The Mathematical Theory of Ito Diffusions on Hypersurfaces, with Applications to NMR Relaxation Problems

A mathematical framework for translational Brownian motion on hypersurfaces is presented, using an imbedding of the surface and Ito diffusions in the ambient space. This includes a survey of Ito calculus and differential geometry. Computational methods for time correlation functions relevant to spin relaxation studies on curved interfaces are given, and explicit calculations of time correlation functions and order parameters for a Rippled surface are presented.     

纳米分辨率的三维形貌检测仪

从光干涉方式，瞳窗关系和光源带宽等基本关系出发，利用空间不变系统理论解析了相移显微干涉检测系统中衍射效应的影响。为了减小横向分辨率对纵向分辨率的影响，利用相关信息提取或数字滤波的方法，获得了纳米分辨率的三维形貌。     

Orbital correlation effects: The independent pair-potential approximation with application to the ground state and first ionized state of boron hydride

A new method is proposed for calculating correlation effects in atomic and molecular systems. The basis of the method is the formulation of a set of partial configuration expansions which yield directly variational orbital correlation corrections which are appropriately summed in order to obtain an estimate of the total correlation energy. This method is applied to the ground state of boron hydride and its cation at the equilibrium distance of BH. The results of the method are compared in detail with independent electron pair results and second order CI results. It is further shown that multiple substitutions are approximately accounted for in this method and the extent to which they are included is compared with other approximations. Finally, three methods of increasing accuracy, aimed at reducing the necessary computational effort, are given for determining the vertical ionization potential. The most economical method yields an IP of 9.70 eV or 0.03 eV less than the experimental IP. Completion of the basis is estimated to improve this value to 9.77 eV.     

Application of coset representations to the construction of symmetry adapted functions

Summary A coset representation (G(/G _i )), which is defined algebraically by a coset decomposition of a finite groupG by its subgroupG _i , is shown to be a method for the decomposition of a regular body into its point group orbits. This proof also shows that each member of theG(/G _i ) orbit belongs to theG _i site-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and thek-value equations of framework groups as special cases. The relationship of the coset representation and the site-symmetry affords a general procedure for obtaining symmetry adapted functions.     

A modified steepest descent method with applications to maximizing likelihood functions

In maximizing a non-linear function G(), it is well known that the steepest descent method has a slow convergence rate. Here we propose a systematic procedure to obtain a 1–1 transformation on the variables , so that in the space of the transformed variables, the steepest descent method produces the solution faster. The final solution in the original space is obtained by taking the inverse transformation. We apply the procedure in maximizing the likelihood functions of some generalized distributions which are widely used in modeling count data. It was shown that for these distributions, the steepest descent method via transformations produced the solutions very fast. It is also observed that the proposed procedure can be used to expedite the convergence rate of the first derivative based algorithms, such as Polak-Ribiere, Fletcher and Reeves conjugate gradient methods as well.     

Variational approximations for renormalization group transformations

Approximate recursion relations which give upper and lower bounds on the free energy are described. Optimal calculations of the free energy can then be obtained by treating parameters within the renormalization equations variationally. As an example, a particularly simple lower bound approximation which preserves the symmetry of the Hamiltonian (the one-hypercube approximation) is described. The approximation is applied to both the Ising model and the Wilson-Fisher model. At the fixed point a parameter is set variationally and critical indices are calculated. For the Ising model the agreement with the exact results atd = 2 is surprisingly good, 0.1%, and is good atd=3 and evend=4. For the Wilson-Fisher model the recursion relation is reduced to a one-dimensional integral equation which can be solved numerically givingv=0.652 atd=3, or by expansion in agreement with the results of Wilson and Fisher to leading order in . The method is also used to calculate thermodynamic functions for thed = 2 Ising model; excellent agreement with the Onsager solution is found.Supported in part by the National Science Foundation under Grants Nos. MPS73-04886A01 and GH-41512 and by the Brown University Materials Research Laboratory supported by the National Science Foundation. M.C.Y. was supported by a grant from the Scientific and Technical Research Council of Turkey.     

桑白皮中21种无机元素的测定及其质量评价

为准确了解不同产区桑白皮中多种元素分布及含量情况,用65%硝酸溶液对其样品进行微波消解处理,后采用电感耦合等离子体质谱（ICP-MS/MS）法测定桑白皮样品中21种元素的含量。得到21种元素的线性关系良好,R2 > 0.999,方法学考察试验的RSD均小于3%,加样回收率在91.32%~108.84%之间,RSD均小于4%;Al、Ga、Sr、Ba相比而言含量较大,且不同产区的特异性较明显;主成分分析中17批样品得到4个主成分,累计方差贡献率达 87.380%,确定V、Co、Al、Li、As、Se、Pb、U、Zn、Cr、Cd、Ga、Ba、Cs、Ni等15种元素为桑白皮药材的特征元素;相关性分析可知,元素间的相关性较强, P≤0.05的元素与主成分分析的特征元素基本一致。参照2020年版《中国药典》中对中药材中有害元素的限量规定,发现桑白皮样品中Pb、Cd、As、Cu有害元素的含量均未超过限量要求。ICP-MS/MS能准确快速测定桑白皮中21种元素含量情况,这对评价不同产区桑白皮质量,完善桑白皮质量标准具有一定参考价值。     

Critical Analysis of Hypothesis Tests in Federal Information Processing Standard (140-2)

This work presents an analysis of the existing dependencies between the tests of the FIPS 140-2 battery. Two main analytical approaches are utilized, the first being a study of correlations through the Pearson’s correlation coefficient that detects linear dependencies, and the second one being a novel application of the mutual information measure that allows detecting possible non-linear relationships. In order to carry out this study, the FIPS 140-2 battery is reimplemented to allow the user to obtain p-values and statistics that are essential for more rigorous end-user analysis of random number generators (RNG).     

Application of Two Novel Acoustic Emission Parameters on Identifying the Instability of Granite

The issue of monitoring and early warning of rock instability has received increasing critical attention in the study of rock engineering. To investigate the damage evolution process of granite under triaxial compression tests, acoustic emission (AE) tests were performed simultaneously. This study firstly introduced two novel parameters, i.e., the coefficient of variation (CoV) of the information entropy and correlation dimension of the amplitude data from the AE tests, to identify the precursor of the failure of granite. Then the relationship between the changes in these parameters and the stress-time curve was compared and analyzed. The results of this study show that: (1) There is a strong correlation between the CoV of the information entropy and the failure process of granite. The granite failed when the CoV curve raised to a plateau, which could be used as an indicator of rock instability. (2) The fluctuation of the correlation dimension indicates the different stages during the loading process, i.e., the initial compaction stage, the linear elastic stage, the yield stage, and the failure stage. Each stage contains a descending and a rising process in the correlation dimension curve, and the exhibited starting point or the bottom point at the correlation dimension curve could be selected as the indicator point for the rock instability. (3) The combined analysis of the Information entropy and Correlation dimension can improve the accuracy of rock instability prediction. This study provides new insights into the prediction of rock instability, which has theoretical implications for the stability of subsurface engineering rock masses.     

A New Entropy Function to Analyze Isentropic Processes of Ideal Gases with Variable Specific Heats

A new entropy function s⁺ is defined in terms of the existing entropy function s° and temperature as s⁺ = s° − R lnT to facilitate the analysis of isentropic processes of ideal gases with variable specific heats. The function s⁺ also makes it possible to calculate the entropy changes of ideal gases during processes when volume information is available instead of pressure information and the variation of specific heats with temperature is to be accounted for. The introduction of the function s⁺ eliminates the need to use the dimensionless isentropic functions relative pressure P_r and relative specific volume v_r of ideal gases and to tabulate their values. The P_r and v_r data are often confused with pressure and specific volume, with an adverse effect on the study of the second law of thermodynamics. The new s⁺ function nicely complements the existing s° function in entropy change calculations: the former is conveniently used when volume information is given while the latter is used when pressure information is available. Therefore, the introduction of the new entropy function s⁺ is expected to make a significant contribution to the thermodynamics education and research by streamlining entropy analysis of ideal gases.