This paper solves the Riemann Problem for a nonstrictly hyperbolic system of conservation laws arising in geometrical optics, in which the initial data containing Dirac delta functions belongs to the space of Borel measures. Under the generalized Rankine-Hugoniot relations and entropy condition, by analyzing the interaction among delta shock wave, contact discontinuity, and vacuum, three kinds of Riemann solutions with different geometric structures are obtained, in which a double-contact delta shock wave travels at constant speed in a vacuum state, and the delta-shock, vacuum and contact discontinuity appear in a solution structure.

Let $R$ be a prime ring of char (R)$\neq 2$ with Utumi quotient ring $U$ and $C=Z(U)$ be the extended centroid of $R$. Suppose that $0\neq p\in R$, $f(x_1,\ldots,x_n)$ is a noncentral multilinear polynomial over $C$ and $F$, $G$ are two nonzero generalized derivations of $R$. If

$$ p\Big(F\big(f(X)\big)f(X)-f(X)G^2\big(f(X)\big)\Big)=0$$

for all $X=(x_1,\ldots,x_n)\in R^n$, then one of the following statements holds:

(i) there exist $\lambda \in C$ and $a\in U$ such that $F(x)=\lambda x$ and $G(x)=xa$ for all $x\in R$, with $\lambda=a^2$;

(ii) there exists $b\in U$ such that $F(x)=xb^2$ and $G(x)=bx$ for all $x\in R$;

(iii) there exist $a,b\in U$ such that $F(x)=ax$ and $G(x)=xb$ for all $x\in R$, with $b^2\in C$ and $p(a-b^2)=0$;

(iv) there exist $a,b\in U$ such that $F(x)=ax$ and $G(x)=xb$ for all $x\in R$, with $f(R)^2\in C$ and $p(a-b^2)=0$;

(v) there exist $a,b,c\in U$ such that $F(x)=ax+xb$ and $G(x)=cx$ for all $x\in R$, with $b-c^2\in C$ and $p(a+b-c^2)=0$.

 BL-algebras were introduced by H\'ajeck. He presented BL-algebras as algebraic structures of type (2,2,2,2,0,0). In this paper, we present the class of BL-algebras as a variety of universal algebras of type (2,2,0,0). We also present a simple description of ideals of a finite BL-algebra and show that the set of all ideals of a finite BL-algebra is a Boolean algebra.

 In this paper, we proposed a new regularized preconditioner for solving generalized saddle point systems from Navier-Stokes equations. This preconditioner is established by transforming the (1,2)-block and (2,2)-block of the original coefficient matrix. We prove that the iteration method generated by the regularized preconditioner is unconditionally convergent. In addition, it is proved that all eigenvalues of the regularized preconditioned matrix are clustered around the points $(0,0)$ and $(1,0)$ as parameter $\alpha$ is close to $+\infty$. Numerical experiments are provided to show the effectiveness of the regularized preconditioner.

 In this paper, we present a logarithmic barrier interior-point method for solving a second-order cone programming problem. Newton's method is used to compute the descent direction, and minorant function are used as an efficient alternative to line search methods to determine the step-size along the direction in order to reduce the computation cost.

 Let $G=(V,E)$ be a graph. A subset $S$ of $V$ is called a \textit{dominating set} of $G$ if every vertex not in $S$ is adjacent to some vertex in $S.$ The \textit{domination number} $\gamma(G)$ of $G$ is the minimum cardinality taken over all dominating sets of $G.$ A set $S \subseteq V$ is a \textit{total dominating set,} abbreviated TDS, of $G$ if every vertex in $V$ is adjacent to a vertex in $S.$ The \textit{total domination number} of a graph $G,$ denoted by $\gamma_{t}(G),$ is the minimum cardinality of a TDS of $G.$ In this paper, we investigate the lower and upper bounds for the total domination number of a graph and its derived graphs like line graph, subdivision graph, square graph, block graph and total graph and characterize such extremal graphs.

 The integral transforms related with improper integrals and special functions play an important role in the study of mathematical physics and applied mathematics. In this paper, we provide the analytical solution of an improper integral $\int_{0}^{\infty}\frac{x^{s-1}}{(a-x)}dx;a>0$ of complex analysis, using series manipulation and Dixon's summation theorem for the Clausen series $_3F_2(1)$. Further, by applying the above-mentioned improper integral together with suitable substitutions, partial fractions and L'H\^opital's rule, we evaluate other improper integrals with suitable convergence conditions, based on Mellin-transform as corollaries and examples.

 Controlled frames have been presented lately to cure the numerical output of iterative algorithms for inverting the frame operator. In this paper we expand systematically these notations, including their polar relationship. In the present paper, we introduce the sense of controlled K-g-frame in Hilbert spaces and get several approximates for identifying controlled K-g-frames. This work will be useful in some questions in sampling theory which are processed by controlled frames.

 In this paper, the gaussian bi-periodic Jacobsthal and gaussian bi-periodic Jacobsthal Lucas sequences are defined. The Binet formula as well as the generating function for these sequences are given. The convergence property of the consecutive terms of this sequence is examined after which the well-known Cassini, Catalan and the D'ocagne identities as well as some related summation formulas are also given.

 The aim of this paper is to establish the boundedness of bilinear Calder\'{o}n-Zygmund operator $BT$ and its commutator $[b_{1}, b_{2}, BT]$ generated by $BT$ and $b_{1}, b_{2}\in$\\$\mathrm{BMO}(\mathbb{R}^{n})$(or $b_{i}\in\dot{\Lambda}_{\beta_{i}}(\mathbb{R}^{n}), i=1, 2)$ on product of generalized fractional Morrey spaces. Under assumption that $\Psi$ satisfies a certain doubling condition, the authors prove that the $BT$ is bounded from the product of generalized fractional Morrey spaces $\mathcal{L}^{p_{1},\eta_{1},\Psi}(\mathbb{R}^{n})\times \mathcal{L}^{p_{2},\eta_{2},\Psi}(\mathbb{R}^{n})$ into space $\mathcal{L}^{p,\eta,\Psi}(\mathbb{R}^{n})$, and it is bounded from the product of spaces $\mathcal{L}^{p_{1},\eta_{1},\Psi}(\mathbb{R}^{n})\times \mathcal{L}^{p_{2},\eta_{2},\Psi}(\mathbb{R}^{n})$ into generalized weak fractional Morrey spaces $W\mathcal{L}^{p,\eta,\Psi}(\mathbb{R}^{n})$, where $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ for $1<p_{i}<\frac{n}{\eta_{i}}$, $i= 1,2$. Furthermore, the boundedness of the commutator $[b_{1},b_{2},BT]$ associated with $\mathrm{BMO}$ functions or Lipschitz functions on product of generalized fractional Morrey spaces is also obtained.