Some exact expressions for the mean and higher moments of functions of sample moments

Examples of exact expressions for the moments (mainly of the mean) of functions of sample moments are given. These provide checks on alternative developments such as asymptotic series for n, and simulation processes. Exact expressions are given for the mean of the square of the sample coefficient of variation, particularly in uniform sampling; Frullani integrals studied by G. H. Hardy arise. It should be kept in mind that exact results for (joint) moment generating functions (mgfs) are of interest as they produce a means of obtaining exact results for (cross) moments—including moments with negative indices. Thus an exact expression for the joint mgf of the 1st two noncentral moments can be used to obtain the mean of the (c.v.)² (but not for the mean of the c..). A general expression is given for the moment generating function of the sample variance. The limitations of Fisher's symbolic formula for the characteristic function of sample moments (or more general statistics) are noted.This research was sponsored by the Applied Mathematical Sciences Research program, Office of Energy Research, U. S. Department of Energy under contract DE-AC0584OR21400 with the Martin Marietta Energy Systems. Inc.     

A long-time tail for random walk in random scenery

Consider a simple random walk on ^d whose sites are colored black or white independently with probabilityq, resp. 1–q. Walk and coloring are independent. Letn _k be the number of steps by the walk between itskth and (k+1) th visits to a black site (i.e., the length of itskth white run), and let _k =E(n _k )–q ^–1. Our main result is a proof that (*) lim _k k ^d/2 _k = (1 –q)q ^{d/2 – 2}(d/2) ^d/2. Since it is known thatq ^{– 1} _k =E(n ₁ n _{k + 1} B) –E(n ₁ B)E(n _{k + 1} B), withB the event that the origin is black, (*) exhibits a long-time tail in the run length autocorrelation function. Numerical calculations of _k (1k100) ind=1, 2, and 3 show that there is an oscillatory behavior of _k for smallk. This damps exponentially fast, following which the power law sets in fairly rapidly. We prove that if the coloring is not independent, but is convex in the sense of FKG, then the decay of _k cannot be faster than (*).     

Partition function zeros for the one-dimensional ordered plasma in Dirichlet boundary conditions

We consider the grand canonical partition function for the ordered one-dimensional, two-component plasma at fugacity in an applied electric fieldE with Dirichlet boundary conditions. The system has a phase transition from a low-coupling phase with equally spaced particles to a high-coupling phase with particles clustered into dipolar pairs. An exact expression for the partition function is developed. In zero applied field the zeros in the plane occupy the imaginary axis from –i to –i_c and i_c to i for some _c. They also occupy the diamond shape of four straight lines from ±i_c to _c and from ±i_c to –_c. The fugacity acts like a temperature or coupling variable. The symmetry-breaking field is the applied electric fieldE. A finite-size scaling representation for the partition in scaled coupling and scaled electric field is developed. It has standard mean field form. When the scaled coupling is real, the zeros in the scaled field lie on the imaginary axis and pinch the real scaled field axis as the scaled coupling increases. The scaled partition function considered as a function of two complex variables, scaled coupling and scaled field, has zeros on a two-dimensional surface in a domain of four real variables. A numerical discussion of some of the properties of this surface is presented.     

压电石英晶体在电解质水溶液中的振荡性能研究(Ⅱ)——频率、频移特性

根据两面电极浸入液体中的压电晶体等效电路模型和Pierce振荡器振荡方程的解,得到晶体在电解质溶液中的振荡频率(F_1)与溶液电导率(x)的关系式:F_1=(1/2π)(1/C_1+1/C_2)(1/X_e)[(R_e~2+X_e~2)x~2+2R_ex+1]。以自行设计的Pierce晶体振荡器。研究了晶体和主振电路参数、输入电压及检测池外壳屏蔽接地对AT切9MHz石英晶体在KCl、Na_2SO_4水溶液中的频率、相对于纯水频移随溶液电导率变化的影响规律,结果与上式相符。     

Numerical solution of minimax problems of optimal control,part 1

This paper contains general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We consider two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).For Problem (Q), we exploit the analogy with a bounded-state problem in combination with a transformation of the Jacobson type. This requires the proper augmentation of the state vectorx(t), the control vectoru(t), and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.For Problem (R), we exploit the analogy with a bounded-control problem in combination with a transformation of the Valentine type. This requires the proper augmentation of the control vectoru(t) and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.In a subsequent paper (Part 2), the transformation techniques presented here are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer; both the single-subarc approach and the multiple-subarc approach are discussed.This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.     

Hydration and dielectric properties of aqueous solutions of protonated diallylmethylammonium polymer: transition from monomer to polymer

Abstract

The high-frequency dielectric relaxation of aqueous solutions of protonated diallylammonium polyelectrolyte, namely poly(diallylmethylammonium trifluoroacetate) has been studied at the maximum water dispersion frequencies, 7.5–25?GHz, and temperatures of 288, 298, and 308?K. Dielectric relaxation parameters have been calculated and compared with similar characteristics of aqueous solutions of monomer, diallylmethylammonium trifluoroacetate, and pyrrolidinium trifluoroacetate salt simulated structure of pyrrolidinium polymer link. It has been concluded that although the monomer features hydrophobic hydration, its polymer exhibits hydrophilic hydration properties. This change is related to conformation of hydrophilic-hydrophobic polycations in aqueous solutions and the change in the structure of polymer links.     

Two-beam interference detection of the changes in fibre structural parameters during low drawing process

A Pluta polarizing interference microscope with a fibre stretching device attached was used to detect the changes in molecular structure that occur during the fibre cold drawing process. Fibres drawn with low draw ratios show different types of deformation mechanisms. The interference patterns recorded at different draw ratios are used to relate the deformation mechanisms with the measured structural parameters. Some optical parameters are measured such as; refractive indices nand n birefringence n, the polarizability per unit volume , the orientation distribution function f(θ), the angle θ between the stretching direction and the chain axis, and the angle θ_m which the transition dipole moment of the molecular species makes with the molecular axis (chain axis). Three polymeric fibres (Polyethylene terephthalate PET, Polypropylene PP and Polyamide PA) were used in this study and their interferograms are illustrated.     

Structure and electrical properties of Pb(Zr<Subscript>0.25</Subscript>Ti<Subscript>0.75</Subscript>)O<Subscript>3</Subscript> thin films on LaNiO<Subscript>3</Subscript>—Coated thermally oxidized Si substrates

Pb(Zr_0.25Ti_0.75)O₃ (PZT25) thin films were prepared on LaNiO₃-coated thermally oxidized silicon substrates by chemical solution deposition method, where LaNiO₃ electrodes were also prepared by a chemical solution deposition technique. The dielectric constant and dielectric loss of the PZT25 thin films were 570 and 0.057, respectively. The remanent polarization and coercive field were 20.11 μC/cm² and 60.7 kV/cm, respectively. The PZT25 thin films on LaNiO₃-coated thermally oxidized silicon substrates showed improved fatigue characteristics compared with their counterparts on plantium-coated silicon substrates.     

Modified AutoDock for accurate docking of protein kinase inhibitors

Protein kinases are an important class of enzymes controlling virtually all cellular signaling pathways. Consequently, selective inhibitors of protein kinases have attracted significant interest as potential new drugs for many diseases. Computational methods, including molecular docking, have increasingly been used in the inhibitor design process [1]. We have considered several docking packages in order to strengthen our kinase inhibitor work with computational capabilities. In our experience, AutoDock offered a reasonable combination of accuracy and speed, as opposed to methods that specialize either in fast database searches or detailed and computationally intensive calculations.However, AutoDock did not perform well in cases where extensive hydrophobic contacts were involved, such as docking of SB203580 to its target protein kinase p38. Another shortcoming was a hydrogen bonding energy function, which underestimated the attraction component and, thus, did not allow for sufficiently accurate modeling of the key hydrogen bonds in the kinase-inhibitor complexes.We have modified the parameter set used to model hydrogen bonds, which increased the accuracy of AutoDock and appeared to be generally applicable to many kinase-inhibitor pairs without customization. Binding to largely hydrophobic sites, such as the active site of p38, was significantly improved by introducing a correction factor selectively affecting only carbon and hydrogen energy grids, thus, providing an effective, although approximate, treatment of solvation.