We prove that for a generic hypersurface in ${{\mathbb{P}}^{2n+1}}$ of degree at least $2\,+\,2/n$, the $n$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing.

The tracial numerical range of operators on a 2-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are obtained.

We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $\text{GL}\left( n,\,\mathbb{C} \right)$ is isomorphic to another. As a consequence we discover families of Littlewood–Richardson coefficients that are non-zero, and a condition on Schur non-negativity.

Given a countable set $S$ of positive reals, we study finite-dimensional Ramsey-theoretic properties of the countable ultrametric Urysohn space ${{\text{Q}}_{S}}$ with distances in $S$.

The consideration of tensor products of 0-Hecke algebramodules leads to natural analogs of the Bessel $J$-functions in the algebra of noncommutative symmetric functions. This provides a simple explanation of various combinatorial properties of Bessel functions.

Let $KG$ be a non-commutative strongly Lie solvable group algebra of a group $G$ over a field $K$ of positive characteristic $p$. In this note we state necessary and sufficient conditions so that the strong Lie derived length of $KG$ assumes its minimal value, namely $\left\lceil {{\log }_{2}}(p\,+\,1) \right\rceil$.

The behavior of the dynamical zeta function ${{Z}_{D}}(s)$ related to several strictly convex disjoint obstacles is similar to that of the inverse $Q(s)\,=\,\frac{1}{\zeta (s)}$ of the Riemann zeta function $\zeta \left( s \right)$. Let $\prod \left( s \right)$ be the series obtained from ${{Z}_{D}}(s)$ summing only over primitive periodic rays. In this paper we examine the analytic singularities of ${{Z}_{D}}(s)$ and $\prod \left( s \right)$ close to the line $\Re s={{s}_{2}},$ where ${{s}_{2}}$ is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions ${{Z}_{D}}(s),$$\prod \left( s \right)$ has a singularity at $s\,=\,{{s}_{2}}$.

We show that limits of CS iterations of the $n$-Silver forcing notion have the $n$-localization property.

An $R$-module $M$ is called a multiplication module if for each submodule $N$ of $M,\,N\,=\,IM$ for some ideal $I$ of $R$. As defined for a commutative ring $R$, an $R$-module $M$ is said to be semiprimitive if the intersection of maximal submodules of $M$ is zero. The maximal spectra of a semiprimitive multiplication module $M$ are studied. The isolated points of $\text{Max}\left( M \right)$ are characterized algebraically. The relationships among the maximal spectra of $M$, $\text{Soc}\left( M \right)$ and $\text{Ass}\left( M \right)$ are studied. It is shown that $\text{Soc}\left( M \right)$ is exactly the set of all elements of $M$ which belongs to every maximal submodule of $M$ except for a finite number. If $\text{Max}\left( M \right)$ is infinite, $\text{Max}\left( M \right)$ is a one-point compactification of a discrete space if and only if $M$ is Gelfand and for some maximal submodule $K$, $\text{Soc}\left( M \right)$ is the intersection of all prime submodules of $M$ contained in $K$. When $M$ is a semiprimitive Gelfand module, we prove that every intersection of essential submodules of $M$ is an essential submodule if and only if $\text{Max}\left( M \right)$ is an almost discrete space. The set of uniform submodules of $M$ and the set of minimal submodules of $M$ coincide. $\text{Ann}\left( \text{Soc}\left( M \right) \right)M$ is a summand submodule of $M$ if and only if $\text{Max}\left( M \right)$ is the union of two disjoint open subspaces $A$ and $N$, where $A$ is almost discrete and $N$ is dense in itself. In particular, $\text{Ann}\left( \text{Soc}\left( M \right) \right)\,=\,\text{Ann}\left( M \right)$ if and only if $\text{Max}\left( M \right)$ is almost discrete.

It is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A. C. M. van Rooij and W. H. Schikhof in 1992.

Let $k$ be a field of characteristic not 2, 3. Let $G$ be an exceptional simple algebraic group over $k$ of type ${{\text{F}}_{4}},$$^{1}{{\text{E}}_{6}}$ or ${{\text{E}}_{7}}$ with trivial Tits algebras. Let $X$ be a projective $G$-homogeneous variety. If $G$ is of type ${{\text{E}}_{7}},$ we assume in addition that the respective parabolic subgroup is of type ${{\text{P}}_{7}}.$ The main result of the paper says that the degree map on the group of zero cycles of $X$ is injective.

In this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra.

Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed.

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by $\text{L}$. Schläfli in 1858. Using this, explicit examples of surfaces of every possible type are given.

Using a canonical linear embedding of the algebra ${{G}^{\infty }}\left( \Omega \right)$ of Colombeau generalized functions in the space of $\overline{\mathbb{C}}$ -valued $\mathbb{C}$-linear maps on the space $D\left( \Omega \right)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class ${{G}^{\infty }}$ . These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

The homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by $\text{B}$. Larose and $\text{C}$. Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs $\left( X,\,A \right)$ where $X$ is a poset and $A$ is a subposet of $X$. We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif.

We show that the class of system proportionally modular numerical semigroups coincides with the class of numerical semigroups having a Toms decomposition.

In this paper we derive formulas for summation of series involving J. Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H. M. Srivastava. These series are expressed in terms of the Riemann $\zeta$ function and Dirichlet functions $\eta$, $\lambda$, $\beta$, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.

Let $R\,=\,\oplus _{i=1}^{\infty }\,{{R}_{i}}$ be a graded nil ring. It is shown that primitive ideals in $R$ are homogeneous. Let $A\,=\,\oplus _{i=1}^{\infty }\,{{A}_{i}}$ be a graded non-PI just-infinite dimensional algebra and let $I$ be a prime ideal in $A$. It is shown that either $I\,=\,\{0\}$ or $I\,=\,A$. Moreover, $A$ is either primitive or Jacobson radical.

In this paper we study the best constant of the Sobolev trace embedding ${{H}^{1}}(\Omega )\,\to \,{{L}^{2}}(\partial \Omega ),$ where $\Omega$ is a bounded smooth domain in ${{\mathbb{R}}^{N}}.$ We find a formula for the first variation of the best constant with respect to the domain. As a consequence, we prove that the ball is a critical domain when we consider deformations that preserve volume.