APPROXIMATION OF GENERALIZED BI－AXIALLY SYMMETRIC POTENTIALS WITH FAST RATES OF GROWTH

The paper deals with growth and approximation of solutions (not necessarily entire) of certain elliptic partial differential equations. These solutions are called Generalized Bi-axially Symmetric Potentials (GBSP's). The GBSP's are taken to be regular in a finite hyperball and influence of the growth of their maximum moduli on the rate of decay of their approximation errors in sup norm is studied. The authors obtain the characterizations of the q-type and lower q-type of a GBSP H ∈ HP,0 < R < ∞, in terms of rate of decay of approximation error E.(H,R0), 0 < R0<R <∞.

Generalized Available Potential Energy

The kinetic energy generation in either the dry or moist atmosphere may be estimated by the same relationships if we introduce the new concept of generalized available potential energy. The largest magnitude of generalized available potential energy and corresponding reference state of either dry or moist atmosphere are calculated in terms of the mitial conditions and entropy variation of the atmosphere. The obtained relationships are applicable for the statically unstable atmosphere as well. The generalized available potential energy associated with reversible processes reaches the maximum with respect to same initial state. While the generation of kinetic energy in irreversible processes is characterized by sudden changes. When the reference state is assumed to be saturated, we may predict the final temperature and moisture fields corresponding to provided initial state and entropy variation.

Potential symmetries and conservation laws for generalized quasilinear hyperbolic equations

Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physi- cally interesting cases are found by using the partial Lagrangian approach.

The scattering states of the generalized Hulthén potential with an improved new approximate scheme for the centrifugal term

This paper finds the approximate analytical scattering state solutions of the arbitrary 1-wave Schrodinger equation for the generalized Hulthen potential by taking an improved new approximate scheme for the centrifugal term. The normalized analytical radial wave functions of the 1-wave SchrSdinger equation for the generalized Hulthen potential are presented and the corresponding calculation formula of phase shifts is derived. Some useful figures are plotted to show the improved accuracy of the obtained results and two special cases for the standard Hulthen potential and Woods-Saxon potential are also studied briefly.

A Brief Report of Graphic Explanations for Generalized Potential Temperature in the Non-Uniformly Saturated Atmosphere

Compared to potential temperature （θ） in the dry atmosphere and equivalent potential temperature （θc） in the saturated atmosphere, generalized potential tem- perature （θ＂） has already proven a better thermodynamic parameter in describing the non-uniformly saturated real atmosphere. To add otherwise absent graphic explanations, this paper first presents the physical definition of θ through a tephigram. Then, the utility of the measurement in identifying and forecasting the locations of precipita- tion maxima and heat wave areas with diagnostic com- parison studies and traditionally used thermodynamic parameters is shown.

Bound State Solutions of Schrodinger Equation for Generalized Morse Potential with Position-Dependent Mass

The effective mass one-dimensional Schroedinger equation for the generalized Morse potential is solved by using Nikiforov-Uvarov method. Energy eigenvalues and corresponding eigenfunctions are computed analytically. The results are also reduced to the constant mass case. Energy eigenvalues are computed numerically for some diatomic molecules. They are in agreement with the ones obtained before.

High-precision downward continuation of potential fi elds algorithm utilizing adaptive damping coeffi cient of generalized minimal residuals

The downward continuation of potential fields is a process of calculating their values in a lower plane based on those of a certain plane.This technology is not only a data processing method for resource exploration but also plays an extremely important role in military applications.However,the downward continuation of potential fields is a typical linear inverse problem that is ill-posed.Generalized minimal residuals(GMRES)is an eff ective solution to ill-posed inverse problems,but it is unstable under the condition wherein the GMRES is directly applied in the calculation process.Moreover,the long-term behavior of its iterative computation is a disordered,divergent result.Therefore,to obtain stable solutions,GMRES is applied to solve the normal equations of the downward continuation of potential fields;it is also used to prequalify for occasional interruptions in the operation process by adding the damping coefficient,thus strengthening the stability conditions of the equations of residual minimization.Finally,the stable downward continuation of the potential fields method is proposed.As indicated by the theoretical data and the measured testing data,the method proposed in this paper has the advantages of high-precision and excellent stability.Furthermore,compared with the Tikhonov iteration method,the proposed method avoids the need to choose regularization parameters.

Numerical verification of similar Cam-clay model based on generalized potential theory

From the mathematical principles, the generalized potential theory can be employed to create constitutive model of geomaterial directly. The similar Cam-clay model, which is created based on the generalized potential theory, has less assumptions,clearer mathematical basis, and better computational accuracy. Theoretically, it is more scientific than the traditional Cam-clay models. The particle flow code PFC3 D was used to make numerical tests to verify the rationality and practicality of the similar Cam-clay model. The verification process was as follows: 1) creating the soil sample for numerical test in PFC3 D, and then simulating the conventional triaxial compression test, isotropic compression test, and isotropic unloading test by PFC3D; 2)determining the parameters of the similar Cam-clay model from the results of above tests; 3) predicting the sample's behavior in triaxial tests under different stress paths by the similar Cam-clay model, and comparing the predicting results with predictions by the Cam-clay model and the modified Cam-clay model. The analysis results show that the similar Cam-clay model has relatively high prediction accuracy, as well as good practical value.

Characterization of event-related potentials in obsessive compulsive disorder patients: Comparison with depression and generalized anxiety disorder patients

BACKGROUND： Study results of event-refated potential in obsessive compulsive disorder （OCD） remain controversial, potentially as a result of different instruments utilized and their differing technical characteristics. OBJECTIVE： To investigate the differences in several common event-related potentials, Le. contingent negative variations, P300, and mismatch negativity （MMN）, in OCD patients, depression patients, generalized anxiety disorder （GAD） patients, and healthy controls. DESIGN, TIME AND SETTING： A case-control study was performed in the Department of Electrophysiology, Shanghai Mental Health Center from May 2002 to December 2005. PARTICIPANTS： A total of 38 OCD patients, 20 depression patients, and 18 GAD patients, who were diagnosed according to the criteria of Chinese Classification of Mental Disorders （Version 3）, formulated by the Chinese Psychiatry Association, were selected from the Outpatient Department of Shanghai Mental Health Center. Patients with two or more the above diseases were excluded. In addition, 28 healthy people, gender and age matched, were selected as controls. METHODS： Contingent negative variations, P300, and MMN were recorded by a Nicolet Spirit Instrument. All electrodes were attached at Cz according to the Intemationa11020 system, with the mastoid leads as reference and Fpz as ground. MAIN OUTCOME MEASURES： Amplitude and latency of contingent negative variations, P300, and MMN. RESULTS： The contingent negative variations, P300, and MMN were different （P 〈 0.01）. OCD patients showed an increased MI amplitude compared with controls, depression, and GAD patients （P 〈 0.01）. Target P300 amplitudes were significantly lower in OCD, depression, and GAD patients compared with controls （P 〈 0.01）. Moreover, N2 latency and latency of MMN were prolonged in OCD and depression groups compared with controls （P 〈 0.05）. CONCLUSION： Event-related potentials were different in depression, GAD, and OCD patients and healthy controls. In particular, OCD patients exhibited unique characteristics.

Travelling Wave in the Generalized Kuramoto Model with Inertia

We study the dynamics of the generalized Kuramoto model with inertia, in which oscillators with positive cou- pling strength are conformists and oscillators with negative coupling strength are contrarians. By numerically simulating the model, we find that the model supports a modulated travelling wave state except for already displayed travelling wave states and π state in previous be characterized by the phase distributions of oscillators. travelling wave state of the model in the parameter space literature. The modulated travelling wave state may Finally, the modulated travelling wave state and the are presented.

Three types of generalized Kadomtsev-Petviashvili equations arising from baroclinic potential vorticity equation

By means of the reductive perturbation method, three types of generalized （2＋l）-dimensional Kadomtsev- Petviashvili （KP） equations are derived from the baroclinic potential vorticity （BPV） equation, including the modified KP （mKP） equation, standard KP equation and cylindrical KP （cKP） equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.

Groundstate with Zero Eigenvalue for Generalized Sombrero-Shaped Potential in N-Dimensional Space

Based on an iterative method for solving the groundstate of Schrodinger equation, it is found that a kind of generalized Sombrero-shaped potentials in N-dimensional space has groundstates with zero eigenvalue. The restrictions on the parameters in the potential are discussed.

A Comparative study on generalized Manev potential and Newtonian potential in perturbed restricted four-body problem

The motion of a test particle within the context of the restricted four-body problem(R4BP)driven by a new kind of potential,called the generalized Manev potential,with perturbations in the Coriolis and centrifugal forces is considered in this study.The system possesses eight libration points which were distributed on its plane of motion in different manner from those of the usual Newtonian potential.Unlike the case of the perturbed R4BP under Newtonian potential,where two of these librations are stable,all of them are unstable in linear sense under Manev potential.We found that a gradual perturbation in the centrifugal force causes the trajectories of motion to drift inward but the Coriolis force was proven to have no effect on the location of the libration points of the system.Using first order Lyapunov characteristic exponents,the dynamical behavior of the system is found irregular.We experimented with a high velocity stellar system(82 G.Eridani)to establish the applicability of the model in astrophysics.

Theorem of Necessity and Sufficiency of Stable Equilibrium for Generalized Potential Equality between System and Reservoir

The literature reports that equality of temperature, equality of potential and equality of pressure between a system and a reservoir are necessary conditions for the stable equilibrium of the system-reservoir composite or, in the opposite and equivalent logical inference, that stable equilibrium is a sufficient condition for equality. The aim and the first novelty of the present study is to prove that equality of temperature, potential and pressure is also a sufficient condition for stable equilibrium, in addition to necessity, implying that stable equilibrium is a condition also necessary, in addition to sufficiency, for equality. The second novelty is that the proof of the sufficiency of equality (or the necessity of stable equilibrium) is attained by means of the generalization of the entropy property, derived from the generalization of exergy property, which is used to demonstrate that stable equilibrium is a logical consequence of equality of generalized potential. This proof is underpinned by the Second Law statement and the Maximum-Entropy Principle based on generalized entropy which depends on temperature, potential and pressure of the reservoir. The conclusion, based on these two novel concepts, consists of the theorem of necessity and sufficiency of stable equilibrium for equality of generalized potentials within a composite constituted by a system and a reservoir.

Exact Computation of Parallel Robot's Generalized Inertia Matrix

According to the definition of the new hypothetical states which have obvious physical significance and are termed as no-gravity static and accelerated states, a method for exact computation of the parallel robot＇s generalized inertia matrix is presented. Based on the matrix theory, the generalized inertia matrix of the parallel robot can be computed on the assumption that the robot is in these new hypothetical states respectively. The approach is demonstrated by the Delta robot as an example. Based on the principle of the virtual work, the inverse dynamics model of the robot is formulized after the kinematics analysis. Finally, a numerical example is given and the element distribution of the Delta robot＇s inertia matrix in the workspace is studied. The method has computationa＇, advantage of numerical accuracy for the Delta robot and can be parallelized easily.

Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential

A bound state solution is a quantum state solution of a particle subjected to a potential such that the particle＇s energy is less than the potential at both negative and positive infinity. The particle＇s energy may also be negative as the potential approaches zero at infinity. It is characterized by the discretized eigenvalues and eigenfunctions, which contain all the necessary information regarding the quantum systems under consideration. The bound state problems need to be extended using a more precise method and approximation scheme. This study focuses on the non-relativistic bound state solutions to the generalized inverse quadratic Yukawa potential. The expression for the non-relativistic energy eigenvalues and radial eigenfunctions are derived using proper quantization rule and formula method, respectively. The results reveal that both the ground and first excited energy eigenvalues depend largely on the angular momentum numbers, screening parameters, reduced mass, and the potential depth. The energy eigenvalues, angular momentum numbers, screening parameters, reduced mass, and the potential depth or potential coupling strength determine the nature of bound state of quantum particles. The explored model is also suitable for explaining both the bound and continuum states of quantum systems.

A Study of Superconducting La₂CuO₄via Generalized BCS Equations Incorporating Chemical Potential

We address the Tc (s) and multiple gaps of La2CuO4 (LCO) via generalized BCS equations incorporating chemical potential. Appealing to the structure of the unit cell of LCO, which comprises sub- lattices with LaO and OLa layers and brings into play two Debye temperatures, the concept of itinerancy of electrons, and an insight provided by Tacon et al.’s recent experimental work concerned with YBa2Cu3O6.6 which reveals that very large electron-phonon coupling can occur in a very narrow region of phonon wavelengths, we are enabled to account for all values of its gap-to-Tc ratio (2Δ0/kBTc), i.e., 4.3, 7.1, ≈8 and 9.3, which were reported by Bednorz and Müller in their Nobel lecture. Our study predicts carrier concentrations corresponding to these gap values to lie in the range 1.3 × 1021 - 5.6 × 1021 cm-3, and values of 0.27 - 0.29 and 1.12 for the gap-to-Tc ratios of the smaller gaps.

Relativistic Study of Spinless Particles for Generalized Hylleraas Potential with Position Dependent Mass

The relativistic study of spinless particles under a special case of equal scalar and vector generalized Hylleraas potential with position dependent mass has been studied. The energy eigenvalues and the corresponding wave functions expressed in terms ofa Jacobi polynomial are obtained using the parametric generalization of NU （Nikiforo-Uvarov） method. In obtaining the solutions for this system, we have used an approximation scheme to evaluate the centrifugal term （potential barrier）. To test the accuracy of the result, we compared the approximation scheme with the centrifugal term and the result shows a good agreement with the centrifugal term for a short-range potential. The results obtained in this work would have many applications in semiconductor quantum well structures, quantum dots, quantum liquids. Under limiting cases, the results could be used to study the binding energy and interaction of some diatomic molecules which is of great applications in nuclear physics, atomic and molecular physics and other related areas. We have also discussed few special cases of generalized Hylleraas potential such as Rosen-Morse, Woods-Saxon and Hulthen potentials.

Relativistic Treatment of Massive Klein-Gordon Particle by Modified Generalized Hulthen Potential

Approximate bound state solutions of spinless particles with a special case of equal scalar and vector modified generalized Hulthen potential has been obtained under the massive Klein-Gordon equation. The energy eigenvalues and the corresponding wave functions expressed in terms of a Jacobi polynomial are also obtained using the parametric generalization of the Nikiforov-Uvarov （NU） method. Under limiting cases our result are in agreement with the existing literature. Our results could be used to study the interactions and binding energies of the central potential for diatomic molecules in the relativistic framework which have many applications in physics and some others related disciplines.

Inertial Entropy and External Validation of Clusterings

Axiomatization of Shannon entropy is a subject that has received lots of attention in the information theory literature.While Shannon entropy is defined on probability distribution,we define a new type of entropy on the set of partitions of finite subsets of metric spaces,which has a rich algebraic structure as a partially ordered set.We propose an axiomatization of an entropy-like measure of partitions of sets of objects located in metric spaces,and we derive an analytic expression of this new type of entropy referred to as inertial entropy.This approach starts with the notion of inertia of a partition and includes a study of the behavior of the sum of square errors of a partition.In this context,we characterize the chain of partitions produced by the Ward hierarchical clustering method.Starting from inertial entropies of partitions,we introduce conditional entropies which,in turn,generate metrics on partitions of finite sets.These metrics are used as external validation tools for clusterings of labeled data sets.The metric generated by inertial entropy can be used to validate data clustering for labeled data sets.This type of validation aims to determine to what extend labeling of the data coincides with the clustering obtained algorithmically,and we obtain a high degree of consistency of the data labeling with the results of several hierarchical clusterings.