We study a second order nonlinear system driven by the vector $p$-Laplacian, with a multivalued nonlinearity and defined on the positive time semi-axis ${{\mathbb{R}}_{+}}.$ Using degree theoretic techniques we solve an auxiliary mixed boundary value problem defined on the finite interval $\left[ 0,\,n \right]$ and then via a diagonalization method we produce a solution for the original infinite time horizon system.

How few three-term arithmetic progressions can a subset $S\,\subseteq \,{{\mathbb{Z}}_{N}}\,:=\,\mathbb{Z}\,/\,N\mathbb{Z}$ have if $|S|\,\ge \,\upsilon N$ (that is, $S$ has density at least $\upsilon$)? Varnavides showed that this number of arithmetic progressions is at least $c(v)\,{{N}^{2}}$ for sufficiently large integers $N$. It is well known that determining good lower bounds for $c\left( \upsilon \right)\,>\,0$ is at the same level of depth as Erdös's famous conjecture about whether a subset $T$ of the naturals where $\sum{_{n\in T}\,1/n}$ diverges, has a $k$-term arithmetic progression for $k\,=\,3$ (that is, a three-term arithmetic progression).

We answer a question posed by B. Green about how this minimial number of progressions oscillates for a fixed density $\upsilon$ as $N$ runs through the primes, and as $N$ runs through the odd positive integers.

We prove that the neighborhood complex $\text{N}\left( G \right)$, the box complex $\text{B}\left( G \right)$, the homomorphism complex $\text{Hom}\left( {{K}_{2}},\,G \right)$ and the Lovász complex $\text{L}\left( G \right)$ have the same simple ${{\mathbb{Z}}_{2}}$ -homotopy type in the sense of Whitehead. We show that these graph complexes are simple ${{\mathbb{Z}}_{2}}$ -universal.

A new semilocal convergence result for the Picard method is presented, where the main required condition in the contraction mapping principle is relaxed.

A Mauldin–Williams graph $M$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a ${{C}^{*}}$-algebra ${{O}_{M}}\left( K \right)$ with a Mauldin–Williams graph $M$ and the invariant set $K$, laying emphasis on the singular points. We assume that the underlying graph $G$ has no sinks and no sources. If $M$ satisfies the open set condition in $K$, and $G$ is irreducible and is not a cyclic permutation, then the associated ${{C}^{*}}$-algebra ${{O}_{M}}\left( K \right)$ is simple and purely infinite. We calculate the $K$-groups for some examples including the inflation rule of the Penrose tilings.

In this paper, we generalize a result recently obtained by the author. We characterize the cyclic vectors in $L_{a}^{p}\,\left( \mathbb{C},\,\phi \right)$ . Let $f\,\in \,L_{a}^{p}\,\left( \mathbb{C},\,\phi \right)$ and $fC$ be contained in the space. We show that $f$ is non-vanishing if and only if $f$ is cyclic.

Let $\Omega$ be a domain in ${{\mathbb{R}}^{n}}\,(n\,\ge \,2).$ We find a necessary and sufficient topological condition on $\Omega$ such that, for any measure $ $ on ${{\mathbb{R}}^{n}}$ , there is a function $u$ with specified boundary conditions that satisfies the Poisson equation $\Delta u\,=\,\mu$ on $\Omega$ in the sense of distributions.

We find a lower bound on the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick.

Let ${{B}_{p}}$ be the unit ball in ${{\mathbb{L}}_{p}}$ , $0\,<\,p\,<\,1$, and let $\Delta _{+}^{s}$ , $s\,\in \,\mathbb{N}$, be the set of all $s$-monotone functions on a finite interval $I$, i.e., $\Delta _{+}^{s}$ consists of all functions $x\,:\,I\,\mapsto \,\mathbb{R}$ such that the divided differences $[x;\,{{t}_{0}},\,...\,,\,{{t}_{s}}]$ of order $s$ are nonnegative for all choices of $\left( s\,+\,1 \right)$ distinct points ${{t}_{0}},\,.\,.\,.\,,{{t}_{s}}\,\in \,I.$ For the classes $\Delta _{+}^{s}{{B}_{P}}\,:=\,\Delta _{+}^{s}\,\cap \,{{B}_{P}},$ we obtain exact orders of Kolmogorov, linear and pseudo-dimensional widths in the spaces ${{\mathbb{L}}_{q}},$$0\,<\,q\,<\,p\,<\,1$:

$${{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{psd}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{kol}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{lin}}\asymp {{n}^{-s}}.$$

For unimodular semidirect products of locally compact amenable groups $N$ and $H$, we show that one can always construct a Følner net of the form $({{A}_{\alpha }}\,\times \,{{B}_{\beta }})$ for $G$, where $({{A}_{\alpha }})$ is a strong form of Følner net for $N$ and $({{B}_{\beta }})$ is any Følner net for $H$. Applications to the Heisenberg and Euclidean motion groups are provided.

In this article we study in detail the expansion of the Riemann $\Xi$ function in Meixner– Pollaczek polynomials. We obtain explicit formulas, recurrence relation and asymptotic expansion for the coefficients and investigate the zeros of the partial sums.

The well-known Falkner–Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to $\text{ }\!\!\lambda\!\!\text{ }\!\!\pi\!\!\text{ /2}$, where $\text{ }\!\!\lambda\!\!\text{ }\,\in \,\mathbb{R}$ is a parameter involved in the equation. It is known that there exists ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,<\,0$ such that the equation with suitable boundary conditions has at least one positive solution for each $\text{ }\!\!\lambda\!\!\text{ }\,\ge \,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ and has no positive solutions for $\text{ }\!\!\lambda\!\!\text{ }\,<\,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ . The known numerical result shows ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,=\,-0.1988$ . In this paper, ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,\in \,[-0.4,\,-0.12]$ is proved analytically by establishing a singular integral equation which is equivalent to the Falkner–Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner–Skan equation.

Let $M$ be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue $\text{ }\!\!\lambda\!\!\text{ }.$ We give upper and lower bounds on the inner radius of the type $C/{{\lambda }^{\alpha }}{{(\log \lambda )}^{\beta }}.$ Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincaré type inequality proved by Maz’ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

We prove, among other things, that if ${{C}_{p}}\left( X \right)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces.

Let ${{b}_{1}}$, ${{b}_{2}}$ be any integers such that $\gcd \left( {{b}_{1}},{{b}_{2}} \right)=1$ and ${{c}_{1}}\left| {{b}_{1}} \right|\,<\,\left| {{b}_{2}} \right|\,\le \,{{c}_{2}}\left| {{b}_{1}} \right|$ , where ${{c}_{1}}$, ${{c}_{2}}$ are any given positive constants. Let $n$ be any integer satisfying $\gcd \left( n,\,{{b}_{i}} \right)\,=\,1$ , $i\,=\,1,\,2$. Let ${{P}_{k}}$ denote any integer with no more than $k$ prime factors, counted according to multiplicity. In this paper, for almost all ${{b}_{2}}$ , we prove (i) a sharp lower bound for $n$ such that the equation ${{b}_{1}}p\,+\,{{b}_{2}}m\,=\,n$ is solvable in prime $p$ and almost prime $m\,=\,{{P}_{k}}$ , $k\,\ge \,3$ whenever both ${{b}_{i}}$ are positive, and (ii) a sharp upper bound for the least solutions $p$, $m$ of the above equation whenever ${{b}_{i}}$ are not of the same sign, where $p$ is a prime and $m\,=\,{{P}_{k}}$ , $k\,\ge \,3$.

We present a construction of singular rearrangement invariant functionals on Marcinkiewicz function/operator spaces. The functionals constructed differ from all previous examples in the literature in that they fail to be symmetric. In other words, the functional $\phi$ fails the condition that if $x\prec \prec \,Y$ (Hardy-Littlewood-Polya submajorization) and $0\,\le \,x,\,y$, then $0\,\le \,\phi \left( x \right)\,\le \,\phi \left( y \right)$. We apply our results to singular traces on symmetric operator spaces (in particular on symmetrically-normed ideals of compact operators), answering questions raised by Guido and Isola.

We use the mutually stationary sets of Foreman and Magidor as a tool to establish the validity of the two-cardinal version of the diamond principle in some special cases.

An $n$-dimensional quantum torus is a twisted group algebra of the group ${{\mathbb{Z}}^{n}}.$ It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational $n$-dimensional quantum tori over any field. Moreover, we show that for $n\,=\,2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

This paper deals with the concepts of condensed and strongly condensed domains. By definition, an integral domain $R$ is condensed (resp. strongly condensed) if each pair of ideals $I$ and $J$ of $R$, $IJ\,=\,\{ab/a\,\in \,I,\,b\,\in \,J\}$ (resp. $IJ\,=\,aJ$ for some $a\,\in \,I\,or\,I\,J\,=\,Ib$ for some $b\,\in \,J$). More precisely, we investigate the ideal theory of condensed and strongly condensed domains in Noetherian-like settings, especially Mori and strong Mori domains and the transfer of these concepts to pullbacks.

The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.