We compute the Nielsen number $N(f)$ of a self homeomorphism $f$ of a three dimensional flat manifold. Furthermore, we determine the possible values of $N(f)$ when $f$ is an arbitrary self-map.

We prove that in a regular category all reflexive and transitive relations are symmetric if and only if every internal category is an internal groupoid. In particular, these conditions hold when the category is $n$-permutable for some $n$.

In this paper we prove Carleman's approximation type theorems in the framework of slice regular functions of a quaternionic variable. Specifically, we show that any continuous function defined on $\mathbb{R}$ and quaternion valued, can be approximated by an entire slice regular function, uniformly on $\mathbb{R}$, with an arbitrary continuous "error" function. As a byproduct, one immediately obtains result on uniform approximation by polynomials on compact subintervals of $\mathbb{R}$. We also prove an approximation result for both a quaternion valued function and its derivative and, finally, we show some applications.

In this paper we consider finite $\beta$-expansions in the field of formal series with Pisot basis $\beta$. We are studying the arithmetic operations on $\beta$-expansions and provide bounds on the number of fractional digits arising in multiplication for arbitrary $\beta$-polynomials noted $L_\odot$. This value is given explicitly for families of Pisot basis. The last part of this paper is devoted to quadratic Pisot series where we will give the exact value for $L_\odot$.

For a locally compact group $G$, Lau and Ghaffari provided many results about $G$-invariant subsets of $G$-modules, and the relationship between $G$-module maps, $L^1(G)$-module maps and $M(G)$- module maps. In both papers their results were specified for one module action. In this paper we extend many of their results to arbitrary Banach $G$-modules and $G$-module maps.

For any locally compact group $G$, we show the existence and uniqueness up to quasi-equivalence of a unitary $C_0$-representation $\pi_0$ of $G$ such that the coefficient functions of $C_0$-representations of $G$ are exactly the coefficient functions of $\pi_0$. The present work, strongly influenced by [4] (which dealt exclusively with discrete groups), leads to new characterizations of the Haagerup property: $G$ has that property if and only if the representation $\pi_0$ induces a $*$-isomorphism of $C^*(G)$ onto $C^*_{\pi_0}(G)$. When $G$ is discrete and countable, we also relate the Haagerup property to relative strong mixing properties in the sense of [9] of the group von Neumann algebra $L(G)$ into finite von Neumann algebras.

Let us consider a Banach function space $X$. The Kӧthe dual space can be characterized as the space of multipliers from $X$ to $L^1$. We extend this characterization to the space of multipliers from $X$ to $L^p$ in order to define the Kӧthe $p$-dual space of $X$. We analyze the properties of this space so as to use it as a tool for studying $p$-th power factorable operators. In particular, we compute $q$-concavity for these spaces and type and cotype when $X$ is an AM-space. As main applications, we give a characterization for Hilbert Banach function spaces, as well as a factorization for $p$-th power factorable operators through an $L^{p,\infty}$-space.

We study the existence of a weak solution for a semilinear elliptic Dirichlet boundary-value problem \begin{align*} Lu(x)-\mu u g_1(x)+h(u)g_2(x)&=f(x)\quad \mbox{in }\Omega,\\ u(x)&=0\qquad \mbox{ on }\partial\Omega, \end{align*} in a suitable weighted Sobolev space, where $\Omega=\mathbb{R} ^n\backslash K,n\geq3$ is an unbounded domain, and where $K$ is a closure of some bounded domain in $\mathbb{R}^n,n\geq3$.

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. In this paper we introduce $\pi$-Rickart modules as a generalization of Rickart modules. $\pi$-Rickart modules are also a dual notion of dual $\pi$-Rickart modules and extends that of generalized right principally projective rings to the module theoretic setting. The module $M$ is called {\it $\pi$-Rickart} if for any $f\in S$, there exist $e^2=e\in S$ and a positive integer $n$ such that $r_M(f^n)=$ Ker$f^n=eM$. We obtain several results about generalized right principally projective rings by using $\pi$-Rickart modules. Moreover, we investigate relations between a $\pi$-Rickart module and its endomorphism ring.

On montre que si l'ensemble $E \subset \Bbb R^n$ est bien approché, à l'échelle $r_0$, par des plans de dimension $d$, il existe une surface lisse $\Sigma_0$ de dimension $d$, qui est proche de $E$ à l'échelle $r_0$. Quand $E$ est Reifenberg-plat, ceci permet d'appliquer un résultat de G. David et T. Toro [Memoirs of the AMS 215 (2012), 1012], et de montrer que $E$ est l'image de $\Sigma_0$ par un homéomorphisme bi-Höldérien de $\Bbb R^n$. Si de plus $d=n-1$ et $E$ est compact et connexe, alors $\Sigma_0$ est orientable, et $\Bbb R^n \sm E$ a exactement deux composantes connexes que la construction ci-dessus permet d'approximer par l'intérieur par des domaines lisses.

We show that if the set $E \subset \Bbb R^n$ is well approximated at the scale $r_0$ by planes of dimension $d$, we can find a smooth surface $\Sigma_0$ of dimension $d$ which is close to $E$ at the scale $r_0$. When $E$ is a Reifenberg flat set, this allows us to apply a result of G. David and T. Toro [Memoirs of the AMS 215 (2012), 1012], and get a bi-Hölder homeomorphism of $\Bbb R^n$ that sends $\Sigma_0$ to $E$. If in addition $d=n-1$ and $E$ is compact and connected, then $\Sigma_0$ is orientable, and $\Bbb R^n \sm E$ has exactly two connected components, which we can approximate from the inside by smooth domains.

We introduce a class of rings, namely the class of left or right $p$-nil rings, for which the adjoint groups behave regularly. Every $p$-ring is close to being left or right $p$-nil in the sense that it contains a large ideal belonging to this class. Also their adjoint groups occur naturally as groups of automorphisms of $p$-groups. These facts and some of their applications are investigated in this paper.

We show that the circle space of a spherical circle plane is a punctured projective 3-space. The main ingredient is a partial solution of the problem of Apollonius on common touching circles.

In this paper, the existence and multiplicity results for semilinear elliptic equations at resonance are obtained by the minimax methods and Morse theory.

Let $G, H$ be finite groups. We asymptotically compute $|\mathrm{Hom}(G, H\wr A_n)|$, thereby establishing a conjecture of T. Müller.