The Ground State Energy of the Massless Spin-Boson Model

We provide an explicit combinatorial expansion for the ground state energy of the massless spin-Boson model as a power series in the coupling parameter. Our method uses the technique of cluster expansion in constructive quantum field theory and takes as a starting point the functional integral representation and its reduction to an Ising model on the real line with long range interactions. We prove the analyticity of our expansion and provide an explicit lower bound on the radius of convergence. We do not need multiscale nor renormalization group analysis. A connection to the loop-erased random walk is indicated.     

M-矩阵最小特征值的上界估计

利用M-矩阵最小特征值与非负矩阵谱半径之间的关系,结合矩阵的迹分两种情况给出M-矩阵最小特征值的上界序列,并且给出数值例子加以说明.     

Approach to Ground State and Time-Independent Photon Bound for Massless Spin-Boson Models

It is widely believed that an atom interacting with the electromagnetic field (with total initial energy well-below the ionization threshold) relaxes to its ground state while its excess energy is emitted as radiation. Hence, for large times, the state of the atom + field system should consist of the atom in its ground state, and a few free photons that travel off to spatial infinity. Mathematically, this picture is captured by the notion of asymptotic completeness. Despite some recent progress on the spectral theory of such systems, a proof of relaxation to the ground state and asymptotic completeness was/is still missing, except in some special cases (massive photons, small perturbations of harmonic potentials). In this paper, we partially fill this gap by proving relaxation to an invariant state in the case where the atom is modelled by a finite-level system. If the coupling to the field is sufficiently infrared-regular so that the coupled system admits a ground state, then this invariant state necessarily corresponds to the ground state. Assuming slightly more infrared regularity, we show that the number of emitted photons remains bounded in time. We hope that these results bring a proof of asymptotic completeness within reach.     

Nash Estimates and Upper Bounds for Non-homogeneous Kolmogorov Equations

We prove a Gaussian upper bound for the fundamental solutions of a class of ultra-parabolic equations in divergence form. The bound is independent on the smoothness of the coefficients and generalizes some classical results by Nash, Aronson and Davies. The class considered has relevant applications in the theory of stochastic processes, in physics and in mathematical finance.     

Bounds and Self-Consistent Estimates for the Overall Properties of Nonlinear Composites

Variational ‘self-consistent’ estimates for nonlinearproblems are formulated, building on a variational formulationpreviously developed by the authors. The formulation employsa linear ‘comparison medium’ for whose propertiessome ‘self-consistent’ choice is made. In contrastto linear problems, three possible self-consistent choices presentthemselves. The results that they give are analysed for twoparticular systems (a nonlinear dielectric and a nonlinear lossycomposite) for which bounds are already available. Estimatesbased on self-consistent embedding of a single inclusion ina homogeneous matrix composed of ‘comparison material’are also developed.     

Estimates for the Upper Bounds of the First Eigenvalue on Submanifolds

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Curvature Estimates for Minimal Submanifolds of Higher Codimension

The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau＇s results and Ecker-Huisken＇s results are generalized to higher codimension. In this way, Hildebrandt-Jost-Widman＇s result for the Bernstein type theorem is improved.     

Upper and Lower Bounds for the Minimal Positive Entropy of Pure Braids

In this paper, the minimal positive entropy of pure braids isshown to be greater than , independent of the braid index. For pure 3-braids, the minimalentropy is shown to be equal to . 2000 Mathematics Subject Classification 37E30 (primary), 37B40(secondary).     

Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees

We give a new lower bound on the length of the minimal Steiner tree with a given topology joining given terminals in Euclidean space, in terms of toroidal images. The lower bound is equal to the length when the topology is full. We use the lower bound to prove bounds on the “error” e in the length of an approximate Steiner tree, in terms of the maximum deviation d of an interior angle of the tree from 120°. Such bounds are useful for validating algorithms computing minimal Steiner trees. In addition we give a number of examples illustrating features of the relationship between e and d, and make a conjecture which, if true, would somewhat strengthen our bounds on the error. J. H. Rubinstein, J. Weng: Research supported by the Australian Research Council N. Wormald: Research supported by the Australian Research Council and the Canada Research Chairs Program. Research partly carried out while the author was in the Department of Mathematics and Statistics, University of Melbourne     

循环矩阵本原指数的上界估计

本文给出了循环矩阵本原指数上界的新的估计及一种由级数较低的循环矩阵的本原指数估计级数较高的循环矩阵的本原指数的方法，解决了一类循环矩阵本原指数的计算问题．     

A Frequency Function and Singular Set Bounds for Branched Minimal Immersions

We establish a frequency function monotonicity formula for two‐valued C^1,α solutions to the minimal surface system on n‐dimensional domains. We also establish the sharp regularity result that such solutions are of class C^{1, 1/2}, and that their branch sets, if nonempty, have Hausdorff dimension equal to n‐2.© 2016 Wiley Periodicals, Inc.     

迭代矩阵谱半径的上界估计

该文对一类广义对角占优矩阵Ｍ,给出了迭代矩阵Ｍ－１Ｎ 的谱半径的上界．特别,当Ｍ是严格对角占优时,证明了所得到的估计值总比通常用作谱半径的估计值要好．     

Curvature Estimates for Stable Minimal Hypersurfaces in R4 and R5

We obtain curvature estimates for certain stable minimalhypersurfaces in R ⁴ and R ⁵without using volume bounds. It follows that if M is acomplete stable minimal hypersurface in R ⁴ orR ⁵, then M is a hyperplane whenM intersects each extrinsic ball in, at most,N-components.     

Some New Error Estimates for Ritz--Galerkin Methods with Minimal Regularity Assumptions

New uniform error estimates are established for finite element approximations of solutions of second-order elliptic equations using only the regularity assumption . Using an Aubin--Nitsche type duality argument we show for example that, for arbitrary (fixed) sufficiently small, there exists an such that for

Here, denotes the norm on the Sobolev space . Other related results are established.

     

Estimates for the Norm of Green's Matrix Based on the Integral Performance Criterion for Dichotomy and Hausdorff Set Bounds

For a system of ordinary differential equations of the form , , we obtain new upper bounds for the second norm of Green's matrix using the integral performance criterion for dichotomy and bounds of the Hausdorff set of the matrix A. These estimates are considerably better for many applications than some well-known ones.     

Minimal Penalties for Gaussian Model Selection

This paper is mainly devoted to a precise analysis of what kind of penalties should be used in order to perform model selection via the minimization of a penalized least-squares type criterion within some general Gaussian framework including the classical ones. As compared to our previous paper on this topic (Birgé and Massart in J. Eur. Math. Soc. 3, 203–268 (2001)), more elaborate forms of the penalties are given which are shown to be, in some sense, optimal. We indeed provide more precise upper bounds for the risk of the penalized estimators and lower bounds for the penalty terms, showing that the use of smaller penalties may lead to disastrous results. These lower bounds may also be used to design a practical strategy that allows to estimate the penalty from the data when the amount of noise is unknown. We provide an illustration of the method for the problem of estimating a piecewise constant signal in Gaussian noise when neither the number, nor the location of the change points are known.     

达尔文模型的有限元后验误差估计

在三维有界单连通区域里,当没有高频现象或电流变化不快时,达尔文模型是麦克斯韦方程组的-个很好的逼近模型.本文考虑达尔文模型的自适应算法,这种方法以有限元后验误差分析为理论基础.本文提供了基于后验误差估计子的上界估计.     

Nonasymptotic Bounds on the L 2 Error of Neural Network Regression Estimates

The estimation of multivariate regression functions from bounded i.i.d. data is considered. The L ₂ error with integration with respect to the design measure is used as an error criterion. The distribution of the design is assumed to be concentrated on a finite set. Neural network estimates are defined by minimizing the empirical L ₂ risk over various sets of feedforward neural networks. Nonasymptotic bounds on the L ₂ error of these estimates are presented. The results imply that neural networks are able to adapt to additive regression functions and to regression functions which are a sum of ridge functions, and hence are able to circumvent the curse of dimensionality in these cases.     

Energy Estimates for the Supersymmetric Nonlinear Sigma Model and Applications

We derive gradient and energy estimates for critical points of the full supersymmetric sigma model and discuss several applications.