Mixtures of Global and Local Edgeworth Expansions and Their Applications

Edgeworth expansions which are local in one coordinate and global in the rest of the coordinates are obtained for sums of independent but not identically distributed random vectors. Expansions for conditional probabilities are deduced from these. Both lattice and continuous conditioning variables are considered. The results are then applied to derive Edgeworth expansions for bootstrap distributions, for Bayesian bootstrap distribution, and for the distributions of statistics based on samples from finite populations. This results in a unified theory of Edgeworth expansions for resampling procedures. The Bayesian bootstrap is shown to be second order correct for smooth positive “priors,” whenever the third cumulant of the “prior” is equal to the third power of its standard deviation. Similar results are established for weighted bootstrap when the weights are constructed from random variables with a lattice distribution.     

Lax pair, Bäcklund transformation and multi-soliton solutions for the Boussinesq-Burgers equations from shallow water waves

Under investigation in this paper is the set of the Boussinesq-Burgers (BB) equations, which can be used to describe the propagation of shallow water waves. Based on the binary Bell polynomials, Hirota method and symbolic computation, the bilinear form and soliton solutions for the BB equations are derived. Bäcklund transformations (BTs) in both the binary-Bell-polynomial and bilinear forms are obtained. Through the BT in the binary-Bell-polynomial form, a type of solutions and Lax pair for the BB equations are presented as well. Propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis. Shock wave and bell-shape solitons are respectively obtained for the horizontal velocity field u and height v of the water surface. In both the head-on and overtaking collisions, the shock waves for the u profile change their shapes, which denotes that the collisions for the u profile are inelastic. However, the collisions for the v profile are proved to be elastic through the asymptotic analysis. Our results might have some potential applications for the harbor and coastal design.     

Point processes of cylinders,particles and flats

Point processesX of cylinders, compact sets (particles), or flats inR ^d are mathematical models for fields of sets as they occur, e.g., in practical problems of image analysis and stereology. For the estimation of geometric quantities of such fields, mean value formulas forX are important. By a systematic approach, integral geometric formulas for curvature measures are transformed into density formulas for geometric point processes. In particular, a number of results which are known for stationary and isotropic Poisson processes of convex sets are generalized to nonisotropic processes, to non-Poissonian processes, and to processes of nonconvex sets. The integral geometric background (including recent results from translative integral geometry), the fundamentals of geometric point processes, and the resulting density formulas are presented in detail. Generalizations of the theory and applications in image analysis and stereology are mentioned shortly.     

Determinant formulas with applications to designing when the observations are correlated

In the general linear model consider the designing problem for the Gauß-Markov estimator or for the least squares estimator when the observations are correlated. Determinant formulas are proved being useful for theD-criterion. They allow, for example, a (nearly) elementary proof and a generalization of recent results for an important linear model with multiple response. In the second part of the paper the determinant formulas are used for deriving lower bounds for the efficiency of a design. These bounds are applied in examples for tridiagonal covariance matrices. For these examples maximin designs are determined.Parts of the paper are based on a part of the author's Habilitationsschrift Bischoff (1993a).     

Nonconforming Galerkin methods for the Helmholtz equation

Nonconforming Galerkin methods for a Helmholtz‐like problem arising in seismology are discussed both for standard simplicial linear elements and for several new rectangular elements related to bilinear or trilinear elements. Optimal order error estimates in a broken energy norm are derived for all elements and in L² for some of the elements when proper quadrature rules are applied to the absorbing boundary condition. Domain decomposition iterative procedures are introduced for the nonconforming methods, and their convergence at a predictable rate is established. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 475–494, 2001     

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half-space are well posed in L₂ for small complex L_∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for k-forms are well posed for small perturbations of block matrices.     

Algorithm for a general discrete k-out-of-n: G system subject to several types of failure with an indefinite number of repairpersons

A discrete k-out-of-n: G system with multi-state components is modelled by means of block-structured Markov chains. An indefinite number of repairpersons are assumed and PH distributions for the lifetime of the units and for the repair time are considered. The units can undergo two types of failures, repairable or non-repairable. The repairability of the failure can depend on the time elapsed up to failure. The system is modelled and the stationary distribution is built by using matrix analytic methods. Several performance measures of interest, such as the conditional probability of failure for the units and for the system, are built into the transient and stationary regimes. Rewards are included in the model. All results are shown in a matrix algorithmic form and are implemented computationally with Matlab. A numerical example of an optimization problem shows the versatility of the model.     

Finding Gateaux-Saddles by a Local Minimax Method

Motivated by quasilinear elliptic PDEs in physical applications, Gateaux-saddles of a class of functionals J:H→{±∞}∪?, which are only Gateaux-differentiable at regular points, are considered. Since mathematical results and numerical methods for saddles of 𝒞¹ or locally Lipschitz continuous functionals in the literature are not applicable, the main objective of this article is to introduce a new mixed norm strong-weak topology approach such that a mathematical framework of a local minimax method is established to handle the singularity issue and to use the Gateaux-derivative of J for finding multiple Gateaux-saddles. Algorithm implementations on weak form and error control are presented. Numerical examples solving quasilinear elliptic problems from physical applications are successfully carried out to illustrate the method. Some interesting solution properties are to be numerically observed and open for analytical verification for the first time.     

Bowen–Franks Groups for Subshifts and Ext-Groups for C*-Algebras

We generalize the Bowen–Franks groups for topological Markov shifts to general subshifts as the Ext-groups for the associated C ^*-algebras. The generalized Bowen–Franks groups for subshifts are shown to be invariant under flow equivalence and, hence, invariant under topological conjugacy. They are regarded as the indices of Fredholm operators related to extensions of the associated C ^*-algebras so that they are described in terms of symbolic dynamical systems. In particular, the group for a sofic subshift is determined by the adjacency matrix of its left Krieger cover graph. The Bowen–Franks groups for some non sofic subshifts are calculated, proving that certain subshifts with the same topological entropy are not flow equivalent.     

Gaugeability and conditional gaugeability

New Kato classes are introduced for general transient Borel right processes, for which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Green-tight measures in the classical Brownian motion case. However, the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.

     

Partition and composition matrices

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another.We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel.We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,…,n}.     

On Legendre functions of imaginary degree and associated integral transforms

New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function P_iξ(cosht). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ. A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler-Fok transforms, since conditions placed on functions are less restrictive.     

An asymptotic theory of the sound dispersion in a binary mixture of gases

The sound propagation in a binary mixture of multiatomic non-relaxing gases is considered using Burnett's equations (for the case of rapid exchanges of the internal and translational energies of the molecules). Asymptotic expressions (for small Knudsen numbers) are obtained for the absorption and dispersion coefficients, which are expressed in terms of the Navier–Stokes and the Navier–Stokes and the Burnett transport coefficients respectively. “Working” expressions of different levels of accuracy for these coefficients are known for the case of a binary mixture of monatomic gases. For this case, the results obtained are compared with known ones and the drawbacks and errors of previous papers are pointed out.     

The Cauchy problem for the 1-D Dirac–Klein–Gordon equation

The Cauchy problem for the Dirac–Klein–Gordon equation are discussed in one space dimension. Time local and global existence for solutions with rough data, especially the solutions for Klein–Gordon equation in the critical and super critical Sobolev norm of [4] are considered. The solutions with general propagation speeds are dealt with.      

Generalized canonical analysis for time series

Canonical correlation analysis is shown to be equivalent to the problem of estimating a linear regression matrix, B₀, of less than full rank. After reparameterizing B₀ some estimates of the new parameters, obtained by solving an eigenvalue problem and closely related to canonical correlations and vectors, are found to be consistent and efficient when the residuals are mutually independent. When the residuals are generated by an autocorrelated, stationary time series these estimates are still consistent and obey a central limit theorem, but they are no longer efficient. Alternative, more general estimates are suggested which are efficient in the presence of serial correlation. Asymptotic theory and iterative computational procedures for these estimates are given. A likelihoodratio test for the rank of B₀ is seen to be an extension of a familiar test for canonical correlations.     

Cash management in a randomly varying environment

We study a cash management system, in which the distribution of the cash flow X_n in period n=1,2,… depends on the state I_n of a randomly varying environment. Sufficient conditions are found for the optimality of a simple transfer rule, generalizing and partially improving the well-known results for the classical case with i.i.d. cash flows. These and further structural results obtained for the cash balance are shown to reduce the computational effort drastically in determining an optimal transfer rule. In addition, structural and computational results w.r.t. the environment are derived. Finally, some examples are given for economic and statistical environments and their interactions with the cash flow process.     

Al2O3-47 nm and Al2O3-36 nm characterizations of nonlinear differential equations for biomedical applications: Magnetized peristaltic transport

Current research models the Al₂O₃ 47nm and Al₂O₃ 36nm nanoparticles transportation through peristalsis with entropy optimization. Conservation laws for mass, momentum and energy are used to model the present flow situation. These equations elaborates the magnetohydrodynamics, Hall, thermal radiation, Joule heating, heat generation and absorption. Convective heat transfer impacts are studied at channel walls. Entropy is modeled in view of thermodynamics second law. Two different expressions for effective viscosity are accounted. Simplification of the modeled equations is done through lubrication assumptions. Solution for momentum equation is obtained analytically and for numerically for temperature equation. Built-in shooting procedure is utilized to obtain the desired numerical results. Later on these obtained results are used to sketch and discussed the flow quantities of interest for the influential parameters accounted in the problem.     

Time shifted aliasing error upper bounds for truncated sampling cardinal series

Time shifted aliasing error upper bound extremals for the sampling reconstruction procedure are fully characterized. Sharp upper bounds are found on the aliasing error of truncated cardinal series and the corresponding extremals are described for entire functions from certain specific Lp, p>1, classes. Analogous results are obtained in multidimensional regular sampling. Truncation error analysis is provided in all cases considered. Moreover, sharpness of bounding inequalities, convergence rates and various sufficient conditions are discussed.     

On optimality conditions for structured families of nonlinear programming problems

Optimality conditions for families of nonlinear programming problems inR ⁿ are studied from a generic point of view. The objective function and some of the constraints are assumed to depend on a parameter, while others are held fixed. Techniques of differential topology are used to show that under suitable conditions, certain strong second-order conditions are necessary for optimality except possibly for parameter values lying in a negligible set.Research sponsored, in part, by the Air Force Office of Scientific Research, under grants number 77-3204 and 79-0120.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.