In this paper we present an approximation of viscosity solution of non linear partial differential equation with dirichlet bounded conditions. Our approach used a fully nonlinear PDE in an unbounded domain. To approximate its unique viscosity solution one needs to localize the PDE under consideration and to define artificial boundary conditions. It is known that backward stochastic differential equations (BSDEs) are a useful tool to estimate the error due to misspecified Dirichlet boundary conditions on the artificial boundary [12], but we perfect in this paper their approximation of localization error.

In this study, we consider the Diophantine equation $x^{2}kxy+ky^{2}+ly=0,$ $l\in \left\{ 1,2,4,8\right\} $ and determine the values of $k$ when the equation has infinitely many positive integer solutions $x$ and $y.$

In this paper we prove the interior controllability of the Linear Beam Equation $$ \left\{ \begin{array}{ll} u_{tt}2\beta\Delta u_t + \Delta^2 u= 1_{\omega}u(t,x), & \mbox{in} \quad (0, \tau) \times \Omega,\\ u = \Delta u = 0, & \mbox{on} \quad (0, \tau) \times \partial \Omega, \end{array} \right. $$ where $\beta>1$, $\Omega$ is a sufficiently regular bounded domain in $\mathbb{R}^{N}$ $(N\geq 1)$, $\omega$ is an open nonempty subset of $\Omega$, $1_{\omega}$ denotes the characteristic function of the set $\omega$ and the distributed control $u\in L^{2}([0,\tau]; L^{2}(\Omega)).$ Specifically, we prove the following statement: For all $\tau >0$ the system is approximately controllable on $[0, \tau]$. Moreover, we exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time $\tau >0$.

In this work we suggest a way to estimate some nonlinear terms appearing in the study of semilinear viscoelastic problems. So far we know how to deal with these terms only when the energy is decreasing. In this case we can estimate parts of these nonlinearities by the initial energy. We solve this issue in the general case with the help of a new differential inequality.

In this article we will use ultrafilter theory to present a modified proof that a locally compact group with a countable basis has a left invariant and a right invariant Haar measure. To facilitate this result, we shall first show that the topological space consisting of all ultrafilters on a nonempty set $X$ is homeomorphic to the topological space of all nonzero multiplicative functionals in the first dual space $\ell_{\infty}^{\star}(X)$.

In this paper we investigate derivations satisfying certain differential identities on 3-prime near-rings, and we provide examples to show that the assumed restrictions cannot be relaxed.

In this paper we generalize some affine completeness properties of abelian groups to modules over commutative domains.

In this paper, we aim to study solutions of reflected generalized BSDEs, involving the integral with respect to a continuous process, which is the local time of the diffusion on the boundary. We consider both a finite random terminal and a infinite horizon. In both case, we establish an existence and uniqueness result. As application, we give a characterization of an American pricing option in infinite horizon; and we also give a probabilistic formula for the viscosity solution of an obstacle problem for elliptic PDEs with a nonlinear Neumann boundary condition.

In this paper, we prove the following: assume that either (i) $T^*$ is $w$-hyponormal and $S$ is $w$-hyponormal such that $\ker(T^*)\subset\ker(T)$ and $\ker(S)\subset\ker(S^*)$ or (ii) $T^*$ is $p$-hyponormal or $\log$-hyponormal and $S$ is $w$-hyponormal such that $\ker(S)\subset\ker(S^*)$ or (iii) $T^*$ is an injective $w$-hyponormal and $S$ is a dominant holds. Then the pair $(T,S)$ satisfy Fuglede-Putnam theorem. Also, other related results are given.

Let $f:A\rightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we study the multiplicationlike conditions in $A\bowtie^fJ$.