Hybrid journal (It can contain Open Access articles) ISSN (Print) 0024-6107 - ISSN (Online) 1469-7750 Published by Oxford University Press[396 journals]

Authors:Smith G. Pages: 667 - 687 Abstract: We use Morse homology to study bifurcation of the solution sets of the Allen–Cahn Equation. PubDate: Sat, 17 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw053 Issue No:Vol. 94, No. 3 (2016)

Authors:Glasner YY; Kitroser DD, Melleray JJ. Pages: 688 - 708 Abstract: Let $G$ be a countable group, ${\rm Sub}(G)$ be the (compact, metric) space of all subgroups of $G$ with the Chabauty topology and ${\rm Is}(G) \subseteq {\rm Sub}(G)$ be the collection of isolated points. We denote by $X!$ the (Polish) group of all permutations of a countable set $X$. Then the following properties are equivalent: (i) ${\rm Is}(G)$ is dense in ${\rm Sub}(G)$; (ii) $G$ admits a ‘generic permutation representation’. Namely, there exists some $\tau ^* \in {\rm Hom}(G,X!)$ such that the collection of permutation representations $\{\varphi \in {\rm Hom}(G,X!) \, \, \varphi \ {\hbox {is permutation isomorphic to}}\ \tau ^*\}$ is co-meager in ${\rm Hom}(G,X!)$. We call groups satisfying these properties solitary. Examples of solitary groups include finitely generated locally extended residually finite groups and groups with countably many subgroups. PubDate: Wed, 28 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw054 Issue No:Vol. 94, No. 3 (2016)

Authors:Raicu C; Weyman J. Pages: 709 - 725 Abstract: We compute the local cohomology modules $\mathcal {H}_Y^{\bullet }(X,\mathcal {O}_X)$ in the case when $X$ is the complex vector space of $n\times n$ symmetric (respectively, skew-symmetric matrices) and $Y$ is the closure of the $\hbox {GL}$-orbit consisting of matrices of any fixed rank, for the natural action of the general linear group $\hbox {GL}$ on $X$. We describe the $\mathcal {D}$-module composition factors of the local cohomology modules, and compute their multiplicities explicitly in terms of generalized binomial coefficients. One consequence of our work is a formula for the cohomological dimension of ideals of even minors of a generic symmetric matrix: in the case of odd minors, this was obtained by Barile in the 1990s. Another consequence of our work is that we obtain a description of the decomposition into irreducible $\hbox {GL}$-representations of the local cohomology modules (the analogous problem in the case when $X$ is the vector space of $m\times n$ matrices was treated in earlier work of the authors). PubDate: Sat, 17 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw056 Issue No:Vol. 94, No. 3 (2016)

Authors:Bénéteau C; Khavinson D, Sola AA, et al. Pages: 726 - 746 Abstract: We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials $p$ minimizing Dirichlet-type norms $\ pf-1\ _{\alpha }$ for a given function $f$. For $\alpha \in [0,1]$ (which includes the Hardy and Dirichlet spaces of the disk) and general $f$, we show that such extremal polynomials are non-vanishing in the closed unit disk. For negative $\alpha $, the weighted Bergman space case, the extremal polynomials are non-vanishing on a disk of strictly smaller radius, and zeros can move inside the unit disk. We also explain how $\mathrm {dist}_{D_{\alpha }}(1,f\cdot \mathcal {P}_n)$, where $\mathcal {P}_n$ is the space of polynomials of degree at most $n$, can be expressed in terms of quantities associated with orthogonal polynomials and kernels, and we discuss methods for computing the quantities in question. PubDate: Tue, 20 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw057 Issue No:Vol. 94, No. 3 (2016)

Authors:Gregory L; Prest M. Pages: 747 - 766 Abstract: We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity.First, we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas, and hence are non-decreasing on Krull–Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty $, and hence, if the algebra is countable, there is a superdecomposable pure-injective representation.It is conjectured that a stronger result is true: that a representation embedding from ${\rm Mod}\hbox {-}S$ to ${\rm Mod}\hbox {-}R$ admits an inverse interpretation functor from its image, and hence that, in this case, ${\rm Mod}\hbox {-}R$ interprets ${\rm Mod}\hbox {-}S$. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings.Finally, we prove that if $R,S$ are finite-dimensional algebras over an algebraically closed field and $I:{\rm Mod}\hbox {-}R\rightarrow {\rm Mod}\hbox {-}S$ is an interpretation functor such that the smallest definable subcategory containing the image of $I$ is the whole of ${\rm Mod}\hbox {-}S,$ then if $R$ is tame, so is $S$ and, similarly, if $R$ is domestic, then $S$ also is domestic. PubDate: Tue, 27 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw055 Issue No:Vol. 94, No. 3 (2016)

Authors:Fiorilli D. Pages: 767 - 792 Abstract: Under a hypothesis that is stronger than the Riemann Hypothesis for elliptic curve $L$-functions, we show that both average analytic and algebraic ranks of elliptic curves in families of quadratic twists are exactly $\frac {1}{2}$. As a corollary we obtain that, under this last hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all curves, and that asymptotically one-half have algebraic rank 0, and the remaining half 1. We also prove an analogous result in the family of all elliptic curves. The proof uses results of Katz–Sarnak and Young on the 1-level density of zeros of elliptic curve $L$-functions. In essence, we show that, under a hypothesis analogous to Montgomery's Conjecture, a density result with limited support on low-lying zeros of $L$-functions is sufficient to determine the average rank. PubDate: Fri, 23 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw058 Issue No:Vol. 94, No. 3 (2016)

Authors:Brown M; Marletta M, Symons F. Pages: 793 - 813 Abstract: We prove a pair of uniqueness theorems for an inverse problem for an ordinary differential operator pencil of second order. The uniqueness is achieved from a discrete set of data, namely, the values at the points $-n^2\ (n\in \mathbb {N})$ of (a physically appropriate generalization of) the Weyl–Titchmarsh $m$-function $m(\lambda )$ for the problem. As a corollary, we establish a uniqueness result for a physically motivated inverse problem inspired by Berry and Dennis (‘Boundary-condition-varying circle billiards and gratings: the Dirichlet singularity’, J. Phys. A: Math. Theor. 41 (2008) 135203).To achieve these results, we prove a limit-circle analogue to the limit-point $m$-function interpolation result of Rybkin and Tuan (‘A new interpolation formula for the Titchmarsh–Weyl $m$-function’, Proc. Amer. Math. Soc. 137 (2009) 4177–4185); however, our proof, using a Mittag-Leffler series representation of $m(\lambda )$, involves a rather different method from theirs, circumventing the $A$-amplitude representation of Simon (‘A new approach to inverse spectral theory, I. Fundamental formalism’, Ann. Math. $(2)$ 150 (1999) 1029–1057). Uniqueness of the potential then follows by appeal to a Borg–Marčenko argument. PubDate: Sat, 17 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw059 Issue No:Vol. 94, No. 3 (2016)

Authors:Bourgain J; Demeter C. Pages: 814 - 838 Abstract: We use decoupling theory to estimate the number of solutions for quadratic and cubic Parsell–Vinogradov systems in two dimensions. PubDate: Thu, 06 Oct 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw063 Issue No:Vol. 94, No. 3 (2016)

Authors:Katsoulis EG; Ramsey C. Pages: 839 - 858 Abstract: A triangular limit algebra $ {\mathcal {A}}$ is isometrically isomorphic to the tensor algebra of a $C^{\ast }$-correspondence if and only if its fundamental relation $ {\mathcal {R}}( {\mathcal {A}})$ is a tree admitting a $ {\mathbb {Z}}^+_0$-valued continuous and coherent cocycle. For triangular limit algebras which are isomorphic to tensor algebras, we give a very concrete description for their defining $ {C}^{\ast }$-correspondence, and we show that it forms a complete invariant for isometric isomorphisms between such algebras. A related class of operator algebras is also classified using a variant of the Aho–Hopcroft–Ullman algorithm from computer-aided graph theory. PubDate: Wed, 28 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw060 Issue No:Vol. 94, No. 3 (2016)

Authors:Infante G; Maciejewski M. Pages: 859 - 882 Abstract: In this paper, we study the existence, localization and multiplicity of positive solutions for parabolic systems with nonlocal initial conditions. In order to do this, we extend an abstract theory that was recently developed by the authors jointly with Radu Precup, related to the existence of fixed points of nonlinear operators satisfying some upper and lower bounds. Our main tool is the Granas fixed-point index theory. We also provide a nonexistence result and some examples to illustrate our theory. PubDate: Sat, 17 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw061 Issue No:Vol. 94, No. 3 (2016)

Authors:Champanerkar A; Kofman I, Purcell JS. Pages: 883 - 908 Abstract: The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal' We show that many families of alternating knots and links simultaneously maximize both ratios. PubDate: Fri, 23 Sep 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw062 Issue No:Vol. 94, No. 3 (2016)

Authors:Mináč J; Tân N. Pages: 909 - 932 Abstract: We show that the absolute Galois group of any field has the vanishing triple Massey product property. Several corollaries for the structure of maximal pro-$p$-quotient of absolute Galois groups are deduced. Furthermore, the vanishing of some higher Massey products is proved. PubDate: Fri, 21 Oct 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw064 Issue No:Vol. 94, No. 3 (2016)

Authors:Durham M. Pages: 933 - 969 Abstract: We build an augmentation of the Masur–Minsky marking complex by Groves-Manning combinatorial horoballs to obtain a graph we call the augmented marking complex, $\mathcal {AM}(S)$. Adapting work of Masur–Minsky, we show that this augmented marking complex is quasiisometric to Teichmüller space with the Teichmüller metric. A similar construction was independently discovered by Eskin–Masur–Rafi. We also completely integrate the Masur–Minsky hierarchy machinery to $\mathcal {AM}(S)$ to build flexible families of uniform quasigeodesics in Teichmüller space. As an application, we give a new proof of Rafi's distance formula for $\mathcal {T}(S)$ with the Teichmüller metric. We have included an appendix, in which we prove a number of facts about hierarchies that we hope will be of independent interest. PubDate: Fri, 21 Oct 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw065 Issue No:Vol. 94, No. 3 (2016)

Authors:Björklund J. Pages: 970 - 992 Abstract: We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in $P\times {\mathbb R}$, where $P$ is a punctured Riemann surface. As an application we show that, for any integer $k$ and any homology class $h\in H_1(P\times {\mathbb R})$, there are $k$ Legendrian knots, all representing $h$, which are pairwise smoothly isotopic through a formal Legendrian isotopy, but which lie in mutually distinct Legendrian isotopy classes. PubDate: Tue, 08 Nov 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw066 Issue No:Vol. 94, No. 3 (2016)

Authors:Borg P. Pages: 993 - 1018 Abstract: A set of sets is called a family. Two families $\mathcal {A}$ and $\mathcal {B}$ are said to be cross-$t$-intersecting if each set in $\mathcal {A}$ intersects each set in $\mathcal {B}$ in at least $t$ elements. An active problem in extremal set theory is to determine the maximum product of sizes of cross-$t$-intersecting subfamilies of a given family. This incorporates the classical Erdös–Ko–Rado (EKR) problem. We prove a cross-$t$-intersection theorem for weighted subsets of a set by means of a new subfamily alteration method, and use the result to provide solutions for three natural families. For $r \in [n] = \{1, 2,\ldots , n\}$, let ${[n] \choose r}$ be the family of $r$-element subsets of $[n]$, and let ${[n] \choose \leq r}$ be the family of subsets of $[n]$ that have at most $r$ elements. Let $\mathcal {F}_{n,r,t}$ be the family of sets in ${[n] \choose \leq r}$ that contain $[t]$. We show that if $g : {[m] \choose \leq r} \rightarrow \mathbb {R}^+$ and $h : {[n] \choose \leq s} \rightarrow \mathbb {R}^+$ are functions that obey certain conditions, $\mathcal {A} \subseteq {[m] \choose \leq r}$, $\mathcal {B} \subseteq {[n] \choose \leq s}$, and $\mathcal {A}$ and $\mathcal {B}$ are cross-$t$-intersecting, then \[ \sum_{A \in \mathcal{A}} g(A) \sum_{B \in \mathcal{B}} h(B) \leq \sum_{C \in \mathcal{F}_{m,r,t}} g(C) \sum_{D \in \mathcal{F}_{n,s,t}} h(D), \] and equality holds if $\mathcal {A} = \mathcal {F}_{m,r,t}$ and $\mathcal {B} = \mathcal {F}_{n,s,t}$. We prove this in a more general setting and characterize the cases of equality. We use the result to show that the maximum product of sizes of two cross-$t$-intersecting families $\mathcal {A} \subseteq {[m] \choose r}$ and $\mathcal {B} \subseteq {[n] \choose s}$ is ${m-t \choose r-t}{n-t \choose s-t}$ for $\min \{m,n\} \geq n_0(r,s,t)$, where $n_0(r,s,t)$ is close to best possible. We obtain analogous results for families of integer sequences and for families of multisets. The results yield generalizations for $k \geq 2$ cross-$t$-intersecting families, and EKR-type results. PubDate: Tue, 08 Nov 2016 00:00:00 GMT DOI: 10.1112/jlms/jdw067 Issue No:Vol. 94, No. 3 (2016)