Clifford algebras,Fourier transforms,and quantum mechanics

In this review, we give an overview of several recent generalizations of the Fourier transform, related to either the Lie algebra or the Lie superalgebra . In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform, and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford–Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these transforms is given, with emphasis on aspects such as eigenfunctions and spectrum of the transform, characterization of the integral kernel, and connection with various special functions. Copyright © 2012 John Wiley & Sons, Ltd.     

Supercharacters and the discrete Fourier,cosine, and sine transforms

Using supercharacter theory, we identify the matrices that are diagonalized by the discrete cosine and discrete sine transforms, respectively. Our method affords a combinatorial interpretation for the matrix entries.     

Class fields,Dirichlet characters, and extended genus fields of global function fields

We study the extended genus field of an abelian extension of a rational function field. We follow the definition of Anglès and Jaulent, which uses the class field theory. First, we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Anglès and Jaulent. Next, we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields. In the absolute abelian case, we compare this approach with the one given by Anglès and Jaulent.     

Asymptotic normality of finite Fourier transforms of stationary generalized processes

This paper indicates a mixing condition under which a net of Fourier transforms, of a stationary generalized process over an abelian locally compact group, has a limiting normal distribution.     

Fourier transform, null variety, and Laplacian's eigenvalues

We consider a quantity κ(Ω)—the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximised, among all convex balanced domains Ω⊂Rd of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians.     

An attempt to differential Galois theory of second order polynomial system and solvable subgroup of M(o)bius transformations

By introducing the conception "relativistic differential Galois group" for the second order polynomial systems, we establish the relation between the conformal relativistic differential Galois group and the subgroup of M(o)bius transformations, and prove that the system is integrable in the sense of Liouville if its conformal relativistic differential Galois group is solvable with a derived length at most 2. Some omissions on the structures of solvable subgroups of M(o)bius transformations at the first author's article published in this journal in 1996 are refreshed in this paper.     

Fourier transforms, generic vanishing theorems and polarizations of abelian varieties

The purpose of this paper is to give two applications of Fourier transforms and generic vanishing theorems: – we give a cohomological characterization of principal polarizations – we prove that if X an abelian variety and a polarization of type (1, ...,1,2), then a general pair is log canonical. Received 14 May, 1999 / Published online September 14, 2000     

Pitfalls of the Fourier Transform Method in Affine Models,and Remedies

We study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models and explain that, in many cases, a calibration procedure based on such a realization will be able to find a “correct parameter set” only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time-consuming. We explain how to construct more accurate and faster pricing and calibration procedures. An important ingredient of our method is the study of the analytic continuation of the solution of the associated system of generalized Riccati equations, and contour deformation techniques. As a byproduct, we show that the straightforward application of the Runge–Kutta method may lead to sizable errors, and suggest certain remedies. In the paper, the method is applied to a wide class of stochastic volatility models with stochastic interest rate and interest rate models of A_n(n) class. The methodology of the paper can be applied to other models (e.g., quadratic term structure models, Wishart dynamics, 3/2-model).     

Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl 3,0

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions Third, we show a set of important properties of the Clifford Fourier transform on Cl_3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl_3,0 multivector functions.     

SL（2,R）上的球型函数与Fourier变换

该文中，我们主要利用对Lie群G＝SL（2，R）上球型函数的一些性质的讨论，给出了G上Fourier变换的一个重要的性质．即对每个γ∈G，存在函数f∈Cc（G），使得f（γ）≠0．     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E(α), α ∈ [2, 3)

We develop a precise analysis of J. O’Hara’s knot functionals E^(α), α ∈ [2, 3), that serve as self‐repulsive potentials on (knotted) closed curves. First we derive continuity of E^(α) on injective and regular H² curves and then we establish Fréchet differentiability of E^(α) and state several first variation formulae. Motivated by ideas of Z.‐X. He in his work on the specific functional E⁽²⁾, the so‐called Möbius Energy, we prove C^∞‐smoothness of critical points of the appropriately rescaled functionals $\tilde{E}^{(\alpha )}= {\rm length}^{\alpha -2}E^{(\alpha )}$ by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.     

延迟Min(N,D)-策略的M/G/1排队系统的队长分布与数值计算

考虑延迟Min(N, D)-策略的M/G/1排队系统. 运用更新过程理论、全概率分解技术和Laplace变换工具, 从任意初始状态出发, 研究了队长的瞬态和稳态性质, 获得了瞬态队长分布的Laplace变换的递推表达式和稳态队长分布的递推表达式, 同时求出了附加队长分布的显示表达式. 进一步讨论了当N \to \infty, 或D \to \infty, 或N=1且P{Y=0}=1, 或P{Y=0}=1时的特殊情形. 最后通过数值实例, 讨论了稳态队长分布对系统参数的敏感性, 并阐述了稳态队长分布的表达式在系统容量优化设计中的重要价值.