Odd Divisor Research Articles

ON THE GIRTH, INDEPENDENCE NUMBER, AND WIENER INDEX OF COPRIME GRAPH OF DIHEDRAL GROUP

The coprime graph of a finite group , denoted by , is a graph with vertex set such that two distinct vertices and are adjacent if and only if their orders are coprime, i.e., where |x| is the order of x. In this paper, we complete the form of the coprime graph of a dihedral group that was given by previous research and it has been proved that if , for some and if . Moreover, we prove that if is even, then the independence number of is , where is the greatest odd divisor of and if is odd, then the independence number of is . Furthermore, the Wiener index of coprime graph of dihedral group has been stated here.

A bias parity slope on the simplest non-periodic binary words

We analyze a natural parity problem on the divisors associated to Sturmian words. We find a clear bias towards the odd divisors and prove a sharp asymptotic estimate for the average of the difference odd-even function tamed by a weight function, which confirms various experimental results.

Değiştirilmiş tek bölen fonksiyonları ve bölen yaprak modeli

In this research, modelling of the leaves with the help of divisor functions are worked on. Given a positive integer k (1 ≤ k ≤ 100), we investigate solutions of the equation σ(n) = σ(n + 2k) with odd square-free integer n. Further, for a positive integer l and odd prime q, there are no results of the equation σ2i(n) = σ2i(q). As an application, we pose the basic structure of the leaves model for real-time virtual ecosystem construction derived from the equation of shifted odd divisor functions. Also, the elliptic, flabellate and five-lobes leaves’s area and the growth process of the leaves were made modelling with the help of divisor functions.

POWERS OF SUBSETS IN FREE PERIODIC GROUPS

It is proved that for every odd $n \ge 1039$ there are two words $u(x, y), v(x,y)$ of length $\le 658n^2$ over the group alphabet $\{x,y\}$ of the free Burnside group $B(2 ,n),$ which generate a free Burnside subgroup of the group $B(2,n)$. This implies that for any finite subset $S$ of the group $B(m,n)$ the inequality $|S^t|>4\cdot 2.9^{[\frac{t}{658s^2}]}$ holds, where $s$ is the smallest odd divisor of $n$ that satisfies the inequality $s\ge1039$.

A PROOF OF MERCA’S CONJECTURES ON SUMS OF ODD DIVISOR FUNCTIONS

AbstractMerca [‘Congruence identities involving sums of odd divisors function’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca’s work, we complete the proof of these conjectures.

An identity for the sum of inverses of odd divisors of n in terms of the number of representations of n as a sum of squares

Let $$\sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}$$ denote the sum of inverses of odd divisors of a positive integer $n$, and let $c_{r}(n)$ be the number of representations of $n$ as a sum of $r$ squares where representations with different orders and different signs are counted as distinct. The aim is of this note is to prove the following interesting combinatorial identity: $$ \sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}=\frac{1}{2}\,\sum_{r=1}^{n}\frac{(-1)^{n+r}}{r}\,\binom{n}{r}\, c_{r}(n). $$

Second Moment Of Dirichlet L-Functions, Character Sums Over Subgroups And Upper Bounds On Relative Class Numbers

Abstract We prove an asymptotic formula for the mean-square average of L-functions associated with subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\mathcal{A}}(p,d)$ recently introduced by E. Elma, where p ≥ 3 is prime and d ≥ 1 is any odd divisor of p − 1. We obtain an asymptotic formula for ${\mathcal{A}}(p,d),$ which holds true for any odd divisor d of p − 1, thus removing E. Elma’s restrictions on the size of d. This answers a question raised in Elma’s paper. Our proof relies on both estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application, in the range $1\leq d\leq\frac{\log p}{3\log\log p}$, we obtain a significant improvement $h_{p,d}^- \leq 2(\frac{(1+o(1))p}{24})^{m/4}$ over the trivial bound $h_{p,d}^- \ll (\frac{dp}{24} )^{m/4}$ on the relative class numbers of the imaginary number fields of conductor $p\equiv 1\mod{2d}$ and degree $m=(p-1)/d$, where d ≥ 1 is odd.

Linear Recurring Sequences and Explicit Factors of x2nd−1 in 𝔽q[x

Let 𝔽q be a finite field of odd characteristic containing q elements, and n be a positive integer. An important problem in finite field theory is to factorize xn − 1 into the product of irreducible factors over a finite field. Beyond the realm of theoretical needs, the availability of coefficients of irreducible factors over finite fields is also very important for applications. In this paper, we introduce second order linear recurring sequences in 𝔽q and reformulate the explicit factorization of [Formula: see text] over 𝔽q in such a way that the coefficients of its irreducible factors can be determined from these sequences when d is an odd divisor of q + 1.

Exact discrete resonances in the Fermi-Pasta-Ulam–Tsingou system

In systems of N coupled anharmonic oscillators, exact resonant interactions play an important role in the exchange of energy between normal modes. In the weakly nonlinear regime, those interactions may be responsible for the equipartition of energy in Fourier space. Here we consider analytically resonant wave-wave interactions for the celebrated Fermi-Pasta-Ulam-Tsingou (FPUT) system. Using a number-theoretical approach based on cyclotomic polynomials, we show that the problem of finding exact resonances for a system of N particles is equivalent to a Diophantine equation whose solutions depend sensitively on the number of particles N, in particular on the set of divisors of N. We provide an algorithm to construct all possible resonances for N particles, based on two basic methods: pairing-off and cyclotomic, which we introduce and use to build up explicit solutions to the 4-wave, 5-wave and 6-wave resonant conditions. Our results shed some light in the understanding of the long-standing FPUT paradox, regarding the sensitivity of the resonant manifolds with respect to the number of particles N and the corresponding time scale of the interactions leading to an eventual thermalisation. In this light we demonstrate that 6-wave resonances always exist for any N, while 5-wave resonances exist if N is divisible by 3 and N ⩾ 9. It is known that in the discrete case 4-wave resonances do not produce energy mixing across the spectrum, so we investigate whether 5-wave resonances can produce energy mixing across a significant region of the Fourier spectrum by looking at the structure of the interconnected network of Fourier modes that can interact nonlinearly via resonances. We obtain that the answer depends on the set of odd divisors of N which are not divisible by 3: the size of this set determines the number of dynamically independent components, corresponding to independent constants of motion (energies). We show that 6-wave resonances connect all these independent components, providing in principle a restoring mechanism for full-scale thermalisation.

Cycle Related Vertex Odd Divisor Cordial Labeling of Graphs

Cycle Related Vertex Odd Divisor Cordial Labeling of Graphs

Some Star Related Vertex Odd Divisor Cordial Labeling of Graphs

Some Star Related Vertex Odd Divisor Cordial Labeling of Graphs

Further Results on Vertex Odd Divisor Cordial Labeling of Some Graphs

Further Results on Vertex Odd Divisor Cordial Labeling of Some Graphs

Inequalities involving the generating function for the number of partitions into odd parts

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of partitions into odd parts and the generating function for the number of odd divisors.

New MDS Self-Dual Codes From Generalized Reed—Solomon Codes

Both MDS and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining existence of $q$-ary MDS self-dual codes for various lengths has been investigated extensively. The problem is completely solved for the case where $q$ is even. The current paper focuses on the case where $q$ is odd. We construct a few classes of new MDS self-dual code through generalized Reed-Solomon codes. More precisely, we show that for any given even length $n$ we have a $q$-ary MDS code as long as $q\equiv1\bmod{4}$ and $q$ is sufficiently large (say $q\ge 2^n\times n^2)$. Furthermore, we prove that there exists a $q$-ary MDS self-dual code of length $n$ if $q=r^2$ and $n$ satisfies one of the three conditions: (i) $n\le r$ and $n$ is even; (ii) $q$ is odd and $n-1$ is an odd divisor of $q-1$; (iii) $r\equiv3\mod{4}$ and $n=2tr$ for any $t\le (r-1)/2$.

Quantum revivals in free field CFT

The recent work by Cardy (arXiv:1603.08267) on quantum revivals and higher dimensional CFT is revisited and enlarged upon for free fields. The expressions for the free energy used here are those derived some time ago. The calculation is extended to spin–half fields for which the power spectrum involves the odd divisor function. An explanation of the rational revivals for odd spheres is given in terms of wrongly quantised fields and modular transformations. Comments are made on the equivalence of operator counting and eigenvalue methods, which is quickly verified.

English

English

Polygon Numbers Associated with the Sum of Odd Divisors Function

ABSTRACTLet n be a positive integer. We investigate the sequences ((Sm(n))m, which concerns the iteration of the odd divisor function S, given byIn this article, we introduced and studied the following invariants: order, m-gonal shape number, type, convexity, and area of n derived from Sm(n). For m = 3, 4, 5, we classify m-gonal shape odd prime integers. According to some numerical computational evidence in the Appendix (see supplemental data), we state conjectures and questions with respect to the existence problem for m-gonal shape numbers. An inductive algorithm for finding m + 1-gonal shape prime integers when an m-gonal shape integer is given. We give some examples and illustrations, for our conjectures and questions.

Two new infinite families of arc-transitive antipodal distance-regular graphs of diameter three with $$\lambda =\mu $$ λ = μ related to groups $$Sz(q)$$ S z ( q ) and $$^2G_2(q)$$ 2 G 2 ( q )

In this paper, we construct two new infinite families of arc-transitive distance-regular graphs, related to Suzuki groups $$Sz(q)$$Sz(q) and Ree groups $$^2G_2(q)$$2G2(q), where $$q>3$$q>3. They are antipodal $$r$$r-covers of complete graphs on $$q^2+1$$q2+1 or $$q^3+1$$q3+1 vertices, respectively, with $$\lambda =\mu $$?=μ and $$r>1$$r>1 being an arbitrary odd divisor of $$q-1$$q-1. We also find that the graph on the set of involutions of $$Sz(q)$$Sz(q) with $$q>3$$q>3, whose edges are the pairs of involutions $$\{u,v\}$${u,v} such that $$|uv|=5$$|uv|=5, is distance-regular.

Bernoulli Identities and Combinatoric Convolution Sums with Odd Divisor Functions

We study the combinatoric convolution sums involving odd divisor functions, their relations to Bernoulli numbers, and some interesting applications.

(2,3,t)-Generations for the Suzuki's sporadic simple group Suz

A group G is called (2,3,t)-generated if it can be generated by an element x of order 2 and an element y of order 3 such that the product xy has order t. In the present article we determine all the (2,3,t)generations for the Suzuki’s sporadic simple group Suz, where t is an odd divisor of |Suz|. This extends the earlier results of Mehrabadi, Ashrafi and Iranmanesh [9]. Mathematics Subject Classification: Primary 20D08, 20F05