Regularized Products and Determinants

Zeta-regularized products are used to define determinants of operators in infinite dimensional spaces. This article provides a general theory of regularized products and determinants which delivers a better approach to their existence and explicit determination. Received: 4 April 2000 / Accepted: 15 January 2001     

Upper Bounds for Regularized Determinants

We conjecture that the zeta-regularized determinant of the Laplace operator with coefficients in a holomorphic vector bundle on a compact K?hler manifold remains bounded when the metric on the bundle varies. This conjecture is shown to be true for certain classes of line bundles on Riemann surfaces. Received: 7 July 1997 / Accepted: 20 April 1998     

Regularized bosonic strings need extrinsic curvature

We show that the string tension in a class of triangulated random surface models with gaussian action does not tend to zero at the critical point. This rules out the existence of a non-trivial continuum limit of these models. Furthermore, the proof seems to lead to the conclusion that the most natural (if not the only) way to avoid this feature of random surface models is by introducing asymptotically free interactions depending e.g. on the extrinsic curvature of the surface.     

Regularized Laplacian determinants of self-similar fractals

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of locally self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the determinant of the discrete graph Laplacian and the regularized one.     

Regularized potentials in nonrelativistic quantum mechanics

Using scaling technique we describe various self-adjoint extensions of the three-dimensiona Schrödinger operator with singular potential as a limit of Schrödinger operators with regularized potentials.I am indebted to Dr. H. Englisch for stimulating discussions. I am also grateful to the Department of Mathematics of the Karl Marx University Leipzig where this work was written.     

Regularized potentials in nonrelativistic quantum mechanics

Using scaling technique, we describe all self-adjoint extensions of Schrödinger operator with a singular potential in one dimension as a limit of Schrödinger operators with regular local potentials.I am indebted to Dr. H. Englisch for stimulating discussions. I am also grateful to the Department of Mathematics at the Karl-Marx-University Leipzig, where this work was made possible.     

Regularized matched-mode processing for source localization

This paper develops a new approach to matched-mode processing (MMP) for ocean acoustic source localization. MMP consists of decomposing far-field acoustic data measured at an array of sensors to obtain the excitations of the propagating modes, then matching these with modeled replica excitations computed for a grid of possible source locations. However, modal decomposition can be ill-posed and unstable if the sensor array does not provide an adequate spatial sampling of the acoustic field (i.e., the problem is underdetermined). For such cases, standard decomposition methods yield minimum-norm solutions that are biased towards zero. Although these methods provide a mathematical solution (i.e., a stable solution that fits the data), they may not represent the most physically meaningful solution. The new approach of regularized matched-mode processing (RMMP) carries out an independent modal decomposition prior to comparison with the replica excitations for each grid point, using the replica itself as the a priori estimate in a regularized inversion. For grid points at or near the source location, this should provide a more physically meaningful decomposition; at other points, the procedure provides a stable inversion. In this paper, RMMP is compared to standard MMP and matched-field processing for a series of realistic synthetic test cases, including a variety of noise levels and sensor array configurations, as well as the effects of environmental mismatch.     

Relativistic degenerate electron plasma

The dielectric response function for a dense, ultra-degenerate, relativistic electron gas is presented. The dispersion relation, damping conditions and screened potential are obtained.     

Bremsstrahlung of degenerate plasma

Conclusions The effect of electron degeneracy on the bremsstrahlung spectrum and total intensity is contained in the functionsQ(A, y) and(A). From the calculations one may deduce that the electron degeneracy plays a significant role for plasmas withA > 1,b1, respectively. From the definition ofb the respective density and temperature ranges can easily be deduced. At any degree of degeneracy (anyb) the main contribution to the bremsstrahlung intensity comes from the low-frequency region,y1. The high frequency contribution (y1) is relatively small; nevertheless in that region the influence of degeneracy is the most significant. The spectral ratioQ(A, y) differs significantly from unity (atA>1) just in the high frequency spectral region.If the electrons are treated as a Boltzmann gas the total radiation intensity has the formI _o =an _i n _e T ^1/2, wherea is a numerical factor. The application of the exact Sommerfeld theory of the bremsstrahlung leads to a more complicated temperature dependence (see [3]). The inclusion of the degeneracy leads to an additional dependence on both electron density and temperature through the correction factor(A) =f(n _e T ^–3/2).     

基于正则化迭代的并行磁共振图像重建算法

为了解决并行磁共振成像过程的病态性和图像信噪比下降问题,降低重建过程中噪声放大和异常值的干扰造成的图像信噪比的损失,提出了一种基于正则化共轭梯度迭代的并行磁共振成像重建算法;该算法基于最小二乘理论,引入正则化,优化方程,进而进行迭代重建;采用了不同加速因子的人脑磁共振K空间欠采样数据以验证该算法的重建性能,仿真结果表明了该算法相较于最小二乘法,能较大限度地降低噪声对重建结果的干扰,具有信噪比更高、误差更小、成像效果更好等特征;重建图像质量得到了较好的改善,对临床诊断更具有适用性。     

Sparse Regularized Optimal Transport with Deformed q-Entropy

Optimal transport is a mathematical tool that has been a widely used to measure the distance between two probability distributions. To mitigate the cubic computational complexity of the vanilla formulation of the optimal transport problem, regularized optimal transport has received attention in recent years, which is a convex program to minimize the linear transport cost with an added convex regularizer. Sinkhorn optimal transport is the most prominent one regularized with negative Shannon entropy, leading to densely supported solutions, which are often undesirable in light of the interpretability of transport plans. In this paper, we report that a deformed entropy designed by q-algebra, a popular generalization of the standard algebra studied in Tsallis statistical mechanics, makes optimal transport solutions supported sparsely. This entropy with a deformation parameter q interpolates the negative Shannon entropy (q=1) and the squared 2-norm (q=0), and the solution becomes more sparse as q tends to zero. Our theoretical analysis reveals that a larger q leads to a faster convergence when optimized with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. In summary, the deformation induces a trade-off between the sparsity and convergence speed.     

Renormalization of the Regularized Relativistic Electron-Positron Field

We consider the relativistic electron-positron field interacting with itself via the Coulomb potential defined with the physically motivated, positive, density-density quartic interaction. The more usual normal-ordered Hamiltonian differs from the bare Hamiltonian by a quadratic term and, by choosing the normal ordering in a suitable, self-consistent manner, the quadratic term can be seen to be equivalent to a renormalization of the Dirac operator. Formally, this amounts to a Bogolubov-Valatin transformation, but in reality it is non-perturbative, for it leads to an inequivalent, fine-structure dependent representation of the canonical anticommutation relations. This non-perturbative redefinition of the electron/positron states can be interpreted as a mass, wave-function and charge renormalization, among other possibilities, but the main point is that a non-perturbative definition of normal ordering might be a useful starting point for developing a consistent quantum electrodynamics. Received: 8 March 2000 / Accepted: 7 July 2000     

Calculated nearly degenerate isospin doublets

It is noted that in several calculations of the odd J states of N = Z nuclei ³⁶Ar, ⁴⁴Ti, and ⁵²Fe, there is a near degeneracy of the lowest T = 0 and lowest T = 1 states, for several J values. Qualitative physical reasons for this behavior are given and the Coulomb mixing is estimated.     

Some observations on degenerate metrics

There are some polynomial formulations of Einstein's equations in which the metric is allowed to become degenerate. We examine some known exact solutions to see whether they may be smoothly joined to solutions with degenerate metrics. If one uses a lapse function which is a spatial scalar, this is very easy. If the lapse function has a small and negative tensor density weight, the joining together may take place across the horizons in the Schwarzschild and Kerr solutions. For large and negative weights, we have been unable to find any examples.