scholarly journals On a Double Series of Chan and Ong

2006 ◽  
Vol 13 (4) ◽  
pp. 793-805
Author(s):  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Explicit Formula ◽  
Double Series

Abstract An arithmetic identity is used to prove a relation satisfied by the double series . As an application an explicit formula is given for the number of representations of the positive integer 𝑛 by the form .

On the Structure of G(H, k)

2014 ◽  
Vol 21 (02) ◽  
pp. 317-330 ◽  
Author(s):  
Guixin Deng ◽  
Pingzhi Yuan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Explicit Formula ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Directed Edge ◽  
Necessary And Sufficient

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k2). We also discuss the problem when G(H1, k) is isomorphic to G(H2, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd (p, k)=1.

scholarly journals On the number of unit solutions of cubic congruence modulo $ n $

2021 ◽  
Vol 6 (12) ◽  
pp. 13515-13524
Author(s):  
Junyong Zhao ◽  
◽  
Keyword(s):  
Positive Integer ◽  
Explicit Formula ◽  
Linear Congruence ◽  
Residue Classes ◽  
Congruence Modulo ◽  
Quadratic Case

<abstract><p>For any positive integer $ n $, let $ \mathbb Z_n: = \mathbb Z/n\mathbb Z = \{0, \ldots, n-1\} $ be the ring of residue classes module $ n $, and let $ \mathbb{Z}_n^{\times}: = \{x\in \mathbb Z_n|\gcd(x, n) = 1\} $. In 1926, for any fixed $ c\in\mathbb Z_n $, A. Brauer studied the linear congruence $ x_1+\cdots+x_m\equiv c\pmod n $ with $ x_1, \ldots, x_m\in\mathbb{Z}_n^{\times} $ and gave a formula of its number of incongruent solutions. Recently, Taki Eldin extended A. Brauer's result to the quadratic case. In this paper, for any positive integer $ n $, we give an explicit formula for the number of incongruent solutions of the following cubic congruence</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ x_1^3+\cdots +x_m^3\equiv 0\pmod n\ \ \ {\rm with} \ x_1, \ldots, x_m \in \mathbb{Z}_n^{\times}. $\end{document} </tex-math></disp-formula></p> </abstract>

On the distribution of zeros of derivatives of the Riemann ξ-function

2020 ◽  
Vol 32 (1) ◽  
pp. 1-22
Author(s):  
Amita Malik ◽  
Arindam Roy
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Real Number ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Density Estimate ◽  
Distribution Of Zeros ◽  
Riemann Zeta ◽  
Derivatives Of

AbstractFor the completed Riemann zeta function {\xi(s)}, it is known that the Riemann hypothesis for {\xi(s)} implies the Riemann hypothesis for {\xi^{(m)}(s)}, where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence {(\alpha\gamma_{m})}, where α is any fixed non-zero real number and {\gamma_{m}} runs over the imaginary parts of the zeros of {\xi^{(m)}(s)}. We also obtain a zero density estimate and an explicit formula for the zeros of {\xi^{(m)}(s)}. In particular, all our results hold uniformly for {0\leq m\leq g(T)}, where the function {g(T)} tends to infinity with T and {g(T)=o(\log\log T)}.

Impulse parameter and a new equivalence between 123- and 132-avoiding permutations

2018 ◽  
Vol 10 (04) ◽  
pp. 1850054
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Positive Integer ◽  
Explicit Formula ◽  
General Class ◽  
Functional Equations ◽  
Kernel Method ◽  
Index Formula ◽  
Combinatorial Proof ◽  
Horizontal Line ◽  
Fixed Positive Integer

In this paper, we enumerate permutations [Formula: see text] according to the number of indices [Formula: see text] such that [Formula: see text], where [Formula: see text] and [Formula: see text] is a fixed positive integer. We term such an index [Formula: see text] an [Formula: see text]-impulse since it marks an occurrence where the bargraph representation of [Formula: see text] rises above (or to the same level as) the horizontal line [Formula: see text]. We find an explicit formula for the distribution as well as a formula for the total number of [Formula: see text]-impulses in all permutations of [Formula: see text]. Comparable distributions are also found for the [Formula: see text]-avoiding permutations of [Formula: see text], where [Formula: see text] is a pattern of length three. Two markedly different distributions emerge, one for [Formula: see text] and another for the remaining patterns in [Formula: see text]. In particular, we obtain a new equidistribution result between 123- and 132-avoiding permutations. To prove our results, we make use of multiple arrays and systems of functional equations, employing the kernel method to solve the system in the case [Formula: see text]. We also provide a combinatorial proof of the aforementioned equidistribution result, which actually applies to a more general class of multi-set permutations.

scholarly journals A Note on Colored Tornheim's Double Series

Integers ◽  
2010 ◽  
Vol 10 (6) ◽  
Author(s):  
Jianqiang Zhao
Keyword(s):  
Explicit Formula ◽  
Short Note ◽  
Double Series ◽  
Numerical Data ◽  
Roots Of Unity ◽  
Double Zeta ◽  
Zeta Values ◽  
Almost All

AbstractIn this short note, we provide an explicit formula to compute every colored double Tornheim's series by using double polylogarithm values at roots of unity. When the colors are given by ±1 our formula is different from that of Tsumura [Proc. AMS 131: 3633–3641, 2003] even though numerical data confirm both are correct in almost all the cases. This agreement can also be checked rigorously by using regularized double shuffle relations of the alternating double zeta values in weights less than eight.

ON THE NUMBER OF SOLUTIONS TO CERTAIN DIAGONAL EQUATIONS OVER FINITE FIELDS

2010 ◽  
Vol 06 (01) ◽  
pp. 1-14 ◽  
Author(s):  
IOULIA BAOULINA
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Explicit Formula ◽  
Finite Fields ◽  
Characteristic P ◽  
Explicit Formulas ◽  
Number Of Solutions

We consider a diagonal equation, which can be reduced to the form [Formula: see text] over a finite field of characteristic p > 2. In 1997, Sun obtained the explicit formula for the number of solutions to an equation of this type when n is even. In this paper, we find explicit formulas for the number of solutions when n is odd, k = 2rh, and there exists a positive integer ℓ such that pℓ ≡ 2m-1h + 1 ( mod 2mh), m = 3 or 4, r ≥ m, h = 1 or 3.

scholarly journals Connectivity and some other properties of generalized Sierpiński graphs

2018 ◽  
Vol 12 (2) ◽  
pp. 401-412
Author(s):  
Sandi Klavzar ◽  
Sara Zemljic
Keyword(s):  
Positive Integer ◽  
Explicit Formula ◽  
Complete Graph ◽  
Building Block ◽  
Connected Components ◽  
Edge Connectivity ◽  
Sierpiński Graphs

If G is a graph and n a positive integer, then the generalized Sierpi?ski graph SnG is a fractal-like graph that uses G as a building block. The construction of SnG generalizes the classical Sierpi?ski graphs Sn p, where the role of G is played by the complete graph Kp. An explicit formula for the number of connected components in SnG is given and it is proved that the (edge-)connectivity of SnG equals the (edge-)connectivity of G. It is demonstrated that SnG contains a 1-factor if and only if G contains a 1-factor. Hamiltonicity of generalized Sierpi?ski graphs is also discussed.

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

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