scholarly journals Solitons of general topological charge over noncommutative tori

2020 ◽  
pp. 2150009
Author(s):  
Ludwik Da̧browski ◽  
Mads S. Jakobsen ◽  
Giovanni Landi ◽  
Franz Luef
Keyword(s):  
Topological Charge ◽  
Vector Bundles ◽  
Energy Functional ◽  
Gabor Frames ◽  
Noncommutative Tori ◽  
Gabor Analysis ◽  
Time Frequency ◽  
Moyal Plane ◽  
Higher Rank ◽  
Vector Valued

We study solitons of general topological charge over noncommutative tori from the perspective of time-frequency analysis. These solitons are associated with vector bundles of higher rank, expressed in terms of vector-valued Gabor frames. We apply the duality theory of Gabor analysis to show that Gaussians are such solitons for any value of a topological charge. Also they solve self/anti-self duality equations resulting from an energy functional for projections over noncommutative tori, and have a reformulation in terms of Gabor frames. As a consequence, the projections generated by Gaussians minimize the energy functional. We also comment on the case of the Moyal plane and the associated continuous vector-valued Gabor frames and show that Gaussians are the only class of solitons there.

scholarly journals Discrete Subspace Multiwindow Gabor Frames and Their Duals

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Yun-Zhang Li ◽  
Yan Zhang
Keyword(s):  
Gabor Frames ◽  
Riesz Bases ◽  
Type I ◽  
Type Ii ◽  
Zak Transform ◽  
Gabor Analysis ◽  
Time Frequency ◽  
Orthonormal Bases ◽  
Gabor Dual ◽  
Discrete Subspace

This paper addresses discrete subspace multiwindow Gabor analysis. Such a scenario can model many practical signals and has potential applications in signal processing. In this paper, using a suitable Zak transform matrix we characterize discrete subspace mixed multi-window Gabor frames (Riesz bases and orthonormal bases) and their duals with Gabor structure. From this characterization, we can easily obtain frames by designing Zak transform matrices. In particular, for usual multi-window Gabor frames (i.e., all windows have the same time-frequency shifts), we characterize the uniqueness of Gabor dual of type I (type II) and also give a class of examples of Gabor frames and an explicit expression of their Gabor duals of type I (type II).

scholarly journals Sampling and periodization of generators of Heisenberg modules

2019 ◽  
Vol 30 (10) ◽  
pp. 1950051
Author(s):  
Mads S. Jakobsen ◽  
Franz Luef
Keyword(s):  
Vector Bundles ◽  
General Setting ◽  
Gabor Frames ◽  
Locally Compact ◽  
Noncommutative Torus ◽  
Gabor Analysis ◽  
Matrix Algebras ◽  
Finite Dimensional ◽  
The Matrix ◽  
Lca Groups

This paper considers generators of Heisenberg modules in the case of twisted group [Formula: see text]-algebras of closed subgroups of locally compact abelian (LCA) groups and how the restriction and/or periodization of these generators yield generators for other Heisenberg modules. Since generators of Heisenberg modules are exactly the generators of (multi-window) Gabor frames, our methods are going to be from Gabor analysis. In the latter setting, the procedure of restriction and periodization of generators is well known. Our results extend this established part of Gabor analysis to the general setting of LCA groups. We give several concrete examples where we demonstrate some of the consequences of our results. Finally, we show that vector bundles over an irrational noncommutative torus may be approximated by vector bundles for finite-dimensional matrix algebras that converge to the irrational noncommutative torus with respect to the module norm of the generators, where the matrix algebras converge in the quantum Gromov–Hausdorff distance to the irrational noncommutative torus.

RATIONAL TIME-FREQUENCY VECTOR-VALUED SUBSPACE GABOR FRAMES AND BALIAN-LOW THEOREM

2013 ◽  
Vol 11 (02) ◽  
pp. 1350013 ◽  
Author(s):  
YUN-ZHANG LI ◽  
YAN ZHANG
Keyword(s):  
Gabor Frames ◽  
Riesz Bases ◽  
Type I ◽  
Type Ii ◽  
Zak Transform ◽  
Time Frequency ◽  
Orthonormal Bases ◽  
Uniqueness Results ◽  
Vector Valued ◽  
General Vector

This paper addresses vector-valued subspace Gabor frames with rational time-frequency product. By introduction of a suitable Zak transform matrix, we characterize vector-valued subspace Gabor frames, Riesz bases and orthonormal bases. We characterize the uniqueness of Gabor duals of type I and type II. Using the uniqueness results, we extend the classical Balian-Low theorem to vector-valued subspace Gabor frames. We also point out Ron-Shen duality principe does not hold for a general vector-valued subspace Gabor frame.

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

Constructing super Gabor frames: the rational time-frequency lattice case

2010 ◽  
Vol 53 (12) ◽  
pp. 3179-3186 ◽  
Author(s):  
ZhongYan Li ◽  
DeGuang Han
Keyword(s):  
Gabor Frames ◽  
Time Frequency

scholarly journals Higher-rank zeta functions andSLn-zeta functions for curves

2020 ◽  
Vol 117 (12) ◽  
pp. 6398-6408
Author(s):  
Lin Weng ◽  
Don Zagier
Keyword(s):  
Zeta Function ◽  
Parabolic Subgroup ◽  
Vector Bundles ◽  
Reductive Group ◽  
Zeta Functions ◽  
Smooth Projective Curve ◽  
Maximal Parabolic Subgroup ◽  
Higher Rank ◽  
Semistable Vector Bundles

In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case whenG=SLnand P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding ton=1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.

The Research of a Two-Directional Vector-Valued Quarternary Wavelet Packets and Applications in Material Engineering

2013 ◽  
Vol 712-715 ◽  
pp. 2487-2492
Author(s):  
Jian Feng Zhou
Keyword(s):  
Functional Analysis ◽  
Frequency Analysis ◽  
Wavelet Packets ◽  
Riesz Bases ◽  
Sufficient Condition ◽  
Subdivision Scheme ◽  
Analysis Method ◽  
Time Frequency ◽  
Dilation Matrix ◽  
Vector Valued

In this paper, we introduce a class of vector-valued four-dimensional wavelet packets according to a dilation matrix, which are generalizations of univariate wavelet packets. The defini -tion of biorthogonal vector four-dimensional wavelet packets is provided and their biorthogonality quality is researched by means of time-frequency analysis method, vector subdivision scheme and functional analysis method. Three biorthogonality formulas regarding the wavelet packets are established. Finally, it is shown how to draw new Riesz bases of space from these wavelet packets. The sufficient condition for the existence of four-dimensional wavelet packets is established based on the multiresolution analysis method.

scholarly journals On a gauge action on sigma model solitons

2018 ◽  
Vol 21 (02) ◽  
pp. 1850008
Author(s):  
Hyun Ho Lee
Keyword(s):  
Noncommutative Geometry ◽  
Sigma Model ◽  
Gabor Frame ◽  
Gabor Frames ◽  
Target Space ◽  
Gauge Action ◽  
Noncommutative Tori ◽  
Linear Formula ◽  
Non Linear

In this paper, we consider a gauge action on sigma model solitons over noncommutative tori as source spaces, with a target space made of two points introduced in [L. Dabrowski, T. Krajewski and G. Landi, Some properties of non-linear [Formula: see text]-models in noncommutative geometry, Int. J. Mod. Phys. B 14 (2000) 2367–2382]. Using new classes of solitons from Gabor frames, we quantify the condition about how to gauge a Gaussian to a prescribed Gabor frame.

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