General Operator Form Research Articles

3-D Numerical Mode Matching Method for Off- Centered Electromagnetic Well Logging Tools in Noncircular Vertical Borehole and Invasion Zones in Multilayered Media

In electromagnetic (EM) well logging for petroleum exploration, off-centered tools within a noncircular borehole and invasion zones in multilayered media represent a challenging 3-D problem for traditional numerical methods. The 3-D finite-element numerical mode-matching (FNMM) method and the spectral numerical mode-matching (SNMM) method are developed to address this problem in this work. The numerical mode-matching (NMM) methods reduce the original 3-D well logging problem into a series of 2-D open waveguide eigenvalue problems plus a 1-D layered medium problem, which can be analytically solved by a recursion procedure. These waveguide eigenvalue problems with anisotropic inhomogeneous media in the horizontal directions are solved numerically by the mixed finite-element method (MFEM) with the flexibility for the complex geometry and the mixed spectral element method (MSEM) with the exponential convergence. A general operator form of Maxwell’s equations is also derived to obtain the source excitation vector independent of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> -axis. Therefore, the NMM solvers can effectively and accurately simulate the response of off-centered EM well logging tools in an irregular borehole in complex formations. The NMM methods are applied to simulate three formations including a five-layer isotropic homogeneous formation, an off-centered logging tool in a noncircular borehole with anisotropic invasion zones in a five-layer anisotropic formation, and the Oklahoma formation. Numerical experiments indicate that the NMM methods are highly efficient and accurate for the EM well logging models.

On The Theory of Field Pattern Synthesis.(Dept.E)

Uniform array synthesis is treated as a problem of approximation in the mean square sense and in the Sobolev sense are considered. Following Tichonov's regularization technique for solving inverse problems , the array excitation current is subjected to a certain constraint in order to make the design less-sensitive to construction errors. A variational formulation of the problem using lagrange multiplier method is presented. The equation determining the excitation current is obtained in a general operator form and is then put in a form amenable to numerical computation. Existence and uniqueness of solution are proved. Some example problems are presented.

General operator form of the non-local three-nucleon force

.This paper describes a procedure to obtain the general form of the three-nucleon force. The result is an operator form where the momentum space matrix element of the three-nucleon potential is written as a linear combination of 320 isospin-spin-momentum operators and scalar functions of momenta. Any spatial and isospin rotation invariant three-nucleon force can be written in this way and in order for the potential to be Hermitian, symmetric under parity inversion, time reversal and particle exchange, the scalar functions must have definite transformation properties under these discrete operations. A complete list of the isospin-spin-momentum operators and scalar function transformation properties is given.

The general operator form for the total-momentum-dependent nucleon-nucleon potential

In this paper we describe a procedure to obtain the general operator form of two-nucleon (2N) potentials and apply it to the case of the 2N potential that has an additional dependence on the total momentum of the system. This violates Galilean invariance but terms including the total momentum appear in some relativistic approaches. In operator form, the potential is expressed as a linear combination of a fixed number of known spin-momentum operators and scalar functions of momenta. Since the scalar functions effectively define the potentials, using the operator form significantly reduces the number of parameters that are needed in numerical implementations. The proposed operator form explicitly obeys the usual symmetries of rotational invariance, particle exchange, time reflection and parity.

Deuteron Disintegration in Three Dimensions

We compare results from the traditional partial wave treatment of deuteron electro-disintegration with a new approach that uses three dimensional formalism. The new framework for the two-nucleon (2N) system using a complete set of isospin - spin states made it possible to construct simple implementations that employ a very general operator form of the current operator and 2N states.

Component-wise positivity of solutions to periodic boundary problem for linear functional differential system

The classical Wazewski theorem claims that the condition p ij ≤ 0, j ≠ i, i, j =1,...,n, is necessary and sufficient for non-negativity of all the components of solution vector to a system of the inequalities , x i (0) ≥ 0, i =1, ..., n. Although this result was extent on various boundary value problems and on delay differential systems, analogs of these heavy restrictions on non-diagonal coefficients p ij preserve in all assertions of this sort. It is clear from formulas of the integral representation of the general solution that these theorems claim actually the positivity of all elements of Green's matrix. The method to compare only one component of the solution vector, which does not require such heavy restrictions, is proposed in this article. Note that comparison of only one component of the solution vector means the positivity of elements in a corresponding row of Green's matrix. Necessary and sufficient conditions of this fact are obtained in the form of theorems about differential inequalities. It is demonstrated that the sufficient conditions of positivity of the elements in the nth row of Green's matrix, proposed in this article, cannot be improved in corresponding cases. The main idea of our approach is to construct a first order functional differential equation for the n th component of the solution vector and then to use assertions, obtained recently for first order scalar functional differential equations. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. Note that in some cases the sufficient conditions, obtained in the article, does not require any smallness of the interval [0, ω], where the system is considered. Mathematics Subject Classification 2000: 34K06; 34K10.

Differential Inequalities for One Component of Solution Vector for Systems of Linear Functional Differential Equations

The method to compare only one component of the solution vector of linear functional differential systems, which does not require heavy sign restrictions on their coefficients, is proposed in this paper. Necessary and sufficient conditions of the positivity of elements in a corresponding row of Green's matrix are obtained in the form of theorems about differential inequalities. The main idea of our approach is to construct a first order functional differential equation for the th component of the solution vector and then to use assertions about positivity of its Green's functions. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. It should be also noted that the sufficient conditions, obtained in this paper, cannot be improved in a corresponding sense and does not require any smallness of the interval , where the system is considered.

On the exact solutions for initial value problems of second-order differential equations

In this paper, the solutions of initial value problems for a class of second-order linear differential equations are obtained in the exact form by writing the equations in the general operator form and finding an inverse differential operator for this general operator form.

Unified Description of Tokamak Ideal MHD Instabilities (I)

By using a coordinate system associated with magnetic surfaces, a unified eigen-mode equation for describing the tokamak ideal MHD instabilities is derived in the shear-Alfven approximation. Based on this equation having a general operator form, the eigen-mode equation governing the large-scale perturbation (such as the kink mode, the low-n ballooning mode and the Alfven mode) and small-scale perturbation (such as the high-n ballooning mode, the local mode) can be further deduced. In the first part of the present study, the small-scale perturbation is discussed in detail.

Two efficient iteration methods for solving inhomogeneous problems of neutron transport theory

This paper concerns the solution of formally inhomogeneous neutron-physical problems in multiplying media, including in nuclear reactors with neutron illumination, for keff close to one. The linear neutron-balance equation with an independent right-hand side is studied in a general operator form. Two unconventional methods of constructing relatively rapidly converging iterations are proposed for calculating the neutron distribution function. A characteristic feature of the iteration algorithms described is that the condition keff<1 need not be satisfied directly in order for the iterations to converge. An analytic proof of this assertion is given. For keff=1, the computational procedure remains meaningful and gives a result that follows from the well-known Fredholm alternative.

Generalized sensitivity analysis of ergodic stochastic systems

The tools for sensitivity analysis of stochastic systems whose behavior is described by ergodic stationary processes are presented. Three kinds of generalized estimates for performance sensitivities of these systems are introduced. These estimates, based on a use of time averages, are given in general operator form. Their convergence conditions and rate of convergence are obtained. It is shown that perturbation analysis estimates are generally preferable to crude Monte Carlo estimates.

Generalized estimates for performance sensitivities of stochastic systems

Three kinds of estimates for performance sensitivities (gradients, Hessians etc.) of stochastic systems are introduced. These estimates are given in general operator form. Their convergence conditions and rate of convergence are presented. Particular attention is given to estimates obtained from a single sample path. Various examples of estimates are considered.

Group theory applied to the hydrogen atom in a strong magnetic field. Derivation of the effective diamagnetic Hamiltonian

The authors present a formalism based on the non-invariance algebra for the Coulomb problems that allows one to deduce an effective Hamiltonian for a wide variety of perturbing potentials. Applications to the problem of the hydrogen atom in magnetic field are performed. They especially derive the exact first- and second-order expressions of the effective diamagnetic Hamiltonian under a general operator form. Some of the consequences and further developments are briefly indicated.

Numerical solution of nonlinear equations by one-parameter imbedding methods

The aim of the paper is a systematic study of one-parameter imbedding methods applied to systems of nonlinear algebraic equations. The algorithms considered are classified as one-loop and multi-loop, respectively. The paper presents several multi-loop algorithms designed for solving equations in a general operator form. Some sufficient conditions for the construction of suitable algorithms with respect to the convergence are given in the paper for the case of algebraic systems, including a detailed discussion of some relations between one-step iterative and imbedding methods. The effectiveness of various imbedding algorithms is tested on a set of problems. It is shown that some of the imbedding methods can be good alternative othe classical Newton's method whilst the majority of methods considered did not give an improvement of the rate of convergence. An advantage of the imbedding methods may be their possib le broadening of the domain of attraction of some solutions.

Thermal conductivity of insulating crystals in the presence of normal processes

The problem of computing the phonon distribution and thermal conductivity in the presence of N processes is examined in the framework of the Boltzmann-Peierls equation. A heat flow may be associated with drift and diffusive distributions respectively; this intuitive concept is given specific mathematical basis as a natural method to solve the Boltzmann equation. A general operator form of this equation is used and the solution is expressed as the sum of an eigenstate of the N-process collision operator having zero eigenvalue plus a second term; the former is a drift term and the latter is diffusive. The amount of drift is determined by making the drift term orthogonal to the diffusive term; this is equivalent to a phonon momentum balance condition. As a special case, the general method may be approximated by relaxation time representation of the collision operators. The Callaway equation follows directly.