• Volume 127, Issue 2

April 2017,   pages  203-392

• Counting rises and levels in $r$-color compositions

An $r$-color composition of a positive integer $n$ is a sequence of positiveintegers, called parts, summing to n in which each part of size $r$ is assigned one of $r$ possible colors. In this paper, we address the problem of counting the $r$-color compositions having a prescribed number of rises. Formulas for the relevant generating functions are computed which count the compositions in question according to a certain statistic. Furthermore, we find explicit formulas for the total number of rises within all of the $r$-color compositions of $n$ having a fixed number of parts. A similar treatment is given for the problem of counting the number of levels and a further generalization in terms of rises of a particular type is discussed.

• Integral pentavalent Cayley graphs on abelian or dihedral groups

A graph is called integral, if all of its eigenvalues are integers. In this paper, we give some results about integral pentavalent Cayley graphs on abelian or dihedral groups.

• An elementary approach to the meromorphic continuation of some classical Dirichlet series

Here we obtain the meromorphic continuation of some classical Dirichlet series by means of elementary and simple translation formulae for these series. We are also able to determine the poles and the residues by this method. The motivation to our work originates from an idea of Ramanujan which he used to derive the meromorphic continuation of the Riemann zeta function.

• $p$-Adic valuation of the Morgan–Voyce sequence and $p$-regularity

We characterize the $p$-adic valuation of the Morgan–Voyce sequence and its companion sequence. Further, we show that the $p$-adic valuation of the Morgan– Voyce sequence is a $p$-regular sequence and we determine its rank explicitly.

• On ($m, n$)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 $\neq$ 0 and $U(R)$ be the set of all unit elements of $R$. Let $m, n$ be positive integers such that $m > n$. In this article, we study a generalization of $n$-absorbing ideals. A proper ideal $I$ of $R$ is called an ($m, n$)-absorbing ideal if whenever $a_1$ · · · $a_m \in I$ for $a_1$, . . . , $a_m \in R$ \ $U(R)$, then there are $n$ of the $a_i$’s whose product is in $I$. We investigate the stability of ($m, n$)-absorbing ideals with respect to various ring theoretic constructions and study ($m, n$)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an ($m, n$)-absorbing ideal if and only if it is an $n$-absorbing ideal, and in an almostDedekind domain every ($m, n$)-absorbing ideal is a product of at most $m − 1$ maximal ideals.

• Nonfiniteness of Hilbert–Kunz functions

Here we answer a question of C. Huneke, by giving an example of a family (parametrized by $\mathbb{A}\frac{1}{\mathbb{F}_p}$) of one-dimensional reduced Cohen–Macaulay rings such thatevery member has the same Hilbert–Kunz multiplicity but there are infinitely many Hilbert–Kunz functions in the family.

• Serre dimension of monoid algebras

Let $R$ be a commutative Noetherian ring of dimension $d$, $M$ a commutative cancellative torsion-free monoid of rank $r$ and $P$ a finitely generated projective $R[M]$-module of rank $t$ . Assume $M$ is $\Phi$-simplicial seminormal. If $M \in C(\Phi)$, then Serre dim $R[M] \leq d$. If $r \leq 3$, then Serre dim $R[int(M)] \leq d$. If $M \subset \mathbb{Z}^2_+$

is a normal monoid of rank 2, then Serre dim $R[M] \leq d$. Assume $M$ is $c$-divisible, $d$ = 1 and $t \geq 3$. Then$P \cong \wedge^{t} P \oplus R[M]^{t−1}$. Assume $R$ is a uni-branched affine algebra over an algebraically closed field and $d$ = 1. Then $P \cong \wedge^{t} P \oplus R[M]^{t−1}$.

• Abelianization of the $F$-divided fundamental group scheme

Let ($X , x_0$) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for ($X , x_0$) produces a homomorphism from the abelianization of the $F$-divided fundamental group scheme of $X$ to the $F$-divided fundamental group of the Albanese variety of $X$. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.

• A non-LEA sofic group

We describe elementary examples of finitely presented sofic groups which are not residually amenable (and thus not initially subamenable or LEA, for short). We ask if an amalgam of two amenable groups over a finite subgroup is residually amenable and answer this positively for some special cases, including countable locally finite groups, residually nilpotent groups and others.

• Permutation representations of the orbits of the automorphism group of a finite module over discrete valuation ring

Consider a discrete valuation ring $R$ whose residue field is finite of cardinality at least 3. For a finite torsion module, we consider transitive subsets $O$ under the action of the automorphism group of the module. We prove that the associated permutation representation on the complex vector space $C[O]$ is multiplicity free. This is achieved by obtaining a complete description of the transitive subsets of $O$ × $O$ under the diagonal action of the automorphism group.

• Analysing the Wu metric on a class of eggs in $\mathbb{C}^{n} – \rm{I}$

We study the Wu metric on convex egg domains of the form $E_{2m}= \{z \in \mathbb{C}^n : \mid z_1\mid ^{2m} + \mid z_2\mid ^2 +\cdots +\mid z_{n−1}\mid^2 + \mid z_n \mid^2$ < $1\}$, where $m \geq 1/2$, $m \neq 1$. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be $C^2$-smooth. Overall however, the Wu metric is shown to be continuous when $m = 1/2$ and even $C^1$-smooth for each $m > 1/2$, and in all cases, a non-K$\ddot{a}$hler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.

• Infinitely many sign-changing solutions of an elliptic problem involving critical Sobolev and Hardy–Sobolev exponent

We study the existence and multiplicity of sign-changing solutions of the following equation \begin{align*} &\left\{ \begin{array}{1} -\delta u=\mu\mid u\mid^{2^\ast-2}u+\frac{\mid u\mid^{2^\ast(t)-2}u}{\mid x\mid^t}+a(x)u\quad {\rm in}\; \Omega ,\\ u=0\quad {\rm on}\; \delta\Omega , \end{array} \right. \end{align*} where $\Omega$ is a bounded domain in $\mathbb R^N$, $0\in \delta\Omega$, all the principal curvatures of $\delta\Omega$ at 0 are negative and $\mu\geq 0$, $a > 0$, $N \geq 7$, 0 < $t$ < 2, $2^\ast = \frac{2N}{N-2}$ and $2^\ast(t)=\frac{2(N-t)}{N-2}$.

• Semigroups of transcendental entire functions and their dynamics

We investigate the dynamics of semigroups of transcendental entire functions using Fatou–Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some condition for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.

• Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

In 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto–Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto–Straus’soriginal result holds for a large class of compact convex sets.

• Eigenvalue estimates for submanifolds with bounded $f$ -mean curvature

In this paper, we obtain an extrinsic low bound to the first non-zero eigenvalue of the $f$ -Laplacian on complete noncompact submanifolds of the weighted Riemannian manifold ($H^{m}(−1), e^{−f} dv$) with respect to the f -mean curvature. In particular, our results generalize those of Cheung and Leung in $\it{Math}. \bf{Z. 236}$ (2001) 525–530.

• Symplectic $S_5$ action on symplectic homotopy K3 surfaces

Let $X$ be a symplectic homotopy K3 surface and $G = S_5$ act on $X$ symplectically. In this paper, we give a weak classification of the $G$ action on $X$ by discussing the fixed-point set structure. Besides, we analyse the exoticness of smoothstructures of $X$ under the action of $G$.

• # Proceedings – Mathematical Sciences

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June 2017