Authors:Domingo García; Manuel Maestre, Ignacio Zalduendo Pages: 609 - 623 Abstract: In the study of the spectra of algebras of holomorphic functions on a Banach space E, the bidual E″ has a central role, and the spectrum is often shown to be locally homeomorphic to E″. In this paper we consider the problem of spectra of subalgebras invariant under the action of a group (functions f such that f ○ g = f). It is natural to attempt a characterization in terms of the space of orbits E″/~ obtained from E″ through the action of the group, so we pursue this approach here and introduce an analytic structure on the spectrum in some situations. In other situations we encounter some obstacles: in some cases, the lack of structure of E″/~ itself; in others, problems of weak continuity and non-approximability of functions in the algebra. We also define a convolution operation related to the spectrum. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000603 Issue No:Vol. 62, No. 3 (2019)

Authors:Thế Cu’ò’ng Nguyễn Pages: 625 - 640 Abstract: The algebraic EHP sequences, algebraic analogues of the EHP sequences in homotopy theory, are important tools in algebraic topology. This note will outline two new proofs of the existence of the algebraic EHP sequences. The first proof is derived from the minimal injective resolution of the reduced singular cohomology of spheres, and the second one follows Bousfield's idea using the loop functor of unstable modules. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000652 Issue No:Vol. 62, No. 3 (2019)

Authors:V. A. Bovdi; A. N. Grishkov Pages: 641 - 654 Abstract: Let F be a field of characteristic two and G a finite abelian 2-group with an involutory automorphism η. If G = H × D with non-trivial subgroups H and D of G such that η inverts the elements of H (H without a direct factor of order 2) and fixes D element-wise, then the linear extension of η to the group algebra FG is called a nice involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000500 Issue No:Vol. 62, No. 3 (2019)

Authors:Víctor García García; Pedro Ortega Salvador Pages: 655 - 665 Abstract: We prove that if p > 1, $w\in A_p^ +$ , b ∈ CMO and $C_b^ + $ is the commutator with symbol b of a Calderón–Zygmund convolution singular integral with kernel supported on (−∞, 0), then $C_b^ + $ is compact from Lp(w) into itself. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S001309151800024X Issue No:Vol. 62, No. 3 (2019)

Authors:Takuzo Okada Pages: 667 - 682 Abstract: The main aim of this paper is to show that a cyclic cover of ℙn branched along a very general divisor of degree d is not stably rational, provided that n ≥ 3 and d ≥ n + 1. This generalizes the result of Colliot-Thélène and Pirutka. Generalizations for cyclic covers over complete intersections and applications to suitable Fano manifolds are also discussed. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000755 Issue No:Vol. 62, No. 3 (2019)

Authors:Victoria Lebed; Leandro Vendramin Pages: 683 - 717 Abstract: This paper explores the structure groups G(X,r) of finite non-degenerate set-theoretic solutions (X,r) to the Yang–Baxter equation. Namely, we construct a finite quotient $\overline {G}_{(X,r)}$ of G(X,r), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if X injects into G(X,r), then it also injects into $\overline {G}_{(X,r)}$ . We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of G(X,r). We show that multipermutation solutions are the only involutive solutions with diffuse structure groups; that only free abelian structure groups are bi-orderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: bi-orderable, left-orderable, abelian, free abelian and torsion free. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000548 Issue No:Vol. 62, No. 3 (2019)

Authors:Jeffrey Giansiracusa Pages: 719 - 731 Abstract: The circle transfer $Q\Sigma (LX_{hS^1})_+ \to QLX_+$ has appeared in several contexts in topology. In this note, we observe that this map admits a geometric re-interpretation as a morphism of cobordism categories of 0-manifolds and 1-cobordisms. Let ð’ž1(X) denote the one-dimensional cobordism category and let Circ(X) ⊂ ð’ž1(X) denote the subcategory whose objects are disjoint unions of unparametrized circles. Multiplication in S1 induces a functor Circ(X) → Circ(LX), and the composition of this functor with the inclusion of Circ(LX) into ð’ž1(LX) is homotopic to the circle transfer. As a corollary, we describe the inclusion of the subcategory of cylinders into the two-dimensional cobordism category ð’ž2(X) and find that it is null-homotopic when X is a point. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000615 Issue No:Vol. 62, No. 3 (2019)

Authors:Be'eri Greenfeld Pages: 733 - 738 Abstract: We prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000718 Issue No:Vol. 62, No. 3 (2019)

Authors:Stefano Vidussi Pages: 739 - 746 Abstract: The fundamental group π of a Kodaira fibration is, by definition, the extension of a surface group $\Pi_b$ by another surface group $\Pi _g$ , i.e. $$1 \rightarrow \Pi_g \rightarrow \pi \rightarrow \Pi_b \rightarrow 1.$$ Conversely, Catanese (2017) inquires about what conditions need to be satisfied by a group of that sort in order to be the fundamental group of a Kodaira fibration. In this short note we collect some restrictions on the image of the classifying map $m \colon \Pi_b \to \Gamma_g$ in terms of the coinvariant homology of $\Pi_g$ . In particular, we observe that if π is the fundamental group of a Kodaira fibration with relative irregularity g−s, then $g \leq 1+ 6s$ , and we show that this effectively constrains the possible choices for π, namely that there are group extensions as above that fail to satisfy this bound, hence it cannot be the fundamental group of a Kodaira fibration. A noteworthy consequence of this construction is that it provides examples of symplectic 4-manifolds that fail to admit a Kähler structure for reasons that eschew the usual obstructions. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000743 Issue No:Vol. 62, No. 3 (2019)

Authors:Filomena Cianciaruso; Gennaro Infante, Paolamaria Pietramala Pages: 747 - 769 Abstract: We prove new results on the existence, non-existence, localization and multiplicity of non-trivial radial solutions of a system of elliptic boundary value problems on exterior domains subject to non-local, nonlinear, functional boundary conditions. Our approach relies on fixed point index theory. As a by-product of our theory we provide an answer to an open question posed by do Ó, Lorca, Sánchez and Ubilla. We include some examples with explicit nonlinearities in order to illustrate our theory. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000706 Issue No:Vol. 62, No. 3 (2019)

Authors:Eduardo Hernández; Jianhong Wu Pages: 771 - 788 Abstract: We study the existence, uniqueness and qualitative properties of global solutions of abstract differential equations with state-dependent delay. Results on the existence of almost periodic-type solutions (including, periodic, almost periodic, asymptotically almost periodic and almost automorphic solutions) are proved. Some examples of partial differential equations with state-dependent delay arising in population dynamics are presented. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S001309151800069X Issue No:Vol. 62, No. 3 (2019)

Authors:Pavel Shumyatsky Pages: 789 - 797 Abstract: Let G be a linear group such that for every g ∈ G there is a finite set ${\cal R}(g)$ with the property that for every x ∈ G all sufficiently long commutators [g, x, x, …, x] belong to ${\cal R}(g)$ . We prove that G is finite-by-hypercentral. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S001309151800072X Issue No:Vol. 62, No. 3 (2019)

K-Stability&rft.title=Proceedings+of+the+Edinburgh+Mathematical+Society&rft.issn=0013-0915&rft.date=2019&rft.volume=62&rft.spage=799&rft.epage=815&rft.aulast=Codogni&rft.aufirst=Giulio&rft.au=Giulio+Codogni&rft_id=info:doi/10.1017/S0013091518000512">Tits Buildings and K-Stability

Authors:Giulio Codogni Pages: 799 - 815 Abstract: A polarized variety is K-stable if, for any test configuration, the Donaldson–Futaki invariant is positive. In this paper, inspired by classical geometric invariant theory, we describe the space of test configurations as a limit of a direct system of Tits buildings. We show that the Donaldson–Futaki invariant, conveniently normalized, is a continuous function on this space. We also introduce a pseudo-metric on the space of test configurations. Recall that K-stability can be enhanced by requiring that the Donaldson–Futaki invariant is positive on any admissible filtration of the co-ordinate ring. We show that admissible filtrations give rise to Cauchy sequences of test configurations with respect to the above mentioned pseudo-metric. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000512 Issue No:Vol. 62, No. 3 (2019)

Authors:Yury Volkov Pages: 817 - 836 Abstract: We prove formulas of different types that allow us to calculate the Gerstenhaber bracket on the Hochschild cohomology of an algebra using some arbitrary projective bimodule resolution for it. Using one of these formulas, we give a new short proof of the derived invariance of the Gerstenhaber algebra structure on Hochschild cohomology. We also give some new formulas for the Connes differential on the Hochschild homology that lead to formulas for the Batalin–Vilkovisky (BV) differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to provide a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000901 Issue No:Vol. 62, No. 3 (2019)

Authors:Ja Kyung Koo; Dong Sung Yoon Pages: 837 - 845 Abstract: Schertz conjectured that every finite abelian extension of imaginary quadratic fields can be generated by the norm of the Siegel–Ramachandra invariants. We present a conditional proof of his conjecture by means of the characters on class groups and the second Kronecker limit formula. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000895 Issue No:Vol. 62, No. 3 (2019)

Authors:Olgur Celikbas; Shiro Goto, Ryo Takahashi, Naoki Taniguchi Pages: 847 - 859 Abstract: A conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000731 Issue No:Vol. 62, No. 3 (2019)

Authors:Suyoung Choi; Shizuo Kaji, Hanchul Park Pages: 861 - 874 Abstract: Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated with the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated with the Weyl chambers of type Cn and Dn, completing the computation for all classical types. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S001309151800086X Issue No:Vol. 62, No. 3 (2019)

Authors:D. Chan; A. Nyman Pages: 875 - 887 Abstract: We study Van den Bergh's non-commutative symmetric algebra ð•Šnc(M) (over division rings) via Minamoto's theory of Fano algebras. In particular, we show that ð•Šnc(M) is coherent, and its proj category ℙnc(M) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that ℙnc(M) is hereditary and there is a structure theorem for sheaves on ℙnc(M) analogous to that for ℙ1. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000871 Issue No:Vol. 62, No. 3 (2019)

Authors:Alejandra Garrido; Jone Uria–Albizuri Pages: 889 - 894 Abstract: We generalize the result about the congruence subgroup property for GGS groups in [3] to the family of multi-GGS groups; that is, all multi-GGS groups except the one defined by the constant vector have the congruence subgroup property. New arguments are provided to produce this more general proof. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000913 Issue No:Vol. 62, No. 3 (2019)

Authors:Laura Ciobanu; Charles Garnet Cox, Armando Martino Pages: 895 - 911 Abstract: In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups and the lamplighter group. PubDate: 2019-08-01T00:00:00.000Z DOI: 10.1017/S0013091518000573 Issue No:Vol. 62, No. 3 (2019)