Performance of Hyperspherical Harmonic Expansionon the Low-lying P and D States of Helium Atom

The wave functions of the n1,3p (n=2, 3, 4) and the n 1,3D (n=3, 4, 5 ) low-lying states of the helium atom are expanded into the complete sets of the symmetrically adapted basis functions from hyperspherical harmonic functions in the angle part and of generalized Laguerre functions in the radial part respectively, and are then augmented by the simplest type of Jastrow correlation factor to incorporate electron-nucleus cusp only. The excellent agreement between the present nonrelativistic eigen-energies and those from the sophisticated configuration interaction (CI) method for the examined states indicates that the hyperspherical harmonic method can also be applied to the P and the D excited states of the helium atom.

ON THE HYPERSPHERICAL HARMONIC METHODTO THREE AND FOUR BODY NUCLEAR PROBLEMS

In this paper, the hyperspherical harmonics used for solving the three and four body problems in nuclear physics are given. The equations of the adiabatic approximation to hyperspherical harmonic method are derived. Some properties of the hypernuclous Heare given as the illustrative example of the application of this method.

Some Features of Hyperspherical Functins

Let P(a,a)n (z) be n-degree hyperspherical polinomials, Q(a,a)n (z) be the second stort of hyperspherical functions, Ea be an elipse on z-plane. In this paper, we found some inequalitys about P(a,a)n (z) and Q(a,a)n (z) on z-plane, and gneralize inequality in paper [2] when a= 0 to when α≥0. In addition ,we also foundwo important expressions of the relation.

Hyperspherical Harmonic Expansions of Potential 1/r_(jt) of Molecular Systems

The expansion coefficient C D |L| of Coulomb potential 1/r jt of molecular systems in hyperspherical harmonics is derived in detail, and the explicit expression is given.

Hyperspherical Study of Ne_2Kr and Ne_2Xe Systems

We have calculated the ground state energies and rotational constants for the Ne2 Kr and Ne2 Xe systems, as well as their corresponding isotopes using the hyperspherical method. We used B-splines as basis function to expand channel functions and make the knot distribution of B-splines characterize the behavior of the channel function precisely. As a result, the extremely slow convergence of quantum mechanic calculation for these van der Waals trimers containing heavy element is improved greatly. The convergent rotational constants and the isotope shifts are obtained and compared with other theoretical and experimental values. Our results using newly proposed Tang-Toennies （TT） pair-wise potentials are consistent with that of Ernesti and Hutson’s calculation using HFD-B potentials, and give an improved agreement with experimental data.

Dominant Correlation Effects in Two-Electron Atoms

Two-electron atoms have been investigated near threshold of double escape within the framework of hyperspherical coordinates. A particularly useful set of hyperspherical angles has been used. It is well known for many years that the hyperradial motion is nearly separable from the hyperspherical angular motion. Therefore, the Born-Oppenheimer separation method should be useful. However, the success of that method in molecular physics is based on the small mass ratio, electron mass to nuclear mass. In the atomic application such a small parameter does not exist. Nevertheless the method works surprisingly well in the lower part of the spectrum. For increasing excitation energy the method becomes shaky. Near ionization threshold, it breaks even down. The author will present elsewhere an improved Born-Oppenheimer method. First pilot developments and comparison with the experimental situation are presented already here. Inclusion of a momentum-momentum radial coupling delivers an improved basis. We show that our extended Born-Oppenheimer approach leads to a deformation of the whole potential energy surface during the collision. In consequence of this deformation we outline a quantum derivation of the Wannier threshold cross section law, and we show that (e, 2e) angular distribution data are strongly influenced by that surface deformation. Finally, we present a mechanism for electron pair formation and decay leading to a supercurrent independent of the temperature. Our framework can be extended to more than two electrons, say 3 or 4. We conclude that our improved Born-Oppenheimer method [1] is expected not only to deliver better numerical data, but it is expected to describe also the Wannier phenomenon. The idea of the new theory together with first qualitative results is presented in this paper.

Charged-Exciton Complexes in Quantum Dots

It is known experimentally that stable charged-exciton complexes can exist in low-dimensional semiconductor nanostructures. Much less is known about the properties of such charged-exciton complexes since three-body problems are very difficult to be solved, even numerically. Here we introduce the correlated hyperspherical harmonics as basis functions to solve the hyperangular equation for negatively and positively charged excitons (trions) in a harmonic quantum dot. By using this method, we have calculated the energy spectra of the low-lying states of a charged exciton as a function of the radius of quantum dot. Based on symmetry analysis, the level crossover as the dot radius increases can be fully explained as the results of symmetry constraint.

Bound States for Hypercentral Singular and Exponential Potentials

With a view to obtaining an exact closed form solution to the Schroedinger equation for a variety of hypercentral potentials, we investigate further application of an ansatz. This method is good enough for many kinds of potentials, but in this article it applies to a type of the hypercentral singular potentials V（x） = ax^2 ＋ bx^-4＋ cx^-6 and exponential hypercentral Morse potential U （x） = Uo （ e^-2ax - 2 e^-ax） for three interacting particles. The Morse potential is used for diatomic molecule while this method will be successfully used for many atomic molecules. The three-body potentials are more easily introduced and treated within the hyperspherical harmonic formalism. The internal particle motion is usually described by means of Jacobi relative coordinates p, A, and R, in terms of three particle positions r1, r2, and r3. We discuss some results obtained by using harmonic and anharmonic oscillators, however the hypercentral potential can be easily generalized in order to allow a systematic anaiysis, which admits an exact solution of the wave equation. This method is also applied to some other types of three-body, four-body, ..., interacting potentials.

Machine learning seismic reservoir prediction method based on virtual sample generation

Seismic reservoir prediction plays an important role in oil exploration and development.With the progress of artificial intelligence,many achievements have been made in machine learning seismic reservoir prediction.However,due to the factors such as economic cost,exploration maturity,and technical limitations,it is often difficult to obtain a large number of training samples for machine learning.In this case,the prediction accuracy cannot meet the requirements.To overcome this shortcoming,we develop a new machine learning reservoir prediction method based on virtual sample generation.In this method,the virtual samples,which are generated in a high-dimensional hypersphere space,are more consistent with the original data characteristics.Furthermore,at the stage of model building after virtual sample generation,virtual samples screening and model iterative optimization are used to eliminate noise samples and ensure the rationality of virtual samples.The proposed method has been applied to standard function data and real seismic data.The results show that this method can improve the prediction accuracy of machine learning significantly.

HIGHER ORDER SINGULAR INTEGRAL EQUATIONS ON COMPLEX HYPERSPHERE

A theory of a class of higher order singular integral under the operator （L f） （u） =[u1 σf/σu1（u） - u1σf/σu1（u） ＋ f（u）] is given. We transform the higher order singular integral to a usual Cauchy integral, extend the permutation formula of the higher order singular integral deduced by Qian and Zhong in [4] to a general case, and discuss the regularization problem of the higher order singular integral equations with Cauchy kernel and variable coefficients on complex hypersphere.

A NEW HYPERSPHERE SUPPORT VECTOR MACHINE ALGORITHM

The hypersphere support vector machine is a new algorithm in pattern recognition. By studying three kinds of hypersphere support vector machines, it is found that their solutions are identical and the margin between two classes of samples is zero or is not unique. In this letter, a new kind of hypersphere support vector machine is proposed. By introducing a parameter n(n>1), a unique solution of the margin can be obtained. Theoretical analysis and experimental results show that the proposed algorithm can achieve better generaliza-tion performance.

Hypersphere World-Universe Model. Tribute to Classical Physics

This manuscript summarizes the results of Classical Physics before Quantum Mechanics and Hypotheses proposed by classical physicists from the 17th until the beginning of 21st century. We then proceed to unify these results into a single coherent picture in frames of the developed Hypersphere World-Universe Model (WUM). The Model proposes 5 types of Dark Matter particles and predicts their masses;models the origin, evolution, and structure of the World and Macroobjects;provides a mathematical framework that ties together a number of Fundamental constants and allows for direct calculation of their values.

Mathematical Overview of Hypersphere World-Universe Model

The Hypersphere World-Universe Model (WUM) provides a mathematical framework that allows calculating the primary cosmological parameters of the World which are in good agreement with the most recent measurements and observations. WUM explains the experimental data accumulated in the field of Cosmology and Astroparticle Physics over the last decades: the age of the World and critical energy density;the gravitational parameter and Hubble’s parameter;temperatures of the cosmic microwave background radiation and the peak of the far-infrared background radiation;the concentration of intergalactic plasma and time delay of Fast Radio Bursts. Additionally, the model predicts masses of dark matter particles, photons, and neutrinos;proposes new types of particle interactions (Super Weak and Extremely Weak);shows inter-connectivity of primary cosmological parameters of the World. WUM proposes to introduce a new fundamental parameter Q in the CODATA internationally recommended values. This paper is the summary of the mathematical results obtained in [1]-[4].

Solar System. Angular Momentum. New Physics

The most widely accepted model of Solar System formation, known as the Nebular hypothesis, does not solve the Angular Momentum problem—why is the orbital momentum of Jupiter larger than rotational momentum of the Sun? The present manuscript introduces a Rotational Fission model of creation and evolution of Macrostructures of the World (Superclusters, Galaxies, Extrasolar Systems), based on Overspinning Cores of the World’s Macroobjects, and the Law of Conservation of Angular Momentum. The Hypersphere World-Universe model is the only cosmological model in existence that is consistent with this Fundamental Law.

High-Energy Atmospheric Physics: Ball Lightning

This article proposes an explanation for High-Energy Atmospheric phenomena through the frames of Hypersphere World-Universe Model (WUM). In WUM, Terrestrial Gamma-Ray Flashes (TGFs) are, in fact, Gamma-Ray Bursts (GRBs). The spectra of TGFs at very high energies are explained by Dark Matter particles annihilation in Geocorona. Lightning initiation problem is solved by GRBs that slam into thunderclouds and carve a conductive path through a thunderstorm. We introduce Multiworld consisting of Macro-World, Large-World, Small-World, and Micro-World, characterized by suggested Gravitational, Extremely-Weak, Super-Weak, and Weak interaction respectively. We propose a new model of Ball Lightning formation based on the Dark Matter Core surrounded by electron-positron plasma in the Small-World.

Analysis of Maxwell’s Equations. Cosmic Magnetism

According to Hypersphere World-Universe Model, dark matter particles DIRACs are magnetic dipoles consisting of two Dirac’s monopoles. We conclude that DIRACs are the subject of Maxwell’s equations. So-called “auxiliary” magnetic field intensity H is indeed current density of magnetic dipoles. The developed approach to magnetic field can explain a wealth of discovered phenomena in Cosmic Magnetism: a dark magnetic field, the large-scale structure of the Milky Way’s magnetic field, and other magnetic phenomena which are only partly related to objects visible in other spectral ranges.

Bipartite Threshold Multi-Secret Sharing Scheme Based on Hypersphere

To address the problem that existing bipartite secret sharing scheme is short of dynamic characteristic, and to solve the problem that each participant can only use secret share once, this paper proposed a bipartite (n1+n2, m1+m2)-threshold multi-secret sharing scheme which combined cryptography and hypersphere geometry. In this scheme, we introduced a bivariate function and a coordinate function over finite field Zp to calculate the derived points of secret share, which can reconstruct the shared secrets by producing the intersection point of hypernormal plane and normal line on the hypertangent plane. At the initial stage the secret dealer distributes to each participant a secret share that can be kept secret based on the intractability of discrete logarithm problem and need not be changed with updating the shared secrets.Each cooperative participant only needs to submit a derived point calculated from the secret share without exposing this secret share during the process of reconstructing the shared secret. Analyses indicate that the proposed scheme is not only sound and secure because of hypersphere geometric properties and the difficulty of discrete logarithm problem, but also efficient because of its well dynamic behavior and the invariant secret share. Therefore, this bipartite threshold multi-secret sharing scheme is easy to implement and is applicable in practical settings.

Astrophysics: Macroobject Shell Model

The model proposes that Nuclei of all macroobjects (Galaxy clusters, Galaxies, Star clusters, Extrasolar systems) are made up of Dark Matter Particles (DMP). These Nuclei are surrounded by Shells composed of both Dark and Baryonic matter. This model is used to explain various astrophysical phenomena: Multi-wavelength Pulsars;Binary Millisecond Pulsars;Gamma-Ray Bursts;Fast Radio Bursts;Young Stellar Object Dippers;Starburst Galaxies;Gravitational Waves. New types of Fermi Compact Stars made of DMP are introduced: Neutralino star, WIMP star, and DIRAC star. Gamma-Ray Pulsars are rotating Neutralino and WIMP stars. Merger of binary DIRAC stars can be a source of Gravitational waves.

Dark Matter Cosmology and Astrophysics

Hypersphere World-Universe Model (WUM) envisions Matter carried from Universe into World from fourth spatial dimension by Dark Matter Particles (DMPs). Luminous Matter is byproduct of Dark Matter (DM) annihilation. WUM introduces Dark Epoch (spanning from Beginning of World for 0.4 billion years) when only DMPs existed, and Luminous Epoch (ever since for 13.8 billion years). Big Bang discussed in standard cosmological model is, in our view, transition from Dark Epoch to Luminous Epoch due to Rotational Fission of Overspinning DM Supercluster’s Cores and annihilation of DMPs. WUM solves a number of physical problems in contemporary Cosmology and Astrophysics through DMPs and their interactions: Angular Momentum problem in birth and subsequent evolution of Galaxies and Extrasolar systems—how do they obtain it;Fermi Bubbles—two large structures in gamma-rays and X-rays above and below Galactic center;Mysterious Star KIC 8462852 with irregular dimmings;Coronal Heating problem in solar physics—temperature of Sun’s corona exceeding that of photosphere by millions of degrees;Cores of Sun and Earth rotating faster than their surfaces;Diversity of Gravitationally-Rounded Objects in Solar system and their Internal Heat;Lightning Initiation problem—electric fields observed inside thunderstorms are not sufficient to initiate sparks;Terrestrial Gamma-Ray Flashes—bursts of high energy X-rays and gamma rays emanating from Earth. Model makes predictions pertaining to Masses of DMPs, proposes New Types of their Interactions. WUM reveals Inter-Connectivity of Primary Cosmological Parameters and calculates their values, which are in good agreement with the latest results of their measurements.