By using the Symmetric Mountain Pass Theorem, we establish some existence criteria to guarantee the second-order discrete $p$-Laplacian systems $\triangle (\varphi_p(\Delta u(n-1)))-a(n)|u(n)|^{p-2}u(n)+\nabla W(n, u(n))=0$ has infinitely many homoclinic orbits, where $p>1, \ n\in {\mathbb{Z}},\ u\in {\mathbb{R}}^{N}$, $a:{\mathbb{Z}}\rightarrow{\mathbb{R}}$ and $W:{\mathbb{Z}}\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}$ are not periodic in $n$. Our conditions on the nonlinear term $W(n,u(n))$ are rather relaxed and we generalize some existing results in the literature.

Recall that a frame $L$ is pseudocompact if $\mathcal{R}L=\mathcal{R}^*L$, where $\mathcal{R}L$ is the $f$-ring of real-valued continuous functions on $L$, and $\mathcal{R}^*L$ its bounded part. Using properties of uniform frames, Walters-Wayland proved that a completely regular frame $L$ is pseudocompact iff the frame homomorphism $\beta L\to L$ is coz-codense. In this note we give a purely ring-theoretic reaffirmation of this characterization by observing that a frame homomorphism $L\to M$ is coz-codense iff the ring homomorphism $\mathcal{R}L\to\mathcal{R}M$ it induces maps non-invertible elements to non-invertible elements, and that $L$ is pseudocompact iff every finitely generated proper ideal of $\mathcal{R}^*L$ is fixed. We also show that if $L$ is not pseudocompact, then $\mathcal{R}^*L$ has a non-maximal free prime ideal -- thus generalizing a 1954 result of Gillman and Henriksen.

We study the (so-called bilinear) factorization problem answered by a weak wreath product (of monads and, more specifically, of algebras over a commutative ring) in the works by Street and by Caenepeel and De Groot. A bilinear factorization of a monad $R$ turns out to be given by monad morphisms $A\to R\leftarrow B$ inducing a split epimorphism of $B$-$A$ bimodules $B\otimes A\rightarrow R$. We prove a biequivalence between the bicategory of weak distributive laws and an appropriately defined bicategory of bilinear factorization structures. As an illustration of the theory, we collect some examples of algebras over commutative rings which admit a bilinear factorization; i.e. which arise as weak wreath products.

In this paper, we investigate the local existence of solutions in $H^s$ for $n$-dimensional nonlinear wave equations with special nonlinear terms, such as $$u_{tt}-\Delta u=u^k|\nabla u|^l,\ \ x\in R^n,\ \ k\in Z^+,\ \ l\geq2.$$ where $\nabla u=(\frac{\partial u}{\partial x_1},\ \frac{\partial u}{\partial x_2},\ \cdots,\ \frac{\partial u}{\partial x_n})$. Meanwhile, we obtain that the regular index $s$ of Sobolev space $H^s$ satisfies $s>\max\{\frac{n+5}4;\ \frac n2+1-\frac1{l-1}\},\ n>3$.

In the paper, we study the uniqueness theorems of meromorphic function concerning nonlinear differential polynomials sharing a nonzero polynomial with finite weight and obtain two results which improve and generalize the results due to X.M Li and L. Gao.

A weakly equivariant Hopf algebra is a Hopf algebra $A$ with an action of a finite group $G$ up to inner automorphisms of $A$. We show that each weakly equivariant Hopf algebra can be replaced by a Morita equivalent algebra $A^{str}$ with a strict action of $G$ and with a coalgebra structure that leads to a tensor equivalent representation category. However, the coproduct of this strictification cannot, in general, be chosen to be unital, so that a strictification of the $G$-action can only be found on a \emph{weak} Hopf algebra $A^{str}$.

We present some new results on the relations between the rotation index of bar billiards of two nested circles $C_R$ and $C_r$, of radii $R$ and $r$ and with distance $d$ between their centers, satisfying Poncelet's porism property. The rational indices correspond to closed Poncelet transverses, without or with self-intersections. We derive an interesting series arising from the theory of special functions. This relates the rotation number $\frac 13$, of a triangle of Poncelet transverses, to a double series involving $R, r$, and $d$. We also provide a Steiner-type formula which gives a necessary condition for a bar billiard to be a pentagon with self-intersections and rotation index $\frac 25$. Finally we show that, close to a pair of circles having Poncelet's porism property for index $\frac{1}{3}$, there exist always circle pairs having indices $\frac{1}{4}$ they and $\frac{1}{6}$; in the case $\frac{1}{4}$ they are even unique.

Using the order structure of the lattice $DP(X)$ of density preserving continuous maps on a Hausdorff space $X$ without isolated points, we describe closed nowhere dense subsets of $X$ and, for a subspace $A$ of $X$, we also deduce topological properties of the space $X-A$ from the lattice theoretic properties of $DP(X,A)$. Finally, we use them to obtain Thrivikraman's results concerning $\beta X-X$ and $K(X)$ and, Magill's result concerning the automorphism group of the lattice $K(X)$.

In this paper, our main purpose is to establish the existence of nontrivial weak solutions to the following systems: $$\left\{ \begin{array}{ll} -\triangle_p u=\lambda V(x)|u|^{r-2}u+F_u(x,u,v),\;\;\; x\in \Omega,\\ -\triangle_p v=\theta V(x)|v|^{r-2}v+F_v(x,u,v),\;\;\; x\in \Omega,\\ u=v=0,\;\;\; x\in \partial\Omega, \end{array} \right.$$ where $\Omega$ is a bounded domain in ${\bf R}^N$, $\lambda,\theta>0$, $\triangle_s u=\mbox{div}(|\nabla u|^{s-2}\nabla u)$ is the s-Laplacian of u. We obtain the existence results in two cases: (i)$1<r<p<N$; (ii)$1<p<r<p^*$. The existence results of solutions are obtained by variational methods.

We show how binary machine instructions can be used to implement fast vector operations over the finite field $\mathbb{F}_3$. Apart from the standard operations of addition, subtraction and dot product, we also consider combined addition and subtraction, weight, Hamming distance, and iteration over all vectors of a given length. Tests show that our implementation can be as much as 10 times faster than the standard method of using modular arithmetic on arrays of bytes. For computing the Hamming distance even a factor of 33 can sometimes be reached, provided a recent CPU is used.

In the paper we provide a few generalizations of Darbo fixed point theorem. Several interconnections among assumptions imposed in the proved theorems are indicated. We also show the applicability of obtained results to the theory of functional integral equations. A concrete example illustrating the mentioned applicability is also included.

We show that the commutator map on $SU(3)$ has order $2^{3}\cdot 3\cdot 5$. As an application, we give an upper bound on the number of homotopy types of gauge groups for principal $SU(3)$-bundles over an $n$-sphere.

This paper concerns with a parabolic system coupled via nonlocal sources, subjecting to homogeneous Dirichlet boundary condition. The main aim of this paper is to study conditions on the global existence and finite time blowup of solutions. By using the super- and sub-solution techniques, the critical exponent of the system is determined. Furthermore, the related classification for the parameters in the model is optimal and complete.