Let $R$ and $S$ be commutative rings with unity, $f\colon R\rightarrow S$ a ring homomorphism and $J$ an ideal of $S$. Then the subring $R\bowtie ^fJ:=\lbrace (a,f(a)+j)\mid a\in R$ and $j\in J\rbrace $ of $R\times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$. In this paper, we determine when $R\bowtie ^fJ$ is a (generalized) filter ring.

In this short note we discuss $N$-supersymmetric worldlines of relativistic massless particles and review the known result that physical spin-$N/2$ fields are in the first BRST cohomology group. For $N=1,2,4$, emphasis is given to particular deformations of the BRST differential, that implement either a covariant derivative for a gauge theory or a metric connection in the target space seen by the particle. In the end, we comment about the possibility of incorporating Ramond-Ramond fluxes in the background.

This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational method to determine the cohomology groups induced by sub-superalgebras. In particular, we show as an example the case of $\mathfrak{osp}(1\mid 4)$ and choose $\mathfrak{osp}(1\mid 2) \times \mathfrak{sp} (2)$ as sub-algebra. We finally comment on some physical applications of such new cohomology classes related to super-branes. The note is a compact version of [10].

The concept of generalized prime $D$-filters is introduced in distributive lattices. Generalized prime $D$-filters are characterized in terms of principal filters and ideals. The notion of generalized minimal prime $D$-filters is introduced in distributive lattices and properties of minimal prime $D$-filters are then studied with respect to congruences. Some topological properties of the space of all prime $D$-filters of a distributive lattice are also studied.

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a $t$-symplectic Lie groupoid; the “$t$" is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $\mathcal{G}\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on $A\mathcal{G}$ (the associated Lie algebroid) and $t$-symplectic Lie groupoid structures on $\mathcal{G}\rightrightarrows M$. In addition, we also introduce the notion of a symplectic Lie group bundle (SLGB) which is a special case of both a $t$-symplectic Lie groupoid and a Lie group bundle. The basic properties of SLGBs are explored.

This short note completes the results of [3] by removing the locality assumption on the operators. After providing a quick survey on (infinitesimally) natural operations, we show that all the bilinear operators classified in [3] can be characterized in a completely algebraic way, even without any continuity assumption on the operations.

This article deals with some results about the existence of solutions and bounded solutions and the attractivity for a class of fractional ${q}$-difference equations. Some applications are made of Schauder fixed point theorem in Banach spaces and Darbo fixed point theorem in Fréchet spaces. We use some technics associated with the concept of measure of noncompactness and the diagonalization process. Some illustrative examples are given in the last section.

One of the important questions that remains after the classification of the finite simple groups is how to recognize a simple group via specific properties. For example, authors have been able to use graphs associated to element orders and to number of elements with specific orders to determine simple groups up to isomorphism. In this paper, we prove that Suzuki groups $Sz(q)$, where $q\pm \sqrt{2q}+1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same order.

The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac{d}{dt}[x(t) - L(x_{t})]= A[x(t)- L(x_{t})]+G(x_{t})+ \frac{1}{\Gamma (\alpha )} \int _{- \infty }^{t} (t-s)^{\alpha - 1} ( \int _{- \infty }^{s}a(s-\xi )x(\xi ) d \xi )ds+f(t)$, ($\alpha > 0$) with the periodic condition $x(0) = x(2\pi )$, where $a \in L^{1}(\mathbb{R}_{+})$ . Our approach is based on the R-boundedness of linear operators $L^{p}$-multipliers and UMD-spaces.

Extending work of many authors we calculate the higher simple structure sets of lens spaces in the sense of surgery theory with the fundamental group of arbitrary order. As a corollary we also obtain a calculation of the simple structure sets of the products of lens spaces and spheres of dimension grater or equal to $3$.