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.

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].

Overview of Hypersphere World-Universe Model

This paper provides an overview of the Hypersphere World-Universe Model (WUM). WUM unifies and simplifies existing cosmological models and results into a single coherent picture, and proceeds to discuss the origin, evolution, structure, ultimate fate, and primary parameters of the World. 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;gamma-ray background and cosmic neutrino background;macrostructure of the world and macroobjects structure. Additionally, the model makes predictions pertaining to masses of dark matter particles, photons, and neutrinos, proposes new types of particle interactions (Super Weak and Extremely Weak), and shows inter-connectivity of primary cosmological parameters of the world and the rise of the solar luminosity during the last 4.6 Byr. The model proposes to introduce a new fundamental parameter Q in the CODATA internationally recommended values.

Hypersphere World-Universe Model

Dirac’s themes were the unity and beauty of Nature. He identified three revolutions in modern physics: Relativity, Quantum Mechanics and Cosmology. In his opinion: “The new cosmology will probably turn out to be philosophically even more revolutionary than relativity or the quantum theory, perhaps looking forward to the current bonanza in cosmology, where precise observations on some of the most distant objects in the universe are shedding light on the nature of reality, on the nature of matter and on the most advanced quantum theories” [Farmelo, G. (2009) The Strangest Man. The Hidden Life of Paul Dirac, Mystic of the Atom. Basic Books, Britain, 661 p]. In 1937, Paul Dirac proposed the Large Number Hypothesis and the Hypothesis of the variable gravitational “constant”;and later added the notion of continuous creation of Matter in the World. The developed Hypersphere World-Universe Model (WUM) follows these ideas, albeit introducing a different mechanism of matter creation. In this paper, we show that WUM is a natural continuation of Classical Physics and it can already serve as a basis for a New Cosmology proposed by Paul Dirac.

Hypersphere World-Universe Model: Basic Ideas

Hypersphere World-Universe Model (WUM) envisions Matter carried from the Universe into the World from the fourth spatial dimension by Dark Matter Particles (DMPs). Luminous Matter is a byproduct of Dark Matter (DM) self-annihilation. WUM introduces Dark Epoch (spanning from the Beginning of the World for 0.45 billion years) and Luminous Epoch (ever since for 13.77 billion years). Big Bang discussed in Standard Cosmology (SC) is, in our view, transition from Dark Epoch to Luminous Epoch due to Rotational Fission of Overspinning DM Supercluster’s Cores and self-annihilation of DMPs. WUM solves a number of physical problems in SC and Astrophysics through DMPs and their interactions: Angular Momentum problem in birth and subsequent evolution of Galaxies and Extrasolar systems;Fermi Bubbles—two large structures in gamma-rays and X-rays above and below Galactic center;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 Heating. Model makes predictions pertaining to Rest Energies of DMPs, proposes New Type 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.

Hypersphere World-Universe Model: Evolution of the World

The main objective of this paper is to discuss the Evolution of a 3D Finite World (that is a Hypersphere of a 4D Nucleus of the World) from the Beginning up to the present Epoch in frames of World-Universe Model (WUM). WUM is the only cosmological model in existence that is consistent with the Law of Conservation of Angular Momentum. To be consistent with this Fundamental Law, WUM introduces Dark Epoch (spanning from the Beginning of the World for 0.45 billion years) when only Dark Matter (DM) Macroobjects (MOs) existed, and Luminous Epoch (ever since for 13.77 billion years) when Luminous MOs emerged due to Rotational Fission of Overspinning DM Superclusters’ Cores and self-annihilation of Dark Matter Particles (DMPs). WUM envisions that DM is created by the Universe in the 4D Nucleus of the World. Dark Matter Particles (DMPs) carry new DM into the 3D Hypersphere World. Luminous Matter is a byproduct of DMPs self-annihilation. By analogy with 3D ball, which has two-dimensional sphere surface (that has surface energy), we can imagine that the 3D Hypersphere World has a “Surface Energy” of the 4D Nucleus. 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;Missing Baryon problem related to the fact that the observed amount of baryonic matter did not match theoretical predictions. WUM reveals Inter-Connectivity of Primary Cosmological Parameters and calculates their values, which are in good agreement with the latest results of their measurements. In 2013, WUM predicted the values of the following Cosmological parameters: gravitational, concentration of intergalactic plasma, and the minimum energy of photons, which were experimentally confirmed in 2015-2018. “The Discovery of a Supermassive Compact Object at the Centre of Our Galaxy” (Nobel Prize in Physics 2020) made by Prof. R. Genzel and A. Ghez is a confirmation of one of the most important predictions of WUM in 2013: “Macroobjects of the World have cores made up of the discussed DM particles. Other particles, including DM and baryonic matter, form shells surrounding the cores”.

On the Physics inside a Closed,Static,Rotating Einsteinian Hypersphere in Due Consideration of the Galaxy

Einstein’s weak equivalence principle suggests that gravity and acceleration (centrifugal force) are indistinguishable from each other and, therefore, equivalent. We maintain that they are not only equivalent, but even identical, or to rephrase the main statement of this work: A gravitational force does not exist. Rather, gravity is a fictitious force, or, more pointedly: Gravity is the centrifugal force which acts upon material bodies within the rotating S3-hypersphere of the Universe. These in turn warp the adjacent space-fabric, shaping it to the well-known field geometry of general relativity.

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.

ABOUT THE FOURIER TRANSFORMS OF NON-SMOOTH MEASURES ON HYPERSPHERES

If dμ is the Fourier transform of a smooth measure,dμ on the hypersphere Sn-1(n≥2)the there exists a constant C dependent only on n such that |dμ(y) |≤C(1+ |y |)-(n-1) /2 for all y∈Rn. In this paper, we show that the above statement is false for non-smooth measures. And we present the corresponding estimations far the Fourier transforms of certain non-smooth measures on Sn-1.

On product affine hyperspheres in R^n+1

In this paper,we study locally strongly convex affine hyperspheres in the unimodular affine space Rn+1 which,as Riemannian manifolds,are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature.As the main result,a complete classification of such affine hyperspheres is established.Moreover,as direct consequences,3-and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.

Application of smoothing technique on twin support vector hypersphere

In order to improve the learning speed and reduce computational complexity of twin support vector hypersphere(TSVH),this paper presents a smoothed twin support vector hypersphere(STSVH)based on the smoothing technique.STSVH can generate two hyperspheres with each one covering as many samples as possible from the same class respectively.Additionally,STSVH only solves a pair of unconstraint differentiable quadratic programming problems(QPPs)rather than a pair of constraint dual QPPs which makes STSVH faster than the TSVH.By considering the differentiable characteristics of STSVH,a fast Newton-Armijo algorithm is used for solving STSVH.Numerical experiment results on normally distributed clustered datasets(NDC)as well as University of California Irvine(UCI)data sets indicate that the significant advantages of the proposed STSVH in terms of efficiency and generalization performance.

Characterization of the Generalized Calabi Composition of Affine Hyperspheres

In this paper, continuing with Hu-Li Vrancken and the recent work of Antid Dillen- Schoels-Vrancken, we obtain a decomposition theorem which settled the problem of how to determine whether a given locally strongly convex aitine hypersurface can be decomposed as a generalized Calabi composition of two affine hyperspheres, based on the properties of its difference tensor K and its affine shape operator S.

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.

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.

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.

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.

Decisive Role of Dark Matter in Cosmology

Hypersphere World-Universe Model (WUM) is an alternative to the prevailing Big Bang Model (BBM). WUM and BBM are principally different Models: 1) Instead of the Initial Singularity with the infinite energy density and the extremely rapid expansion of the space (Inflation) in BBM;in WUM, there was a Fluctuation (4D Nucleus of the World with an extrapolated radius equals to a basic unit of size a) in the Eternal Universe with a finite extrapolated energy density (four orders of magnitude less than the nuclear density) and a finite expansion of the Nucleus in its fourth spatial dimension with speed c that is the gravitodynamic constant;2) Instead of the Infinite Homogeneous and Isotropic Universe around the Initial Singularity in BBM;in WUM, the 3D Finite Boundless World (the Hypersphere of the 4D Nucleus) presents a Patchwork Quilt of different Luminous Superclusters (≳103), which emerged in various places of the World at different Cosmological times. The Medium of the World is Homogeneous and Isotropic. The distribution of Macroobjects in the World is spatially Inhomogeneous and Anisotropic and temporally Non-simultaneous. The Absolute Age of the entire World (determined by the parameters of the Medium) is 14.22 Gyr.