We consider the Poincaré's classical problem of approximation for second order linear differential equations in the class of almost periodic type functions. We obtain an explicit form for solutions of these equations by studying a Riccati equation associated with the logarithmic derivative of a solution. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions of the Riccati equation. A decomposition property of the perturbations induces a decomposition on the Riccati equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also $p$-almost periodic solutions to the Riccati equation.

In this paper, we consider a class of special subsets of a $BCK$-algebra and investigate some related properties. We describe the intersection and union of two of these subsets in a commutative $BCK$-lattice. We consider the intersection of all special subsets in a $BCK$-algebra and investigate some related properties. Finally, we introduce the connection between the class of special subsets and the set of all congruence classes induced by an ideal in a $BCK$-algebra.

We define weakly regular rings by a condition characterizing the rings $C(X)$ for weak almost $P$-spaces $X$. A Tychonoff space $X$ is called a weak almost $P$-space if for every two zero-sets $E$ and $F$ of $X$ with $\text{int } E\subseteq \text{int } F$, there is a nowhere dense zero-set $H$ of $X$ such that $E\subseteq F\cup H$. We show that a reduced $f$-ring is weakly regular if and only if every prime $z$-ideal in it which contains only zero-divisors is a $d$-ideal. Frames $L$ for which the ring $\mathcal{R}L$ of real-valued continuous functions on $L$ is weakly regular are characterized. We show that if the coproduct of two Lindelöf frames is of this kind, then so is each summand. Also, a continuous Lindelöf frame is of this kind if and only if its Stone-Čech compactification is of this kind.

For $-1 \leq B < A \leq 1$, conditions on $A$, $B$, $a$, $c$ are determined that ensure the confluent hypergeometric function $\Phi(a;c;z)$ satisfies the subordination $\Phi(a; c; z) \prec (1+Az)/ (1+Bz)$. This gives rise to conditions for $(c/a)( \Phi(a; c; z)-1)$ to be close-to-convex, $\Phi(a; c; z)$ to be Janowski convex, and $z\Phi(a; c; z)$ to be Janowski starlike.

We prove that if $\pi:Z \rightarrow X$ is a locally semi-proper morphism between two complex spaces and $X$ is $q$-complete, then $Z$ is $(q+r)$-complete, where $r$ is the dimension of the fiber.

We generalize the construction by Pytlik and Szwarc of uniformly bounded representations for free groups to groups acting on trees. We deduce a new version of the proof (by Alain Valette and the author, 1983) of the fact that locally compact groups acting on trees with amenable stabilizers are amenable in $K$-theory.

In the present paper, using the concept of the Abel convergence method, we give a Korovkin type approximation theorem for a sequence of positive linear operators acting from $L_{p}\left[ a,b\right] $ into itself. We also study some quantitative estimates for $L_{p}$ approximation via Abel convergence.

Given a metric space $(X,d)$, the wobbling group of $X$ is the group of bijections $g:X\rightarrow X$ satisfying $\sup\limits_{x\in X} d(g(x),x)<\infty$. We study algebraic and analytic properties of $W(X)$ in relation with the metric space structure of $X$, such as amenability of the action of the lamplighter group $ \bigoplus_{X} \Z/2\Z \rtimes W(X)$ on $\bigoplus_{X} \Z/2\Z$ and property~(T).

Using Banach algebra structure of the Hardy space, we describe all finite codimensional invariant subspaces of a cyclic convolution operator on the Hardy space $H^p$ of the unit disc for $1 \leq p \leq \infty$. We also observe that every operator in the commutant of such operators is not weakly supercyclic.

A new, simple and unified approach in the theory of contractive mappings was recently given by Samet \emph{et al.} (Nonlinear Anal. 75, 2012, 2154-2165) by using the concepts of $\alpha$-$\psi$-contractive type mappings and $\alpha$-admissible mappings in metric spaces. The purpose of this paper is to present a new class of contractive pair of mappings called generalized $\alpha$-$\psi$ contractive pair of mappings and study various fixed point theorems for such mappings in complete metric spaces. For this, we introduce a new notion of $\alpha$-admissible w.r.t $g$ mapping which in turn generalizes the concept of $g$-monotone mapping recently introduced by Ćirić et al. (Fixed Point Theory Appl. 2008(2008), Article ID 131294, 11 pages). As an application of our main results, we further establish common fixed point theorems for metric spaces endowed with a partial order as well as in respect of cyclic contractive mappings. The presented theorems extend and subsumes various known comparable results from the current literature. Some illustrative examples are provided to demonstrate the main results and to show the genuineness of our results.

In this note we propose a concise proof of David Williams' decomposition of the Bessel Process of dimension 3 (BES(3)), starting from $r>0$ at its ultimate minimum. An ultimate minimum of a stochastic process may be seen as a state of a process at a last hitting time. This discussion is strongly motivated by our interest in properties of last hitting times in general, and here specifically, directly linked with the reading guide of Nikeghbali and Platen (2013).

In this note the integrality and the reducedness for the symmetric algebra of the edge ideals of simple graphs are analyzed. More precisely, we will extend a result on the integrality of the symmetric algebra given for connected graphs, stating a characterization for any simple graph. Moreover, through the Gröbner bases theory, we investigate simple graphs for which the symmetric algebra is reduced, in particular we prove its reducedness for graphs with $n$ vertices consisting of a cycle of length $n$.

The Cesàro operator $C_p : L^p ([0,1]) \to L^p ([0,1]) $, a classical kernel operator, induces the vector measure $m_p : A \mapsto C_p (\big.\chi_A)$ which generates a factorization of $C_p$ through $L^1 (m_p) $ via the integration map $I_{m_p} : f \mapsto \int_0^1 f \; d m_p$, for $f \in L^1 (m_p)$. This provides a technique to investigate various operator theoretic properties of $C_p$. Even though the variation measure $|m_p|$ of $m_p$ is finite it turns out, atypically for a kernel operator, that the restriction of $I_{m_p} $ to $L^1(|m_p|) \subseteq L^1 (m_p)$ is not an extension of $C_p$, that is, $C_p $ fails to factorize through the more traditional space $L^1(|m_p|)$.