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## On Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group

Abreu, Luís Daniel
Tipo: Pré-impressão
ENG
Relevância na Pesquisa
66.05%
Gabor frames with Hermite functions are equivalent to Fock frames with monomials windows and to sampling sequences in true poly-Fock spaces. In the L2 case, such an equivalence results from the unitarity of the so-called true poly- Bargmann transform. We will extend the equivalence to Banach spaces, applying Feichtinger-Gr¨ochenig coorbit theory to the Fock representation of the Heisenberg group. This task requires Lp estimates for the true poly-Bargmann transform which are obtained using the theory of modulation spaces. In the L2 case we will also revisit the complex variables approach and obtain an explicit formula for the interpolation problem in true poly-Fock spaces, which yields Gabor frames with Hermite functions by a duality argument.

## Simetrias de Lie de equações diferenciais parciais semilineares envolvendo o operador de Kohn-Laplace no grupo de Heisenberg; Lie point synmetrics of semilinear partial differential equations involving the Kohn-Laplace operator on the Heisenberg group

Igor Leite Freire
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Tese de Doutorado Formato: application/pdf
Relevância na Pesquisa
56.18%
Neste trabalho provamos um teorema que faz a classificacão completa dos grupos de simetrias de Lie da equação semilinear de Kohn - Laplace no grupo de Heisenberg tridimensional. Uma vez que tal equação possui estrutura variacional, determinamos quais são suas simetrias de Noether e a partir delas estabelecemos suas respectivas leis de conservação; In this work, we carry out a complete group classification of Lie point symmetries of semilinear Kohn - Laplace equations on the three-dimensional Heisenberg group. Since this equation has variational structure, we determine which of its symmetries are Noether's symmetries. Then we establish their respectives conservation laws

## Asymptotic behavior of the Riemannian Heisenberg group and its horoboundary

Donne, Enrico Le; Golo, Sebastiano Nicolussi; Sambusetti, Andrea
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.15%
The paper is devoted to the large scale geometry of the Heisenberg group $\mathbb H$ equipped with left-invariant Riemannian distances. We prove that two such distances have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian distance $d$ on $\mathbb H$ there is a unique subRiemanniann metric $d'$ for which $d-d'$ goes to zero at infinity, and we estimate the rate of convergence. As a first immediate consequence we get that the Riemannian Heisenberg group is at bounded distance from its asymptotic cone. The second consequence, which was our aim, is the explicit description of the horoboundary of the Riemannian Heisenberg group.; Comment: 26 pages

## Deformations of the discrete Heisenberg group

Barmeier, Severin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.08%
We study deformations of the discrete Heisenberg group acting properly discontinuously on the Heisenberg group from the left and right and obtain a complete description of the deformation space.; Comment: 8 pages

## Mexican Hat Wavelet on the Heisenberg Group

Mayeli, Azita
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.08%
In this article wavelets (admissible vectors) on the Heisenberg group are studied from the point of view of Calderon's formula. Further we shall show that for the class of Schwartz functions the Calderon admissibility condition is equivalent to the usual admissibility property which will be introduced in this work. Furthermore motivated by a well-known example on the real line, the Mexican-Hat wavelet, we demonstrate the existence and construction of an analogous wavelet on the Heisenberg Lie group with 2 vanishing moments, which together with all of its derivatives has Gaussian decay.; Comment: 8 pages, no figures

## The C*-alegbras of the Heisenberg Group and of thread-like Lie groups

Ludwig, Jean; Turowska, Lyudmila
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.98%
We describe the C*-algebras of the Heisenberg group H_n, n\geq 1, and the thread-like Lie group G_N, N\geq 3, in terms of C*-algebras of operator fields.; Comment: 31 pages

## Vertical versus horizontal Poincar\'e inequalities on the Heisenberg group

Lafforgue, Vincent; Naor, Assaf
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.96%
Let $\H= < a,b | a[a,b]=[a,b]a \wedge b[a,b]=[a,b]b>$ be the discrete Heisenberg group, equipped with the left-invariant word metric $d_W(\cdot,\cdot)$ associated to the generating set ${a,b,a^{-1},b^{-1}}$. Letting $B_n= {x\in \H: d_W(x,e_\H)\le n}$ denote the corresponding closed ball of radius $n\in \N$, and writing $c=[a,b]=aba^{-1}b^{-1}$, we prove that if $(X,|\cdot|_X)$ is a Banach space whose modulus of uniform convexity has power type $q\in [2,\infty)$ then there exists $K\in (0,\infty)$ such that every $f:\H\to X$ satisfies {multline*} \sum_{k=1}^{n^2}\sum_{x\in B_n}\frac{|f(xc^k)-f(x)|_X^q}{k^{1+q/2}}\le K\sum_{x\in B_{21n}} \Big(|f(xa)-f(x)|^q_X+\|f(xb)-f(x)\|^q_X\Big). {multline*} It follows that for every $n\in \N$ the bi-Lipschitz distortion of every $f:B_n\to X$ is at least a constant multiple of $(\log n)^{1/q}$, an asymptotically optimal estimate as $n\to\infty$.

## Discrete Heisenberg group and its automorphism group

Osipov, D. V.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.02%
In this note we give more easy and short proof of a statement previously proved by P. Kahn that the automorphism group of the discrete Heisenberg group ${\rm Heis}(3, \mathbb{Z})$ is isomorphic to the group $(\mathbb{Z} \oplus \mathbb{Z}) \rtimes GL(2,\mathbb{Z})$. The method which we suggest to construct this isomorphism gives far more transparent picture of the structure of the automorphism group of the group ${\rm Heis}(3, \mathbb{Z})$.; Comment: 5 pages, to appear in Mathematical Notes (2015)

## Continued fractions on the Heisenberg group

Lukyanenko, Anton; Vandehey, Joseph
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.13%
We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued fractions. We then discuss dynamical properties of the associated Gauss map, comparing them with base-$b$ expansions on the Heisenberg group and continued fractions on the complex plane.; Comment: Version 2 introduces significant changes to the text, especially Section 4 on the dynamical properties of the Heisenberg Gauss map

## Bracket map for Heisenberg group and the characterization of cyclic subspaces

Barbieri, Davide; Hernandez, Eugenio; Mayeli, Azita
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.96%
The bracket map was originally considered for locally compact abelian groups. In this work we extend the study of bracket maps to the noncommutative setting, providing characterizations of bases and frames for cyclic subspaces of the Heisenberg group. We also indicate how to generalize these results to a class of non-abelian nilpotent Lie groups whose irreducible representations are square integrable modulo the center.

## Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

Austin, Tim; Naor, Assaf; Tessera, Romain
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.96%
Let $\H$ denote the discrete Heisenberg group, equipped with a word metric $d_W$ associated to some finite symmetric generating set. We show that if $(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function $f:\H\to X$ there exist $x,y\in \H$ with $d_W(x,y)$ arbitrarily large and $$\label{eq:comp abs} \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim \left(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\right)^{1/p}.$$ We also show that any embedding into $X$ of a ball of radius $R\ge 4$ in $\H$ incurs bi-Lipschitz distortion that grows at least as a constant multiple of $$\label{eq:dist abs} \left(\frac{\log R}{\log\log R}\right)^{1/p}.$$ Both~\eqref{eq:comp abs} and~\eqref{eq:dist abs} are sharp up to the iterated logarithm terms. When $X$ is Hilbert space we obtain a representation-theoretic proof yielding bounds corresponding to~\eqref{eq:comp abs} and~\eqref{eq:dist abs} which are sharp up to a universal constant.

## Counting Irreducible Representations of the Discrete Heisenberg Group Over the Integers of a quadratic number field

Ezzat, Shannon
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.96%
We calculate the representation growth zeta function of the discrete Heisenberg group over the integers of a quadratic number field. This is done by forming equivalence classes of representations, called twist iso-classes, and explicitly constructing a representative from each twist iso-class. Our method of construction involves studying the eigenspace structure of the elements of the image of the representation and then picking a suitable basis for the representation.; Comment: 18 pages

## Representation growth of the Heisenberg group over $\mathcal{O}[x]/(x^n)$

Dung, Duong Hoang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.96%
We present a conjectured formula for the representation zeta function of the Heisenberg group over $\mathcal{O}[x]/(x^n)$ where $\mathcal{O}$ is the ring of integers of some number field. We confirm the conjecture for $n\leq 3$ and raise several questions.

## Watson-Crick pairing, the Heisenberg group and Milnor invariants

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.2%
We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as well as in terms of lattice paths. We show that the Heisenberg invariant gives a lower bound on the number of unpaired bases in an RNA secondary structure. We also show that the Heisenberg invariant can predict \emph{allosteric structures} for RNA. Namely, if the Heisenberg invariant is large, then there are widely separated local maxima (i.e., allosteric structures) for the number of Watson-Crick pairs found.; Comment: 18 pages; to appear in Journal of Mathematical Biology

## Fine asymptotic geometry in the Heisenberg group

Duchin, Moon; Mooney, Christopher
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.09%
For every finite generating set on the integer Heisenberg group H(Z), Pansu showed that the word metric has the large-scale structure of a Carnot-Caratheodory Finsler metric on the real Heisenberg group H(R). We study the properties of those limit metrics and obtain results about the geometry of word metrics that reflect the dependence on generators. For example we will study the probability that a group element has all of its geodesic spellings sublinearly close together, relative to the size of the element. In free abelian groups of rank at least two, that probability is 0; in infinite hyperbolic groups, the probability is 1. In H(Z) it is a rational number strictly between 0 and 1 that depends on the generating set; with respect to the standard generators, the probability is precisely 19/31.; Comment: v2: minor edits, improved figures

## Rational growth in the Heisenberg group

Duchin, Moon; Shapiro, Michael
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.01%
A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. A long-standing open question asks whether the Heisenberg group has rational growth for all finite generating sets, and we settle this question affirmatively. We also establish almost-convexity for all finite generating sets. Previously, both of these properties were known to hold for hyperbolic groups and virtually abelian groups, and there were no further examples in either case. Our main method is a close description of the relationship between word metrics and associated Carnot-Caratheodory Finsler metrics on the ambient Lie group. We provide (non-regular) languages in any word metric that suffice to represent all group elements.; Comment: Version 2 contains bug fixes and an added application to almost-convexity

## The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

Bridson, Martin R.; de la Harpe, Pierre; Kleptsyn, Victor
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.98%
When equipped with the natural topology first defined by Chabauty, the closed subgroups of a locally compact group $G$ form a compact space $\Cal C(G)$. We analyse the structure of $\Cal C(G)$ for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group $H$. We prove that $\Cal C(H)$ is a 6-dimensional space that is path--connected but not locally connected. The lattices in $H$ form a dense open subset $\Cal L(H) \subset \Cal C(H)$ that is the disjoint union of an infinite sequence of pairwise--homeomorphic aspherical manifolds of dimension six, each a torus bundle over $(\bold S^3 \smallsetminus T) \times \bold R$, where $T$ denotes a trefoil knot. The complement of $\Cal L(H)$ in $\Cal C(H)$ is also described explicitly. The subspace of $\Cal C(H)$ consisting of subgroups that contain the centre $Z(H)$ is homeomorphic to the 4--sphere, and we prove that this is a weak retract of $\Cal C(H)$.; Comment: Minor edits. Final version. To appear in the Pacific Journal. 41 pages, no figures

## Revisiting the Fourier transform on the Heisenberg group

Lakshmi Lavanya, R.; Thangavelu, S.
Tipo: Artigo de Revista Científica Formato: application/pdf
Relevância na Pesquisa
66.13%
A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on Rn as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. In this note we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group.

## Note on coarea formulae in the Heisenberg group

Magnani, Valentino
Tipo: Article; info:eu-repo/semantics/article; info:eu-repo/semantics/publishedVersion Formato: application/pdf