Hybrid journal (It can contain Open Access articles) ISSN (Print) 0024-6093 - ISSN (Online) 1469-2120 Published by Oxford University Press[369 journals]

Authors:Keller M; Lenz D, Münch F, et al. Abstract: AbstractWe present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the combinatorial distance. This is very different from the classical Varadhan-type behavior on manifolds. Moreover, this also gives that short-time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded. PubDate: 2016-11-03

Authors:Krashen D. Abstract: AbstractWe use constructions of versal cohomology classes based on a new notion of ‘presentable functors’, to describe a relationship between the problems of bounding symbol length in cohomology and of finding the minimal degree of a splitting field. The constructions involved are then also used to describe generic splitting varieties for degree 2 cohomology with coefficients in a commutative algebraic group of multiplicative type. PubDate: 2016-11-03

Authors:Walker A. Abstract: AbstractThe classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper, we consider the analogous multiplicative setting of the cyclic group $(\mathbb {Z}/ q\mathbb {Z})^{\times }$ and prove a similar result. For all suitably large primes $q$ we define $P_\eta $ to be the set of primes less than $\eta q$, viewed naturally as a subset of $(\mathbb {Z}/ q\mathbb {Z})^{\times }$. Considering the $k$-fold product set $P_\eta ^{(k)}=\{p_1p_2\cdots p_k:p_i\in P_\eta \}$, we show that, for $\eta \gg q^{-{1}/{4}+\epsilon },$ there exists a constant $k$ depending only on $\epsilon $ such that $P_\eta ^{(k)}=(\mathbb {Z}/ q\mathbb {Z})^{\times }$. Erdös conjectured that, for $\eta = 1,$ the value $k=2$ should suffice: although we have not been able to prove this conjecture, we do establish that $P_1 ^{(2)}$ has density at least $\frac {1}{64}(1+o(1))$. We also formulate a similar theorem in almost-primes, improving on existing results. PubDate: 2016-10-25

Authors:Hindes W. Abstract: AbstractGiven a polynomial $\phi $ over a global function field $K/\mathbb {F}_q(t)$ and a wandering base point $b\in K$, we give a geometric condition on $\phi ,$ ensuring the existence of primitive prime divisors for almost all points in the orbit ${\mathcal O}_\phi (b):=\{\phi ^n(b)\}_{n\geq 0}$. As an application, we prove that the Galois groups (over $K$) of the iterates of many quadratic polynomials are large and use this to compute the density of prime divisors of ${\mathcal O}_\phi (b)$. PubDate: 2016-10-19

Authors:Macpherson D; Tent K. Abstract: AbstractWe consider profinite groups as 2-sorted first-order structures, with a group sort, and a second sort that acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of all open subgroups, then the first-order theory of such a structure is NIP (that is, does not have the independence property) precisely if the group has a normal subgroup of finite index that is a direct product of finitely many compact $p$-adic analytic groups, for distinct primes $p$. In fact, the condition NIP can here be weakened to NTP${}_2$. We also show that any NIP profinite group, presented as a 2-sorted structure, has an open prosoluble normal subgroup. PubDate: 2016-10-19

Authors:Akita T. Abstract: AbstractGiven an odd prime number $p$ and a Coxeter group $W$ such that the order of the product $st$ is prime to $p$ for all Coxeter generators $s,t$ of $W$, we prove that the $p$-local homology groups $H_k(W,\mathbb {Z}_{{(p)}})$ vanish for $1\leq k\leq 2(p-2)$. This generalizes a known vanishing result for symmetric groups due to Minoru Nakaoka. PubDate: 2016-10-19

Authors:Oliver B. Abstract: AbstractWe prove that if ${\mathcal {E}}\trianglelefteq {\mathcal {F}}$ are saturated fusion systems over $p$-groups $T\trianglelefteq S$, such that $C_S({\mathcal {E}})\le T$, and either ${\rm Aut}_{{\mathcal {F}}}(T)/{\rm Aut}_{\mathcal {E}}(T)$ or ${\rm Out}({\mathcal {E}})$ is $p$-solvable, then ${\mathcal {F}}$ can be ‘reduced’ to ${\mathcal {E}}$ by alternately taking normal subsystems of $p$-power index or of index prime to $p$. In particular, this is the case whenever ${\mathcal {E}}$ is simple and ‘tamely realized’ by a known simple group $K$. This answers a question posed by Michael Aschbacher, and is useful when analyzing involution centralizers in simple fusion systems, in connection with his program for reproving parts of the classification of finite simple groups by classifying certain 2-fusion systems. PubDate: 2016-10-06

Authors:Kropholler R; Wilkes G. Abstract: AbstractIn this paper, we prove that right-angled Artin groups (RAAGs) are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that right-angled Coxeter groups (RACGs) are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore, we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion. PubDate: 2016-09-16

Authors:Amoroso F; Masser D. Abstract: AbstractWe prove close to sharp lower bounds for the height of an algebraic number in a Galois extension of $\mathbb {Q}$. PubDate: 2016-09-16

Authors:Conder M; Širáň J. Abstract: AbstractWe prove that for every integer $d\ge 3$ and every group $U$ of units mod $d,$ there exists an orientably-regular map of valency $d$ with exponent group $U$. PubDate: 2016-09-16

Authors:Baba S; Granath H. Abstract: AbstractThe Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic $(2,2,2,3)$ quadrilateral group. We derive the differential structure of the ring of modular forms for the Shimura curve and relate the ring generators to explicit Heun functions for the quadrilateral group. We also show that the Picard–Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms. These results are used to completely describe the exceptional sets of the Heun functions, and we show how to find examples like \[ Hl\left(\frac{27}{2},\frac{7}{36}; \frac{1}{12},\frac{7}{12},\frac{2}{3},\frac{1}{2}; -\frac{96}{25}\right)=\frac{2^{1/2}5^{2/3}}{3^{4/3}}. \] PubDate: 2016-09-16

Authors:Losert V. Abstract: AbstractFor a discrete group $G$ with Fourier algebra $A(G)$, we study the topological centre $\mathfrak {Z}_t$ of the bidual $A(G)''$. If $G$ is amenable, then $\mathfrak {Z}_t=A(G)$. But if $G$ contains a free group $F_r$ ($r\ge 2$), we show that $\mathfrak {Z}_t$ is strictly larger than $A(G)$. Furthermore, it is shown that the subalgebra $A_\# (G)$ of radial functions in $A(G)$ is Arens regular. PubDate: 2016-09-16

Authors:Martínez Torres D. Abstract: AbstractSemisimple (co)adjoint orbits through real hyperbolic elements are well known to be symplectomorphic to cotangent bundles. We provide a new proof of this fact based on elementary results on both the Lie theory and symplectic geometry. Our proof establishes a new connection between the Iwasawa horospherical projection and the symplectic geometry of real hyperbolic (co)adjoint orbits. PubDate: 2016-09-16

Authors:Gillespie J. Abstract: AbstractWe discuss some recent developments in the theory of abelian model categories. The emphasis is on the hereditary condition and applications to homotopy categories of chain complexes and stable module categories. PubDate: 2016-09-13