Structural Foundation and Geometry of the Material Singularity (and Its Quantum Entanglement)

In this paper we develop and study, as the second part of one more general development, the energy transmutation equation for the material singularity, previously obtained through the symmetrisation of a wave packet, that is, we develop the correlation between the terms of this equation, which accounts for the formation of matter from a previous vibrational state, and the different possible energy species. These energetic species are ascribed, in a simplified form, to the equation E¯ω=E¯k+E¯f, which allows us, through its associated phase factor, to gain an insight into the wave character of the kinetic energy and thus to attain the basis of the matter-wave, and all sorts of related phenomenologies, including that concerning quantum entanglement. The formation of the matter was previously identified as an energetic process, analogous to the kinetic one, in which finally the inertial mass is consolidated as a mass in a different phase, now, in addition, the mass of the material singularity is identified as a volumetric density of waves of toroidal geometry created in the process of singularisation or energy transfer between species, which makes it possible to establish the real relation or correspondence between the corpuscular and photonic energy equation (E=mc2=hν), i.e. to explain through m the intimate sense of the first equivalence, which explains what νis in the second one.

On the Vacuum Hydrodynamics of Moving Bodies—The Theory of General Singularity

The Theory of General Singularity is presented, unifying quantum field theory, general relativity, and the standard model. This theory posits phonons as fundamental excitations in a quantum vacuum, modeled as a Bose-Einstein condensate. Through key equations, the role of phonons as intermediaries between matter, energy, and spacetime geometry is demonstrated. The theory expands Einsteins field equations to differentiate between visible and dark matter, and revises the standard model by incorporating phonons. It addresses dark matter, dark energy, gravity, and phase transitions, while making testable predictions. The theory proposes that singularities, the essence of particles and black holes, are quantum entities ubiquitous in nature, constituting the very essence of elementary particles, seen as micro black holes or quantum fractal structures of spacetime. As the theory is refined with increasing mathematical rigor, it builds upon the foundation of initial physical intuition, connecting the spacetime continuum of general relativity with the hydrodynamics of the quantum vacuum. Inspired by the insights of Tesla and Majorana, who believed that physical intuition justifies the infringement of mathematical rigor in the early stages of theory development, this work aims to advance the understanding of the fundamental laws of the universe and the perception of reality.

New Raw Materials Ignite the Singularity of Cosmetic Innovation

As traced by the Big Bang theory,the starting point of the universe,is called the“Singularity”in Da Ci Hai,an unabridged,comprehensive dictionary.According to cosmological reasoning,the singularity has an infinite density of matter,an infinite curvature of space and time,and it is invisible and infinite.These characteristics are analogous to the human imagination at the level of innovation.For the innovation of cosmetic raw materials,there is also the possibility of infinite evolution.For example,in recent years,the scientific research in cosmetic industry the for promoting upgrade in raw materials is quite proactive.From the raw material enterprises,down to the brand company,the investment in raw material innovation is also strengthened at a visible rate.

Some Results for Formal Local Cohomology Modules

Let （R, m） be a commutative Noetherian local ring, I an ideal of R and M a finitely generated R-module. Let limnHm^i（M/I^nM）be the ith formal local cohomology module of M with respect to I.In this paper, we discuss some properties of formal local cohomology modules limnHm^i（M/I^nM）,which are analogous to the finiteness and Artinianness of local cohomology modules of a finitely generated module.

Integral Representation of the Manifold Solution for New Class of the Volterra Type Integral Equation with a Boundary Singularity in the Case, When Kernel Contain Logarithmic Singularity and its Power

In this work we suggestion new methods investigation the model Volterra type integral equation with logarithmic singularity, kernel which consisting from composition polynomial function with logarithmic singularity and function with singular point. The problem investigation this type integral equation at n = 2m reduce to problem investigate the Volterra type integral equation （1） for n = 2 the theory for which was constructed in [2]. In this work, we investigation integral equation （1） at = 2m ＋ 1 In this case, we investigate integral equation （1） reduction it＇s to m integral equation type [2] φ（x）＋∫xa[p1＋p2 ln（x-a/t-a）]φ（t）/t-a dt=f（x）and one the following integral equation [1] ω（x）＋p3∫xω（t）/ a t-adt=g（x）.

The Urban Singularity-Envisioning the Future of Cities

This paper is a hypothetical exploration of the connections between teleological evolution,the Omega Singularity,and the future of cities,weaving together insights from a diverse array of disciplines.Our investigation delves into the possibility that cities are evolving towards a Singularity,a state characterized by infinite knowledge,intelligence,and adaptability,which would bring about a radical transformation of urban environments and their underlying dynamics in the 21st century and beyond.At the heart of this exploration lies the role of language and time as crucial dimensions of the Urban Singularity.Moreover,we examine how linguistic developments and cross-cultural exchanges can foster more inclusive,adaptable,and resilient urban environments,while also highlighting the need for advanced technologies and communication modalities that can support the dynamic needs of future cities.Furthermore,the paper investigates the profound implications and transformative potential of merging human consciousness with the urban Singularity.By examining the interplay between these concepts,we seek for a deeper understanding of the potential trajectories and implications of these concepts for the transformation of human society and our relationship with the built environment.

COUPLING EFFECT OF FLEXIBLE JOINT AND FLEXIBLE LINK ON DYNAMIC SINGULARITY OF FLEXIBLE MANIPULATOR

The coupling effect of the flexible joint and the flexible link on the dynamic singularity of the flexible manipulator is addressed. Firstly, the dynamic equations of a flexible manipulator with a flexible joint and a flexible link are derived. Secondly, the relationship and property between the flexible joint and the flexible link are analyzed. It shows that the flexible joint＇s amplitude will increase abruptly, thereby the dynamic singularity occurs if the frequency of a flexible joint is near or equal to some natural frequency of a flexible link. Finally, some numerical simulations which will verify the correctness of the theoretical analysis, are carded out. The results are fundamental for the design of a flexible manipulator and for the avoidance of the dynamic singularity.

THE JUMPING PHENOMENON OF THE DIMENSIONS OF COHOMOLOGY GROUPS OF TANGENT SHEAF

Let X be a compact complex manifold. Consider a small deformation π ： X → B of X, the dimensions of the cohomology groups of tangent sheaf Hq（xt, Txt ） may vary under this deformation. This article studies such phenomena by studying the obstructions to deform a class in Hq（X, 5TX） with parameter t and gets a formula for the obstructions.

Difference Discrete Variational Principles, Euler?Lagrange Cohomology and Symplectic, Multisymplectic Structures III: Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II, we have studied the difference discrete variational principle and the Euler?Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference discrete Euler?Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler?Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangian and Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler?Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonian schemes or Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler?Lagrange cohomological conditions are satisfied.

Difference Discrete Variational Principle,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures II：Euler—Lagrange Cohomology

In this second paper of a series of papers, we explore the difference discrete versions for the Euler?Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. In terms of the difference discrete Euler?Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler?Lagrange or canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler?Lagrange cohomological conditions are satisfied.

Solution with Singularity of Order One for Singular Integral Equation with Hilbert Kernal

The basic sets of solutions in classH(orH *) for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively. Thus the expressions of solutions and its solvable conditions are simplified. On this basis the solutions and the solvable conditions in classH 1 * as well as the generalized Noether theorem for the complete equation are obtained. Key words Hilbert kernel - solution with singularity of order one - basic set of solutions - Noether theorem - characteristic equation and its adjoint equation CLC number O 175.5 Foundation item: Supported by the National Natural Science Foundation of China (19971064) and Ziqiang Invention Foundation of Wuhan University (201990336)Biography: Zhong Shou-guo(1941-), male, Professor, research direction: singular integral equations and their applications.

QUANTUM COHOMOLOGY OF BLOWUPS OF SURFACES AND ITS FUNCTORIALITY PROPERTY

In this article, using the WDVV equation, the author first proves that all Gromov-Witten invariants of blowups of surfaces can be computed from the Cromov- Witten invariants of itself by some recursive relations. Furthermore, it may determine the quantum product on blowups. It also proves that there is some degree of functoriality of the big quantum cohomology for a blowup.

PROPERTY IDENTIFICATION OF SINGULARITY LOCI OF GOUGHSTEWART MANIPULATOR

The problem of identifying the property of singularity loci of Gough-Stewart manipulators is addressed. After constructing the Jacobian matrix of the Gough-Stewart manipulator, a cubic polynomial expression in the mobile platform position parameters, which represents the constantorientation singularity locus of the manipulator, is derived. Graphical representations of the singularity locus of the manipulator for different orientations are illustrated with examples. Further, the singularity locus of the manipulator in the principal-section, where the mobile platform lies, is analyzed. It shows that singularity loci of the ,manipulator in parallel principal-sections are all quadratic expressions including a parabola, four pairs of intersecting straight lines and infinite hwerbolas. Their geometric and kinematic properties are also researched as well.

COMPLETELY BOUNDED COHOMOLOGY OF NON-SELFADJOINT OPERATOR ALGEBRAS

The authors prove that all n-th completely bounded cohomology groups of a nest algebra T（N） acting on a separable Hilbert space are trivial when the coefficients lie in any ultraweakly closed T（N）-bimodule containing the nest algebra. They also prove that Hcb^n（A, M） ≌ Hcb^n（A, M） for all n ≥ 1 and a CSL algebra .A with an ultraweakly closed .A-bimodul.M containing A.

Stress singularity analysis of anisotropic multi-material wedges under antiplane shear deformation using the symplectic approach

Symplectic approach has emerged a popular tool in dealing with elasticity problems especially for those with stress singularities. However, anisotropic material problem under polar coordinate system is still a bottleneck. This paper presents a subfield method coupled with the symplectic approach to study the anisotropic material under antiplane shear deformation. Anisotropic material around wedge tip is considered to be consisted of many subfields with constant material properties which can be handled by the symplectic approach individually. In this way, approximate solutions of the stress and displacement can be obtained. Numerical examples show that the present method is very accurate and efficient for such wedge problems. Besides, this paper has extended the application of the symplectic approach and provides a new idea for wedge problems of anisotropic material.

On the Cohomology of Associative Superalgebras

This paper presents a sufficient condition for the cohomology groups of an associative superalgebra to vanish. As its application, we prove that the cohomology groups H n(L,M) vanish when L is a strongly semisimple Lie superalgebra and M is an irreducible faithful L module.

ON THE SINGULARITY OF LEAST SQUARES ESTIMATOR FOR MEAN-REVERTING α-STABLE MOTIONS

We study the problem of parameter estimation for mean-reverting α-stable motion, dXt = （a0 - θ0Xt）dt ＋ dZt, observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case （a0, θ0） = （0, 0）. If a0 = 0, then the mean-reverting α-stable motion becomes Ornstein-Uhlenbeck process and is studied in [7] in the ergodic case θ0 〉 0. For the Ornstein-Uhlenbeck process, asymptotics of the least squares estimators for the singular case （θ0 = 0） and for ergodic case （θ0 〉 0） are completely different.

Targeted Sub-Attomole Cancer Biomarker Detection Based on Phase Singularity 2D Nanomaterial-Enhanced Plasmonic Biosensor

Detection of small cancer biomarkers with low molecular weight and a low concentration range has always been challenging yet urgent in many clinical applications such as diagnosing early-stage cancer,monitoring treatment and detecting relapse.Here,a highly enhanced plasmonic biosensor that can overcome this challenge is developed using atomically thin two-dimensional phase change nanomaterial.By precisely engineering the configuration with atomically thin materials,the phase singularity has been successfully achieved with a significantly enhanced lateral position shift effect.Based on our knowledge,it is the first experimental demonstration of a lateral position signal change>340μm at a sensing interface from all optical techniques.With this enhanced plasmonic effect,the detection limit has been experimentally demonstrated to be 10^(-15) mol L^(−1) for TNF-α cancer marker,which has been found in various human diseases including inflammatory diseases and different kinds of cancer.The as-reported novel integration of atomically thin Ge_(2)Sb_(2)Te_(5) with plasmonic substrate, which results in a phase singularity and thus a giant lateral position shift, enables the detection of cancer markers with low molecular weight at femtomolar level. These results will definitely hold promising potential in biomedical application and clinical diagnostics.

Discrete vortex bound states with a van Hove singularity in the vicinity of the Fermi level

A theoretical study on discrete vortex bound states is carried out near a vortex core in the presence of a van Hove singularity(VHS) near the Fermi level by solving Bogoliubov–de Gennes(Bd G) equations. When the VHS lies exactly at the Fermi level and also at the middle of the band, a zero-energy state and other higher-energy states whose energy ratios follow integer numbers emerge. These discrete vortex bound state peaks undergo a splitting behavior when the VHS or Fermi level moves away from the middle of the band. Such splitting behavior will eventually lead to a new arrangement of quantized vortex core states whose energy ratios follow half-odd-integer numbers.

Analyzing Some Elements of Technological Singularity Using Regression Methods

Technological advancement has contributed immensely to human life and society.Technologies like industrial robots,artificial intelligence,and machine learning are advancing at a rapid pace.While the evolution of Artificial Intelligence has contributed significantly to the development of personal assistants,automated drones,smart home devices,etc.,it has also raised questions about the much-anticipated point in the future where machines may develop intelligence that may be equal to or greater than humans,a term that is popularly known as Technological Singularity.Although technological singularity promises great benefits,past research works on Artificial Intelligence(AI)systems going rogue highlight the downside of Technological Singularity and assert that it may lead to catastrophic effects.Thus,there is a need to identify factors that contribute to technological advancement and may ultimately lead to Technological Singularity in the future.In this paper,we identify factors such as Number of scientific publications in Artificial Intelligence,Number of scientific publications in Machine Learning,Dynamic RAM(Random Access Memory)Price,Number of Transistors,and Speed of Computers’Processors,and analyze their effects on Technological Singularity using Regression methods(Multiple Linear Regression and Simple Linear Regression).The predictive ability of the models has been validated using PRESS and k-fold cross-validation.Our study shows that academic advancement in AI and ML and Dynamic RAM prices contribute significantly to Technological Singularity.Investigating the factors would help researchers and industry experts comprehend what leads to Technological Singularity and,if needed,how to prevent undesirable outcomes.