In this paper, we propose some ``measurements" of the ``non-stopping timeness" of ends $\mathcal{G}$ of previsible sets, such that $\mathcal{G}$ avoids stopping times, in an ambiant filtration. We then study several explicit examples, involving last passage times of some remarkable martingales.

Let $X$ be a Banach space, $Y$ a nonempty locally weakly compact closed convex subset of $X$, $(\Omega,\Sigma,\mu)$ a complete positive $\sigma$-finite measure space and $\Sigma_0$ a sub-$\sigma$-algebra of $\Sigma$. This paper gives existence results of best simultaneous approximations to two functions in $L_p(\Omega,\Sigma,X)$ from $L_p(\Omega,\Sigma,Y)/L_p(\Omega,\Sigma_0,Y)$ if $\overline{{\rm span}\, Y}$/and $\overline{{\rm span}\, Y}^*$ has/have the Radon-Nikodym property.

In this paper, we apply the contraction mapping theorem to obtain some bounded and unbounded perturbation theorems concerning local integrated cosine functions. Some growth conditions of perturbations of local integrated cosine functions are also established.

In this paper, we study the second initial boundary value problem for strongly hyperbolic systems in cylinders with Lipschitz base. We investigate the unique solvability of the problem and the smoothness with respect to time of the generanized solution.

We investigate the $L^p$ boundedness for a class of parametric Marcinkiewicz integral operators associated to submanifolds under the $L(\log L)^{\alpha}({S}^{n-1})$ or Block space condition on the kernel functions. Our results improve the recent results by Al-Qassem and Pan in Studia Mathematica.

In this paper, we study the problem of finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the generalized mixed vector equilibrium problem and the solution set of a variational inequality problem with a monotone Lipschitz continuous mapping in Hilbert spaces. We first consider an auxiliary problem for the generalized mixed vector equilibrium problem and prove the existence and uniqueness of the solution for the auxiliary problem. We then introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the generalized mixed vector equilibrium problem and the solution set of a variational inequality problem with a monotone Lipschitz continuous mapping. The results presented in this paper can be considered as a generalization of some known results due to Peng and Yao [16, 17].

This paper is concerned with the solution semigroup of a quasi-linear Lasota equation. We show the existence and uniqueness of a solution semigroup for the quasi-linear Lasota equation. We also find a necessary and sufficient condition for the solution semigroup of the equation to be chaotic.

In this paper, a geodesic $\alpha$-invex subset of a Riemannian manifold is introduced. Geodesic $\alpha$-invex and $\alpha$-preinvex functions on a geodesic $\alpha$-invex set with respect to particular maps are also defined. Further, we study the relationships between geodesic $\alpha$-invex and $\alpha$-preinvex functions on Riemannian manifolds. Some results of a non smooth geodesic $\alpha$-preinvex function are also discussed using proximal subdifferentiation. At the end, mean value inequality and the mean value theorem in $\alpha$-invexity analysis are extended to Cartan-Hadamard manifolds. Our results extend and generalize the known results in the literature.

Our aim in this paper is to deal with the growth properties at infinity for the solutions of the stationary Schrödinger equation in an $n$-dimensional cone. Meanwhile, the geometrical properties of the exceptional sets are also discussed.

Lutwak, Yang and Zhang introduced $L_p$-projection bodies and Ludwig defined general $L_p$-projection bodies. In this paper, a solution to the Shephard problem for general $L_p$-projection bodies is established.

In this paper, using Banach limits, we study attractive points and fixed points of nonlinear mappings in Hilbert spaces. Then we obtain attractive point theorems and fixed point theorems for nonlinear mappings in Hilbert spaces. Using these results, we finally prove a nonlinear ergodic theorem for 2-generalized hybrid mappings in Hilbert spaces.

In a Hilbert space, we study the asymptotic behavior of the subgradient method for solving a variational inequality, under the presence of computational errors. Most results known in the literature establish convergence of optimization algorithms, when computational errors are summable. In the present paper, the convergence of the subgradient method to the solution of a variational inequalities is established for nonsummable computational errors. We show that the the subgradient method generates good approximate solutions, if the sequence of computational errors is bounded from above by a constant.

In this paper, we investigate superstability and the Hyers-Ulam stability of $n$-Jordan $*$-derivations in $C^{*}$-algebras and $JC^{*}$-algebras for the generalized Jensen-type functional equation $$r f(\frac{a+b}{r}) + r f(\frac{a-b}{r}) = 2 f(a).$$

In this paper, we investigate $n$-Jordan $*$-homomorphisms in $C^{*}$-algebras associated with the following functional inequality $$\left\|f\left(\frac{b-a}{3} \right) + f\left( \frac{a-3c}{3} \right) + f\left( \frac{3a+3c-b}{3} \right)\right\|\leq \|f(a)\|.$$ We moreover prove the superstability and the Hyers-Ulam stability of $n$-Jordan $*$-homomorphisms in $C^{*}$-algebras associated with the following functional equation $$f\left( \frac{b-a}{3} \right) + f\left( \frac{a-3c}{3} \right) + f\left(\frac{3a+3c-b}{3} \right) = f(a).$$

Let $G$ be a locally compact group and let $1 \leq p \lt \infty$. We characterize supercyclic weighted translation operators on the Lebesgue space $L^p(G)$ in terms of the weight. Using this result, the characterization for Cesàro hypercyclic weighted translation operators is given. We also determine when scalar multiples of weighted translation operators are hypercyclic and topologically mixing, and show, for any weighted translation operator $T$, $\beta T$ is mixing for all $\beta \in (1,4)$ if $T$ and $4T$ are mixing.

Generally, the coefficients $p(x)$ and $q(x)$ of quadratic pencils of Sturm-Liouville operators are uniquely determined by two spectra or one spectrum and norming constants. In the present paper we show if $% p(x)$ and $q(x)$ are known on half of the domain interval, then one spectrum suffices to determine them uniquely on the other half.

Let $R$ be a prime ring, $Z(R)$ its center, $U$ its right Utumi quotient ring, $C$ its extended centroid, $G$ a non-zero generalized derivation of $R$, $f(x_1,\ldots,x_n)$ a non-zero polynomial over $C$ and $I$ a non-zero right ideal of $R$. If $f(x_1,\ldots,x_n)$ is not central valued on $R$ and $[G(f(r_1,\ldots,r_n)), f(r_1,\ldots,r_n)] \in C$, for all $r_1,\ldots,r_n \in I$, then either there exist $a \in U$, $\alpha \in C$ such that $G(x) = ax$ for all $x \in R$, with $(a-\alpha)I = 0$ or there exists an idempotent element $e \in soc(RC)$ such that $IC = eRC$ and one of the following holds:

1. $f(x_1,\ldots,x_n)$ is central valued in $eRCe$;

2. $char(R) = 2$ and $eRCe$ satisfies the standard identity $s_4$;

3. $char(R) = 2$ and $f(x_1,\dots,x_n)^2$ is central valued in $eRCe$;

4. $f(x_1,\ldots,x_n)^2$ is central valued in $eRCe$ and there exist $a, b \in U$, $\alpha \in C$ such that $G(x) = ax+xb$, for all $x \in R$, with $(a-b+\alpha)I = 0$.

In this paper, we introduce the notions of the almost h-semi-invariant submersion and the h-semi-invariant submersion which may be the extended version of the notion of the semi-invariant submersion [18]. Using them, we obtain some properties. Finally, we give some examples for them.

This paper is concerned with the existence of mild solutions, strong solutions and strict solutions for a class of neutral integrodifferential equations with nonlocal conditions in Banach space. Since the nonlinear terms of the systems involve spacial derivatives, the theory of fractional power and $\alpha$-norm is used to discuss the problem. In the end an example is provided to illustrate the applications of the obtained results.

Almost self-centered graphs were recently introduced as the graphs with exactly two non-central vertices. In this paper we characterize almost self-centered graphs among median graphs and among chordal graphs. In the first case $P_4$ and the graphs obtained from hypercubes by attaching to them a single leaf are the only such graphs. Among chordal graph the variety of almost self-centered graph is much richer, despite the fact that their diameter is at most 3. We also discuss almost self-centered graphs among partial cubes and among $k$-chordal graphs, classes of graphs that generalize median and chordal graphs, respectively. Characterizations of almost self-centered graphs among these two classes seem elusive.

In this paper, we prove the boundedness of all solutions for the equation $x'' + n^2x + \phi(x) + g''(x) q(t) = 0$, where $n \in \mathbb{N}$, $q(t) = q(t+2\pi)$, $\phi(x)$ and $g(x)$ are bounded.