A multi-phase level set framework for source reconstruction in bioluminescence tomography

We propose a novel multi-phase level set algorithm for solving the inverse problem of bioluminescence tomography. The distribution of unknown interior source is considered as piecewise constant and represented by using multiple level set functions. The localization of interior bioluminescence source is implemented by tracing the evolution of level set function. An alternate search scheme is incorporated to ensure the global optimal of reconstruction. Both numerical and physical experiments are performed to evaluate the developed level set reconstruction method. Reconstruction results show that the proposed method can stably resolve the interior source of bioluminescence tomography.     

Solution of a Generalized Inverse Spectral Problem for Weakly Resolved Spectra Using a Coefficient of Correlation

We consider in a new statement the inverse problem of the theory of vibrational spectra of molecules, in which a theoretical spectral curve directly gets closer to the experimental one. A coefficient of correlation between the curves is used as a criterion of the closeness of the curves and as a functional of the inverse problem. The procedure developed for solving the inverse problem makes it possible to dispense with the preliminary referencing of frequencies in an experimental spectrum and with its expansion into elementary absorption bands.     

基于迭代Tikhonov正规化的三刺激值重建光谱方法研究

光谱图像中的反射率光谱数据维数高,且与光源、设备均无关,能够比较全面、真实、客观地描述图像中物体的颜色信息。针对三色相机的光谱图像获取系统中三维色度数据重建多维光谱数据产生的光谱信息丢失、以及伴随而生的颜色信息丢失问题,提出了迭代Tikhonov正规化的光谱重建方法。首先依据色度学理论中色度值与反射率光谱之间的关系,构建反射率光谱重建方程建立起相机所获三维色度数据与高维反射率光谱数据的映射关系;然后,通过反射率光谱重建方程的病态分析,在Moore-Penrose伪逆矩阵求解思想的基础上构建迭代Tikhonov正规化方法求解反射率光谱,并利用训练样本数据通过L-曲线方法训练获取迭代Tikhonov正规化的最优正规化参数,以有效控制并改善反射率光谱重建方程求解的病态、减少重建光谱的光谱信息丢失。实验通过选取样本数据对光谱重建方法进行验证。验证实验的结果表明所提出的光谱重建方法改善了三色相机的光谱图像获取系统中重建光谱的光谱信息丢失程度,使得重建光谱的光谱误差和色度误差较其他光谱重建方法均有明显降低。     

On isospectral sets of Jacobi operators

We consider the inverse spectral problem for a class of reflectionless bounded Jacobi operators with empty singularly continuous spectra. Our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum. The corresponding isospectral set of Jacobi operators is explicitly determined in terms of Dirichlet-type data.     

On an inverse spectral problem

We study an inverse spectral problem for the Sturm–Liouville problem. Such a problem arises in the study of micro- and nano-rods.     

Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.     

Inverse Spectral Problem and Peakons of an Integrable Two-component Camassa-Holm System

In this paper, we are concerned with the explicit construction of peakon solutions of the integrable twocomponent system with cubic non-linearity. We establish the spectral and inverse spectral problem associated to the Lax pairs of the system. The inverse problem is solved by the classical results of Stieltjes continued fractions, which also contributes a lot to the spectral problem. The explicit formulas are obtained from solutions of the inverse problem. The positivity of the spectral measures is implied by J. Moser's work on the Jacobi spectral problem.     

Matter-wave solitons in the presence of collisional inhomogeneities: Perturbation theory and the impact of derivative terms

We study the dynamics of bright and dark matter-wave solitons in the presence of a spatially varying nonlinearity. When the spatial variation does not involve zero crossings, a transformation is used to bring the problem to a standard nonlinear Schrödinger form, but with two additional terms: an effective potential one and a non-potential term. We illustrate how to apply perturbation theory of dark and bright solitons to the transformed equations. We develop the general case, but primarily focus on the non-standard special case whereby the potential term vanishes, for an inverse square spatial dependence of the nonlinearity. In both cases of repulsive and attractive interactions, appropriate versions of the soliton perturbation theory are shown to accurately describe the soliton dynamics.     

Formulas for q-Spherical Functions Using Inverse Scattering Theory of Reflectionless Jacobi Operators

We study the spectral problem associated to a Ruijsenaars-type (q-)difference version of the one-dimensional Schrödinger operator with Pöschl-Teller potential. The eigenfunctions are constructed explicitly with the aid of the inverse scattering theory of reflectionless Jacobi operators. As a result, we arrive at combinatorial formulas for basic hypergeometric deformations of zonal spherical functions on odd-dimensional hyperboloids and spheres.     

On the Yang-Mills mass gap problem

Theorem 4.1 of the author’s paper “Quantum Yang-Mills-Weyl dynamics in the Schroedinger paradigm”, RJMP 21 (2), 169–188 (2014) claims the relative ellipticity of cutoff Yang-Mills quantum energy-mass operators in von Neumann algebras with regular traces. This implies that the spectra of cutoff self-adjoint Yang-Mills energy-mass operators in a nonperturbative quantum Yang-Mills theory (with an arbitrary compact simple gauge group) are nonnegative sequences of the eigenvalues converging to +∞. The spectra are self-similar in the inverse proportion to the running coupling constant. In particular, they have self-similar positive spectral mass gaps. Presumably, this is a solution of the Yang-Mills Millennium problem. The present note shows that the fundamental spectral value of a cutoff quantum Yang-Mills energy-mass operator is the simple zero eigenvalue with the vacuum eigenvector. The direct proof (without von Neumann algebras) is based on the domination over the number operator (with simple fundamental eigenvalue) and the standard spectral variational principle.     

Self-induced transparency with the Stark pulse technique

A theory based on the method of the inverse scattering problem is presented describing propagation of an ultrashort light pulse under the condition that resonant interaction with atoms of the medium is initially switched on by an additional Stark pulse. A Hamiltonian form of the Maxwell-Bloch equation is found without redefining the spatiotemporal variables. The method of the inverse scattering problem is based on a special spectral problem. Equations of the Gel’fand-Levitan-Marchenko type and a soliton solution are obtained.     

Meblin transforms associated with Julia sets and physical applications

We introduce the Mellin transform of the balanced invariant measure associated to the Julia set generated by a rational transformation. We show that its analytic continuation is a meromorphic function, the poles of which are on a semi-infinite periodic lattice. This allows one to have an understanding of the behavior of the measure near a repulsive fixed point. Trace identities corresponding to the fact that the analytically continued Mellin transform vanishes at negative integers are derived for the polynomial case. The quadratic map is first analyzed in detail, and the analytic properties of the inverse of the Green's function are exhibited. Of interest is the appearance of a dense set of spikes at dyadic points when the Julia set is disconnected. These results are used to study the residues of the Mellin transform. A certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems, the excitation spectrum of which displays unusual oscillations. The appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion.Supported in part by a N.A.T.O. Postdoctoral fellowship.     

Localization of quasiparticles in a disordered vortex

We study the diffusive motion of low-energy normal quasiparticles along the core of a single vortex in a dirty, type-II, s-wave superconductor. The physics of this system is argued to be described by a one-dimensional supersymmetric non-linear σ model, which differs from the σ models known for disordered metallic wires. For an isolated vortex and quasiparticle energies less than the Thouless energy E_Th, we recover the spectral correlations that are predicted by random matrix theory for the universality class C. We then consider the transport problem of transmission of quasiparticles through a vortex connected to particle reservoirs at both ends. The transmittance at zero energy exhibits a weak localization correction reminiscent of quasi-one-dimensional metallic systems with symmetry index β = 1. Weak localization disappears with increasing energy over a scale set by E_Th This crossover should be observable in measurements of the longitudinal heat conductivity of an ensemble of vortices under mesoscopic conditions. In the regime of strong localization, the localization length is shown to decrease by a factor of 8 as the quasiparticle energy goes to zero.     

On a model for quantum friction III. Ergodic properties of the spin-boson system

We investigate the dynamics of a 2-level atom (or spin 1/2) coupled to a mass-less bosonic field at positive temperature. We prove that, at small coupling, the combined quantum system approaches thermal equilibrium. Moreover we establish that this approach is exponentially fast in time. We first reduce the question to a spectral problem for the Liouvillean, a self-adjoint operator naturally associated with the system. To compute this operator, we invoke Tomita-Takesaki theory. Once this is done we use complex deformation techniques to study its spectrum. The corresponding zero temperature model is also reviewed and compared. From a more philosophical point of view our results show that, contrary to the conventional wisdom, quantum dynamics can be simpler at positive than at zero temperature.     

Method for retrieving the refractive index of ordered particles from data on the photonic band gap

A method for retrieving the refractive index of spherical particles arranged into ordered structures is proposed. It is based on the solution of the inverse problem using data on the photonic band gap. The solution has been obtained within the quasi-crystalline approximation of the multiple wave scattering theory and the transfer-matrix method. Quantitative results are presented for systems of silicon oxide particles. The effective refractive indices of synthetic opal particles have been found from the available experimental data on the spectral position of the photonic band gap. The described technique is applicable for retrieving not only the refractive index of particles but also other characteristics of ordered particulate structures from the coherent transmittance spectra.     

Inverse problems for matrix Sturm-Liouville operators

Inverse spectral problems for nonselfadjoint matrix Sturm-Liouville differential operators on a finite interval and on the half-line are studied. As a main spectral characteristic, we introduce the so-called Weyl matrix and prove that the specification of the Weyl matrix uniquely determines the matrix potential and the coefficients of the boundary conditions. Moreover, for a finite interval, we also study the inverse problems of recovering matrix Sturm-Liouville operators from discrete spectral data (eigenvalues and “weight” numbers) and from a system of spectra. The results thus obtained are natural generalizations of the classical results in inverse problem theory for scalar Sturm-Liouville operators. Dedicated to the memory of B. M. Levitan     

Regularization methods for near-field acoustical holography.

The reconstruction of the pressure and normal surface velocity provided by near-field acoustical holography (NAH) from pressure measurements made near a vibrating structure is a linear, ill-posed inverse problem due to the existence of strongly decaying, evanescentlike waves. Regularization provides a technique of overcoming the ill-posedness and generates a solution to the linear problem in an automated way. We present four robust methods for regularization; the standard Tikhonov procedure along with a novel improved version, Landweber iteration, and the conjugate gradient approach. Each of these approaches can be applied to all forms of interior or exterior NAH problems; planar, cylindrical, spherical, and conformal. We also study two parameter selection procedures, the Morozov discrepancy principle and the generalized cross validation, which are crucial to any regularization theory. In particular, we concentrate here on planar and cylindrical holography. These forms of NAH which rely on the discrete Fourier transform are important due to their popularity and to their tremendous computational speed. In order to use regularization theory for the separable geometry problems we reformulate the equations of planar, cylindrical, and spherical NAH into an eigenvalue problem. The resulting eigenvalues and eigenvectors couple easily to regularization theory, which can be incorporated into the NAH software with little sacrifice in computational speed. The resulting complete automation of the NAH algorithm for both separable and nonseparable geometries overcomes the last significant hurdle for NAH.     

INVERSE SCATTER1 NG TRANSFORM AND REGULAR RIEMANN-HILBERT PROBLEM

It is shown that the inverse scattering transform method for solving the Lax pair of given nonlinear evolution equation can be reduced to a kind of Riemann-Hilbert (RH) problem of meromorphic functions with respect to the complex spectral parameter. The RH problem is generally regular no matter solitons are involved or not. The linear singular integral equation associated with the RH problem has been derived, which is essential1y equivalent to the Gel fand-Levitan-Marchenko equation.Furthermore, the regtllar RH problem satisfied by the Sacklund transformation from a fundamental solution set of the eigenvalue equations of Lax pair to a new set has Fen given as well. The RH problem reduced from the inverse scattering transform is in fact a special case of that satisfied by the Backlund transformation.     

反散射变换和正规Riemann-Hilbert问题

本文指出,非线性演化方程Lax表示的反散射交换可以化为关于亚纯函数的正规Riemann-Hilbert问题,并导出了相应的积分方程,后者在实质上与Gel’fand-Levitan-Marchenko方程等价。本文进而导出Lax表示的不同基本解组之间的Darboux-B?cklund变换所满足的正规Riemann-Hilbert问题。与反散射变换直接联系的正规Riemann-Hilbert问题是其特殊形式。 关键词：