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

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.

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.

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 consider when a single submodule and also when every submodule of a module $M$ over a general ring $R$ has a unique closure with respect to a hereditary torsion theory on Mod-$R$.

In this paper, we give two characterizations of the exactness of operator spaces.

An integral domain $R$ is IDPF (Irreducible Divisors of Powers Finite) if, for every non-zero element $a$ in $R$, the ascending chain of non-associate irreducible divisors in $R$ of ${{a}^{n}}$ stabilizes on a finite set as $n$ ranges over the positive integers, while $R$ is atomic if every non-zero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension $S$ of $R$ is a root extension or radical extension if for each $s$ in $S$, there exists a natural number $n\left( s \right)$ with ${{s}^{n\left( s \right)}}$ in $R$. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $\left( R,\,S \right)$ is governed by the relative sizes of the unit groups $\text{U}\left( R \right)$ and $\text{U}\left( S \right)$ and whether $S$ is a root extension of $R$. The following results are deduced from these considerations: An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let $R$ be a Noetherian domain with integral closure $S$. Suppose the conductor of $S$ into $R$ is non-zero. Then $R$ is IDPF if and only if $S$ is a root extension of $R$ and $\text{U}\left( S \right)/\text{U}\left( R \right)$ is finite.

We prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple ${{C}^{*}}$-algebras.

We give weighted norm inequalities for the maximal fractional operator ${{\mathcal{M}}_{q}},\beta $ of Hardy–Littlewood and the fractional integral ${{I}_{\gamma }}$. These inequalities are established between ${{\left( {{L}^{q}},\,{{L}^{p}} \right)}^{\alpha }}\left( X,\,d,\,\mu \right)$ spaces (which are superspaces of Lebesgue spaces ${{L}^{\alpha }}\left( X,\,d,\,\mu \right)$ and subspaces of amalgams $\left( {{L}^{q}},\,{{L}^{p}} \right)\left( X,d,\mu \right)$) and in the setting of space of homogeneous type $\left( X,d,\mu \right)$. The conditions on the weights are stated in terms of Orlicz norm.

Let $X$ and $Y$ be Banach spaces isomorphic to complemented subspaces of each other with supplements $A$ and $B$. In 1996, W. T. Gowers solved the Schroeder–Bernstein (or Cantor–Bernstein) problem for Banach spaces by showing that $X$ is not necessarily isomorphic to $Y$. In this paper, we obtain a necessary and sufficient condition on the sextuples $\left( p,\,q,\,r,\,s,\,u,\,v \right)$ in $\mathbb{N}$ with $p\,+\,q\,\ge \,1$, $r+s\ge 1$ and $u,\,v\,\in \,{{\mathbb{N}}^{*}}$, to provide that $X$ is isomorphic to $Y$, whenever these spaces satisfy the following decomposition scheme

$${{A}^{u}}\,\sim \,{{X}^{p}}\,\oplus \,{{Y}^{q}},\,{{B}^{v}}\,\sim \,{{X}^{r}}\,\oplus \,{{Y}^{s}}.$$

Namely, $\Phi \,=\,\left( p\,-\,u \right)\left( s\,-\,v \right)-\left( q\,+\,u \right)\left( r\,+\,v \right)$ is different from zero and $\Phi $ divides $p\,+\,q$ and $r\,+\,s$. These sextuples are called Cantor–Bernstein sextuples for Banach spaces. The simplest case $\left( 1,0,0,1,1,1 \right)$ indicates the well-known Pełczyński's decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder– Bernstein problem become evident.

We extend the scope of B. Shapirovskii's results on the order of $\pi $-bases in compact spaces and answer some questions of V. Tkachuk.

The existence of the global attractor of a damped forced Hirota equation in the phase space ${{H}^{1}}\left( \mathbb{R} \right)$ is proved. The main idea is to establish the so-called asymptotic compactness property of the solution operator by energy equation approach.

We prove that if the $\left( 1,\,1 \right)$-current of integration on an analytic subvariety $V\,\subset \,D$ satisfies the uniform Blaschke condition, then $V$ is the zero set of a holomorphic function $f$ such that $\log \,\left| f \right|$ is a function of bounded mean oscillation in $bD$. The domain $D$ is assumed to be smoothly bounded and of finite d’Angelo type. The proof amounts to non-isotropic estimates for a solution to the $\overline{\partial }$-equation for Carleson measures.

Let $R$ be a unit-regular ring and let $\sigma $ be an endomorphism of $R$ such that $\sigma \left( e \right)\,=\,e$ for all ${{e}^{2}}\,=\,e\,\in \,R$ and let $n\,\ge \,0$. It is proved that every element of $R[x;\,\sigma ]/\left( {{x}^{n+1}} \right)$ is equivalent to an element of the form ${{e}_{0}}\,+\,{{e}_{1}}x\,+\,\cdots \,+\,{{e}_{n}}{{x}^{n}}$, where the ${{e}_{i}}$ are orthogonal idempotents of $R$. As an application, it is proved that $R[x;\,\sigma ]/\left( {{x}^{n+1}} \right)$ is left morphic for each $n\,\ge \,0$.

We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality

$${{\left( \int_{E}{\Phi {{\left( {{T}_{k}}f\left( x \right) \right)}^{q}}u\left( x \right)dx} \right)}^{1/q}}\,\le \,C{{\left( \int_{E}{\Phi {{\left( f\left( x \right) \right)}^{p}}v\left( x \right)dx} \right)}^{1/p}},$$

where $0\,<\,p,\,q\,<\,\infty $, $\Phi $ is a logarithmically convex function, and ${{T}_{k}}$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin–Cochran-Lee type inequalities.

We discuss properties of the regular part ${{S}_{\text{reg}}}$ of a subcartesian space $S$. We show that ${{S}_{\text{reg}}}$ is open and dense in $S$ and the restriction to ${{S}_{\text{reg}}}$ of the tangent bundle space of $S$ is locally trivial.

We consider semilinear periodic problems with the right-hand side nonlinearity satisfying a double resonance condition between two successive eigenvalues. Using a combination of variational and degree theoretic methods, we prove the existence of at least two nontrivial solutions.

In an $\text{H}$-closed, Urysohn space, disjoint $\text{H}$-sets can be separated by disjoint open sets. This is not true for an arbitrary H-closed space even if one of the $\text{H}$-sets is a point. In this paper, we provide a systematic study of those spaces in which disjoint $\text{H}$-sets can be separated by disjoint open sets.

This paper studies the existence and uniqueness of almost periodic solutions of nonlinear impulsive functional differential equations with infinite delay. The results obtained are based on the Lyapunov–Razumikhin method and on differential inequalities for piecewise continuous functions.