The Experimental Mathematician: the Pleasure of Discovery and the Role of Proof

The emergence of powerful mathematical computing environments, the growing availability of correspondingly powerful (multi-processor) computers and the pervasive presence of the internet allow for mathematicians, students and teachers, to proceed heuristically and ‘quasi-inductively’. We may increasingly use symbolic and numeric computation, visualization tools, simulation and data mining. The unique features of our discipline make this both more problematic and more challenging. For example, there is still no truly satisfactory way of displaying mathematical notation on the web; and we care more about the reliability of our literature than does any other science. The traditional role of proof in mathematics is arguably under siege – for reasons both good and bad.     

一类Smale空间上的C~*-代数自同构的熵

Ian Putnam利用Smale空间上的渐近等价关系定义了广群C~*-代数及其典则自同构.本文在零维Smale空间的情形下,计算此类C~*-自同构的逼近熵,证明了相应C~*-动力系统关于CNT熵和逼近熵的"变分原理"成立.由此推演出此类Smale空间上的Bowen测度诱导的C~*-代数上的态是此典则自同构的唯一平衡态.     

The Higher Dimensional Hermite Transform: A New Approach

In this article we expand the filter functions of the classical Hermite transform into the Clifford-Hermite polynomials. Furthermore, we construct a new higher dimensional Hermite transform within the framework of Clifford analysis using the radial and generalized Clifford-Hermite polynomials. Finally we compare this newly introduced Clifford-Hermite transform with the Clifford-Hermite Continuous Wavelet transform.     

On the Roots of a Random System of Equations. The Theorem of Shub and Smale and Some Extensions

We give a new proof of a theorem of Shub and Smale on the expectation of the number of roots of a system of m random polynomial equations in m real variables, having a special isotropic Gaussian distribution. Further, we present a certain number of extensions, including the behavior as m → +∞ of the variance of the number of roots, when the system of equations is also stationary.     

The multiconfiguration methods in quantum chemistry: Palais–Smale condition and existence of minimizersLes méthodes de multiconfiguration en chimie quantique : condition de Palais–Smale et existence de minima

In this Note, we propose a new proof for the existence of a minimum in the multiconfiguration methods in Quantum Chemistry. We use a Palais–Smale condition with Morse-type information, whose proof is based on the Euler–Lagrange equations, written in a simple and useful way. To cite this article: M. Lewin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 299–304.     

The Bahadur-Kiefer Representation of the Two Dimensional Spatial Medians

We consider the asymptotic behavior, both in distribution and almost sure, of the Bahadur-Kiefer representation of the two dimensional spatial medians. The rates appearing in this expansion are non-standard. The rate in the almost sure expansion is n(2 log n)-1/2(2 log log n)-1. The set of clusters points in the almost sure representation is obtained. The distribution of the Bahadur-Kiefer representation of the two dimensional spatial medians converges with rate n(2 log n)-1/2 to a limit that is determined precisely.     

The Continuity of Barrier Function with Respect to the Parameter

The authors study the continuity of barrier function Be（x） with respect to the parameter. A sufficient condition which makes Be（x） be continuous with respect to c is obtained, and an example of discontinuity when the condition is not satisfied is also constructed.     

The Ground State Energy of the Three Dimensional Ginzburg-Landau Functional Part I: Bulk Regime

We consider the Ginzburg-Landau functional defined over a bounded and smooth three dimensional domain. Supposing that the magnetic field is comparable with the second critical field and that the Ginzburg-Landau parameter is large, we determine a precise asymptotic formula for the minimizing energy. In particular, this shows how bulk superconductivity decreases in average as the applied magnetic field approaches the second critical field from below. Other estimates are also obtained which allow us to obtain, in a subsequent paper [19 Fournais , S. , Kachmar , A. , Persson , M. The ground state energy of the three dimensional Ginzburg-Landau functional. Part II: Surface regime. Preprint.  [Google Scholar]], a fine characterization of the second critical field. The approach relies on a careful analysis of several limiting energies, which is of independent interest.     

The Use of Dimensional and Similarity Analyses in the Propagation of Uncertainties: A Physical Example

Dimensional and similarity analyses are used in physics and engineering, specially in fluid mechanics, to reduce the dimension of the input variable space with no loss of information. Here, we apply these techniques to the propagation of uncertainties for computer codes by the Monte Carlo method, in order to reduce the variance of the estimators of the parameters of the output variable distribution. In the physics and engineering literature, dimensional analysis is often formulated intuitively in terms of physical quantities or dimensions such as time, longitude, or mass; here we use the more rigorous and more abstract generalized dimensional analysis of Moran and Marshek. The reduction of dimensionality is only successful in reducing estimator variance when applying variance-reduction techniques and not when using ordinary random sampling. In this article we use stratified sampling, and the key point of the success of the reduction in dimensionality in improving the precision of the estimates is a better measurement of the distances betwen the outputs, for given inputs. We illustrate the methodology with an application to a physical problem, a radioactive contaminant transport code. A substantial variance reduction is achieved for the estimators of the mean, variance, and distribution function of the output. Last, we present a discussion on which conditions are necessary for the method to be successful.     

The Higher Dimensional Continuum Dislocation Dynamics Theory: A Runge-Kutta Discontinuous Galerkin Approach

The reformulation in conservative form of the higher-dimensional continuum dislocations dynamics(hdCDD) theory of Hochrainer (Ph.D. thesis, 2006) is presented together with a framework for elasto-plasticity problem based on this theory. A Runge-Kutta discontinuous Galerkin(RKDG) method is used for the evolution of hdCDD to obtain information of the micro-structure which is coupled with a finite element method for the stress computation. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A New Inclusion Function for Optimization: Kite – The One Dimensional Case

In this paper the kite inclusion function is presented for branch-and-bound type interval global optimization using at least gradient information. The basic idea comes from the simultaneous usage of the centered forms and the linear boundary value forms. We will show that the new technique is not worse and usually considerably better than these two. The best choice for the center of the kite inclusion will be given. The isotonicity and at least quadratical convergence hold and there is a pruning effect of the kite which is derived from the construction of the inclusion, thus more function evaluations are not needed to use it. A numerical investigation on large standard multiextremal test functions has been done to show the performance.