• Volume 128, Issue 4

September 2018

• Universal formulas for the number of partitions

In this paper, a formula that generalizes the total number of partitions of a natural number and the number of all possible decompositions of a certain number of parts can be united in the same formula. An advantage of this formula compared to similar ones is that it is given as a finite sum. Another advantage is that this amount may be expressed as a polynomial whose coefficients can be computed explicitly in an elementary form. The most important advantage of this approach is the fact that it is possible to express the results obtained in the general form, which so far in all similarattempts was not the case. From the general form we will prove as follows: (a) Partition functions, can be written with one fractal polynomial. (b) In partition functions, $p (n)$is the first half of its coefficients with the highest degree which are in common with all unified polynomials that form it. (c) The remaining coefficients vary. The first variablecoefficient can have two values; the next coefficient can have up to six values, etc. The variability of coefficients increases as the degree of polynomials decreases up to a free member whose variability is up to LCM(2, 3, . . . , $n$).

• On the diophantine equation $y^{2} = \prod_{i \leq 8}(x + k_{i})$

This paper improves the result of Tengely (Periodica Math. Hung., 72(1) (2016) 23–28).

• Certain Somos’s $P–Q$ type Dedekind $\eta$-function identities

In this paper, we provide a new proof for the Dedekind $\eta$-function identities discovered by Somos. During this process, we found two new Dedekind $\eta$-function identities. Furthermore, we extract interesting partition identities from some of the $\eta$-function identities.

• A note on signs of Fourier coefficients of two cusp forms

Kohnen and Sengupta (Proc. Am. Math. Soc. 137(11) (2009) 3563–3567) showed that if two Hecke eigencusp forms of weight $k_{1}$ and $k_{2}$ respectively, with 1 < $k_{1}$ < $k_{2}$ over $\Gamma_{0}(N)$, have totally real algebraic Fourier coefficients $\{a(n)\}$ and$\{b(n)\}$ respectively for $n \geq 1$ with $a(1) = 1 = b(1)$, then there exists an element $\sigma$ of the absolute Galois group Gal($\mathbb{\bar{Q}/Q}$) such that $a(n)^{\sigma} b(n)^{\sigma}$ < 0 for infinitely many $n$. Later Gun et al. (Arch. Math. (Basel) 105(5) (2015) 413–424) extended their result by showing that if two Hecke eigen cusp forms, with 1 < $k_{1}$ < $k_{2}$, have real Fourier coefficients $\{a(n)\}$ and $\{b(n)\}$ for $n \geq 1$ and $a(1)b(1) \neq 0$, then there exists infinitely many $n$ such that $a(n)b(n)$ > 0 and infinitely many $n$ such that $a(n)b(n)$ < 0. When $k_{1} = k_{2}$, the simultaneous sign changes of Fourier coefficients of two normalized Hecke eigen cusp forms follow from an earlier work of Ram Murty (Math. Ann. 262 (1983) 431–446). In this note, we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup $\Gamma_{0}(N)$ where the coefficients lie in an arithmetic progression. Next, we consider an analogous question for the particular sparse sequences of Fourier coefficients of normalized Hecke eigencusp forms for the full modular group.

• Inhomogeneous diophantine approximation with prime constraints

We study the problem of inhomogeneous diophantine approximation under certain primality restrictions.

• Special properties of Hurwitz series rings

In this paper, we study some properties of the Hurwitz series ring $H R$ (resp. Hurwitz polynomial ring $h R$), such as the flatness or the faithful flatness of $H R/(f)$ (resp. $h R/(f)$), the strongly Hopfian property and the radical property of $H R$ (resp. $h R$). We give some sufficient and necessary conditions for $H R/(f)$ (resp. $h R/(f)$) to be flat or faithful flat. We also prove that the strongly Hopfian property transfer between$R$ and $H R$ (resp. $h R$), and some radicals of $H R$ can be determined in terms of those of $R$, in case $R$ satisfies some additional conditions.

• Semifinite bundles and the Chevalley–Weil formula

In our previous paper (Commun. Algebra, 45(8) (2017) 3422–3448), we studied the category of semifinite bundles on a proper variety defined over a field of characteristic 0. As a result, we obtained the fact that for a smooth projective curve defined over an algebraically closed field of characteristic 0 with genus $g$ > 1, Nori fundamental group acts faithfully on the unipotent fundamental group of its universal covering. However, it was not mentioned about any explicit module structure. In this paper, we prove that the Chevalley–Weil formula gives a description of it.

• Space of invariant bilinear forms

Let $\mathbb{F}$ be a field, $V$ a vector space of dimension $n$ over $\mathbb{F}$. Then the set of bilinear forms on $V$ forms a vector space of dimension $n^{2}$ over $\mathbb{F}$. For char $\mathbb{F} \neq 2$, if $T$ is an invertible linear map from $V$ onto $V$ then the set of $T$ -invariant bilinear forms, forms a subspace of this space of forms. In this paper, we compute the dimension of $T$ -invariant bilinear forms over $\mathbb{F}$. Also we investigate similar type of questions for the infinitesimally $T$ -invariant bilinear forms ($T$ -skew symmetric forms). Moreover, we discuss the existence of nondegenerate invariant (resp. infinitesimally invariant) bilinear forms.

• Augmentation quotients for real representation rings of cyclic groups

Denote by $C_{m}$ the cyclic group of order $m$. Let $\mathcal{R}(C_{m})$ be its real representation ring, and $\Delta(C_{m})$ its augmentation ideal. In this paper, we give an explicit $\mathbb{Z}$-basis for the $n$-th power $\Delta^{n}(C_{m})$ and determine the isomorphism class of the $n$-th augmentation quotient $\Delta^{n}(C_{m})/\Delta^{n+1}(C_{m})$ for each positive integer $n$.

• Characterization of finite $p$-groups by their Schur multiplier

Let $G$ be a finite $p$-group of order $p^{n}$ and $M(G)$ be its Schur multiplier. Itis a well known result by Green that $\mid{M(G)}\mid = p^{\frac{1}{2}n(n−1)−t(G)}$ for some $t(G) \geq 0$. In this article, we classify non-abelian $p$-groups $G$ of order $p^{n}$ for $t(G) = log_{p}(\mid{G}\mid) + 1$.

• Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

In this article, we consider the following fractional Hamiltonian systems:$$_{t}D^{\alpha}_{\infty}(_{−\infty}D^{\alpha}_{t} u) + \lambda L(t)u = \Delta W(t, u), t \in \mathbb{R},$$where $\alpha \in (1/2, 1), \lambda$ > 0 is a parameter, $L \in C (\mathbb{R, R}^{n\times n})$ and $W \in C^{1}(\mathbb{R \times R}^{n}, \mathbb{R})$. Unlike most other papers on this problem, we require that $L(t)$ is a positive semi-definite symmetric matrix for all $t \in \mathbb{R}$, that is, $L(t) \equiv 0$ is allowed to occur in some finite interval $\mathbb{I}$ of $\mathbb{R}$. Under some mild assumptions on $W$, we establish the existence of nontrivial weak solution, which vanish on $\mathbb{R \backslash I}$ as $\lambda \rightarrow \infty$, and converge to $\tilde{u}$ in $H^{\infty}(\mathbb{R})$; here $\tilde{u} \in E^{\alpha}_{0}$ is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval $\mathbb{I}$. Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.

• A constructive approach to the finite wavelet frames over prime fields

In this article, we present a constructive method for computing the frame coefficients of finite wavelet frames over prime fields using tools from computational harmonic analysis and group theory.

• On the convergence of a new iterative algorithm of three infinite families of generalized nonexpansive multi-valued mappings

In this paper,we establish some weak and strong convergence theorems for a new iterative algorithm under some suitable conditions to approximate the common fixed point of three infinite families of multi-valued generalized non expansive mappings in a uniformly convex Banach spaces. Our results generalize and improve several previously known results of the existing literature.

• Proceedings – Mathematical Sciences

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Volume 129 | Issue 4
September 2019