A minimal surface intersecting the boundary of a smooth bounded domain $\subset\mathbb{R}^3$, when it is non-degenerate, gives rise to a family of transition layer solutions of the Allen-Cahn equation. The stability properties of the transition layer solution are determined by the eigenvalues of the Jacobi operator on the minimal surface with Robin type boundary conditions which encode the geometric information of the domain boundary.

For bounded sequences in a normed linear space $X$, we introduce a notion of limit, called the $s\sigma$-limit, and discuss some interesting properties related to $\sigma$-limit and $s\sigma$-limit. It is shown that the space $X_{s\sigma}$ (resp. $X_\sigma$) of all $s\sigma$-convergent (resp. $\sigma$-convergent) sequences in $X$ is a Banach space, and the space $\mathbb{C}_{s\sigma}$ is a unital Banach subalgebra of $\ell^\infty$ such that every Banach limit restricted to $\mathbb{C}_{s\sigma}$ is a multiplicative linear functional. We also use $s\sigma$-limit to characterize continuity of functions and prove two versions of the dominated convergence theorem in terms of $\sigma$-limit and $s\sigma$-limit.

In this paper, we first show that any reflexive $\mathcal{A}$-submodule $\mathcal{U}$ of a unital operator algebra $\mathcal{A}$ in $\mathcal{B(H)}$ is precisely of the following form: $$\mathcal{U} = \{ T \in \mathcal{B(H)}: TE \subseteq \phi(E) \quad \forall E \in \mathrm{Lat}\mathcal{A} \},$$ where $\phi$ is an order homomorphism of $\mathrm{Lat}\mathcal{A}$ into itself. Furthermore we investigate the density of the rank-one submodule of a reflexive $\mathcal{A}$-submodule in the $w^{*}$-topology and in certain pointwise approximation, and obtain several equivalent conditions by means of the order homomorphism $\phi$.

Let R be a commutative ring with identity (zero-divisors admitted). Various properties of submodules a multiplication module are considered. In fact, our aim here is to generalize some of the results in the paper listed as [1], from finitely generated faithful multiplication ideals to finitely generated faithful multiplication modules.

We study some measures which are related to the notion of the $\epsilon$-complexity. We prove that measure of $\epsilon$-complexity defined on the base of the notion of $\epsilon$-separability is equivalent to the dual measure that is defined through $\epsilon$-nets.

In this article, the Schur-convexities of the weighted arithmetic mean of function and the extended mean values are proved. Moreover, some inequalities involving the arithmetic mean, the harmonic mean, the logarithmic mean, and comparison between the extended mean values and the generalized weighted mean with two parameters and constant weight are obtained.

In 1983, T.T. Moh gave a computer-search algorithm to find the possible counter-example of Jacobian conjecture for polynomials of degree less than or equal to 100. His algorithm claimed that there are only six possible counter-examples. When we re-code his algorithm, there seems to be exceptions other than his six cases. We modified Moh’s algorithm with his approximate root theory to fulfill his claim.

It is well known that a semigroup $S$ is a Clifford semigroup if and only if $S$ is a strong semilattice of groups. In this paper, we extend this important result from semigroups to semirings by showing that a semiring $S$ is a Clifford semiring if and only if $S$ is a strong distributive lattice of skew-rings. Also, as a further generalization, we prove that a semiring $S$ is a genneralized Clifford semiring if and only if $S$ is a strong b-lattice of skew-rings. Some results which have been recently obtained in the literature [2] are strengthened and extended.

In this article we characterize the projective special linear group $L_{11}(2)$, by the set of the order of its elements.

Let $(z_n)$ be a sequence of points in the open unit disc $D$ and $\rho_n = \prod_{m \ne n}|(z_n - z_m)(1 - \bar{z}_mz_n)^{-1}| \gt 0$. Let $a = (a_j)^\infty_{j = 1}$ be a sequence of positive numbers and $\ell^s(a) = \{(w_j);~(a_jw_j) \in \ell^s\}$ where $1 \le s \le \infty$. When $1 \le p \le \infty$ and $1/p + 1/q = 1$, we show that $\{(f(z_n));~f \in H^p\} \supset \ell^s(a)$ if and only if there exists a finite positive constant $\gamma$ such that $\left\{{\sum^\infty_{n = 1}} (a_n \rho_n)^{-t}(1 - |z_n|^2)^t|f(z_n)|^t \right\}^{1/t} \le \gamma \|f\|_q ~(f \in H^q)$, where $1/s + 1/t = 1$. As results, we show that $\{(f(z_j));~f \in H^p\} \supset \ell^1(a)$ if and only if ${\sup_n}(a_n \rho_n)^{-1}(1 - |z_n|^2)^{1/p} \lt \infty$, and $\{(f(z_n));~f \in H^1\} \supset \ell^\infty(a)$ if and only if ${\sum_n}(a_n \rho_n)^{-1}(1 - |z_n|^2)\delta_{z_n}$ is finite measure on $D$. These are also proved in the case of weighted Hardy spaces.

In this paper, we prove that the constant factor in the Hardy-Carleman’s type inequality is the best possible. A related integral inequality with a best constant factor is considered.

In this paper, we established an equivalence between the category of Braided Crossed Modules of Groups and the category of Simplicial Groups with Moore Complex of length 2.

We establish a local and a global existence results for a Dirac-Klein-Gordon type system in $1+1$ dimensions with a pseudoscalar bilinear form.

Many types of convex operators which take values in some complete lattices can be represented by convex integrands. We consider a certain structure of faces of convex sets, and give a new proof of the representation theorem which is applicable in infinite-dimensional cases. As an application of such representations, we consider the conjugate duality of convex operators.

We give a simple proof of the fact that the following property: any injective transition function of a cellular automaton on a group $G$ is surjective, holds for any group $G$ approximable by amenable groups. For finitely generated groups this was proved by Gromov in [1]

In this paper, we use the boundary measurements of normalized eigenfunctions to estimate the variation of the corresponding eigenvalues. With this form, we can show that some eigenvalues, as functions of domain, possess monotonicity as the domain varies according some constraints.