The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research such as cohomology theory, representation theory, category theory and homological algebra. In this paper, we extend this classical ring construction by associating a ring to a ring~$R$ and a family $M=(M_i)_{i=1}^{n}$ of $n$ $R$-modules for a given integer $n\geq 1$. We call this new ring construction an $n$-trivial extension of $R$ by $M$. In particular, the classical trivial extension will merely be the $1$-trivial extension. Thus, we generalize several known results on the classical trivial extension to the setting of $n$-trivial extensions, and we give some new ones. Various ring-theoretic constructions and properties of $n$-trivial extensions are studied, and a detailed investigation of the graded aspect of $n$-trivial extensions is also given. We finish the paper with an investigation of various divisibility properties of $n$-trivial extensions. In this context, several open questions arise.

Laxton introduced a group structure on the set of equivalence classes of linear recurrence sequences of degree~2. This result yields much information on the divisibilities of such sequences. In this paper, we introduce other equivalence relations for the set of linear recurrence sequences $(G_n)$, which are defined by $G_0, G_1 \in \mathbb{Z} $ and $G_n=TG_{n-1}-NG_{n-2}$ for fixed integers~$T$ and $N=\pm 1$. The relations are given by certain congruences modulo~$p$ for a fixed prime number~$p$, which are different from Laxton's without modulo $p$ equivalence relations. We determine the initial terms $G_0$ and $G_1$ of all of the representatives of the equivalence classes $\overline {(G_n)}$ satisfying $p\nmid G_n$ for any integer~$n$ and give the number of equivalence classes. Furthermore, we determine the representatives of Laxton's without modulo~$p$ classes from our modulo~$p$ classes.

We settle an open question proposed by Banjade to generalize Wolff's ideal theorem on certain uniformly closed subalgebras of $H^{\infty }(\mathbb{D} )$. Also, we discuss some subalgebras where Wolff's ideal theorem holds without the additional condition $ F(0) \neq 0$.

In a previous paper the author studied the compactness of multiplication, composition and weighted composition operators among some sequence spaces. We were motivated by these results and present two different approaches for obtaining some of the results in a previous paper. The first approach is to apply the theory of matrix transformations and the Hausdorff measure of noncompactness, and the second one is to use known results on multiplier spaces and the Hausdorff measure of noncompactness. We also use our techniques and methods from our proofs of the existing results to establish some new results related to the class of Fredholm operators and some classes of operators considered here and in the previous paper.

We study the structure of the partially or\-dered set of the elementary domains of partial (linear or projective) representations of groups. This provides an important information on the lattice of all domains. Some of these results are obtained through structural facts on the ideals of the semigroup $\mathcal{S} _3(G)$, a quotient of Exel's semigroup $\mathcal{S} (G)$, which plays a crucial role in the theory of partial projective representations. We also fill a gap in the proof of an earlier result on the structure of partial group representations.

A result of Graber, Harris and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field will have a rational point. Here, we show that rationally connected varieties over the maximally unramified extension of the $p$-adics usually, in a precise sense, have rational points. This result is in the spirit of Ax and Kochen's result, which states that the $p$-adics are usually $C_{2}$ fields. The method of proof utilizes a construction from mathematical logic called the ultraproduct.

In this paper, we propose a new application of genetic-type algorithms to find monic, irreducible, non-cyclotomic integer polynomials with \textit {small degree} and Mahler measure less than $1.3$, which do not appear in Mossinghoff's list of all known polynomials with degree at most 180 and Mahler measure less than 1.3 {Mossinghoff}. The primary focus lies in finding such polynomials of small degree. In particular, the list referred to above is known to be complete through degree 44, and we show that it is not complete from degree 46 on by supplying two new polynomials of small Mahler measure, of degrees 46 and 56. We also provide a large list of polynomials of small Mahler measure of degrees up to 180 which, although discovered by us through the use of a method described in Boyd and Mossinghoff {Boyd and Mossinghoff} based on limit points of small Mahler measures, do not appear on Mossinghoff's list \cite {Mossinghoff 07}. Finally, we verify that our new polynomials of degrees 46 and 56 cannot be produced from the known small limit points.

In this paper, two Brunn-Minkowski type inequalities for mixed chord-integrals of index~$i$ are established, which are related to the radial Blaschke-Minkowski homomorphisms of star bodies. Moreover, two inequalities similar to Giannopoulos, Hartzoulaki and Paouris's inequality are also considered.

We characterize Ding modules and complexes over Ding-Chen rings. We show that, over a Ding-Chen ring $R$, the Ding projective (respectively, Ding injective, respectively, Ding flat) $R$-modules coincide with the Gorenstein projective (respectively, Gorenstein injective, respectively, Gorenstein flat) modules, which, in turn, are noth\-ing more than modules appearing as a cycle of an exact complex of projective (respectively, injective, respectively, flat) modules. We prove a similar characterization for chain complexes of $R$-modules: a complex~$X$ is Ding projective (respectively, Ding injective, respectively, Ding flat) if and only if each component $X_n$ is Ding projective (respectively, Ding injective, respectively, Ding flat). Along the way, we generalize some results of Stovicek and Bravo, Gillespie and Hovey to obtain other interesting corollaries. For example, we show that, over any Noetherian ring, any exact chain complex with Gorenstein injective components must have all cotorsion cycle modules, that is, $Ext ^1_R(F,Z_nI) = 0$ for any such complex $I$ and flat module $F$. On the other hand, over any coherent ring, the cycles of any exact complex $P$ with projective components must satisfy $Ext ^1_R(Z_nP,A) = 0$ for any absolutely pure module~$A$.

Results of Haiman and Sturmfels \cite {HS04} on multigraded Hilbert schemes are used to establish a quasi-projective scheme which parametrizes all left homogeneous ideals in the Weyl algebra having a fixed Hilbert function with respect to a given grading by an abelian group.

In this note, we construct a BSE-algebra of type I which is isomorphic to no C$^*$-algebras. This affirmatively solves the problem posed by Takahasi and Hatori, ``is there a BSE-algebra of type I isomorphic to no C$^*$-algebras?''

Let $(A,\alpha )$ be a system consisting of a $C^*$-algebra $A$ and an automorphism $\alpha $ of~$A$. We describe the primitive ideal space of the partial-isometric crossed product $A\times _{\alpha }^{piso }\mathbb{N} $ of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.

We obtain three finite generalizations of Gauss's square exponent identity. For example, we prove that, for any non-negative integer $m$, \[ \sum _{k=-m}^{m}(-1)^k \left [{3m-k+1}\atop {m+k}\right ] (-q;q)_{m-k}q^{k^2}=1, \] where \[ \left [{\vphantom {3}n}\atop {\vphantom {k}m}\right ] =\prod _{k=1}^m\frac {1-q^{n-k+1}}{1-q^k} \quad \mbox {and}\quad (a;q)_n=\prod _{k=0}^{n-1}(1-aq^k). \] These identities reduce to Gauss's famous identity $$\sum _{k=-\infty }^{\infty }(-1)^kq^{k^2}=\frac {(q;q)_{\infty }}{(-q;q)_{\infty }}$$ by letting $m\to \infty $.

We find an adequate interpretation of the stationary Lam\'{e} operator within the framework of elliptic complexes and study the first mixed problem for the nonstationary Lam\'{e} system.

A completely regular topological space $X$ is called a $P$-space if every zero-set in $X$ is open. An intermediate ring is a ring $A(X)$ of real-valued continuous functions on $X$ containing all the bounded continuous functions. In this paper, we find new characterizations of $P$-spaces $X$ in terms of properties of correspondences between ideals in $A(X)$ and $z$-filters on $X$. We also show that some characterizations of $P$-spaces that are described in terms of properties of $C(X)$ actually characterize $C(X)$ among intermediate rings on $X$.

We consider the asymptotic probability that integers chosen according to a binomial distribution will have certain properties: (i) that such an integer is not divisible by the $k$th power of a prime, (ii) that any $k$ of $s$ chosen integers are relatively prime and (iii) that a chosen integer is prime. We also prove an analog of the Dirichlet divisor problem for the binomial distribution. We show how these results yield corresponding facts concerning the number of points on a smooth complete intersection over a finite field.