Bi-Sobolev mappings and Kp-distortions in the plane

In this paper we introduce the class of the inner p-quasiconformal mappings, that are homeomorphisms $f:\mathbb{D}\stackrel{onto}{?}\mathbb{D}$, $f\in W_{\mathrm{loc}}^{1,1}(\mathbb{D};\mathbb{D})$, where $\mathbb{D}?{\mathbb{R}}^2$ is the unit disk, such that there exists a constant ${\mathbb{K}}_p\ge 0$ for which the following distortion inequality

$|Df(x){|}^p\le {\mathbb{K}}_p|J_f(x){|}^{p?1}\text{a.e.}x\in \mathbb{D}$

is satisfied. The study of such mappings is motivated by the fact that their inverses satisfy the distortion inequality introduced in [11]. Here we give a characterization of them so that their components solve a suitable uniformly elliptic p-harmonic system. Moreover, for mappings satisfying the previous distortion inequality with ${\mathbb{K}}_p={\mathbb{K}}_{p,f}(x)$ not necessarily constant, we identify the homeomorphism f whose p-distortion function ${\mathbb{K}}_{p,f}(x)$ is minimal in $L^1$ norm.     

On Ozaki's condition for p-valency

Let f be an analytic function in a convex domain $D?\mathbb{C}$. A well-known theorem of Ozaki states that if f is analytic in D, and is given by $f(z)=z^p+{\sum }_{n=p+1}^{\infty }{a}_n{z}^n$ for $z\in D$, and

$\mathfrak{Re}\{{\mathrm{e}}^{\mathrm{i}\alpha }{f}^{(p)}(z)\}>0,(z\in D),$

for some real α, then f is at most p-valent in D. Ozaki's condition is a generalization of the well-known Noshiro–Warschawski univalence condition. The purpose of this paper is to provide some related sufficient conditions for functions analytic in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ to be p-valent in $\mathbb{D}$, and to give an improvement to Ozaki's sufficient condition for p-valence when $z\in \mathbb{D}$.     

The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.     

The Ramsey property for Banach spaces and Choquet simplices,and applications

We show that the class of finite-dimensional Banach spaces and the class of finite-dimensional Choquet simplices have the Ramsey property. As an application, we show that the group $\mathrm{Aut}(\mathbb{G})$ of surjective linear isometries of the Gurarij space $\mathbb{G}$ is extremely amenable, and that the canonical action $\mathrm{Aut}(\mathbb{P})?\mathbb{P}$ is the universal minimal flow of the group $\mathrm{Aut}(\mathbb{P})$ of affine homeomorphisms of the Poulsen simplex $\mathbb{P}$. This answers questions of Melleray–Tsankov and Conley–Törnquist.     

Topological invariants for semigroups of holomorphic self-maps of the unit disk

Let $({\phi }_t)$, $({?}_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disk $\mathbb{D}?\mathbb{C}$. Let $f:\mathbb{D}\to \mathbb{D}$ be a homeomorphism. We prove that, if $f°{?}_t={\phi }_t°f$ for all $t\ge 0$, then f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disk.     

Transcendence degree one function fields over a finite field with many automorphisms

Let $\mathbb{K}$ be the algebraic closure of a finite field ${\mathbb{F}}_q$ of odd characteristic p. For a positive integer m prime to p, let $F=\mathbb{K}(x,y)$ be the transcendence degree 1 function field defined by $y^q+y=x^m+x^{?m}$. Let $t=x^{m(q?1)}$ and $H=\mathbb{K}(t)$. The extension $F|H$ is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus $\mathfrak{g}(K)=(qm?1)(q?1)$, p-rank (Hasse–Witt invariant) $\gamma (K)={(q?1)}^2$ and a $\mathbb{K}$-automorphism group of order at least $2q^{2}m(q?1)$. In this paper we prove that this subgroup is the full $\mathbb{K}$-automorphism group of K; more precisely ${\text{Aut}}_{\mathbb{K}}(K)=\mathrm{\Delta }?D$ where Δ is an elementary abelian p-group of order $q^2$ and D has an index 2 cyclic subgroup of order $m(q?1)$. In particular, $\sqrt{m}|{\text{Aut}}_{\mathbb{K}}(K)|>\mathfrak{g}{(K)}^{3/2}$, and if K is ordinary (i.e. $\mathfrak{g}(K)=\gamma (K)$) then $|{\text{Aut}}_{\mathbb{K}}(K)|>{\mathfrak{g}}^{3/2}$. On the other hand, if G is a solvable subgroup of the $\mathbb{K}$-automorphism group of an ordinary, transcendence degree 1 function field L of genus $\mathfrak{g}(L)\ge 2$ defined over $\mathbb{K}$, then $|{\text{Aut}}_{\mathbb{K}}(K)|\le 34{(\mathfrak{g}(L)+1)}^{3/2}<68\sqrt{2}\mathfrak{g}{(L)}^{3/2}$; see [15]. This shows that K hits this bound up to the constant $68\sqrt{2}$.Since ${\text{Aut}}_{\mathbb{K}}(K)$ has several subgroups, the fixed subfield $F^N$ of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in ${\text{Aut}}_{\mathbb{K}}(K)$ is large enough. This possibility is worked out for subgroups of Δ.     

Starlike trees whose maximum degree exceed 4 are determined by their Q-spectra

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of a graph $G$, respectively. The matrix $D(G)+A(G)$ is called the signless Laplacian matrix of $G$. The spectrum of the matrix $D(G)+A(G)$ is called the Q-spectrum of $G$. A graph is said to be determined by its Q-spectrum if there is no other non-isomorphic graph with the same Q-spectrum. In this paper, we prove that all starlike trees whose maximum degree exceed $4$ are determined by their Q-spectra.     

Bilinear forms on homogeneous Sobolev spaces

In this paper we characterize the boundedness of the bilinear form defined by

$(f,g)\in {\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})\to \underset{\mathbb{R}}{\int }{(?\mathrm{\Delta })}^{s/2}(fg)(x){(?\mathrm{\Delta })}^{s/2}(b)(x)dx,$

in the product of homogeneous Sobolev spaces ${\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})$, $0<s<1/2$. We deduce a characterization of the space of pointwise multipliers from ${\dot{H}}^s(\mathbb{R})$ to its dual ${\dot{H}}^{?s}(\mathbb{R})$ in terms of trace measures.     

Primeness property for graded central polynomials of verbally prime algebras

Let F be an infinite field. The primeness property for central polynomials of $M_n(F)$ was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for $M_n(F)$ and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider $M_n(R)$, where R admits a regular grading, with a grading such that $M_n(F)$ is a homogeneous subalgebra and provide sufficient conditions – satisfied by $M_n(E)$ with the trivial grading – to prove that $M_n(R)$ has the primeness property if $M_n(F)$ does. We also prove that the algebras $M_{a,b}(E)$ satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property.     

Lehmer's totient problem over Fq[x]

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in {\mathbb{F}}_q[x]$ and $\phi (q,p(x))$ be the Euler's totient function of $p(x)$ over ${\mathbb{F}}_q[x]$, where ${\mathbb{F}}_q$ is a finite field with q elements. We prove that $\phi (q,p(x))|(q^{\mathrm{deg}(p(x))}?1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3$, $p(x)$ is the product of any 2 non-associate irreducibles of degree 1; or (iii) $q=2$, $p(x)$ is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.