A novel algorithm for evaluating cement azimuthal density based on perturbation theory in horizontal well

Cement density monitoring plays a vital role in evaluating the quality of cementing projects,which is of great significance to the development of oil and gas.However,the presence of inhomogeneous cement distribution and casing eccentricity in horizontal wells often complicates the accurate evaluation of cement azimuthal density.In this regard,this paper proposes an algorithm to calculate the cement azimuthal density in horizontal wells using a multi-detector gamma-ray detection system.The spatial dynamic response functions are simulated to obtain the influence of cement density on gamma-ray counts by the perturbation theory,and the contribution of cement density in six sectors to the gamma-ray recorded by different detectors is obtained by integrating the spatial dynamic response functions.Combined with the relationship between gamma-ray counts and cement density,a multi-parameter calculation equation system is established,and the regularized Newton iteration method is employed to invert casing eccentricity and cement azimuthal density.This approach ensures the stability of the inversion process while simultaneously achieving an accuracy of 0.05 g/cm^(3) for the cement azimuthal density.This accuracy level is ten times higher compared to density accuracy calculated using calibration equations.Overall,this algorithm enhances the accuracy of cement azimuthal density evaluation,provides valuable technical support for the monitoring of cement azimuthal density in the oil and gas industry.

Perturbation Theory of Fractional Lagrangian System and Fractional Birkhoffian System

Perturbation to symmetry and adiabatic invariants are studied for the fractional Lagrangian system and the fractional Birkhoffian system in the sense of Riemann-Liouville derivatives.Firstly,the fractional Euler-Lagrange equation,the fractional Birkhoff equations as well as the fractional conservation laws for the two systems are listed.Secondly,the definition of adiabatic invariant for fractional mechanical system is given,then perturbation to symmetry and adiabatic invariants are established for the fractional Lagrangian system and the fractional Birkhoffian system under the special and general infinitesimal transformations,respectively.Finally,two examples are devoted to illustrate the results.

On solitary waves.Part 2 A unified perturbation theory for higher-order waves

A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on （i） the accuracy of the theoretically predicted wave properties and （ii） the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property （e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy）, on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ （n） （the relative mean-square difference between successive orders n of wave elevations） reaching a minimum, δm at a specific order, n = n both depending on the base adopted, e.g. nm,α= 11-12 based on parameter α （wave amplitude）, nm,δ = 15 on δ （amplitude-speed square ratio）, and nm.ε= 17 on ε （ wave number squared）. The asymptotic range is brought to completion by the highest order of n = 18 reached in this work.

Generalization of iterative perturbation theory and coherent potential approximation (ITP+CPA) to double exchange model with orbital degeneracy

A combination of the iterative perturbation theory （ITP） of the dynamical mean field theory （DMFT） and coherentpotential approximation （CPA） is generalized to the double exchange model with orbital degeneracy. The Hubbard interaction and the off-diagonal components for the hopping matrix tij^mn（m ≠ n） are considered in our calculation of spectrum and optical conductivity. The numerical results show that the effects of the non-diagonal hopping matrix elements are important.

Derivation of Martin-Hou Equation of State from Hard-particle Perturbation Theory

In this paper, the Martin-Hou equation of state is derived by using a power series representation of radial distribution function and an analytic representation of multi-section potential based on the Barker-Henderson hard-particle perturbation theory including high-order terms. In the derivation, a theoretical form of Martin-Hou equation was obtained. It had a similar form and the same capability to predict P-V-T properties as the Martin-Hou equation and no additional data were required for evaluating the constants. The characteristic constants of the theoretical expression have certain relationships with the molecular parameters.

Thermoelastic Analysis of Non-uniform Pressurized Functionally Graded Cylinder with Variable Thickness Using First Order Shear Deformation Theory(FSDT) and Perturbation Method

Recently application of functionally graded materials（FGMs） have attracted a great deal of interest. These materials are composed of various materials with different micro-structures which can vary spatially in FGMs. Such composites with varying thickness and non-uniform pressure can be used in the aerospace engineering. Therefore, analysis of such composite is of high importance in engineering problems. Thermoelastic analysis of functionally graded cylinder with variable thickness under non-uniform pressure is considered. First order shear deformation theory and total potential energy approach is applied to obtain the governing equations of non-homogeneous cylinder. Considering the inner and outer solutions, perturbation series are applied to solve the governing equations. Outer solution for out of boundaries and more sensitive variable in inner solution at the boundaries are considered. Combining of inner and outer solution for near and far points from boundaries leads to high accurate displacement field distribution. The main aim of this paper is to show the capability of matched asymptotic solution for different non-homogeneous cylinders with different shapes and different non-uniform pressures. The results can be used to design the optimum thickness of the cylinder and also some properties such as high temperature residence by applying non-homogeneous material.

Practical φ_0-stability of stochastic differential equations and corresponding stochastic perturbation theory

The notions of practical φ0-stability were introduced for stochastic differential equations. Sufficient conditions on such practical properties were obtained by using the comparison principle and the cone-valued Lyapunov function methods. Based on an extended comparison theorem, a perturbation theory of stochastic differential systems was given.

Perturbation Theory Combined with Boundary Element Method for Analysis of Gear Contact Problems

This paper combines the perturbation theory with the boundary element methodfor contact problems of three-dimensional elasticity mechanism to analyse the effect oferrors on the shape of the contact area and pressure distribution in gear drive through theperturbation of a cubic order geometry,there by greatly bringing down both computationwork volume and cost and providing a powerful tool for engineering study on the effectof errors on structural strength.

Characteristics of spectral-hole burning in Tm^(3+):YAG based on the perturbation theory

In this paper, the physical mechanism of the interaction between electromagnetic wave and spectral-hole burning crystal material is investigated in detail. In the small signal regime, a perturbation theory model is used to analyze the mechanism of spectral-hole burning. By solving the Liouville equation, three-order perturbation results are obtained. From the theoretic analysis, spectral-hole burning can be interpreted as a photon echo of the zero-order diffraction echo when the first optical pulse and the second optical pulse are overlapped in time. According to the model, the spectral-hole width is dependent on the chirp rate of the reading laser. When the chirp rate is slow with respect to the spectral features of interest, the spectral hole is closely mapped into time domain. For a fast chirp rate, distortions are observed. The results follow Maxwell-Bloch model and they are also in good agreement with the experimental results.

Study on lattice vibrational properties and Raman spectra of Bi_2Te_3 based on density-functional perturbation theory

We present a variational density-functional perturbation theory （DFPT） to investigate the lattice dynamics and vibra- tional properties of single crystal bismuth telluride material. The phonon dispersion curves and phonon density of states （DOS） of the material were obtained. The phonon dispersions are divided into two fields by a phonon gap. In the lower field, atomic vibrations of both Bi and Te contribute to the DOS. In the higher field, most contributions come from Te atoms. The calculated Born effective charges and dielectric constants reveal a great anisotropy in the crystal. The largest Born effective charge generates a significant dynamic charge transferring along the c axis. By DFPT calculation, the greatest LO-TO splitting takes place in the infrared phonon modes and reaches 1.7 THz in the Brillouin zone center. The Raman spectra and peaks corresponding to respective atomic vibration modes were found to be in good agreement with the experimental data.

A Power Allocation Scheme Using Updated SLNR Value Based on Perturbation Theory

The performance of downlink multiple-input multiple-output (MIMO) cellular networks is limited by co-channel interference (CCI). In this paper, we propose a linear precoding scheme based on signal-to-leakage-and-noise ratio (SLNR) criteria which can reduce the CCI significantly. Since each user’s SLNR value is corresponding to the largest eigenvalue of the generalized matrix which indicates the channel quality that we propose a scheme to do a dynamic power allocation as an auxiliary way to improve SLNR precoding scheme. We use the perturbation theory to update each user’s SLNR value each time step in time-varying channels rather than directly decompose the channel matrix so as to reduce the amount of calculation. The simulation results show that the proposed scheme offers about 0.3 bps/Hz additional capacity gain and 0.5 dB BER gain over conventional SLNR precoding method with lower computational complexity. And it also obtains about 0.5 bps/Hz additional capacity gain and 1 dB BER gain compared to the scheme only update the preceding vectors.

Molecular Application of a State Specific Multi-Reference Brillouin-Wigner Perturbation Theory

The single reference second order Brillouin-Wigner perturbation theory recently developed, which eliminates its size-extensivity error, has been generalized to state-specific, multi-reference （SS-MR）, BWPT2 providing a size-extensive correction to the electron correlation problem for systems that demand the use of a multi-reference function. Illustrative numerical tests of the size-extensivity corrections are made for widely used molecules in their ground states, which are pronounced multi-reference characteristics. We have implemented two-reference and three-reference cases for CH2, BH and bond breaking process in the ground states of HF molecules. The results are compared with the rigorously size-extensive methods such as the M^ller-Plesset perturbation theory, i.e., MP2, full configuration interaction （Full-CI） and allied methods using the same basis sets.

Three-Dimensional Static Analysis of Nanoplates and Graphene Sheets by Using Eringen’s Nonlocal Elasticity Theory and the Perturbation Method

A three-dimensional(3D)asymptotic theory is reformulated for the static analysis of simply-supported,isotropic and orthotropic single-layered nanoplates and graphene sheets(GSs),in which Eringen’s nonlocal elasticity theory is used to capture the small length scale effect on the static behaviors of these.The perturbation method is used to expand the 3D nonlocal elasticity problems as a series of two-dimensional(2D)nonlocal plate problems,the governing equations of which for various order problems retain the same differential operators as those of the nonlocal classical plate theory(CST),although with different nonhomogeneous terms.Expanding the primary field variables of each order as the double Fourier series functions in the in-plane directions,we can obtain the Navier solutions of the leading-order problem,and the higher-order modifications can then be determined in a hierarchic and consistent manner.Some benchmark solutions for the static analysis of isotropic and orthotropic nanoplates and GSs subjected to sinusoidally and uniformly distributed loads are given to demonstrate the performance of the 3D nonlocal asymptotic theory.

Comparison of the Linear Sigma Model and Chiral Perturbation Theory for Nucleon Properties

We compare the static nucleon properties in the Chiral Perturbation Theory (χPT) and the Linear Sigma Model (LSM). We consider a chiral model for the nucleon which is based on the linear sigma model with scalar-isoscalar and scalarisovector mesons coupled to quarks. We have solved the field equations in the mean field approximation for the hedgehog baryon state with different sets of model parameters. A good investigation of some static nucleon properties is obtained by the LSM.

Discussion on the Energy Parallax and the Relationship to Perturbation Theory in Mathematical Physics

This work is a discussion on the energy parallax theory developed in [1] [2] based on the multiplicity of the solutions theorem. This theory is compared with the perturbation theory in mathematical physics. The perturbation theory uses the increment of a solution which can be formalized with a Taylor series development. With the energy parallax theory, the convergence property of the Taylor series of the energy of a system is the key to decide to include additional solutions, defined on the so-called energy spaces [2]. The development is supported using various examples in quantum mechanics (i.e. Rayleigh-Schrödinger perturbation theory) and wave theory with the Electromagnetic (EM) energy density (i.e. evanescent waves within the skin layer of a dielectric material). Finally, we discuss the Woodward effect [3] and the application of the energy parallax when assuming that the variations of EM energy density can trigger such effect within asymmetric cavities.

Perturbation Theory for Open Two-Level Nonlinear Quantum Systems

Perturbation theory is an important tool in quantum mechanics. In this paper, we extend the traditional perturbation theory to open nonlinear two-level systems, treating decoherence parameter 7 as a perturbation. By this virtue, we give a perturbative solution to the master equation, which describes a nonlinear open quantum system. The results show that for small decoherence rate 7, the ratio of the nonlinear rate C to the tunneling coefficient V (i.e., r=C/V) determines the validity of the perturbation theory. For small ratio r, the perturbation theory is valid, otherwise it yields wrong results.

Circular Scale of Time as a Guide of the Schrödinger’s Perturbation Theory

The paper is a kind of a review which considers an investigation of the scale of time suggested by an application of the Schrödinger perturbation method, especially when the perturbation of a non-degenerate quantum state is examined. In fact the method was applied in numerous cases—also by Schrödinger himself—without any use of the notion of time. Simultaneously, because of the development of computers, their use in solving the perturbation problems gradually decreased. However, the point of importance in the paper became the time. We demonstrate that collisions of a quantum system with the perturbation potential can be arranged along a circular scale of time whose properties provide us precisely with the energy terms obtained by the Schrödinger perturbation theory. This validity of results is checked till the perturbation order N = 7.

Masses and Widths of Light Scalar Mesons in Chiral Perturbation Theory

基于 diquark 模型，我们假设轻分级的介子是 q 22 状态而非 q。chiral 为轻分级的介子的有效 Lagrangian 被构造，并且集体关系被获得：isotriplet (0) 和 isosinglet (f 0) 是最重并且是退化的， isodoublets (κ ) 更重，另外的 isosinglet (σ ) 最轻；并且 2M κ2= M a0 2+ M σ2 。用为 0 和 σ 质量的试验性的价值，我们获得 M κ= 794 兆电子伏，它与试验性的价值一致。然后拿 Γ (00→ ηπ0)= 90 兆电子伏和 Γ (f 00→ π 0π0)= 20 兆电子伏，我们得到 σ 的宽度：Γ (σ → π +π-)= 150 兆电子伏。