We prove the following sub- and super-additive properties of the psi function. \begin{enumerate} \item[(i)] The inequality \[ \psi\bigl( (x+y)^{\alpha} \bigr) \leq \psi(x^{\alpha})+\psi(y^{\alpha}) \quad{(\alpha \in\mathbf{R})} \] holds for all $x,y>0$ if and only if $\alpha\leq \alpha_0=-1.0266\ldots$\,. Here, $\alpha_0$ is given by \[ 2^{\alpha_0}=\inf_{t>0} \frac{\psi^{-1}\bigl( 2\psi(t) \bigr)}{t}=0.4908\ldots\, . \] \item[(ii)] The inequality \[ \psi(x^{\beta})+\psi(y^{\beta}) \leq \psi\bigl( (x+y)^{\beta} \bigr) \quad(\beta \in\mathbf{R}) \] is valid for all $x,y>0$ if and only if $\beta=0$. \end{enumerate}

We develop a new route through which to explore $\ker \Psi_{X}$, the kernel of the $\pi_{1}$-shape group homomorphism determined by a general space $X$, and establish, for each locally path connected, paracompact Hausdorff space $% X$, $\ker\Psi_{X}$ is precisely the Spanier group of $X$.

We study the structure of arbitrary split involutive Lie algebras. We show that any of such algebras $L$ is of the form $L={\mathcal U} +\sum_{j}I_{j}$ with ${\mathcal U}$ a subspace of the involutive abelian Lie subalgebra $H$ and any $I_{j}$ a well described involutive ideal of $L$ satisfying $[I_j,I_k]=0$ if $j\neq k$. Under certain conditions, the simplicity of $L$ is characterized and it is shown that $L$ is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive Lie algebra.

In 1956 Yu.I.~Manin published an elementary proof of Helmut Hasse's 1933 result stating that the Riemann hypothesis holds in the case of an elliptic function field over a finite field. We briefly explain how Manin's proof relates to more modern proofs of the same result. This enables us to present an analogous elementary proof for the case of finite fields of characteristic two, which was excluded in the original argument.

For each nonzero cardinal number $\kappa$, there is a field extension $\mathbf{Q} \subseteq L$ and a maximal chain $\mathcal{H}_{\kappa}$ of intermediate fields going from $\mathbf{Q}$ to $L$ such that the cardinal number of $\mathcal{H}_{\kappa}$ is $\kappa$. If $\kappa$ is infinite, then for all infinite cardinals $\nu%_1 \lt \nu_2 \leq \lt \kappa$, it can be arranged that $\mathcal{H}_{\nu%_1 } \subset %\mathcal{H}_{\nu_2} \subseteq \mathcal{H}_{\kappa}$. However, there exists an infinite cardinal for which there does not exist a field extension $L/K$ such that $\kappa$ is the supremum of the cardinalities of chains of intermediate fields going from $K$ to $L$. For a field extension $L/K$, this supremum of cardinalities has been denoted $\lambda(L/K)$. If $L/K$ is infinitely generated, we reduce its calculation to set theory, as follows. Let $\ala$ be the infimum of the cardinalities of generating sets of $L/K$. Let $\Omega(\aleph_{\alpha})$ be the supremum of the cardinalities of chains %that consist of subsets of a set of cardinality $\ala$. ($\Omega(\aleph_{\alpha})$ is equal to what has been called $\mbox{ded\,}(\aleph_{\alpha})$ in the literature.) Then $\ala\lt \oa\leq \p$; and if $\alpha > 0$ (but not necessarily if $\alpha =0$), then $\lambda(L/K)=\Omega(\aleph_{\alpha})$.

This paper will obtain an asymptotic formula of the finite-time ruin probability in a generalized risk model with constant interest force, in which the claim sizes are pairwise quasi-asymptotically independent and their arrival process is an arbitrary counting process. In particular, when the claim inter-arrival times follow a certain dependence structure, the result obtained can also include an asymptotic formula for the infinite-time ruin probability.

This is a survey of some of the results which were obtained in the last 12 years on the non-linear geometry of Banach spaces. We focus on the contribution of the late Nigel Kalton.

We prove some versions of the Schwarz lemma for real harmonic functions in certain Riemann surfaces with positive Gauss curvature.

In this paper we generalize the complete convergence for $\rho^*$-mixing random fields given by Kuczmaszewska et al.~\cite{kuclag} to the case of weight sums.

The harmonic index of a graph $G$ is defined as the sum of weights $\frac{2}{d(v_i)+d(v_j)}$ of all edges $v_iv_j$ of $G$, where $d(v_i)$ denotes the degree of the vertex $v_i$ in $G$. In this paper, we study how the harmonic index behaves when the graph is under perturbations. These results are used to provide a simpler method for determining the unicyclic graphs with maximum and minimum harmonic index among all unicyclic graphs, respectively. Moreover, a lower bound for harmonic index is also obtained.

Due to Alexander, every closed oriented $3$-manifold has an open book decomposition. In this note, we review two proofs of this theorem, one which constructs a planar open book decomposition. We then present an algorithmic way for determining the monodromy of this planar open book decomposition.

In this paper we continue our study of property (RD) for Hecke pairs initiated in \cite{s1}. We study the permanence of property (RD) under different commensurability relations of subgroups in Hecke pairs. As an application, we prove that if $H$ is a normal subgroup of a group $G$ and $K$ is a subgroup of $G$ commensurable to $H$, then the Hecke pair $(G,K)$ has (RD) if and only if the quotient group $G/H$ has (RD). This is used to investigate an infinite number of non-elementary examples of Hecke pairs with property (RD). In particular, we introduce a class of groups all of whose subgroups are almost normal and all such Hecke pairs having property (RD). We also discuss property (RD) of certain Hecke pairs arising from group extensions.

The main objective of this paper is to discuss starlikeness of order $\beta$ of the solutions of a differential equation and, as a consequence, to obtain conditions on the kernel function $g$ such that the function defined by \[ f(z)=\int_0^1 \int_0^1 g(r,s,z)\,dr\,ds \] is a starlike function of the same order.

In this paper the authors investigate a power mean inequality for a special function which is defined by the complete elliptic integrals.

In this paper, given $0\lt p\lt \infty$, we define a logarithmic Hardy-Bloch type space \begin{multline*} BH_{p,L}=\left\{f(z)\in H(D):||f||_{p,L}\right.\\ \left.=\sup_{z\in D}(1-|z|)\log\frac{e}{1-|z|} M_p(|z|,f')\lt \infty\right\}. \end{multline*} Then we mainly study the relation between $BH_{p,L}$ and three classical spaces: Hardy space, Dirichlet type space and VMOA. We also obtain some estimates on the growth of $f\in BH_{p,L}$.

In this paper, we establish 4- and 8-dissections for some infinite products. In particular, we generalize Hirschhorn's formulas for 8-dissections of a continued fraction of Gordon and its reciprocal. Our results also imply a theorem on the periodicity of signs of the coefficients of an infinite product given by Chan and Yesilyurt.

The purpose of this paper is to derive several new transformation formulas between bilateral and unilateral basic hypergeometric series. From these transformations, some $q$-series identities are obtained.

This paper continues our study on the existence and multiplicity of periodic solutions to delay differential equations of the form \[ \dot{z}(t)=-f(z(t-1))-f(z(t-2))-\cdots -f(z(t- n+1)), \] where $z\in\br^N$, $f\in C(\br^N, \br^N)$ and $n>1$ is an odd number. By using the Galerkin approximation method and the $S^1$-index theory in the critical point theory, some known results for Kaplan-Yorke type differential delay equations are generalized to the higher-dimensional case. As a result, the Kaplan-Yorke conjecture is proved to be true in the case of higher-dimensional systems.