scholarly journals Positive definite *-spherical functions, property (T) and C*-completions of Gelfand pairs

2015 ◽  
Vol 160 (1) ◽  
pp. 77-93
Author(s):  
NADIA S. LARSEN ◽  
RUI PALMA
Keyword(s):  
Banach Algebra ◽  
General Result ◽  
Open Subgroup ◽  
Positive Definite ◽  
Good Choice ◽  
Gelfand Pairs ◽  
Compact Open Subgroup ◽  
Simple Algebraic Group ◽  
Property T

AbstractThe study of existence of a universal C*-completion of the *-algebra canonically associated to a Hecke pair was initiated by Hall, who proved that the Hecke algebra associated to (SL2($\mathbb{Q}$p), SL2($\mathbb{Z}$p)) does not admit a universal C*-completion. Kaliszewski, Landstad and Quigg studied the problem by placing it in the framework of Fell–Rieffel equivalence, and highlighted the role of other C*-completions. In the case of the pair (SLn($\mathbb{Q}$p), SLn($\mathbb{Z}$p)) for n ⩾ 3 we show, invoking property (T) of SLn($\mathbb{Q}$p), that the C*-completion of the L1-Banach algebra and the corner of C*(SLn($\mathbb{Q}$p)) determined by the subgroup are distinct. In fact, we prove a more general result valid for a simple algebraic group of rank at least 2 over a $\mathfrak{p}$-adic field with a good choice of a maximal compact open subgroup.

scholarly journals Bounding the covolume of lattices in products

2019 ◽  
Vol 155 (12) ◽  
pp. 2296-2333
Author(s):  
Pierre-Emmanuel Caprace ◽  
Adrien Le Boudec
Keyword(s):  
Open Subgroup ◽  
Discrete Groups ◽  
Discrete Subgroups ◽  
Compact Open Subgroup ◽  
Locally Finite ◽  
Formula Group ◽  
Property T ◽  
Dense Projections ◽  
Closed Normal Subgroup ◽  
Compactly Generated

We study lattices in a product $G=G_{1}\times \cdots \times G_{n}$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_{i}$ is non-compact and every closed normal subgroup of $G_{i}$ is discrete or cocompact (e.g. $G_{i}$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $\unicode[STIX]{x1D6E4}$ with dense projections is finite. The same result holds if $\unicode[STIX]{x1D6E4}$ is non-uniform, provided $G$ has Kazhdan’s property (T). We show that for any compact subset $K\subset G$, the collection of discrete subgroups $\unicode[STIX]{x1D6E4}\leqslant G$ with $G=\unicode[STIX]{x1D6E4}K$ and dense projections is uniformly discrete and hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $\operatorname{Aut}(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.

scholarly journals Role of sexual imprinting in assortative mating and premating isolation in Darwin’s finches

2018 ◽  
Vol 115 (46) ◽  
pp. E10879-E10887 ◽  
Author(s):  
Peter R. Grant ◽  
B. Rosemary Grant
Keyword(s):  
Natural Habitat ◽  
Good Choice ◽  
Premating Isolation ◽  
Sexual Imprinting ◽  
Darwin's Finches ◽  
Darwin’S Finches ◽  
Adaptive Radiations ◽  
Beak Size ◽  
Species Specific

Global biodiversity is being degraded at an unprecedented rate, so it is important to preserve the potential for future speciation. Providing for the future requires understanding speciation as a contemporary ecological process. Phylogenetically young adaptive radiations are a good choice for detailed study because diversification is ongoing. A key question is how incipient species become reproductively isolated from each other. Barriers to gene exchange have been investigated experimentally in the laboratory and in the field, but little information exists from the quantitative study of mating patterns in nature. Although the degree to which genetic variation underlying mate-preference learning is unknown, we provide evidence that two species of Darwin’s finches imprint on morphological cues of their parents and mate assortatively. Statistical evidence of presumed imprinting is stronger for sons than for daughters and is stronger for imprinting on fathers than on mothers. In combination, morphology and species-specific song learned from the father constitute a barrier to interbreeding. The barrier becomes stronger the more the species diverge morphologically and ecologically. It occasionally breaks down, and the species hybridize. Hybridization is most likely to happen when species are similar to each other in adaptive morphological traits, e.g., body size and beak size and shape. Hybridization can lead to the formation of a new species reproductively isolated from the parental species as a result of sexual imprinting. Conservation of sufficiently diverse natural habitat is needed to sustain a large sample of extant biota and preserve the potential for future speciation.

scholarly journals New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms

2006 ◽  
Vol 13 (4) ◽  
pp. 687-691
Author(s):  
Guram Gogishvili
Keyword(s):  
Quadratic Form ◽  
General Result ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Precise Estimate ◽  
Definite Integral ◽  
Singular Series ◽  
Prime Factors ◽  
Quaternary Quadratic Form ◽  
Best Estimates

Abstract Let 𝑚 ∈ ℕ, 𝑓 be a positive definite, integral, primitive, quaternary quadratic form of the determinant 𝑑 and let ρ(𝑓,𝑚) be the corresponding singular series. When studying the best estimates for ρ(𝑓,𝑚) with respect to 𝑑 and 𝑚 we proved in [Gogishvili, Trudy Tbiliss. Univ. 346: 72–77, 2004] that where 𝑏(𝑘) is the product of distinct prime factors of 16𝑘 if 𝑘 ≠ 1 and 𝑏(𝑘) = 3 if 𝑘 = 1. The present paper proves a more precise estimate where 𝑑 = 𝑑0𝑑1, if 𝑝 > 2; 𝑕(2) ⩾ –4. The last estimate for ρ(𝑓,𝑚) as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.

scholarly journals Homomorphisms into totally disconnected, locally compact groups with dense image

2019 ◽  
Vol 31 (3) ◽  
pp. 685-701 ◽  
Author(s):  
Colin D. Reid ◽  
Phillip R. Wesolek
Keyword(s):  
Normal Subgroup ◽  
Open Subgroup ◽  
Locally Compact Groups ◽  
Group Homomorphism ◽  
Compact Groups ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Dense Image ◽  
Profinite Completions ◽  
Totally Disconnected

Abstract Let {\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair {(G,\phi^{-1}(L))} , where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.

scholarly journals Totally disconnected locally compact groups locally of finite rank

2015 ◽  
Vol 158 (3) ◽  
pp. 505-530 ◽  
Author(s):  
PHILLIP WESOLEK
Keyword(s):  
Lie Groups ◽  
Finite Rank ◽  
Open Subgroup ◽  
Simple Groups ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Elementary Group ◽  
Finite Set ◽  
Finite Direct Product ◽  
Totally Disconnected

AbstractWe study totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contain a compact open subgroup with finite rank. We show such groups that additionally admit a pro-π compact open subgroup for some finite set of primes π are virtually an extension of a finite direct product of topologically simple groups by an elementary group. This result, in particular, applies to l.c.s.c. p-adic Lie groups. We go on to obtain a decomposition result for all t.d.l.c.s.c. groups containing a compact open subgroup with finite rank. In the course of proving these theorems, we demonstrate independently interesting structure results for t.d.l.c.s.c. groups with a compact open pro-nilpotent subgroup and for topologically simple l.c.s.c. p-adic Lie groups.

scholarly journals Przewlekłe leczenie przeciwkrzepliwe

2018 ◽  
Vol 49 (2) ◽  
pp. 47-49 ◽  
Author(s):  
Krystyna Zawilska
Keyword(s):  
Pulmonary Embolism ◽  
Oral Anticoagulants ◽  
Direct Oral Anticoagulants ◽  
Compression Therapy ◽  
Good Choice ◽  
Anticoagulant Treatment ◽  
Efficacy And Safety ◽  
Risk Of Bleeding

AbstractUnprovoked venous thromboembolism (VTE) - proximal venous thrombosis or pulmonary embolism - should be treated either 3 months or indefinitely if the risk of bleeding is low. This article summarizes the efficacy and safety of extended therapy of VTE with direct oral anticoagulants (DOAC) in comparison with warfarin, as well as the role of of acetylsalicylic acid (ASA) for the long-term prevention of recurrent VTE. As the Survet study showed, for some patients who have already completed at least 6 months of anticoagulant treatment for their index VTE event, an oral glycosaminoglycan - sulodexide associated with compression therapy is a good choice, because it decreases the incidence of recurrences of VTE without detectable risks for the patients’ safety.

Early bird or versioning

2019 ◽  
Vol 32 (3) ◽  
pp. 769-792
Author(s):  
Mingchun Chen ◽  
Zhiying Liu ◽  
Chaoliang Ma
Keyword(s):  
Dynamic Game ◽  
Good Choice ◽  
Pricing Strategy ◽  
Pricing Strategies ◽  
Content Type ◽  
Early Bird ◽  
Bayesian Equilibrium ◽  
Dynamic Game Model ◽  
Selection Of

Purpose Crowdfunding, especially reward-based crowdfunding, has quickly evolved into a commonly used vehicle for innovating entrepreneurs to develop their products. Many crowdfunding platforms allow creators maximum flexibility in terms of the prices and rewards offered in a project to gain sufficient capital. However, creators need to understand how to design project rewards and how to select a pricing strategy, in addition to whether the creator should spend resources on designing multiple rewards of varying quality. The purpose of this paper is to address these issues by answering whether and why there are significant differences in the application of early-bird and versioning pricing strategies in crowdfunding. Design/methodology/approach This paper develops a two-stage dynamic game model with incomplete information, proposes a corollary calculated by analyzing a perfect Bayesian equilibrium, and then tests Corollary 1 by empirical analysis. Findings Contrary to the findings of other studies, the results show that an early-bird pricing strategy is likely better than a versioning pricing strategy for earning greater revenue in a crowdfunding context, on average. This finding means that creators do not have to spend as much in designing rewards of various qualities; rather, they should only provide multiple price options for high-quality rewards. However, if the heterogeneity of target backers’ valuations and the quality difference between two types of products are adequately high, a versioning pricing strategy may be a good choice for creators. Practical implications This paper provides a reference for creators regarding the selection of pricing strategies and the design of reward quality when launching crowdfunding projects. Originality/value This paper explains an interesting and practical issue in the design of reward quality and the selection of a pricing strategy after fully considering the role of the crowdfunding all-or-nothing mechanism and special backer behavior.

scholarly journals DISTALITY OF CERTAIN ACTIONS ON -ADIC SPHERES

2019 ◽  
Vol 109 (2) ◽  
pp. 250-261
Author(s):  
RIDDHI SHAH ◽  
ALOK KUMAR YADAV
Keyword(s):  
Compact Group ◽  
Unit Sphere ◽  
Open Subgroup ◽  
Linear Action ◽  
Compact Open Subgroup ◽  
The Real ◽  
Affine Maps

Consider the action of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on the $p$-adic unit sphere ${\mathcal{S}}_{n}$ arising from the linear action on $\mathbb{Q}_{p}^{n}\setminus \{0\}$. We show that for the action of a semigroup $\mathfrak{S}$ of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on ${\mathcal{S}}_{n}$, the following are equivalent: (1) $\mathfrak{S}$ acts distally on ${\mathcal{S}}_{n}$; (2) the closure of the image of $\mathfrak{S}$ in $\operatorname{PGL}(n,\mathbb{Q}_{p})$ is a compact group. On ${\mathcal{S}}_{n}$, we consider the ‘affine’ maps $\overline{T}_{a}$ corresponding to $T$ in $\operatorname{GL}(n,\mathbb{Q}_{p})$ and a nonzero $a$ in $\mathbb{Q}_{p}^{n}$ satisfying $\Vert T^{-1}(a)\Vert _{p}<1$. We show that there exists a compact open subgroup $V$, which depends on $T$, such that $\overline{T}_{a}$ is distal for every nonzero $a\in V$ if and only if $T$ acts distally on ${\mathcal{S}}_{n}$. The dynamics of ‘affine’ maps on $p$-adic unit spheres is quite different from that on the real unit spheres.

SHIFT INVARIANT SPACES FOR LOCAL FIELDS

2011 ◽  
Vol 09 (03) ◽  
pp. 417-426 ◽  
Author(s):  
A. AHMADI ◽  
A. ASKARI HEMMAT ◽  
R. RAISI TOUSI
Keyword(s):  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Open Subgroup ◽  
Local Fields ◽  
Locally Compact Abelian Group ◽  
Compact Open Subgroup ◽  
Parseval Frame ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces

This paper is an investigation of shift invariant subspaces of L2(G), where G is a locally compact abelian group, or in general a local field, with a compact open subgroup. In this paper we state necessary and sufficient conditions for shifts of an element of L2(G) to be an orthonormal system or a Parseval frame. Also we show that each shift invariant subspace of L2(G) is a direct sum of principle shift invariant subspaces of L2(G) generated by Parseval frame generators.

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