scholarly journals A Gyrogeometric Mean in the Einstein Gyrogroup

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1333 ◽  
Author(s):  
Takuro Honma ◽  
Osamu Hatori
Keyword(s):  
Alternative Proof ◽  
Left Translation ◽  
Elementary Calculation ◽  
Translation Invariant

In this paper, we define a gyrogeometric mean on the Einstein gyrovector space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation.

FINE DISINTEGRATION OF THE LEFT REGULAR REPRESENTATION

2005 ◽  
Vol 04 (06) ◽  
pp. 683-706 ◽  
Author(s):  
JEAN LUDWIG ◽  
CARINE MOLITOR-BRAUN
Keyword(s):  
Regular Representation ◽  
Unitary Irreducible Representation ◽  
Irreducible Representations ◽  
Left Translation ◽  
Direct Integral ◽  
Translation Invariant ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Left Regular Representation ◽  
Left Regular

Let Hn be the (2n + 1)-dimensional Heisenberg group. We decompose L2(Hn) as the closure of a direct sum of infinitely many left translation invariant eigenspaces (for certain systems of partial differential equations). The restriction of the left regular representation to each one of these eigenspaces disintegrates into a direct integral of unitary irreducible representations, such that each infinite dimensional unitary irreducible representation appears with multiplicity 0 or 1 in this disintegration.

scholarly journals Solving equations in dense Sidon sets

2021 ◽  
pp. 1-10
Author(s):  
SEAN PRENDIVILLE
Keyword(s):  
Linear Equations ◽  
Alternative Proof ◽  
Translation Invariant ◽  
Sidon Sets ◽  
Solving Equations

Abstract We offer an alternative proof of a result of Conlon, Fox, Sudakov and Zhao [CFSZ20] on solving translation-invariant linear equations in dense Sidon sets. Our proof generalises to equations in more than five variables and yields effective bounds.

On invariant convex subsets in algebras defined on a locally compact group G

2012 ◽  
Vol 49 (3) ◽  
pp. 301-314
Author(s):  
Ali Ghaffari
Keyword(s):  
Compact Group ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Left Translation ◽  
Measure Algebra ◽  
Locally Compact ◽  
Translation Invariant ◽  
Second Dual ◽  
Necessary And Sufficient ◽  
Convex Subsets

Suppose that A is either the Banach algebra L1(G) of a locally compact group G, or measure algebra M(G), or other algebras (usually larger than L1(G) and M(G)) such as the second dual, L1(G)**, of L1(G) with an Arens product, or LUC(G)* with an Arenstype product. The left translation invariant closed convex subsets of A are studied. Finally, we obtain necessary and sufficient conditions for LUC(G)* to have 1-dimensional left ideals.

scholarly journals On L1-kernels of unitary representations of semisimple Lie groups

1992 ◽  
Vol 112 (2) ◽  
pp. 343-348
Author(s):  
Mohammed E. B. Bekka ◽  
Jean Ludwig
Keyword(s):  
Linear Functional ◽  
Locally Compact Group ◽  
General Question ◽  
Left Translation ◽  
Invariant Mean ◽  
Compact Groups ◽  
Semisimple Lie Groups ◽  
Translation Invariant ◽  
Amenable Groups ◽  
Positive Linear Functional

Let G be a locally compact group with fixed left Haar measure dx. Recall that G is said to be amenable if there exists a left translation invariant mean on the space L∞(G), i.e. if there exists a positive, linear functional M on L∞(G) such that M(lG) = 1 and M(xø) = Mø for all ø∈L∞(G), x∈G, where xø denotes the left translate xø(y) = ø(xy). The class of amenable groups includes all soluble and all compact groups (concerning the theory of amenable groups we refer to [9]). It is easy to see that G is amenable if and only if ℂ1G, the space of the constant functions on G, has a closed left translationinvariant complement in L∞(G). This reformulation of amenability leads to the following more general question.

WEIGHTED ORLICZ ALGEBRAS ON LOCALLY COMPACT GROUPS

2015 ◽  
Vol 99 (3) ◽  
pp. 399-414 ◽  
Author(s):  
ALEN OSANÇLIOL ◽  
SERAP ÖZTOP
Keyword(s):  
Linear Subspace ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Ideal Space ◽  
Left Translation ◽  
Compact Groups ◽  
Young Function ◽  
Locally Compact ◽  
Closed Linear Subspace ◽  
Translation Invariant

For a locally compact group $G$ with left Haar measure and a Young function ${\rm\Phi}$, we define and study the weighted Orlicz algebra $L_{w}^{{\rm\Phi}}(G)$ with respect to convolution. We show that $L_{w}^{{\rm\Phi}}(G)$ admits no bounded approximate identity under certain conditions. We prove that a closed linear subspace $I$ of the algebra $L_{w}^{{\rm\Phi}}(G)$ is an ideal in $L_{w}^{{\rm\Phi}}(G)$ if and only if $I$ is left translation invariant. For an abelian $G$, we describe the spectrum (maximal ideal space) of the weighted Orlicz algebra and show that weighted Orlicz algebras are semisimple.

scholarly journals Factorisation of Lipschitz functions on zero dimensional groups

1981 ◽  
Vol 23 (2) ◽  
pp. 215-225 ◽  
Author(s):  
Walter R. Bloom
Keyword(s):  
Lipschitz Functions ◽  
Suitable Choice ◽  
Left Translation ◽  
Invariant Metric ◽  
Lipschitz Spaces ◽  
Locally Compact ◽  
Translation Invariant ◽  
Dimensional Group ◽  
Image Position ◽  
Range Of Values

Let G denote a locally compact metrisable zero dimensional group with left translation invariant metric d. The Lipschitz spaces are defined bywhere af: x → f(ax) and α > 0; when r = ∞ the members of Lip(α; r) are taken to be continuous. For a suitable choice of metric it is shown that , where 1 ≤ p ≤ 2, α > q−1, p, q are conjugate indices and . It is also shown that for G infinite the range of values of α cannot be extended.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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