We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's theorem due to DeTurck and Gordon.

We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\delta$ arbitrarily small and positive, the nearest neighbor spacings between integers $n$ with $\left| \omega \left( n \right)\,-\,\log \log n \right|\,<\,{{\left( \log \log n \right)}^{\delta }}$ obey the Poisson distribution law.

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.

In this paper we shall discuss nonlinear multipoint boundary value problems for second order differential equations when deviating arguments depend on the unknown solution. Sufficient conditions under which such problems have extremal and quasi-solutions are given. The problem of when a unique solution exists is also investigated. To obtain existence results, a monotone iterative technique is used. Two examples are added to verify theoretical results.

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.

Classical results of Kalton and techniques of Feder are used to study the complementation of the space $W\left( X,\,Y \right)$ of weakly compact operators and the space $K\left( X,\,Y \right)$ of compact operators in the space $L\left( X,\,Y \right)$ of all bounded linear maps from $X$ to $Y$.

In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the Hardy–Littlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type (1, 1) estimates for these maximal functions. The ${{L}^{p}}$ estimates for $p\,>\,1$ follow fromthe interpolation. Some applications are included.

The classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of ${{l}_{p}}$ spaces. In a previous paper, an interpolation space, defined via the real method and using ${{l}_{p}}$ spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm.

We exhibit infinitely many hyperbolic 3-manifold groups that are not right-orderable.

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.

In this paper, we find a class of $\left( \alpha ,\,\beta \right)$ metrics which have a bounded Cartan torsion. This class contains all Randers metrics. Furthermore, we give some applications and obtain two corollaries about curvature of this metrics.

Using the time change method we show how to construct a solution to the stochastic equation $d{{X}_{t}}\,=\,b({{X}_{t}}\_)d{{Z}_{t}}\,+\,a({{X}_{t}})dt$ with a nonnegative drift a provided there exists a solution to the auxililary equation $d{{L}_{t}}=[{{a}^{-1/\alpha }}b]({{L}_{t}}\_)d\overline{{{Z}_{t}}}+dt$ where $Z,\,\overline{Z}$ are two symmetric stable processes of the same index $\alpha \,\in \,(0,\,2]$. This approach allows us to prove the existence of solutions for both stochastic equations for the values $0\,<\,\alpha \,<\,1$ and only measurable coefficients $a$ and $b$ satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.

We consider the problem of simultaneous extension of continuous convex metrics defined on subcontinua of a Peano continuum. We prove that there is an extension operator for convex metrics that is continuous with respect to the uniform topology.

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.

We prove that the least perimeter $P(n)$ of a partition of a smooth, compact Riemannian surface into $n$ regions of equal area $A$ is asymptotic to $n/2$ times the perimeter of a planar regular hexagon of area $A$. Along the way, we derive tighter estimates for flat tori, Klein bottles, truncated cylinders, and Möbius bands.

We give a further decay estimate for the Dziubański–Hernández wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley–Wiener theorem for ultradifferentiable functions.

A Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial — one with $\{0,\,1\}$-coefficients — and shows that they form a strictly increasing sequence with limit $\left( 1\,+\,\sqrt{5} \right)/2$. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.

We consider semilinear evolution equations with some locally Lipschitz nonlinearities, perturbed by Banach space valued, continuous, and adapted stochastic process. We show that under some assumptions there exists a solution to the equation. Using the result we show that there exists a mild, continuous, global solution to a semilinear Itô equation with locally Lipschitz nonlinearites. An example of the equation is given.

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 $n\,\ge \,0$.

The fiber ${{W}_{n}}$ of the double suspension ${{S}^{2n-1}}\,\to \,{{\Omega }^{2}}{{S}^{2n+1}}$ is known to have a classifying space $B{{W}_{n}}$. An important conjecture linking the $EPH$ sequence to the homotopy theory of Moore spaces is that $B{{W}_{n}}\,\simeq \,\Omega {{T}^{2np+1}}(p)$, where ${{T}^{2np+1}}(p)$ is Anick's space. This is known if $n\,=\,1$. We prove the $n\,=\,p$ case and establish some related properties.