• Volume 124, Issue 4

November 2014,   pages  471-612

• Irrational factor races

We investigate the behavior of the sum of the irrational factor function over arithmetic progressions. We first establish a general asymptotic formula for such a sum, and then obtain some further results in the case of arithmetic progressions $3n\pm 1$.

• Lenstra theorem in number fields

In this paper, we present a number field version of the celebrated result of Lenstra (Math. Comp. 42(165) (1984) 331–340) in 1984. Also, this result allows us to improve a result of Wikstrőm (On the 𝑙-ary GCD-algorithm in rings of integers (2005) pp. 1189–1201).

• Meromorphic connections on vector bundles over curves

We give a criterion for filtered vector bundles over curves to admit a filtration preserving meromorphic connection that induces a given meromorphic connection on the corresponding graded bundle.

• An Engel condition with an additive mapping in semiprime rings

The main purpose of this paper is to prove the following result: Let $n \gt 1$ be a fixed integer, let 𝑅 be a $n!$-torsion free semiprime ring, and let $f : R \to R$ be an additive mapping satisfying the relation $[f (x), x]_{n} = [[... [[f(x),x],x],...], x] = 0$ for all $x \in R$. In this case $[f(x), x] = 0$ is fulfilled for all $x \in R$. Since any semisimple Banach algebra (for example, $C^{\ast}$ algebra) is semiprime, this purely algebraic result might be of some interest from functional analysis point of view.

• Finite groups all of whose minimal subgroups are $NE^{\ast}$-subgroups

Let 𝐺 be a finite group. A subgroup 𝐻 of 𝐺 is called an $NE$-subgroup of 𝐺 if it satisfies $H^G \cap N_{G}(H) = H$. A subgroup 𝐻 of 𝐺 is said to be a $NE^{\ast}$-subgroup of 𝐺 if there exists a subnormal subgroup 𝑇 of 𝐺 such that $G = HT$ and $H \cap T$ is a $NE$-subgroup of 𝐺. In this article, we investigate the structure of 𝐺 under the assumption that subgroups of prime order are $NE^{\ast}$-subgroups of 𝐺. The finite groups, all of whose minimal subgroups of the generalized Fitting subgroup are $NE^{\ast}$-subgroups are classified.

• A complete classification of minimal non-$PS$-groups

Let 𝐺 be a finite group. A subgroup 𝐻 of 𝐺 is called 𝑠-permutable in 𝐺 if it permutes with every Sylow subgroup of 𝐺, and 𝐺 is called a $PS$-group if all minimal subgroups and cyclic subgroups with order 4 of 𝐺 are 𝑠-permutable in 𝐺. In this paper, we give a complete classification of finite groups which are not $PS$-groups but their proper subgroups are all $PS$-groups.

• On 𝐴-nilpotent abelian groups

Let 𝐺 be a group and $A = \text{Aut}(G)$ be the group of automorphisms of 𝐺. Then, the element $[g, \alpha] = g^{-1}\alpha(g)$ is an autocommutator of $g \in G$ and $\alpha \in A$. Hence, for any natural number 𝑚 the 𝑚-th autocommutator subgroup of 𝐺 is defined as

$K_{m}(G)=\langle [g,\alpha_{1},\ldots,\alpha_{m}]|g\in G,\alpha_{1},\ldots,\alpha_{m}\in A\rangle$,

where $[g, \alpha_{1}, \alpha_{2},\ldots, \alpha_{m}] = [[g,\alpha_{1},\ldots,\alpha_{m−1}], \alpha_{m}]$. In this paper, we introduce the new notion of 𝐴-nilpotent groups and classify all abelian groups which are 𝐴-nilpotent groups.

• A double inequality for bounding Toader mean by the centroidal mean

In this paper, the authors find the best numbers 𝛼 and 𝛽 such that $\overline{C}(\alpha a+ (1 - \alpha)b$, $\alpha b + (1 - \alpha)a) \lt T (a, b) \lt \overline{C} (\beta a + (1 - \beta)b, \beta b + (1 - \beta)a)$ for all 𝑎, $b \gt 0$ with $a \neq b$, where $\overline{C}(a, b) = \frac{2(a^{2}+ab+b^{2})}{3(a+b)}$ and $T(a, b) = \frac{2}{\pi} \int^{\pi/2}_{0}\sqrt{a^{2}\cos^{2}\theta + b^{2}\sin^{2} \theta} {\rm d}\theta$ denote respectively the centroidal mean and Toader mean of two positive numbers 𝑎 and 𝑏.

• Zeros and uniqueness of 𝑄-difference polynomials of meromorphic functions with zero order

In this paper, we investigate the value distribution of 𝑞-difference polynomials of meromorphic function of finite logarithmic order, and study the zero distribution of difference-differential polynomials $[f^{n}(z)f (qz + c)]^{(k)}$ and $[f^{n}(z)(f (qz + c) - f (z))]^{(k)}$, where $f(z)$ is a transcendental function of zero order. The uniqueness problem of difference-differential polynomials is also considered.

• Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps

We give an asymptotic formula of the Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps $f_{\lambda}(z) = z^{d} + \lambda/z^{d}$, where $d \geq 3$ and 𝜆 is small.

• On two functional equations originating from number theory

Reducing the functional equations introduced in Proc. Indian Acad. Sci. (Math. Sci.) 113(2) (2003) 91–98 and in Appl. Math. Lett. 21 (2008) 974–977 to equations in complex variables and quaternions, we find general solutions of the equations. We also obtain the stability of the equations.

• A sharp Rogers–Shephard type inequality for Orlicz-difference body of planar convex bodies

In this paper, we prove a sharp Rogers–Shephard type inequality for the Orlicz-difference body of planar convex bodies, which extend the works of Bianchini and Colesanti (Proc. Amer. Math. Soc. 138(7) (2008) 2575–2582).

• Nash equilibria via duality and homological selection

Given a multifunction from 𝑋 to the 𝑘-fold symmetric product Sym$_{k}(X)$, we use the Dold–Thom theorem to establish a homological selection theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash equilibria are traditionally multilinear in the mixed strategies. The main aim of this paper is to relax the hypothesis of multilinearity. We use basic intersection theory, Poincaré duality in addition to the Dold–Thom theorem.

• Limit distributions of random walks on stochastic matrices

Problems similar to Ann. Prob. 22 (1994) 424–430 and J. Appl. Prob. 23 (1986) 1019–1024 are considered here. The limit distribution of the sequence $X_{n}X_{n−1}\ldots X_{1}$, where $(X_{n})_{n\geq 1}$ is a sequence of i.i.d. $2 \times 2$ stochastic matrices with each $X_{n}$ distributed as 𝜇, is identified here in a number of discrete situations. A general method is presented and it covers the cases when the random components $C_{n}$ and $D_{n}$ (not necessarily independent), $(C_{n}, D_{n})$ being the first column of $X_{n}$, have the same (or different) Bernoulli distributions. Thus $(C_{n}, D_{n})$ is valued in $\{0, r\}^{2}$, where 𝑟 is a positive real number. If for a given positive real 𝑟, with $0 \lt r \leq \frac{1}{2}$, $r^{-1}C_{n}$ and $r^{-1}D_{n}$ are each Bernoulli with parameters $p_{1}$ and $p_{2}$ respectively, $0 &lt; p_{1}$, $p_{2} \lt 1$ (which means $C_{n}\sim p_{1}\delta_{\{r\}} + (1 - p_{1})\delta_{\{0\}}$ and $D_{n} \sim p_{2}\delta_{\{r\}} + (1 - p_{2})\delta_{\{0\}}$), then it is well known that the weak limit 𝜆 of the sequence $\mu^{n}$ exists whose support is contained in the set of all $2 \times 2$ rank one stochastic matrices. We show that $S(\lambda)$, the support of 𝜆, consists of the end points of a countable number of disjoint open intervals and we have calculated the 𝜆-measure of each such point. To the best of our knowledge, these results are new.