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Preview: The Quarterly Journal of Mathematics - current issue

The Quarterly Journal of Mathematics Current Issue





Published: Tue, 11 Apr 2017 00:00:00 GMT

Last Build Date: Tue, 03 Oct 2017 12:50:46 GMT

 



The asymptotic behavior of the minimal pseudo-ANOSOV dilatations in the hyperelliptic handlebody groups *

2017-04-11

Abstract
We consider the hyperelliptic handlebody group on a closed surface of genus g. This is the subgroup of the mapping class group on a closed surface of genus g consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the handlebody. We prove that the logarithm of the minimal dilatation (that is, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus g is comparable to 1/g. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of the subgroup of genus g in question is the same as that of the ambient mapping class group of genus g. We also determine finite presentations of the hyperelliptic handlebody groups.



Isometric isomorphisms of the annihilator of C 0 ( G ) in LUC ( G ) *

2017-03-09

Abstract
Let LUC (G) denote the C*-algebra of left uniformly continuous functions with the uniform norm and let C0(G)⊥ denote the annihilator of C0(G) in LUC(G)*. In this article, among other results, we show that if G is a locally compact group and H is a discrete group then whenever there exists a weak-star continuous isometric isomorphism between C0(G)⊥ and C0(H)⊥, G is isomorphic to H as a topological group. In particular, when H is discrete C0(H)⊥ determines H within the class of locally compact topological groups.



An order theoretic characterization of spin factors

2017-03-01

Abstract
The famous Koecher–Vinberg theorem characterizes the Euclidean Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. Recently, Walsh gave an alternative characterization of the Euclidean Jordan algebras. He showed that the Euclidean Jordan algebras correspond to the finite dimensional order unit spaces (V, C, u) for which there exists a bijective map g:C◦→C◦ with the property that g is antihomogeneous, that is, g(λx)=λ−1g(x) for all λ>0 and x∈C◦, and g is an order-antimorphism, that is, x≤Cy if and only if g(y)≤Cg(x). In this paper, we make a first step towards extending this order theoretic characterization to infinite dimensional JB-algebras. We show that if (V, C, u) is a complete order unit space with a strictly convex cone and dimV≥3, then there exists a bijective antihomogeneous order-antimorphism g:C◦→C◦ if and only if (V, C, u) is a spin factor.



Sur le nombre de matrices aléatoires à coefficients rationnels

2017-03-01

Abstract
We prove an asymptotic relation for the number of stochastic matrices which have rational coefficients with height less than B when B tends to infinity. It solves a recent question of Shparlinski.



A model for phase transitions with competing terms

2017-03-01

Abstract
In this paper, we study, via Γ-convergence techniques, the asymptotic behaviour of a family of coupled singular perturbations of a non-convex functional of the type ∫Ωf(u(x),∇u(x),ρ(x))dx as a variational model to address two-phase transition problems under the volume constraints ∫Ωu(x)dx=Vf, ∫Ωρ(x)dx=Vs, and where the additional unknown ρ interplays with ∇u in the formation of interfaces.



Lattice point counting in sectors of Hyperbolic 3-space

2017-02-27

Abstract
Let Γ be a cocompact discrete subgroup of PSL2(C) and denote by H the three-dimensional upper half-space. For a p∈H, we count the number of points in the orbit Γp, according to their distance, arccoshX, from a totally geodesic hyperplane. The main term in n dimensions was obtained by Herrmann for any subset of a totally geodesic submanifold. We prove a pointwise error term of O(X3/2) by extending the method of Huber and Chatzakos–Petridis to three dimensions. By applying Chamizo's large sieve inequalities, we obtain the conjectured error term O(X1+ε) on an average in the spatial aspect. We prove a corresponding large sieve inequality for the radial average and explain why it only improves on the pointwise bound by 1/6.



On the equation X1 + X2 = 1 infinitely generated multiplicative groups in positive characteristic

2017-02-27

Abstract
Let K be a field of characteristic p>0 and let G be a subgroup of K*×K*with dimℚ(G⊗Zℚ)=r finite. Then Voloch proved that the equation ax+by=1in(x,y)∈G for given a,b∈K⁎ has at most pr(pr+p−2)/(p−1) solutions (x,y)∈G, unless (a,b)n∈G for some n≥1. Voloch also conjectured that this upper bound can be replaced by one depending only on r. Our main theorem answers this conjecture positively. We prove that there are at most 31×19r+1 solutions (x, y) unless (a,b)n∈G for some n≥1 with (n, p)  = 1. During the proof of our main theorem, we generalize the work of Beukers and Schlickewei to positive characteristic, which heavily relies on diophantine approximation methods. This is a surprising feat on its own, since usually these methods cannot be transferred to positive characteristic.



Erratum

2017-02-14




Chow Ring of the Moduli Space of Stable Sheaves Supported on Quartic Curves

2017-02-14

Abstract
Motivated by the computation of the BPS-invariants on a local Calabi–Yau 3-fold suggested by S. Katz, we compute the Chow ring and the cohomology ring of the moduli space of stable sheaves of Hilbert polynomial 4m+1 on the projective plane. As a byproduct, we obtain the total Chern class and Euler characteristics of all line bundles, which provide numerical data for the strange duality on the plane.



The Hasse Principle for Systems of Quadratic and Cubic Diagonal Equations

2017-02-10

Abstract
Employing Brüdern's and Wooley's new complification method, we establish an asymptotic Hasse principle for the number of solutions to a system of r3 cubic and r2 quadratic diagonal forms, where r3≥2r2>0, in s≥6r3+⌊(14/3)r2⌋+1 variables.



Invariants for bi-Lipschitz equivalence of ideals

2017-02-09

Abstract
We introduce the notion of bi-Lipschitz equivalence of ideals and derive numerical invariants for such equivalence. In particular, we show that the log canonical threshold of ideals is a bi-Lipschitz invariant. We apply our method to several deformations ft:(Cn,0)→(C,0) and show that they are not bi-Lipschitz trivial, specially focusing on several known examples of non-μ*-constant deformations.



The cocycle identity holds under Stopping

2017-02-09

Abstract
In recent work of the authors, it was shown how to use any finite quantum stop time to stop the CCR flow and its strongly continuous isometric cocycles (Q. J. Math. 65:1145–1164, 2014). The stopped cocycle was shown to satisfy a stopped form of the cocycle identity, valid for deterministic increments of the time used for stopping. Here, a generalization of this identity is obtained, where both cocycle parameters are replaced with finite quantum stop times.



On generalized notions of amenability and operator homology of the Fourier algebra

2017-02-08

Abstract
Let G be a locally compact group, A (G) its Fourier algebra and Acb (G) the closure of A (G) in the space of completely bounded multipliers of A (G). We show that the Fourier algebras of weakly amenable, non-amenable groups are not approximately amenable. We also prove that A (G) is operator approximately biprojective if and only if G is discrete. Finally, we study various (operator) cohomological properties and its related (operator) homological properties of the algebra Acb (G).



The Structure of Triple Derivations on Semisimple Banach *-Algebras

2017-02-03

Abstract
Triple derivations on C*-algebras and JB*-triples had been extensively studied in the literature. In this paper, we characterize the structure of triple derivations on semisimple complex Banach *-algebras. In particular, we show that every triple derivation on a semisimple complex Banach *-algebra is automatically continuous and is a special kind of generalized derivations. Our theorems improve and generalize some known results for C*-algebras obtained in Barton and Friedman (Bounded derivations of JB*-triples, Quart. J. Math. Oxford Ser.41 (1990), 255–268), Burgos et al. (Local triple derivations on C*-algebras and JB*-triples, Bull. Lond. Math. Soc.46 (2014), 709–724) and Burgos et al. (Local triple derivations on C*-algebras, Comm. Algebra42 (2014), 1276–1286). The analogous result for standard operator *-algebras on Hilbert spaces is also described.



A lower bound for the least prime in an Arithmetic progression

2017-01-31

Abstract
Fix k a positive integer, and let be coprime to k. Let p(k,ℓ) denote the smallest prime equivalent to ℓ(modk), and set P(k) to be the maximum of all the p(k,ℓ). We seek lower bounds for P(k). In particular, we show that for almost every k one has P(k)≫ϕ(k)logklog2klog4k/log3k, answering a question of Ford, Green, Konyangin, Maynard and Tao. We rely on their recent work on large gaps between primes. Our main new idea is to use sieve weights to capture not only primes, but also small multiples of primes. We also give a heuristic which suggests that liminfkP(k)ϕ(k)log2k=1.



Quaternionic Bott–Chern Cohomology and existence of HKT metrics

2017-01-18

Abstract
We study quaternionic Bott–Chern cohomology on compact hypercomplex manifolds and adapt some results from complex geometry to the quaternionic setting. For instance, we prove a criterion for the existence of hyperkähler with torsion metrics on compact hypercomplex manifolds of real dimension 8 analogous to the one given by Teleman [34] and Angella et al. [3] for compact complex surfaces.