Left Translation Research Articles

Fourier transforms of convolution operators on orlicz spaces

AbstractIn this paper, we study convolution operators on an Orlicz spaceLΦ(G) commuting with left translations, where Φ is an N-function andGis a locally compact group. We also present some basic properties of the Fourier transform of a Φ-convolution operator in the context of locally compact abelian groups.

Convergence of almost orbits of semigroups

In this paper we establish the following general theorem. Let S be a semitopological reversible semigroup and let Y be a linear subspace of $${\ell }^\infty (S)$$ which contains the constants and is left translation invariant. Suppose that Y has a left invariant mean. Let $$(E, \Vert \cdot \Vert _E)$$ be a uniformly convex Banach space and let C be a nonempty, bounded, closed and convex subset of E. Assume that C has nonempty interior, is locally uniformly rotund and $$\mathcal {T}=\left\{ T\left( s\right) \right\} _{s \in S}$$ is an asymptotically nonexpansive semigroup which acts on C. Assume also that $$\mathcal {T}$$ has a unique fixed point $$\tilde{x}$$ and that, in addition, this point $$\tilde{x}$$ lies on the boundary $$\partial C$$ of C. Then $$\{u(s) \}_{s \in S}$$ converges strongly to $$\tilde{x}$$ for each almost orbit $$u \in AO_Y (\mathcal {T})$$ .

Body postural realignment in the first 2 months after orthognathic surgery

Introduction This study aimed to assess the static posture in patients with Angle Class II and III malocclusions in the first 2 months after orthognathic surgery. Methods This was a longitudinal observational study. Eligible participants were adult patients who had an indication of orthognathic surgery (bilateral sagittal split osteotomy of the maxilla and/or mandible, can be associated or not with genioplasty). Thirty-five patients were evaluated from the orthognathic surgery group (OSG) and control group (CG). Measurements in OSG were performed at 3 time points: preoperative orthognathic surgery (P0), first postoperative month (P1), and second postoperative month (P2). Static posture was evaluated using the PostureScreen Mobile (PostureCo Inc, Trinity, Fla) application in 4 views. Results Patients with Angle Class II malocclusion in the OSG evidenced a tendency to a left hip translation at P1 with a significant difference at P2 in the anterior view (P = 0.052). In the right lateral view, patients with Angle Class II malocclusion in the OSG at P1 presented an accentuated anterior shoulder translation when compared with CG (P Conclusions These findings suggest a realignment of static posture in the first 2 months after orthognathic surgery.

Flat affine manifolds and their transformations

We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of affine transformations is $n$, this orbit has discrete isotropy. For any given Lie subgroup $H$ of affine transformations of the manifold, we show the existence of an associative envelope of the Lie algebra of $H$, relative to the connection. The case when $M$ is a Lie group and $H$ acts on $G$ by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transformations admits a flat affine bi-invariant structure. The paper is illustrated with several examples.

Γ-supercyclicity of families of translates in weighted Lp-spaces on locally compact groups

Let ω be a weight function defined on a locally compact group G, 1⩽p<+∞, S⊂G and let us assume that for any s∈S, the left translation operator Ts is continuous from the weighted Lp-space Lp(G,ω) into itself. For a given set Γ⊂C, a vector f∈Lp(G,ω) is said to be (Γ,S)-dense if the set {λTsf:λ∈Γ,s∈S} is dense in Lp(G,ω). In this paper, we characterize the existence of (Γ,S)-dense vectors in Lp(G,ω) in terms of the weight and the set Γ.

Involutive latin solutions of the Yang-Baxter equation

Wolfgang Rump showed that there is a one-to-one correspondence between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras in which all left translations Lx are bijections, the squaring map is a bijection, and the identity (xy)(xz)=(yx)(yz) holds. We call these algebras rumples in analogy with quandles, another class of binary algebras giving solutions of the Yang-Baxter equation. We focus on latin rumples, that is, on rumples in which all right translations are bijections as well.We prove that an affine latin rumple of order n exists if and only if n=p1p1k1⋯pmpmkm for some distinct primes pi and positive integers ki. A large class of affine solutions is obtained from nonsingular near-circulant matrices A, B satisfying [A,B]=A2. We characterize affine latin rumples as those latin rumples for which the displacement group generated by LxLy−1 is abelian and normal in the group generated by all translations.We develop the extension theory of rumples sufficiently to obtain examples of latin rumples that are not affine, not even isotopic to a group. Finally, we investigate latin rumples in which the dual identity (zx)(yx)=(zy)(xy) holds as well, and we show, among other results, that the generators LxLy−1 of their displacement group have order dividing four.

Characterization of eigenfunctions of the Laplace–Beltrami operator using Fourier multipliers

Let X be a rank one Riemannian symmetric space of noncompact type and Δ be the Laplace–Beltrami operator of X. The space X can be identified with the quotient space G/K where G is a connected noncompact semisimple Lie group of real rank one with finite center and K is a maximal compact subgroup of G. Thus G acts naturally on X by left translations. Through this identification, a function or measure on X is radial (i.e. depends only on the distance from eK), when it is invariant under the left-action of K. We consider right-convolution operators Θ on functions f on X defined by, Θ:f↦f⁎μ where μ is a radial (possibly complex) measure on X. These operators will be called multipliers. In particular Θ is a radial average when μ is a radial probability measure. Notable examples of radial averages are ball, sphere and annular averages and the heat operator. In this paper we address problems of the following type: Fix a multiplier, in particular an averaging operator Θ. Suppose that {fk}k∈Z is a bi-infinite sequence of functions on X such that for all k∈Z, Θfk=Afk+1 and ‖fk‖<M for some constants A∈C, M>0 and a suitable norm ‖⋅‖. From this hypothesis, we try to infer that f0, hence every fk, is an eigenfunction of Δ.

Relatively uniformly continuous semigroups on vector lattices

In this paper we introduce the notions of relatively uniform continuity and strong continuity with respect to the relatively uniform topology for semigroups on general vector lattices. These notions allow us to study semigroups on non-locally convex spaces such as Lp(R) for 0<p<1 and non-complete spaces such as Lip(R), UC(R), and Cc(R). We provide examples of relatively uniform continuous semigroups such as the (left) translation semigroup and the Ornstein-Uhlenbeck semigroup.

A MINIMAL CONGRUENCE LATTICE REPRESENTATION FOR

Let$p$be an odd prime. The unary algebra consisting of the dihedral group of order$2p$, acting on itself by left translation, is a minimal congruence lattice representation of $\mathbb{M}_{p+1}$.

Analytic linear Lie rack structures on Leibniz algebras

A linear Lie rack structure on a finite dimensional vector space V is a Lie rack operation pointed at the origin and such that for any x, the left translation is linear. A linear Lie rack operation is called analytic if for any where is an n + 1-multilinear map symmetric in the n first arguments. In this case, is exactly the left Leibniz product associated to Any left Leibniz algebra has a canonical analytic linear Lie rack structure given by where In this paper, we show that a sequence of n + 1-multilinear maps on a vector space V defines an analytic linear Lie rack structure if and only if is a left Leibniz bracket, the are invariant for and satisfy a sequence of multilinear equations. Some of these equations have a cohomological interpretation and can be solved when the zero and the 1-cohomology of the left Leibniz algebra are trivial. On the other hand, given a left Leibniz algebra we show that there is a large class of (analytic) linear Lie rack structures on which can be built from the canonical one and invariant multilinear symmetric maps on A left Leibniz algebra on which all the analytic linear Lie rack structures are build in this way will be called rigid. We use our characterizations of analytic linear Lie rack structures to show that and are rigid. We conjecture that any simple Lie algebra is rigid as a left Leibniz algebra.

V.V. Nabokov and F.I. Tyutchev: Double Being

The article attempts to find common ground between the artistic systems divided by a century – the lyrics of the Russian poet and philosopher F. I. Tyutchev and prose of the outstanding Russian-American writer V. V. Nabokov. The first part of the article deals with the ideas of domestic and foreign researchers of the peculiarity of the Nabokov’s aesthetic system. Nabokov did not write a special article and did not publish a single lecture on Tyutchev’s lyrics, although he repeatedly turned to the work of the poet, in particular, while teaching in America, left excellent translations of his poems into English. The authors of the article show the common perception of life of nature and man for both authors. The inner closeness of Nabokov and Tyutchev is manifested not so much in direct statements as indirectly. This is an acknowledgement of the narrator in the General Ledger, Nabokov’s Other shores (“Pushkin and Tolstoy, Tyutchev, Gogol stood at the four corners of my world”); and accented by the intertextuality in the prose of the author with specific Tyutchen’s reminiscences, sometimes shocking-pun (short story “Cloud, lake, tower”); and focusing on the motifs of memory, gift, abyss (in Schellingian understanding); and Nabokov’s use of the key opposition “dream – chaos”, which runs through the entire lyrics of Tyutchev. The second part of the article is devoted to the analysis of Tyutchev’s poetry in the light of researches conducted by Russian philosophers and literary critics. The article notes the little-known fact that Nabokov and Tyutchev were related in kinship. You can find out at what point the kinship of the two great masters of the word crossed by reading the article.

Topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups

In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of SU(2), U(2), SO(3), SO(3) × S1 and Spinℂ(3). We define the rotation vectors (numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors (numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism f : L(p, −1) × S1 → L(p, −1) × S1, the induced isomorphism (π ◦ f ◦ i)* maps each element in the fundamental group of L(p, −1) to itself or its inverse, where i : L(p, −1) → L(p, −1) × S1 is the natural inclusion and π : L(p, −1) × S1 → L(p, −1) is the projection.

Semi Bornological Groups

The motivation for this work is to find the sufficient conditions to bornologize any group and solve the problem of boundedness for any group, by constructing new structure semi bornological groups. In particular, we show that every left (right) translations in semi bornological groups are bornological isomorphisms and therefore the semi bornological groups structures are homogeneous, and this property from the semi bornological group is not shared with bornological semigroups. Further, the boundedness of a semi bornological group can be deduced from its boundedness at the identity. So, the problem of boundedness for any group can be solved.

A full description of property T of unital C⁎-crossed products

Let Γ be a discrete group that acts on a unital C⁎-algebra A through an action β. We study property T as well as its stronger variant of the full and the reduced crossed products. We show that if A has a β-invariant tracial state, then the reduced crossed product A⋊β,rΓ has strong property T if and only if Γ has property T and (A⋊β,rΓ,A) has strong property T. If Γ has the Haagerup property, the strong property T of (A⋊β,rΓ,A) in the above statement can be replaced by that of A. Furthermore, the strong property T of the full crossed product is shown to coincide with that of the reduced one. In the case of property T, we obtain the corresponding result when the group is either ICC or abelian. As an application, a non-compact locally compact group G is amenable if and only if Cb(G)⋊lt,rGd does not have strong property T, where Gd is the group G equipped with the discrete topology and lt is the left translation action.

Compact actions whose orbit equivalence relations are not profinite

Let Γ↷(X,μ) be a measure preserving action of a countable group Γ on a standard probability space (X,μ). We prove that if the action Γ↷X is not profinite and satisfies a certain spectral gap condition, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets). As a consequence, we show that if Γ is a countable dense subgroup of a compact non-profinite group G such that the left translation action Γ↷G has spectral gap, then Γ↷G is antimodular and not orbit equivalent to any, not necessarily free, profinite action. This provides the first such examples of compact actions, partially answering a question of Kechris and answering a question of Tsankov.

Insufficient Balance Recovery Following Unannounced External Perturbations in Persons With Stroke

Background. Persons with stroke (PwS) are at increased risk of falls, especially toward the paretic side, increasing the probability of a hip fracture. The ability to recover from unexpected loss of balance is a critical factor in fall prevention. Objectives. We aimed to compare reactive balance capacity and step kinematics between PwS and healthy controls. Methods. Thirty subacute PwS and 15 healthy controls were exposed to forward, backward, right, and left unannounced surface translations in 6 increasing intensities while standing. Single step threshold, multiple step threshold, and fall threshold (ie, perturbation intensity leading to a fall into harness system) were recorded as well as reactive step initiation time, step length, and step velocity. Results. Twenty-five PwS fell into harness system during the experiment while healthy controls did not fall. Fourteen out of 31 falls occurred in response to surface translations toward the nonparetic side, that is, falling toward the paretic side. Compared with healthy controls, PwS demonstrated significantly lower fall threshold and multiple step threshold in response to forward, backward, and lateral surface translations. Impairments were more pronounced in response to forward surface translation and toward the nonparetic side (ie, loss of balance toward the paretic side). A trend toward significant shorter step length in response to lateral surface translations was found in PwS compared with healthy controls. Conclusions. Findings highlight the importance of assessing reactive balance capacity in response to perturbations in different directions and intensities in addition to the routine assessment in PwS.

Analysis of Brain Lesion Impact on Balance and Gait Following Stroke.

Falls are a leading cause of serious injury and restricted participation among persons with stroke (PwS). Reactive balance control is essential for fall prevention, however, only a few studies have explored the effects of lesion characteristics (location and extent) on balance control in PwS. We aimed to assess the impact of lesion characteristics on reactive and anticipatory balance capacity, gait, and hemiparetic lower limb function, in PwS. Forty-six subacute PwS were exposed to forward, backward, right and left unannounced horizontal surface translations in six increasing intensities while standing. Fall threshold (i.e., perturbation intensity that results in a fall into the harness system) was measured. In addition, the Berg Balance Scale (BBS), 6 Minute Walk Test (6MWT) and Lower Extremity Fugl-Meyer (LEFM) were measured. Lesion effects were analyzed separately for left and right hemisphere damaged (LHD, RHD) patients, using voxel-based lesion-symptom mapping (VLSM). Our results show that voxel clusters where damage exerted a significant impact on balance, gait and lower-limb function were found in the corticospinal tract (CST), in its passage in the corona radiata and in the posterior limb of the internal capsule. An additional significant impact was found to lesions affecting the putamen and the external capsule (EC). Balance, gait, and hemiparetic lower limb function showed much overlap of the corresponding “significant” voxel clusters. Test scores of RHD and LHD patients were affected largely by damage to homologous regions, with the LHD group showing a wider distribution of “significant” voxels. The study corroborates and extends previous findings by demonstrating that balance control, gait, and lower limb function are all affected mainly by damage to essentially the same brain structures, namely—the CST and adjacent structures in the capsular-putaminal region.

Cocycle superrigidity for translation actions of product groups

Let G be either a profinite or a connected compact group, and Γ, Λ be finitely generated dense subgroups. Assuming that the left translation action of Γ on G is strongly ergodic, we prove that any cocycle for the left-right translation action of Γ × Λ on G with values in a countable group is virtually cohomologous to a group homomorphism. Moreover , we prove that the same holds if G is a (not necessarily compact) connected simple Lie group provided that Λ contains an infinite cyclic subgroup with compact closure. We derive several applications to OE-and W *-superrigidity. In particular, we obtain the first examples of compact actions of F2 × F2 which are W *-superrigid.

On density of infinite subsets II: Dynamics on homogeneous spaces

Let $G$ be a noncompact semisimple Lie group, $\Gamma$ be an irreducible cocompact lattice in $G$, and $P<G$ be a minimal parabolic subgroup. We consider the dynamics of $P$ acting on $G/\Gamma$ by left translation. For any infinite subset $A\subset G/\Gamma$, we show that, for any $\epsilon>0$, there is a $g\in P$ such that $gA$ is $\epsilon$-dense. We also prove a similar result for certain discrete group actions on $\mathbb T^n$.

Simply connected latin quandles

A (left) quandle is connected if its left translations generate a group that acts transitively on the underlying set. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to constant quandle cocycles of Andruskiewitsch and Graña. A connected quandle is simply connected if it has no nontrivial coverings, or, equivalently, if all its second constant cohomology sets with coefficients in symmetric groups are trivial. In this paper, we develop a combinatorial approach to constant cohomology. We prove that connected quandles that are affine over cyclic groups are simply connected (extending a result of Graña for quandles of prime size) and that finite doubly transitive quandles of order different from [Formula: see text] are simply connected. We also consider constant cohomology with coefficients in arbitrary groups.