Indices of Kummer extensions of prime degree

Let p be a prime and let ${\zeta }_p$ be a primitive p-th root of unity. For a finite extension k of $\mathbb{Q}$ containing ${\zeta }_p$, we consider a Kummer extension $L/k$ of degree p. In this paper, we show that if $k=\mathbb{Q}({\zeta }_p)$ and the class number of k is one, the index of $L/k$ is one. We also show that if $L/k$ is tamely ramified with a normal integral basis, the index is at most a power of p. In the last section, we show that there exist infinitely many cubic Kummer extensions of $\mathbb{Q}({\zeta }_3)$ for both wildly and tamely ramified cases, whose integer rings do not have a power integral basis over that of $\mathbb{Q}({\zeta }_3)$.     

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.     

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.     

Sampling in a weighted Sobolev space

We show that functions f in some weighted Sobolev space are completely determined by time-frequency samples ${\{f(t_n)\}}_{n\in \mathbb{Z}}\cup {\{\stackrel{?}{f}({\lambda }_k)\}}_{k\in \mathbb{Z}}$ along appropriate slowly increasing sequences ${\{t_n\}}_{n\in \mathbb{Z}}$ and ${\{{\lambda }_n\}}_{n\in \mathbb{Z}}$ tending to ±∞ as $n\to \pm \infty$.     

Approximation of maximum of Gaussian random fields

This contribution is concerned with Gumbel limiting results for supremum ${\mathbb{M}}_{\mathit{n}}={\sup }_{\mathit{t}\in [\mathbf{0},{\mathit{T}}_{\mathit{n}}]}?|X_{\mathit{n}}(\mathit{t})|$ with $X_{\mathit{n}},\mathit{n}\in {\mathbb{N}}^2$ centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for ${\mathbb{M}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and show a second-order approximation for $\mathbb{E}{\{{\mathbb{M}}_{\mathit{n}}^p\}}^{1/p}$ for any $p\ge 1$.     

The Generalized Riemann Hypothesis and the discriminant of number fields

Under the Generalized Riemann Hypothesis for the Dedekind zeta-function ${\zeta }_{\kappa }$, we obtain a formula for the discriminant $D_{\kappa /\mathbb{Q}}$ of the algebraic number field κ in terms of an integral of ${\zeta }_{\kappa }$ on the critical line.     

Efimov spaces and the separable quotient problem for spaces Cp(K)

The classic Rosenthal–Lacey theorem asserts that the Banach space $C(K)$ of continuous real-valued maps on an infinite compact space K has a quotient isomorphic to c or ${?}_2$. More recently, Ka?kol and Saxon [20] proved that the space $C_p(K)$ endowed with the pointwise topology has an infinite-dimensional separable quotient algebra iff K has an infinite countable closed subset. Hence $C_p(\beta \mathbb{N})$ lacks infinite-dimensional separable quotient algebras. This motivates the following question: (?) Does$C_p(K)$admit an infinite-dimensional separable quotient (shortly SQ) for any infinite compact space K? Particularly, does $C_p(\beta \mathbb{N})$ admit SQ? Our main theorem implies that $C_p(K)$ has SQ for any compact space K containing a copy of $\beta \mathbb{N}$. Consequently, this result reduces problem (?) to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of $\beta \mathbb{N}$). Although, it is unknown if Efimov spaces exist in ZFC, we show, making use of some result of R. de la Vega (2008) (under ?), that for some Efimov space K the space $C_p(K)$ has SQ. Some applications of the main result are provided.     

On integrally closed simple extensions of valuation rings

Let v be a Krull valuation of a field with valuation ring $R_v$. Let θ be a root of an irreducible trinomial $F(x)=x^n+a{x}^m+b$ belonging to $R_v[x]$. In this paper, we give necessary and sufficient conditions involving only $a,b,m,n$ for $R_v[\theta ]$ to be integrally closed. In the particular case when v is the p-adic valuation of the field $\mathbb{Q}$ of rational numbers, $F(x)\in \mathbb{Z}[x]$ and $K=\mathbb{Q}(\theta )$, then it is shown that these conditions lead to the characterization of primes which divide the index of the subgroup $\mathbb{Z}[\theta ]$ in $A_K$, where $A_K$ is the ring of algebraic integers of K. As an application, it is deduced that for any algebraic number field K and any quadratic field L not contained in K, we have $A_{KL}=A_K{A}_L$ if and only if the discriminants of K and L are coprime.     

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.     

Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between ?-quasi-cyclic codes over a finite field ${\mathbb{F}}_q$ and linear codes over a ring $R={\mathbb{F}}_q[Y]/(Y^m?1)$. Using this correspondence, we prove that every ?-quasi-cyclic self-dual code of length m? over a finite field ${\mathbb{F}}_q$ can be obtained by the building-up construction, provided that $\mathrm{char}({\mathbb{F}}_q)=2$ or $q\equiv 1(\mathrm{mod}4)$, m is a prime p, and q is a primitive element of ${\mathbb{F}}_p$. We determine possible weight enumerators of a binary ?-quasi-cyclic self-dual code of length p? (with p a prime) in terms of divisibility by p. We improve the result of Bonnecaze et al. (2003) [3] by constructing new binary cubic (i.e., ?-quasi-cyclic codes of length 3?) optimal self-dual codes of lengths $30,36,42,48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40,20,12]$ code over ${\mathbb{F}}_3$ and a new 6-quasi-cyclic self-dual $[30,15,10]$ code over ${\mathbb{F}}_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28,14,9]$ code over ${\mathbb{F}}_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over ${\mathbb{F}}_4$.