We prove global well-posedness of the 3D fractional porous medium equation with critical dissipative for small initial data in the Triebel-Lizorkin spaces $F^{s}_{p,q},~s>\frac{3}{p},~p,q\in(1,\infty)$. In particular, our result implies the global existence for small initial data in the Sobolev spaces $H^{s}_{p},~s>\frac{3}{p},~p\in(1,\infty)$ since $H^{s}_{p}(\mathbb{R}^{3})=F^{s}_{p,2}(\mathbb{R}^{3})$. The key ingredients of our proof are the pointwise exponential decay estimate of the semigroup $e^{-t\alpha(-\Delta)^{\frac{1}{2}}}$ and maximum principle.

This paper is concerned with the so-called Bouchut-Boyaval relaxation model, which is a fully linearly degenerate $4\times4$ hyperbolic system of conservation laws. The well-known Suliciu relaxation model is its particular situation. The Riemann problem is completely solved with four different structures. The first is composed of the contact discontinuities, the second contains a single overcompressible $\delta$-shock wave, and the last two involve a contact discontinuity and a non-overcompressible $\delta$-shock wave. As the main feature, the Bouchut-Boyaval relaxation model as well as the Suliciu relaxation model admits not only the overcompressible $\delta$-shock waves but also non-overcompressible $\delta$-shock waves.

Let $R$ be a noncommutative prime ring of char $(R)\neq 2$, $Q_r$ be its right Martindale quotient ring and $C$ be its extended centroid. Suppose that $f(r_1,\ldots,r_n)$ is a noncentral multilinear polynomial over $C$ and $F:R\rightarrow R$ is an $X$-generalized skew derivations of $R$. We describe all possible forms of the maps $F$ when $R$ satisfies the condition

$$a[F(f(r)),f(r)]-[F(f(r)),f(r)]a\in C$$

for all $r=(r_1,\ldots,r_n)\in R^n$.

As an application of this result, we also describe the possible forms of the maps $F$ and $G$ satisfying the conditions

(i) $[F(f(r)),f(r)]\in C$ for all $r=(r_1,\ldots,r_n)\in R^n$;

(ii) $[[F(u),u], [G(v),v]]=0$ for all $u,v\in f(R)$,\end{lieju}

where $F$ and $G$ are both $X$-generalized skew derivations of $R$.

A sequential closure relative to a given topology $\tau$ provides a method for constructing a new topology $\tau_{S}$, the associated sequential topology, which is the finest topology possessing the same convergent sequences. In this paper, we give the elementary properties of this topology in relation with many topological concepts such as convergence, continuity, capacity and density. We will also be concerned with the problem of the stability of the associated sequential topologies. We shall explore the relationship between the intrinsic associated sequential topology and the induced associated sequential topology relative to inductive topology, projective topology and disjoint union topology.

In this paper, we investigate the symmetric matrix $H = [H_{\max(i,j)}]_{i,j=1}^n$ and its Hadamard exponential $e^{\circ H} = [\, e^{H_{\max(i,j)}} \,]$, where \(H_k\) denotes the \(k\)th harmonic number. We derive explicit formula for the determinants and inverses of these matrices, and we establish expressions for their Euclidean norms as well as bounds for their spectral norms. Illustrative examples and tables are provided to highlight and verify these properties.

This study concerns the existence of positive solutions for the following nonlinear boundary value problem

$$

\left\{\begin{array}{ll}

-\Delta u = a m(x)u -bu^{2}- c\frac{u^{p}}{u^{p}+1}-K & \textrm{ in }\Omega,\\

\textbf{n}. \nabla u + g(u) = 0\ , \ & x \in \partial \Omega .

\end{array}\right.

$$

where $ \Delta u= \text{div}(\nabla u)$ is the Laplacian of $u$, while $a,b,c,p,K$ are positive constants with $ p\geq 2 $, $\Omega$ is a bounded smooth domain of $\R^N$ with $\partial\Omega$ in $ C ^{2} $and $ g \in C^1\left( [0,\infty),[0,\infty)\right) $ is decreasing for some $\theta > 0$. The weight function $m$ satisfies $m\in C(\Omega)$ and $ m(x)\geq m_{0}>0 $ for $ x\in \Omega $, also $\|m\|_{\infty}= l<\infty$. This model describes the dynamics of the fish populations. Our existence results are established via the

well-known sub-super solution method.

Domination parameters are defined by combining domination with another graph-theoretical property. This paper considers some of the five fundamental domination parameters in domination theory like total domination and paired domination. Initially, we determine the domination number $\gamma$ for bishop's and rook's graph, then find the total domination number $\gamma_t$ and paired domination $\gamma_{pr}$ for queen's graph, bishop's graph, and rook's graph on a hexagonal board of order $N$ ($H_N$).

In this paper, a characterization of normed barrelled spaces is given, as well as a similar characterization of rings of sets with the Nikodym Property. Finally, a condition that is equivalent to a Banach space having a separable quotient is discussed.

In this paper, we introduced bivariate Chlodowsky variant of Bernstein-Schurer operators based on (p,q)-integers. We also studied the estimates of moments of these operators, the weighted approximation theorem, and Korovkin-type approximation theorems. Besides these, the error of approximation using full modulus of continuity and partial modulus of continuity, the rate of convergence, and a convergence theorem for Lipschitz continuous functions were also presented. Furthermore, we generalised the Chlodowsky variant of Bernstein-Schurer operators based on (p,q)-integers. Lastly, the numerical results for the defined operators are comparatively illustrated to show the minimised error of approximation to some functions.

This paper investigates the initial value problems for a class of magnetohydrodynamics Euler equations. First, the elementary wave interactions including the overtaking and collisions of shock waves and central rarefaction waves are studied. Then, one special Riemann problem of the magnetohydrodynamics Euler equations with three constant initial values is considered, and the global solution structure is obtained. Finally, a class of Cauchy problems with general initial values for the magnetohydrodynamics Euler equations is studied, and the global existence of weak entropy solution is established.