This paper discusses the connection between the local cohomology modules and the Serre classes of $R$-modules. This connection has provided a common language for expressing some results regarding the local cohomology $R$-modules that have appeared in different papers.

Consider a finite morphism $f\,:\,X\,\to \,Y$ of smooth, projective varieties over a finite field $\mathbb{F}$. Suppose $X$ is the vanishing locus in ${{\mathbb{P}}^{N}}$ of $r$ forms of degree at most $d$. We show that there is a constant $C$ depending only on $(N,\,r,\,d)$ and $\deg (f)$ such that if $\left| \mathbb{F} \right|\,>\,C$, then $f\,(\mathbb{F})\,:\,X(\mathbb{F})\,\to Y(\mathbb{F})$ is injective if and only if it is surjective.

We establish a flexible oscillation criterion based on an averaging technique that improves upon a result due to C. G. Philos.

Given a nonnegative integer $m$ and a finite collection $A$ of linear forms on ${{\mathbb{Q}}^{d}}$, the arrangement of affine hyperplanes in ${{\mathbb{Q}}^{d}}$ defined by the equations $\alpha \left( x \right)\,=\,k$ for $\alpha \,\in \,A$ and integers $k\,\in \,\left[ -m,\,m \right]$ is denoted by ${{A}^{m}}$. It is proved that the coefficients of the characteristic polynomial of ${{A}^{m}}$ are quasi-polynomials in $m$ and that they satisfy a simple combinatorial reciprocity law.

Let $f\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be ${{C}^{\infty }}$ and let $h\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be positive and continuous. For any unbounded nondecreasing sequence $\{{{c}_{k}}\}$ of nonnegative real numbers and for any sequence without accumulation points $\{{{x}_{m}}\}$ in ${{\mathbb{R}}^{n}}$, there exists an entire function $g\,:\,{{\mathbb{C}}^{n}}\,\to \,\mathbb{C}$ taking real values on ${{\mathbb{R}}^{n}}$ such that

$$\left| {{g}^{\left( \alpha \right)}}\left( x \right)-{{f}^{\left( \alpha \right)}}\left( x \right) \right|\text{ }<h\left( x \right),\left| x \right|\ge {{c}_{k}},\left| \alpha \right|\le k,k=0,1,2,...,$$

$${{g}^{\left( \alpha \right)}}\left( {{x}_{m}} \right)\,=\,{{f}^{\left( \alpha \right)}}\left( {{x}_{m}} \right),\,\,\,\left| {{x}_{m}} \right|\,\ge \,{{c}_{k}},\,\left| \alpha \right|\,\le \,k,\,m,\,k\,=\,0,\,1,\,2,\,.\,.\,.\,.$$

This is a version for functions of several variables of the case $n\,=\,1$ due to $L$. Hoischen.

By $\text{d(X,Y)}$ we denote the (multiplicative) Banach–Mazur distance between two normed spaces $X$ and $Y$. Let $X$ be an $n$-dimensional normed space with $\text{d(X,}\,l_{\infty }^{n}\text{)}\,\le \,\text{2}$, where $l_{\infty }^{n}$ stands for ${{\mathbb{R}}^{n}}$ endowed with the norm $\parallel ({{x}_{1}},\,.\,.\,.\,,\,{{x}_{n}}){{\parallel }_{\infty }}\,:=\,\max \{|{{x}_{1}}|,\,.\,.\,.\,,\,|{{x}_{n}}|\}$. Then every metric space $(S,\,\rho )$ of cardinality $n+1$ with norm $\rho $ satisfying the condition $\max D/\min D\,\le \,2/\,\text{d(}X,\,l_{\infty }^{n}\text{)}$ for $D\,:=\,\{\rho (a,\,b)\,:\,a,\,b\,\in \,S,\,a\,\ne \,b\}$ can be isometrically embedded into $X$.

We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are Morita equivalent, then so are the quasi-Baer right ring hulls ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(R)$ and ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(S)$ of $R$ and $S$, respectively. As an application, we prove that if unital ${{C}^{*}}$-algebras $A$ and $B$ are Morita equivalent as rings, then the bounded central closure of $A$ and that of B are strongly Morita equivalent as ${{C}^{*}}$-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring $A[G]$ of a torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous ring $A$. Examples that illustrate and delimit the results of this paper are provided.

In this paper, we define and study the geometry of semi-slant submanifolds of an almost paracontact metric manifold. We give some characterizations for a submanifold to be semi-slant submanifold to be semi-slant product and obtain integrability conditions for the distributions involved in the definition of a semi-slant submanifold.

We characterize those linear projections represented as a convex combination of two surjective isometries on standard Banach spaces of continuous functions with values in a strictly convex Banach space.

The bigraded Hilbert function and the minimal free resolutions for the diagonal coinvariants of the dihedral groups are exhibited, as well as for all their bigraded invariant Gorenstein quotients.

In this paper, the boundedness from below of multiplication operators between $\alpha$-Bloch spaces ${{B}^{\alpha }},\,\alpha \,>\,0$, on the unit disk $D$ is studied completely. For a bounded multiplication operator ${{M}_{u}}\,:\,{{B}^{\alpha }}\,\to \,{{B}^{\beta }}$, defined by ${{M}_{u}}f\,=\,uf$ for $f\,\in \,{{B}^{\alpha }}$, we prove the following result:

(i) If $0<\beta <\alpha ,\,\text{or}\,\text{0}<\alpha \le \text{1}\,\text{and}\,\alpha <\beta \text{,}\,{{M}_{u}}$ is not bounded below;

(ii) if $0\,<\,\alpha \,=\,\beta \,\le \,1,\,{{M}_{u}}$ is bounded below if and only if lim ${{\inf }_{z\to \partial D}}\,\left| u\left( z \right) \right|\,>\,0;$

(iii) if $1\,<\,\alpha \,\le \,\beta ,\,{{M}_{u}}$ is bounded below if and only if there exist a $\delta \,>\,0$ and a positive $r\,<\,1$ such that for every point $z\,\in \,D$ there is a point ${{z}^{'}}\,\in \,D$ with the property $d\left( {{z}^{'}},\,z \right)\,<\,r$ and ${{\left( 1\,-\,{{\left| {{z}^{'}} \right|}^{2}} \right)}^{\beta -\alpha }}\left| u\left( {{z}^{'}} \right) \right|\,\ge \,\delta$, where $d\left( \cdot ,\,\cdot \right)$ denotes the pseudo-distance on $D$.

Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants, and the direct summand property holds if there is a surjective $k{{[V]}^{G}}$-linear map $\pi \,:\,k[V]\,\to \,k{{[V]}^{G}}$.

The following Chevalley–Shephard–Todd type theorem is proved. Suppose $G$ is abelian. Then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds.

Let $P$ be a maximal proper parabolic subgroup of a connected simple linear algebraic group $G$, defined over $\mathbb{C}$, such that $n\,:=\,{{\dim}_{\mathbb{C}}}\,G/P\,\ge \,4$. Let $\iota :\,Z\,\to \,G/P$ be a reduced smooth hypersurface of degree at least $\left( n\,-\,1 \right)\,.\,\deg \text{ree}\left( T\left( G/P \right) \right)/n$. We prove that the restriction of the tangent bundle ${{\iota }^{*}}\,TG/P$ is semistable.

We describe the representation theory of ${{C}^{*}}$-crossed-products of a unital ${{C}^{*}}$-algebra $A$ by the cyclic group of order 2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to $A$ is irreducible and those who are the sum of two unitarily unequivalent representations of $A$. We characterize each class in term of the restriction of the representations to the fixed point ${{C}^{*}}$-subalgebra of $A$. We apply our results to compute the $K$-theory of several crossed-products of the free group on two generators.

For a given convex body $K$ in ${{\mathbb{R}}^{d}}$, a random polytope ${{K}^{(n)}}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of ${{K}^{(n)}}$ and $K$ as $n$ tends to infinity. For a simplicial polytope $P$, a precise asymptotic formula for the difference of the mean widths of ${{P}^{(n)}}$ and $P$ is obtained.

Let $R$ be a dense subring of $\text{End}\left( _{D}V \right)$, where $V$ is a left vector space over a division ring $D$. If $\dim{{\,}_{D}}V\,=\,\infty $, then the range of any nonzero polynomial $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ on $R$ is dense in $\text{End}\left( _{D}V \right)$. As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\,\ne \,a\,\in \,R$. If $af{{\left( {{x}_{1}},\ldots ,{{x}_{m}} \right)}^{n\left( {{x}_{i}} \right)}}\,=\,0$ for all ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$, where $n\left( {{x}_{i}} \right)$ is a positive integer depending on ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$, then $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.

We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field.

In this paper, we investigate a proper $\text{CAT(0)}$ space $(X,\,d)$ that is homeomorphic to ${{\mathbb{R}}^{2}}$ and we show that the asymptotic dimension asdim$(X,\,d)$ is equal to 2.

We show that if the general real method ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$.