(G1; x) is logconcave and (a2i ??ai??1ai+1)b21 _ aiai??1b2 for all 1 _ i _ _(G1), then I(G1[G2]; x) is log-concave; (ii) if ai??1 _ b1ai for 1 _ i _ _(G1), then I(G1[G2]; x) is unimodal. In particular, if ai is increasing in i, then I(G1[G2]; x) is unimodal. We also give two su_cient conditions when the independence polynomial of a complete multipartite graph is unimodal or log-concave. Finally, for every odd positive integer _ > 3, we find a connected graph G not a tree, such that _(G) = _, and I(G; x) is symmetric and has only real zeros. This answers a problem of Mandrescu and Miric?a.

A Note on the Particular Set with Size Three

For a fixed integer k , a P k – set is defined as a set of n positive integers { x 1 , x 2 , x 3 ,..., x n } with the property that x i , x j + k is a perfect square, whenever i ≠ j . In this paper, we prove P 11 = {1,14,25}, P −11 = {1,15,36}, and P −11 {4,9,23} sets cannot be extendable. It is also proved that P 11 sets don’t contain any multiple of 3 and P − 11 sets don’t include any multiple of 7. Moreover, it is demonstrated that all of the elements of the sets P − 11 of size three cannot be odd positive integer. Mathematics Subject Classifications: 11A07, 11D45, 11A15.

Cyclic codes of odd length over ℤ4 [u] / 〈u k 〉

Let R=ź4u/źukź=ź4+uź4+ź+ukź1ź4$R=\mathbb{Z}_{4}[u]/ \langle u^k \rangle=\mathbb{Z}_{4}+u \mathbb{Z}_{4}+\ldots+u^{k-1}\mathbb{Z}_{4}$ (uk=0$u^{k}=0$), where k ź 2 is an positive integer. For any odd positive integer n, it is known that cyclic codes of length n over R are identified with ideals of the ring Rx/źxnź1ź$R[x]/\langle x^{n}-1\rangle$. In this paper, an explicit representation for each cyclic code over R of length n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes of length n over R is obtained. Precisely, the dual code of each cyclic code and self-dual cyclic codes of length n over R are investigated. As an application, some good quasi-cyclic codes of length 7k and index k over ź4 are obtained from cyclic codes over R = ź4 [u] / źukź when k = 2, 3, 4.

Lyapunov-type inequality for a higher order dynamic equation on time scales.

The purpose of this work is to establish a Lyapunov-type inequality for the following dynamic equation \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n^\ riangle (t,x(t))+u(t)x^{p}(t)=0$$\\end{document}Sn▵(t,x(t))+u(t)xp(t)=0on some time scale T under the anti-periodic boundary conditions S_k(a,x(a))+S_k(b,x(b))=0, (0le kle n-1), where S_0(t,x(t))=x(t), S_k(t,x(t))=a_k(t)S^triangle _{k-1}(t,x(t)) for 1le kle n-1 and S_n(t,x(t))=a_n(t)[S_{n-1}^Delta (t,x(t))]^p, a_kin C_{rd}({mathbf{T}},(-infty ,0)cup (0,infty )),(1le kle n) with a_n(a)=a_n(b) and uin C_{rd}({mathbf{T}}, {mathbf{R}}), p is the quotient of two odd positive integers and a,bin {mathbf{T}} with a<b.

On the cohomology and their torsion of real toric objects

Abstract In this paper, we do the following two things: (i) We present a formula to compute the rational cohomology ring of a real topological toric manifold, and thus that of a small cover or a real toric manifold, which implies the formula of Suciu and Trevisan. Furthermore, the formula also works for an arbitrary coefficient ring G in which 2 is a unit. (ii) We construct infinitely many real toric manifolds and small covers whose integral cohomology rings have a q-torsion for any positive odd integer q.

A generalization of the Erdös–Surányi problem

Erdös–Surányi and Prielipp suggested to study the following problem: For any integers k>0 and n, are there an integer N and a map ϵ:{1,…,N}→{−1,1} such that (0.1)n=∑j=1Nϵ(j)jk? Mitek and Bleicher independently solved this problem affirmatively.In this paper we consider the case that for some positive odd integer L the numbers ϵ(j) are L-th roots of unity. We show that the answer to the corresponding question is negative if and only if L is a prime power.

Oscillation criteria for third order neutral Emden–Fowler delay dynamic equations on time scales

This paper is concern with a class of third-order neutral Emden–Fowler dynamic equation $$\begin{aligned} (a((rz^\Delta )^\Delta )^\alpha )^\Delta (t)+q(t)x^\alpha (\delta (t))=0, \end{aligned}$$ where \(z(t):=x(t)+p(t)x(\tau (t)), \alpha \) is a quotient of odd positive integers. By generalized Riccati transformation and comparison principles, some new criteria which ensure that every solution is oscillatory are established, which improve and supplement some known results in literatures.

EGYPTIAN FRACTIONS WITH ODD DENOMINATORS

The number of solutions of the diophantine equation $\sum_{i=1}^k \frac{1}{x_i}=1,$ in particular when the $x_i$ are distinct odd positive integers is investigated. The number of solutions $S(k)$ in this case is, for odd $k$: \[\exp \left( \exp \left( c_1\, \frac{k}{\log k}\right)\right) \leq S(k) \leq \exp \left( \exp \left(c_2\, k \right)\right) \] with some positive constants $c_1$ and $c_2$. This improves upon an earlier lower bound of $S(k) \geq \exp \left( (1+o(1))\frac{\log 2}{2} k^2\right)$.

Improved oscillation criteria for second order nonlinear delay difference equations with non-positive neutral term

Abstract In this paper, we present some oscillation criteria for second order nonlinear delay difference equation with non-positive neutral term of the form $$\Delta (a_n (\Delta z_n )^\alpha ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;n \ge n_0 &gt; 0,$$ where zn = xn − pnxn−τ, and α is a ratio of odd positive integers. Examples are provided to illustrate the results. The results obtained in this paper improve and complement to some of the existing results.

Oscillation Results for Second Order Nonlinear Differential Equation with Delay and Advanced Arguments

In this paper we study the oscillation criteria for the second order nonlinear difierential equation with delay and advanced arguments of the form ((x(t) + a(t)x(t i ae1) + b(t)x(t + ae2)) fi ) 00 + q(t)x fl (t i ?1) + p(t)x ∞ (t + ?2) = 0; t ‚ t0 where ae1; ae2; ?1 and ?2 are nonnegative constants and fi; fl and ∞ are the ratios of odd positive integers. Examples are provided to illustrate the main results.

SETS WITH EVEN PARTITION FUNCTIONS AND CYCLOTOMIC NUMBERS

Let $P\in \mathbb{F}_{2}[z]$ be such that $P(0)=1$ and degree $(P)\geq 1$. Nicolas et al. [‘On the parity of additive representation functions’, J. Number Theory73 (1998), 292–317] proved that there exists a unique subset ${\mathcal{A}}={\mathcal{A}}(P)$ of $\mathbb{N}$ such that $\sum _{n\geq 0}p({\mathcal{A}},n)z^{n}\equiv P(z)~\text{mod}\,2$, where $p({\mathcal{A}},n)$ is the number of partitions of $n$ with parts in ${\mathcal{A}}$. Let $m$ be an odd positive integer and let ${\it\chi}({\mathcal{A}},.)$ be the characteristic function of the set ${\mathcal{A}}$. Finding the elements of the set ${\mathcal{A}}$ of the form $2^{k}m$, $k\geq 0$, is closely related to the $2$-adic integer $S({\mathcal{A}},m)={\it\chi}({\mathcal{A}},m)+2{\it\chi}({\mathcal{A}},2m)+4{\it\chi}({\mathcal{A}},4m)+\cdots =\sum _{k=0}^{\infty }2^{k}{\it\chi}({\mathcal{A}},2^{k}m)$, which has been shown to be an algebraic number. Let $G_{m}$ be the minimal polynomial of $S({\mathcal{A}},m)$. In precedent works there were treated the case $P$ irreducible of odd prime order $p$. In this setting, taking $p=1+ef$, where $f$ is the order of $2$ modulo $p$, explicit determinations of the coefficients of $G_{m}$ have been made for $e=2$ and 3. In this paper, we treat the case $e=4$ and use the cyclotomic numbers to make explicit $G_{m}$.

Some Nonexistence and Asymptotic Existence Results for Weighing Matrices

Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking, and communication. In this paper, we first show that if positive integer k cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order 4n and weight k, where n is an odd positive integer. Then we show that, for any square k, there is an integer N(k) such that, for each n≥N(k), there is a symmetric weighing matrix of order n and weight k. Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita, and Seberry.

On pretzel knots and Conjecture ℤ

Conjecture [Formula: see text] is a knot theoretical equivalent form of the Kervaire conjecture. We show that Conjecture [Formula: see text] is true for all the pretzel knots of the form [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are odd positive integers.

A randomized version of the Collatz [formula omitted] problem

We consider a Markov chain on the positive odd integers, which can be viewed as a stochastic version of the Collatz 3x+1 Problem. We show that, no matter its initial value, the chain visits 1 infinitely often. Its values, however, are unbounded.

On pairs of equations in one prime, two prime squares and powers of 2

It was proved that, for k=332, every pair of large positive odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two prime squares and k powers of 2. In this paper, we sharpen the value of k to 128.

Sums of generalized harmonic series

We examine the remarkable connection, first discovered by Beukers, Kolk and Calabi, between ζ(2n), the value of the Riemann Zeta-function at an even positive integer, and the volume of some 2n-dimensional polytope. It can be shown that this volume is equal to the trace of a compact self-adjoint operator. We provide an explicit expression for the kernel of this operator in terms of Euler polynomials. This explicit expression makes it easy to calculate the volume of the polytope and hence ζ(2n). In the case of odd positive integers, we rediscover an integral representation for ζ(2n + 1), obtained by a different method by Cvijovic and Klinowski. Finally, we indicate that the origin of the miraculous Beukers–Kolk–Calabi change of variables in the multidimensional integral, which is at the heart of this circle of ideas, can be traced to the amoeba associated with the certain Laurent polynomial. The article is dedicated to the memory of Vladimir Arnold (1937–2010).

Rational period functions in higher level cases

Extending Knopp's results [9,10] we investigate examples and properties of rational period functions in higher level cases, including location of poles and behavior under the action of Hecke operators. More precisely, we prove that a rational period function may have poles only at 0 or at real quadratic irrationalities. Moreover by applying the action of Hecke operators we prove that for positive odd integer k and p∈{2,3}, the space of all rational period functions of weight 2k for Γ0+(p) is infinite dimensional.

Global finite-time stabilization of a class of switched nonlinear systems with the powers of positive odd rational numbers

In this paper, we will present new results on global finite-time stabilization for a class of switched strict-feedback nonlinear systems, whose subsystems have chained integrators with the powers of positive odd rational numbers (i.e., numerators and denominators of the powers are all positive odd integers but not necessarily relatively prime). All the powers in each equation of subsystems of the switched systems can be different. Based on the technique of adding a power integrator, the global finite-time stabilizers of individual subsystems are first systematically constructed to guarantee global finite-time stability of the closed-loop smooth switched system under arbitrary switchings, and then a co-design of stabilizers and a state-dependent switching law is proposed to achieve global finite-time stabilization of the closed-loop non-smooth switched systems. In the controller design, a common coordinate transformation of all subsystems is exploited to avoid using individual coordinate transformations for individual subsystems. We also give some sufficient conditions that enable our design by characterizing the powers of the chained integrators of the considered switched systems. Numerical examples are provided to demonstrate the effectiveness of the proposed results.

Perfect power Riesel numbers

A Riesel number k is an odd positive integer such that k⋅2n−1 is composite for all integers n≥1. In 2003, Chen proved that there are infinitely many Riesel numbers of the form kr, when r≢0,4,6,8(mod12), and he conjectured that such Riesel powers exist for all positive integers r. In 2008, Filaseta, Finch and Kozek extended Chen's theorem slightly by constructing Riesel numbers of the form k4 and k6. In 2009, Wu and Sun provided more evidence to support Chen's conjecture by showing that there exist infinitely many Riesel numbers of the form kr for any positive integer r that is coprime to 15015. In this article, we extend the results of Wu and Sun by proving that there exist infinitely many Riesel numbers of the form kr for any positive integer r that is coprime to 105.