In this paper, we prove some tripled fixed point theorems for Meir-Keleer condensing operator in a Banach space by using $L$-functions. We apply these results to establish the existence of solutions for a system of functional integral equations of Volterra type.

In this paper, we give computational examples of string operations over the rational numbers field on Gorenstein spaces introduced by Félix and Thomas. Especially, we determine the structure of rational string operations on the classifying space of a compact connected Lie group and the Borel construction associated to an action of $S^{1}$ to $S^{2}$.

We consider a hinged elastic beam subject to a nonlinear restoring force. We show that the equilibrium position of the beam under the action of a vertical load displays an oscillating profile. We compare the nonlinear model with the linear model and we study the dependence of the solution with respect to the physical parameters involved.

Based on the semantic concepts developed by M.~Dunn and N.~Belnap, a four-valued language containing only two logical symbols is proposed. We show that this language is functionally complete with regard to the given semantics. Specifically, we prove that every truth-function is expressed by a formula of the language. To do this, we define two concepts akin to the disjunctive and conjunctive normal forms. Using these concepts, we establish that every truth-function for a four-valued semantics can be represented by a formula in a disjunctive form or in a conjunctive form.

Integer multiplication in Jacobians of genus $2$ curves over a finite field $\mathbb{F}_q$ is a fundamental operation for hyperelliptic curve cryptography. Algorithmically, the result of this operation is given by the very well known algorithms of Cantor. One method to reverse duplication in these cases consists in associating, to every preimage of the desired doubled divisor defined over $\mathbb{F}_q$, a root in $\mathbb{F}_q$ of the so called bisection polynomial. We generalize this approach to genus $2$ curves with two points at infinity, both in even and odd characteristics. We attach a bisection polynomial to each possible type of Mumford coordinate, we show the factorization of these in terms of Galois orbits in the set of bisections, and we compare the efficiency of our approach versus brute-force adaptations of the existing methods to our setting.

Let ${\cal L}(E)$ be the algebra of all bounded linear operators on a Banach space $E$. For $A,B\in{\cal L}(E)$, define the basic elementary operator $M_{A,B}$ by $M_{A,B}(X)=AXB$, ($X\in{\cal L}(E)$). If $\cal S$ is a symmetric norm ideal of ${\cal L}(E)$, we denote $M_{{\cal S},A,B}$ the restriction of $M_{A,B}$ to $\cal S$. In this paper, the norm equality $\|I+M_{{\cal S},A,B}\|=1+\|A\|\|B\|$ is studied. In particular, we give necessary and sufficient conditions on $A$ and $B$ for this equality to hold in the special case when $E$ is a Hilbert space and $\cal S$ is a Schatten $p$-ideal of ${\cal L}(E)$.

In this paper, we study a class of semilinear elliptic equations with critical weighted Hardy-Sobolev exponents and superlinear nonlinearity. By means of the variational methods and some analysis techniques, positive solution is obtained.

In this paper, we consider the existence and multiplicity of solutions for perturbed $p$-Laplacian equation problems with critical nonlinearity in $\mathbb{R}^N$: $- \varepsilon^p\Big[g\Big(\displaystyle\int_{\mathbb{R}^N}|\nabla_A u|^pdx\Big)\Big]\Delta_{p,A}u + V(x)|u|^{p-2}u = |u|^{p^\ast-2}u + h(x, |u|^p)|u|^{p-2}u$ for all $(t, x) \in \mathbb{R} \times \mathbb{R}^N$, where $V(x)$ is a nonnegative potential, $\Delta_{p,A}u(x) :=\mathop{\rm div}(|\nabla u+iA(x)u|^{p-2}(\nabla u + iA(x)u)$ and $\nabla_Au := (\nabla + iA)u$. By using Lions' second concentration compactness principle and concentration compactness principle at infinity to prove that the $(PS)_c$ condition holds locally and by variational method, we show that this equation has at least one solution provided that $\varepsilon < \mathcal {E}$, for any $m \in \mathbb{N}$, it has $m$ pairs of solutions if $\varepsilon < \mathcal {E}_m$, where $\mathcal {E}$ and $\mathcal {E}_m$ are sufficiently small positive numbers.

Let ${\mathcal A}$ be a Banach algebra and let $\phi$ be a non-zero character on ${\mathcal A}$. We introduce the notion of $\phi$-ergodic property for a Banach right ${\mathcal A}$-module $X$. This concept considerably generalizes the existence of $\phi$-means of norm one on ${\mathcal A}^*$. We also show that the $\phi$-ergodic property of $X$ is related to some other properties such as a Hahn-Banach type extension property and the existence of $\phi$-means of norm one on a certain subspace of ${\mathcal A}^*$. Finally, we give some characterizations for $\phi$-amenability of a Banach algebra in terms of its closed ideals.

In this paper, the shearlet theory is extended from Euclidean spaces to locally compact groups. More precisely, the abstract shearlet group is defined as a 3-fold semidirect product and the abstract shearlet transform is constructed by means of a quasiregular representation of the semidirect product group. Its properties are investigated and results are illustrated by some examples.

We prove that if two division $p$-algebras of prime degree share an inseparable field extension of the center then they also share a cyclic separable one. We show that the converse is in general not true. We also point out that sharing all the inseparable field extensions of the center does not imply sharing all the cyclic separable ones.