We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, a concept recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion, we count the number of orbits for the action of the subgroup of fixed points (respectively periodic) points on the set of infinite fixed (respectively periodic) points. Finally, we study the dynamics of infinite points: for automorphisms and some endomorphisms, defined in a precise way, fitting a classification given by Delgado and Ventura, we prove that every infinite point is either periodic or wandering, which implies that the dynamics is asymptotically periodic.

Given groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered.

We establish multiplicity results for the following class of quasilinear problems P

\begin{equation*} \left\{ \begin{array}{@{}l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in} \quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right. \end{equation*}

$\Delta _{\Phi }u=\text {div}(\varphi (x,|\nabla u|)\nabla u)$

$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$

$\Omega \subset \mathbb {R}^{N}$

$\Omega _N$

$\Omega _p$

$\overline {\Omega _N}\cap \overline {\Omega _p}=\emptyset$

$(P)$

$-\Delta _{\Phi }$

$-\Delta _N$

$\Omega _N$

$-\Delta _p$

$\Omega _p$

$f:\Omega \times \mathbb {R}\to \mathbb {R}$

$e^{\alpha |t|^{\frac {N}{N-1}}}$

$\Omega _N$

$|t|^{p^{*}-2}t$

$\Omega _p$

$|t|$

$\alpha >0$

$p^{*}=\frac {Np}{N-p}$

$1< p< N$

$f$

$f$

$\Omega$

$(P)$

Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of $n$ points of an algebraic surface is algebraic at a CM point $\tau$ and rational numbers $z_1$ and $z_2$. Our result gives a refinement of the algebraicity on Betti numbers.

Relations between the Atiyah–Patodi–Singer rho invariant and signatures of links have been known for a long time, but they were only partially investigated. In order to explore them further, we develop a versatile cut-and-paste formula for the rho invariant, which allows us to manipulate manifolds in a convenient way. With the help of this tool, we give a description of the multivariable signature of a link $L$ as the rho invariant of some closed three-manifold $Y_L$ intrinsically associated with $L$. We study then the rho invariant of the manifolds obtained by the Dehn surgery on $L$ along integer and rational framings. Inspired by the results of Casson and Gordon and Cimasoni and Florens, we give formulas expressing this value as a sum of the multivariable signature of $L$ and some easy-to-compute extra terms.

Let $X$ be a connected complex manifold and let $Z$ be a compact complex subspace of $X$. Assume that ${\rm Aut}(Z)$ is strongly Jordan. In this paper, we show that the automorphism group ${\rm Aut}(X,\, Z)$ of all biholomorphisms of $X$ preserving $Z$ is strongly Jordan. A similar result has been proved by Meng et al. for a compact Kähler submanifold $Z$ of $X$ instead of a compact complex subspace $Z$ of $X$. In addition, we also show some rigidity result for free actions of large groups on complex manifolds.

Let $\mathcal {O}(\pi )$ denote the number of odd parts in an integer partition $\pi$. In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$, where $\pi '$ is the conjugate of $\pi$. Let $p(r,\,m;n)$ denote the number of partitions of $n$ with srank congruent to $r$ modulo $m$. Generating function identities, congruences and inequalities for $p(0,\,4;n)$ and $p(2,\,4;n)$ were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for $p(r,\,m;n)$ with $m=16$ and $24$. These results are refinements of some inequalities due to Swisher.

In this paper, we study the Lieb's translation lemma in Coulomb–Sobolev space and then apply it to investigate the existence of Pohožaev type ground state solution for elliptic equation with van der Waals type potential.

We study the asymptotic behaviour of the periodically mixed Zaremba problem. We cover the part of the boundary by a chess board with a small period (square size) $\varepsilon$ and impose the Dirichlet condition on black and the Neumann condition on white squares. As $\varepsilon \to 0$, we get the effective boundary condition which is always of the Dirichlet type. The Dirichlet data on the boundary, however, depend on the ratio between the magnitudes of the two boundary values.

We prove that certain families of homogenous affine iterated function systems in $\mathbb {R}^{d}$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters. These examples demonstrate that in the theorems of Falconer and Jordan–Pollicott–Simon on the almost sure dimensions of self-affine sets and measures, the set of exceptional translation parameters can be a dense set. The proof combines results from the literature on self-affine tilings of $\mathbb {R}^{d}$ with an adaptation of a classic argument of Erdős on the singularity of certain Bernoulli convolutions. This result encompasses a one-dimensional example due to Kenyon which arises as a special case.

In this paper, we consider the following two-component Schrödiner system

\[ \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} & \text{in}\;\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 & \text{in}\;\mathbb{R}^{3}. \end{cases} \]

A thin Lie algebra is a Lie algebra $L$, graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$, and such that each non-zero ideal of $L$ lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of $L$ (that is, the next diamond past $L_1$) occurs in degree $k$. We prove that if $k>5$, then $[Lyy]=0$ for some non-zero element $y$ of $L_1$. In characteristic different from two this means $y$ is a sandwich element of $L$. We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.

We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$, a Sylow $p$-subgroup of one complement is conjugate to a Sylow $p$-subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only if, for each prime $p$, there exists a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $S\cap N$ in $S$ are conjugate in $G$. In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of $S\cap N$ in $S$ be conjugate within $S$. We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma.

Let $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$-grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$, $b\in \mathbb {N}$, we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$, as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$-algebras which are not PI equivalent. Actually, we prove that the $T_{G}$-ideal of the former algebra is contained in the $T$-ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

We prove a version of the Fatou theorem for bounded functions with a bounded $\overline \partial _J$ part of the differential on wedge-type domains in an almost complex manifold.

When $p$ is an odd prime, Delbourgo observed that any Kubota–Leopoldt $p$-adic $L$-function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on $p$, and without the Euler factor when the character is non-trivial, by computing the periods of appropriate measures. As an application, we will reprove the Ferrero–Greenberg formula for the derivative $L_p'(0,\chi )$. We will also discuss the convergence of sum expressions in terms of elementary $p$-adic analysis, as well as their relation to Stickelberger elements; such discussions in turn give alternative proofs of the validity of sum expressions.

We characterize the generalized Auslander–Reiten duality on the category of finitely presented modules over some certain Hom-finite category. Examples include the category FI of finite sets with injections, and the one VI of finite-dimensional vector spaces with linear injections over a finite field.

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Motivated by the works of Andrews and Merca, Guo and Zeng deduced truncated versions for two other classical theta series identities of Gauss. Very recently, Xia et al. proved new truncated theorems of the three classical theta series identities by taking different truncated series than the ones chosen by Andrews–Merca and Guo–Zeng. In this paper, we provide a unified treatment to establish new truncated versions for the three identities of Euler and Gauss based on a Bailey pair due to Lovejoy. These new truncated identities imply the results proved by Andrews–Merca, Wang–Yee, and Xia–Yee–Zhao.

A group is said to have rational growth with respect to a generating set if the growth series is a rational function. It was shown by Parry that certain torus bundle groups of even trace exhibits rational growth. We generalize this result to a class of torus bundle groups with odd trace.

We prove the existence of exponentially and superexponentially localized breather solutions for discrete nonlinear Klein–Gordon systems. Our approach considers $d$-dimensional infinite lattice models with general on-site potentials and interaction potentials being bounded by an arbitrary power law, as well as, systems with purely anharmonic forces, cases which are much less studied particularly in a higher-dimensional set-up. The existence problem is formulated in terms of a fixed-point equation considered in weighted sequence spaces, which is solved by means of Schauder's Fixed-Point Theorem. The proofs provide energy bounds for the solutions depending on the lattice parameters and its dimension under physically relevant non-resonance conditions.