We study hypersurfaces of constant mean curvature immersed into warped product spaces of the form $\mathbb{R}\times_\varrho\mathbb{P}^n$, where $\mathbb{P}^n$ is a complete Riemannian manifold. In particular, our study includes that of constant mean curvature hypersurfaces in product ambient spaces, which have recently been extensively studied. It also includes constant mean curvature hypersurfaces in the so-called pseudo-hyperbolic spaces. If the hypersurface is compact, we show that the immersion must be a leaf of the trivial totally umbilical foliation $t\in\mathbb{R}\mapsto\{t\}\times\mathbb{P}^n$, generalizing previous results by Montiel. We also extend a result of Guan and Spruck from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel and to derive height estimates for compact constant mean curvature hypersurfaces with boundary in a leaf.

A terrace for $\mathbb{Z}_m$ is a particular type of sequence formed from the $m$ elements of $\mathbb{Z}_m$. For $m$ odd, many procedures are available for constructing power-sequence terraces for $\mathbb{Z}_m$; each terrace of this sort may be partitioned into segments, of which one contains merely the zero element of $\mathbb{Z}_m$, whereas every other segment is either a sequence of successive powers of an element of $\mathbb{Z}_m$ or such a sequence multiplied throughout by a constant. We now refine this idea to show that, for $m=n-1$, where $n$ is an odd prime power, there are many ways in which power-sequences in $\mathbb{Z}_n$ can be used to arrange the elements of $\mathbb{Z}_n\setminus\{0\}$ in a sequence of distinct entries $i$, $1\le i\le m$, usually in two or more segments, which becomes a terrace for $\mathbb{Z}_m$ when interpreted modulo $m$ instead of modulo $n$. Our constructions provide terraces for $\mathbb{Z}_{n-1}$ for all prime powers $n$ satisfying $0\ltn\lt300$ except for $n=125$, $127$ and $257$.

The symmetric inverse monoid $\mathcal{I}_{n}$ is the set of all partial permutations of an $n$-element set. The largest possible size of a $2$-generated subsemigroup of $\mathcal{I}_{n}$ is determined. Examples of semigroups with these sizes are given. Consequently, if $M(n)$ denotes this maximum, it is shown that $M(n)/|\mathcal{I}_{n}|\rightarrow1$ as $n\rightarrow\infty$. Furthermore, we deduce the known fact that $\mathcal{I}_{n}$ embeds as a local submonoid of an inverse $2$-generated subsemigroup of $\mathcal{I}_{n+1}$.

Let $\varphi(\cdot)$ be the Euler function and let $\sigma(\cdot)$ be the sum-of-divisors function. In this note, we bound the number of positive integers $n\le x$ with the property that $s(n)=\sigma(n)-n$ divides $\varphi(n)$.

We study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.

We consider the quasilinear elliptic variational system

\begin{alignat*} {2} -\Delta_pu\amp=\lambda F_u(x,u,v)+\mu H_u(x,u,v) \quad\amp \amp\text{in }\varOmega, \\ -\Delta_qv\amp=\lambda F_v(x,u,v)+\mu H_v(x,u,v) \amp \amp \text{in }\varOmega, \\ u\amp=v=0 \amp \amp \text{on }\partial\varOmega, \end{alignat*}

where $\varOmega$ is a strip-like domain and $\lambda$ and $\mu$ are positive parameters. Using a recent two-local-minima theorem and the principle of symmetric criticality, existence and multiplicity are proved under suitable conditions on $F$.

We establish two separation theorems in which the classic interior is replaced by the quasi-relative interior.

In this paper, we obtain some results on the existence of solutions for the system

$$ (-\Delta+q_i)u_i=\mu_im_iu_i+f_i(x,u_1,\dots,u_n)\text{ in }\mathbb{R}^{N},\quad i=1,\dots,n, $$

where each of the $q_i$ are positive potentials satisfying $\lim_{|x|\rightarrow+\infty}q_i(x)=+\infty$, each of the $m_i$ are bounded positive weights and each of the $\mu_i$ are real parameters. Depending upon the hypotheses on $f_i$, we use either the method of sub- and supersolutions or a bifurcation method.

The operator Cauchy dual to a $2$-hyperexpansive operator $T$, given by $T'\equiv T(T^*T)^{-1}$, turns out to be a hyponormal contraction. This simple observation leads to a structure theorem for the $C^*$-algebra generated by a $2$-hyperexpansion, and a version of the Berger–Shaw theorem for $2$-hyperexpansions.

As an application of the hyperexpansivity version of the Berger–Shaw theorem, we show that every analytic $2$-hyperexpansive operator with finite-dimensional cokernel is unitarily equivalent to a compact perturbation of a unilateral shift.

We work with interpolation methods for $N$-tuples of Banach spaces associated with polygons. We compare necessary conditions for interpolating closed operator ideals with conditions required to interpolate compactness. We also establish a formula for the measure of non-compactness of interpolated operators.

We study the membership of derivatives of Blaschke products in Hardy and Bergman spaces, especially for the the interpolating Blaschke products and for those whose zeros lie in a Stolz domain. We obtain new and very simple proofs of some known results and prove new theorems that complement or extend the earlier works of Ahern, Clark, Cohn, Kim, Newman, Protas, Rudin, Vinogradov and others.

We study the problem of finding conditions which guarantee that given operator data propagate under the control of unitary operators. This is a kind of moment problem and the way we solve it is based on unitary dilations. Comparing it with other results in this area, our approach has the advantages of

allowing the data to be unbounded operators,

considering arbitrary dilations, not necessarily regular ones.

The relationship between the moment problem and harmonizable multivariate discrete processes is indicated.

The asymptotic behaviour of a certain integral is investigated. The investigation involves a hypergeometric function of a type for which the asymptotics have not previously been considered.

Asymptotic expansions for an incomplete Bessel function of large argument are derived when the parametric point (a) is well away from any saddle point, (b) coincides with a saddle point and (c) is in the neighbourhood of a saddle point.

We characterize algebra homomorphisms from the Lebesgue algebra $L^1_\omega(\mathbb{R})$ into a Banach algebra $\mathcal{A}$. As a consequence of this result, every bounded algebra homomorphism $\varPhi:L^1_\omega(\mathbb{R})\to\mathcal{A}$ is approached through a uniformly bounded family of fractional homomorphisms, and the Hille–Yosida theorem for $C_0$-groups is proved.

In this paper, the dual $L_p$ John ellipsoids, which include the classical Löwner ellipsoid and the Legendre ellipsoid, are studied. The dual $L_p$ versions of John's inclusion and Ball's volume-ratio inequality are shown. This insight allows for a unified view of some basic results in convex geometry and reveals further the amazing duality between Brunn–Minkowski theory and dual Brunn–Minkowski theory.