Time-Frequency Analysis.

Reconstructing curves and surfaces from sets of points is a common activity in medical imaging, computer graphics, and engineering. Given a physical object, we often seek to build a digital representation of it from a set of discrete samples. This digital representation can then be visualized, analyzed and manipulated on the computer. To create this digital representation from samples collected from the physical object, we must employ algorithms to reconstruct the shape of the object. This text discusses algorithms for performing these reconstructions for both two-dimensional curves and three-dimensional surfaces. The text presents these algorithms in the context of the mathematical basis upon which they are derived and their properties proven. The mathematical content of the text is drawn from

Curve and surface reconstruction: algorithms with mathematical analysis by Tamal K. Dey Cambridge University Press

We construct quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert state space of rotational states of a rigid body. These codes, which protect against both drift in the body's orientation and small changes in its angular momentum, may be well suited for robust storage and coherent processing of quantum information using rotational states of a polyatomic molecule. Extensions of such codes to rigid bodies with a symmetry axis are compatible with rotational states of diatomic molecules, as well as nuclear states of molecules and atoms. We also describe codes associated with general nonabelian compact Lie groups and develop orthogonality relations for coset spaces, laying the groundwork for quantum information processing with exotic configuration spaces.

Robust encoding of a qubit in a molecule

A new proposal for how to encode quantum information in the rotational states of individual molecules could protect these qubits from losing information as a result of noise.

https://link.aps.org/pdf/10.1103/PhysRevX.10.031050

Robust Encoding of a Qubit in a Molecule

We describe connections between the localization technique introduced by I. B. Simonenko and operator covariant transform produced by nilpotent Lie groups.

/pdf/operator-covariant-transform-and-local-principle-1kwk62lln6.pdf

Operator Covariant Transform and Local Principle

In this article we define the continuous Gabor transform for second countable, non-abelian, unimodular and type I groups and also we investigate a Plancherel formula and an inversion formula for our definition. As an example we show that how these formulas work for the Heisenberg group and also the matrix group ${SL(2,\mathbb{R})}$.

/pdf/continuous-gabor-transform-for-a-class-of-non-abelian-groups-60r280sk2c.pdf

Continuous Gabor transform for a class of non-Abelian groups

This paper presents a systematic study for structure of abstract Banach function $*$-algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $\mu$ be the normalized $G$-invariant measure over the homogeneous space $G/H$ associated to the Weil’s formula and $1\leq p<\infty$. Then we introduce the notions of convolution and involution for the Banach function spaces $L^{p}(G/H,\mu)$.

/pdf/abstract-convolution-function-algebras-over-homogeneous-51pv7y3z5c.pdf

Abstract convolution function algebras over homogeneous spaces of compact groups

In this paper, we introduce the notion of finite wave packet groups for positive prime integers, as groups of cyclic dilations, translations and modulations. We will present a unified group theoretical approach for the cyclic wave packet transform on finite Abelian groups of prime order. Then, we study basic properties of this windowed transform.

Cyclic wave packet transform on finite Abelian groups of prime order

Let G be a locally compact group and also let H be a compact subgroup of G. It is shown that, if m is a relatively invariant measure on G=H then there is a well-defined convolution on L 1 (G=H;m) such that the Banach space L 1 (G=H;m) becomes a Banach al- gebra. We also find a generalized definition of this convolution for other L p -spaces. Finally, we show that various types of involutions can be considered on G=H. 2010 Mathematics Subject Classification: Primary 43A15, 43A85

/pdf/convolution-and-involution-on-function-spaces-of-homogeneous-4z540xverf.pdf

Convolution and Involution on Function Spaces of Homogeneous Spaces

In this article we introduce the notion of finite wave packet groups over finite fields as the finite group of dilations, translations, and modulations. Then we will present a unified theoretical linear algebra approach to the theory of wave packet transform (WPT) over finite fields. It is shown that each vector defined over a finite field can be represented as a coherent sum of finite wave packet group elements as well.

/pdf/wave-packet-transform-over-finite-fields-281iuou26k.pdf

Arash Ghaani Farashahi

Papers

Continuous Gabor transform for a class of non-Abelian groups

Abstract convolution function algebras over homogeneous spaces of compact groups

Cyclic wave packet transform on finite Abelian groups of prime order

Convolution and Involution on Function Spaces of Homogeneous Spaces

Wave Packet Transform over Finite Fields