We define a spinor-Minkowski metric for SL(4,C). It is not a trivial generalization of the SL(2,C) metric and it involves the Minkowskian one. We define 4x4 version of the Pauli matrices and eight 4-component associated generalized eigenvectors that can be regarded as undotted covariant spinors. The 4-component spinors can be grouped into four categories. Each category transforms in its own way. The outer products of pairwise combinations of 4-component spinors can be associated with 4-vectors. Including the dotted covariant, undotted and dotted contravariant forms totally we have sixteen pairs of spinors. Eight of them live in the conjugate space which has no countepart in SL(2,C).

Recent advances in understanding the Planck scale have led to a new, straightforward method for quantizing the general theory of relativity. This results in a theory of gravity that predicts the same outcomes as general relativity with greater fundamental comprehension connected directly to the Planck scale.

The separability problem is one of the basic and emergent problems in present and future quantum information processing. The latter focuses on information and computing based on quantum mechanics and uses quantum bits as its basic information units. In this paper we present an overview of the progress in the separability problem in bipartite systems, more specifically in two quantum bits (qubits) system, from the criterion based on the Bell’s inequalities in 1964 to the Li-Qiao Criterion and the enhanced entanglement criterion based on the symmetric informationally complete positive operator valued measures (SIC POVMs) in 2018.

We analyse recent contradictory results and conclusions about the spectrum of the screened Coulomb potential $V(r)=-r^{-1}e^{-C/r}$. The well known Hellmann-Feynman theorem shows that all the bound states of the Coulomb potential ($C=0$) remain bounded as $C$ increases. We derive a simple approximate analytical expression for the eigenvalues for sufficiently small values of the screening parameter $C$ and an approximate asymptotic expression for the asymptotic behaviour of the s-state eigenvalues when $C\rightarrow \infty $. Present results are expected to resolve the discrepancy about the spectrum of the quantum-mechanical model just mentioned.

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex space $\mathbb C^d$ with random steps parametrized by $N$ Ising variables $\sigma_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number $r$, related by $\sum \sigma_i = q r$. This equivalence provides a \textbf{dual} theory describing a strong turbulent phase of the Navier-Stokes flow in $\mathbb R_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a \textbf{quantum} statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N \rightarrow \infty$ or its chemical potential $\mu \rightarrow 0$. The system with fixed $N$ has different asymptotics at odd and even $N\rightarrow \infty$, but the limit $\mu \rightarrow 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $\mu$ using methods of number theory. It grows as $\nu/\mu^2$ in the continuum limit $\mu \rightarrow 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu} \to 0$. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-\lambda}$, where the anomalous dimensions $\lambda$ satisfy the spectral equation \eqref{spectralEq}. The spectrum becomes a continuum in the statistical limit $N \to \infty$, presumably leading to multifractal phenomena.

The natural units of measure lauded by Max Planck more than 100 years ago are rarely used today. But many physical constants, including Planck's constant, the gravitational constant, the speed of light, vacuum permittivity, and vacuum permeability, consist of natural units in their unit dimensions. The natural units are present in all formulas containing these constants. The defining characteristic of the natural units is an alignment of unit values at the Planck scale. This alignment gives a computational basis of proportionality from which the correlated properties and dynamics of elementary particles, including wavelength, period, mass, momentum, and energy, manifest in equal or inversely proportional ratios of the Planck scale. These correlations explain many of the defining equations of quantum mechanics, classical gravity, and electromagnetism.

In recent, Chae found that, at weak gravitational acceleration g_N≲10⁻⁹ms⁻², the observed orbit of the wide binary stars is not accordant with current theory of binary star. Here, we present, this finding should be an observational evidence for that Newtonian theory of orbit perturbation is still valid in galactic scale.

The compatibility between the de Sitter Swampland conjecture and Ho\v{r}ava-Lifshitz $F(\bar{R})$ theories with a flat FLRW metric is studied. We first study the standard $f(R)$ theories and show that the only way in which the dS conjecture can be made independent of $R$ is by considering a power law of the form $f(R)\sim R^{\gamma}$. The conjecture and the consistency of the theory puts restrictions on $\gamma$ to be greater but close to $1$. For the $F(\bar{R})$ theories described by its two parameters $\lambda$ and $\mu$ we use the equations of motion to construct the function starting with an ansatz for the scale factor in the Jordan frame of the power law form. By doing a conformal transformation on the three metric to the Einstein frame we can obtain an action of gravity plus a scalar filed by relating the parameters of the theory. The non-projectable and projectable cases are studied and the differences are outlined. The $F(\bar{R})$ function obtained consists of terms of the form $\bar{R}^{\gamma}$ with the possibility of having negative power terms. The dS conjecture leads to inequalities for the $\lambda$ parameter, in both versions it gets restricted to be greater but close to $1/3$. For the general case in which $\mu$ and $\lambda$ are considered as independent, the action contains an extra term but we propose that the conjecture is still applicable. Once again the non-projectable and projectable cases are studied. The $F$ function obtained has the same form as before, the consistency of the theory and the dS conjecture lead to a set of inequalities on both parameters that are studied numerically. In all cases $\lambda$ is restricted by $\mu$ around $1/3$ and we obtain $\lambda\to1/3$ if $\mu\to0$. Finally we consider the $f(R)$ limit $\mu,\lambda\to1$ and we obtain consistent results.

Cosmic strings can arise from spontaneous phase transitions during the inflationary period and are describable as disclination topological defects in spacetime. Analytical integration of the geodesic equations for a particle around a cosmic string reveals distinct orbit patterns dependent on the disclination intensity $\beta_{0}$. After its interaction with the cosmic string, the particle's behavior becomes random. To categorize these patterns, we describe the geodesic motion as iterated maps of a discrete parameter. Through the stabilization values of these functions, we correlate the scattering angle and the number of loops for the particle with $\beta_{0}$. Therefore, we interpret the gravitational field mathematically as an instructor of a particle's next position, given its previous one. We find that the radial function in the small displacement limit is similar to the logistic map. This leads us to conjecture the potential of the discrete parameter being a multiple of Planck's time, suggesting a method for the geometric discretization of spacetime around a cosmic string.

Staudinger taught us that macromolecules were made up of the covalently bonded monomer repeat units chaining up as polymer chains. The chemical nature of the monomer directed the type of covalent bonds conferring most of the specific properties of the polymer. The more the number of repeat units the longer the chains and the more the possibility for the chains to assume a variety of shapes, from an extended elongated one to a more compact coiled one. Also, the chemical process that resulted in the synthesis of macromolecules produced many chains, often not with the same shape or size. The properties of the polymers improved when the chains became longer but it was more difficult to process them: their viscosity increased with molecular weight; viscosity was no longer an intensive property like it was for small liquids. The main question raised in polymer physics remains: how do these long chains interact and move as a group when submitted to shear deformation at high temperature when they are viscous liquids? This question is debated in a field of polymer physics called RHEOLOGY, whose purpose is to understand the viscoelastic aspects of polymer melts deformation. The current consensus is that we need to distinguish two cases: the deformation of “un-entangled chains” for macromolecules with molecular weight, M, smaller than Me, “the entanglement molecular weight”, and the deformation of “entangled” chains for M > Me. Several eminent scientists have extensively studied these 2 cases over the last 70 years. Paul J. Flory, in 1974, and Pierre-Gilles de Gennes, in 1991, have been awarded the Nobel Price in Chemistry and Physics, respectively, for their significant theoretical contribution to understand these challenging problems. For both of these authors the properties of polymers derive from the statistical characteristics of the macromolecule itself, the designated statistical system that defines the thermodynamic state of the polymer. Me, the molecular weight between entanglements, is defined from the rubber elasticity theory and is known to be equal to Mc/2 where Mc is the molecular weight for the entanglements when viscosity measurements are made. The current paradigm is that the viscoelasticity of un-entangled melts ( M < Mc) is well described by the Rouse model and that the entanglement issues raised by the impact of the increase of the length of the macromolecules on the melt viscoelasticity, when M > Mc, are well understood by the reptation model introduced by de Gennes and co-workers. Both models can be classified in the category of “chain dynamics statistics”. In this paper we examine in details the failures and the current challenges facing the current paradigm of polymer rheology: the Rouse model for M < Mc, the reptation model for M > Mc, the time-temperature superposition principle, the strain induced time dependence of viscosity, shear-refinement and sustained-orientation. The basic failure of the current paradigm and its inherent inability to fully describe the experimental reality is documented in this paper. In the discussion and conclusion of the paper we suggest that a different solution to explain the viscoelasticity of polymer chains and of their “entanglement” is needed. This requires a change of paradigm to describe the dynamics of the interactions within the chains and across the chains. A brief description of our currently proposed open dissipative statistical approach, “the Grain-Field Statistics”, is presented.

This article is the first part of a scientific project under the general title "Geometrized Vacuum Physics". This study is based on two main postulates: 1) independence of the propagation velocity of electromagnetic waves in vacuum from their frequency; 2) the constancy of the averaged zero vacuum balance, associated with the assertion that only mutually opposite formations are born from vacuum, so that on average they completely compensate for each other's manifestations. In this part of the "Geometrized Vacuum Physics" the foundations of the Algebra of Stignatures are laid, which is the mathematical and logical foundation of the entire project. In the next articles of this project it will be shown that the Algebra of Stignatures can be used for the development of "zero" (vacuum) technologies, algebraic genetics, vacuum cosmology, the vacuum standard model of "elementary particles", vacuum gravity and levitation, vacuum energy, ethics and aesthetics and many other branches of knowledge. At the end of this article, one of the many possible applications of the Algebra of Stignatures is given, in particular, the basics of the stignature-spectral analysis are outlined, with the help of which the possibilities of communication channels can be significantly expanded.

Here we will model the curvature and contraction of space-time using as a basis the equation of state of an ideal gas and the Hawking´s equation for the temperature of a black hole. We will use this mathematical model to hypothesize that the Boltzmann´s constant depends on the state of matter, that is, there is a known Boltzmann´s constant for flat space-time and an effective Boltzmann´s constant for curved space-time. This model will allow us to quantify the structure of space-time and will serve as a basis to determine the origin of gravity and the origin of elementary particles. Using the Shannon-Boltzmann-Gibbs entropy relation, we will demonstrate that information is not lost and depends on the state of matter, the information is encoded and depends on the effective Boltzmann´s constant.

A new coordinate system based on the proper times of rest frames in the Schwarzschild metric is developed to show that a radially falling worldline becomes light-like at the event horizon in the Schwarzschild metric. This system has no coordinate singularity at the event horizon and light travels on 45 degree lines everywhere in the space. The light-like nature of the worldline is confirmed by analyzing it in Kruskal-Szekeres (KS) coordinates. It is shown that the worldline in KS coordinates has an undefined derivative at the horizon at any point other than the origin of the KS coordinates. It is then demonstrated that the origin of the KS coordinate system can be reached in the falling frame by exploiting the time symmetry of the manifold and that the falling worldline is indeed light-like there. The light-like nature of the worldline means that the horizon is length contracted to a point in the falling frame, such that the event horizon is proven to be the source of the Schwarzschild metric and the end point of gravitational collapse.

Motivated by assumptions that Quantum Mechanics (QM) is not the basement of reality and that additional dimensionality (XD) is required to complete and extend the Standard Models (SM) of Particle Physics and Cosmology, experimental protocols utilizing putative properties of Kaluza-Klein-like (KK) cyclicality embedded within Dirac’s electron hypertube model in conjunction with a Dirac covariant polarized vacuum is developed to produce a new type of low-energy tabletop cross section based on topological phase transitions rather than high energy collision sprays as in the LHC. The model is presented in a highly theoretical axiomatic manner utilizing off-the-shelf parameters generally marginalized by the mainstream physics community. When successful the protocol leads to obsolescence of the highly efficacious 100-year supercollider run producing a paradigm shift with an inherent new set of Unified Field (UF) transformations beyond the Galilean Lorentz-Poincaré transformations of the SM to 3rd regime natural science: Classical – Quantum - UF.

Arguments have been made that violation of the CHSH and similar inequalities shows that reality at the quantum level must be non-local. The derivation of Bell inequality is re-examined and it is shown that violations of these inequalities merely demonstrates the existence of contextuality - they say nothing about the causal influences underlying such contextuality. It is argued that contextual systems do not possess enduring (propositional) properties, merely contingent properties. An example is presented of a classical situation, a two player co-operative game, whose random variables are consistently connected in the sense of Dzhafarov, and which is contextual, violating the CHSH inequality. In fact it also violates the Tsierl'son bound. The key is that this system is generated and its properties are disposed, not determined.

We consider a real scalar field in de Sitter background and compute its thermal propagators. We propose that in a dS/CFT context, non-trivial thermal effects as seen by an `out' observer can be encoded in the anomalous dimensions of the d=3 Ising model. One of these anomalous dimensions, the critical exponent eta, fixes completely a number of cosmological observables, which we compute.

The present paper reports an exact approach quantifying the electromagnetic contribution to the anomalous magnetic moment occurring in isolated system comprised of non-composite particle carrying elementary electric charge. Essential averaging procedure and regularization of the electromagnetic field potentials necessary when quantifying the electromagnetic self-interactions and when deriving equations of motion without singularities and obeying the conservation laws are thoroughly discussed. The study shows that the dynamics of the considered system is associated to a unique classical transcendental equations of motion satisfied by the particle’s velocity and the electromagnetic contribution to the anomalous g-factor known from the quantum electrodynamics. The equations of motion predict a value of the anomalous g-factor that agrees with the experimentally measured one reported in the literature and that calculated with the aid of quantum electrodynamics. In the present study the computational accuracy is restricted to match one part in a billion, obtaining ae=0.001159652(23), thus reveling the potential of non-perturbative methods in predicting the electron’s anomalous g-factor.

By developing the previously proposed method of combining continuum mechanics with Einstein Field Equations, it has been shown that the classic relativistic description, curvilinear description, and quantum description of the physical system may be reconciled. For a system with an electromagnetic field, the Lagrangian density equal to the invariant of the electromagnetic field was obtained, vanishing four-divergence of canonical four-momentum appears to be consequence of the Poyinting theorem, and explicit form of one of gauges of the electromagnetic four-potential was introduced. The proposed method allows for further development with additional fields.

This article is the third part of a scientific project under the general title "Geometrized vacuum physics based on the Algebra of Signatures". In the first two papers [1,2], the ideal (i.e., non-curved and immobile) local region of vacuum was studied and the foundations of the Algebra of Signatures were laid. This article considers the possibilities of describing the curved and moving state of the same vacuum region on the basis of the mathematical apparatus of the Algebra of Signatures. The reasons for the multilateral consideration of vacuum and twisting of intra-vacuum processes into spiral bundles are disclosed. The 4-tensor is introduced for two-sided and 16-sided consideration of the curvature of the local vacuum region. On the basis of kinematic models, the following assumptions were made: about the inert properties of vacuum layers; about the possibility of displacement of vacuum layers relative to each other at a speed significantly exceeding the speed of light; about the possibility of "rupture" of the local region of vacuum. The proposed kinematic models of the movement of vacuum layers can be a theoretical basis for the development of "zero" (i.e., vacuum) technologies.

Louis de Broglie in the beginning 20th century voices his theory of a double solution, according to which a pilot wave accompanies a particle, simulated as a point singularity, along the most optimal path from its creation on a source up to the detection. The pilot wave is a real hidden wave, which is similar to the wave function resulting from the solution of the Schrödinger equation. This theory is in agreement with de Broglie’s postulate about the matter waves. In this article we are based on the Helmholtz decomposition theorem according to which any velocity may be represented as a sum of two velocities – irrotational and solenoidal ones. The first velocity stems from the gradient of the scalar field. The second occurs from a pseudo-vector field. We proclaim that the gradients of the scalar field define guiding paths of the pilot wave. While the pseudo-vector field defines a particle solenoidal filling. We give mathematical models of the irrotational and solenoidal flows simulating the position of a particle in a guiding wave. Modified Navier-Stokes equation in pair with the continuity equation resulting in the Schrödinger equation gives such solutions consisting of superposition of the irrotational and solenoidal flow. It is declared that the guiding wave forms from the irrotational flows. In turn, the solenoidal flows underlie forming particles that travel along optimal paths slave by the guiding wave. There are described mathematical solutions of the spherical particle washed by the superfluid representing the guiding wave along which it travels.

On the basis of the findings of this paper, the theories of quantum mechanics and general relativity can be expressed in a comprehensive theory. The events were surveyed length of the time, in the Euclidean six-dimensional space-time (R6=x1+x2+x3+t1+t2+t3). Previous studies have not studied the relationship between the origin of matter's inherent properties, fundamental constants, and quantum mechanics phenomenon with space-time geometry. This theory prediction the with certainty obtained probabilistic results of quantum mechanics. This prediction is on the basis of metric resulting from the Eccentricity of the Ellipse due to the object density, in the length of time. The cause and nature of dark matter and dark energy are the findings of this article.

We show that the entropy or information flow of a gravitational system constrains the dependence of gravitational force on radial distance r which for very large r is equivalent to gravity in two dimensions. This explains the "origin" of dark matter and provides a natural solution to the problem of flat rotation curves of galaxies at very large distances and a larger deflection angle for a gravitational lens.

This study aims to assess trends and significant changes in precipitation and river flow in the Senegal River basin and its upper basin over the period 1982-2021. Eight hydroclimate indicators, namely maximum river discharge (QMAX), standardized flow index, mean daily rainfall intensity index (SDII), maximum 5-day consecutive precipitation (RX5DAY), annual precipitation exceeding the 95th percentile (R95p), annual precipitation exceeding the 99th percentile (R99P), annual flows exceeding the 95th percentile (Q95p), and annual flows exceeding the 99th percentile (Q95p) were considered.. The modified Mann-Kendall test (MMK) was used to analyze trends in extremes while standard normal homogeneity and Pettit’s tests were employed to detect potential breakpoints in these trends. The results indicated an irregular precipitation pattern, with high values of extreme precipitation indices (R95p, R99p, SDII, and RX5DAY) in the southern part, while the northern part exhibits lower values. Inter-annual analysis revealed a significant increase in extreme precipitation events between 1982 and 2021, with a notable breakpoint around the years 2006 and 2007. This breakpoint marks a transition to a much wetter period starting from 2008. Regarding extreme flows, a significant increase was observed between 1982 and 2021. This study provides insights into past hydrological extremes in the study area and can serve as a basis for future research in this field.

The present analysis deals with the study of the $f(Q, T )$ theory of gravity which was recently considered by many cosmologists. In this theory of gravity, the action is taken as an arbitrary function $f(Q, T )$ where $Q$ is non-metricity and $T$ is the trace of energy-momentum tensor for matter fluid. In this study, we have taken three different forms of the function $f(Q, T )$ as $f(Q,T)=a_1Q+a_2 T$ and $f(Q,T)=a_3 Q^2+a_4 T$ and discussed some physical properties of the same. Also, we have obtained the various cosmological parameters towards Friedmann-Lemaitre-Robertson-walker (FLRW) Universe by defining the transit form of scale factor which yields the Hubble parameter in redshift form as $H(z)=\frac{H_{0}}{(\lambda+1)} \left(\lambda+ (1+z)^{\delta}\right)$. %We have obtained the approx best-fit values of model parameters using the least square method for observational constraints on available datasets like Hubble dataset $H(z)$, Supernova dataset SNe-Ia, etc., by applying the Root Mean Squared Error formula (RMSE). By applying the Root Mean Squared Error formula (RMSE), we were able to determine the approximate best-fit values of model parameters using the least square approach for observational constraints on the datasets Hubble dataset $H(z)$, Supernova dataset SNe-Ia, etc. We have observed that the deceleration parameter $q(z)$ exhibits a signature-flipping (transition) point within the range $0.623 \le z_{0} \le 1.668$ through which it changes its phase from the decelerated to the accelerated expanding universe with $\omega = -1$ at $z=-1$ for the approximate best-fit values of the model parameters. %For obtained approx best-fit values of model parameters we have observed that the deceleration parameter $q(z)$ shows a signature-flipping (transition) point within the range $0.623 \le z_{0} \le 1.668$ through which it changes its phase from decelerated to the accelerated expanding universe with $\omega = -1$ at $z=-1$.

G4v is a four-vector theory of gravitation proposed by Mead (2015, 2023a,b) which reinterprets several major results commonly celebrated for the tensor-based approach of General Relativity (GR). Here, we show that G4v obeys the correspondence principle, an important test of any relativistic theory. We derive the non-relativistic Lagrangian and Energy expressions at the limit of low speeds and potentials, complete with the correct terms for potential energy (something traditional Special Relativity is incapable of achieving). G4v could represent a new interpretation of the gravitational field with essentially the same mathematics as electromagnetism (Mead 2023a). It is also possible to interpret G4v as a very interesting (and useful) approximation of the Einstein field equations (Mead 2023b).

Quantum physics through the lenses of Bayesian statistics consider probabilities to be degree of believes or knowledge and subjective. A Bayesian derivation of the probability density function in phase space is presented. The entropy proposed by Geiger and Kedem \cite{GeiKed2022a,GeiKed2022b} in phase space is expanded to define quantitatively interference and entanglement. Comparisons between each of these two quantities and the entropy are made. A brief presentation of entanglement in phase space to the spin degree of freedom and to mixed states completes the work.

There are gravitational field and baryonic matter (star, planet, gases etc.) in our world; gravitational field is much more fundamental for that the relativistic mass of the gravitational field of a neutron star is almost 10²¹ times the mass of the neutron star. And, it is astonishing that the relativistic mass density of the gravitational field on the surface of a neutron star can be larger than the mass density of the neutron star. It is well known that, in the weak gravitational field a light line appears a straight line while in the strong field it is a curved line. It is stressed that the result measured in the strong field is more fundamental for that in a galaxy the measurable object is generally with strong field. It results in that the way that the result measured on the Earth with weak field is used as the universal standard need be revised as gravity is studied in the scale of a galaxy. It seems that a new theory of gravity need be developed for the new observations.

In this research, we investigate the bending of photon light and/or time-like geodesics within the context of four-dimensional space-time geometry, specifically in the presence of topological defects. These topological defects manifest as cosmic string space-time configurations, encompassing screw dislocation, spiral dislocation, and spacelike dislocation. Our primary objective is to determine the extent of deflection experienced by light-like and time-like geodesics and analyze how this deflection is affected by the cosmic string parameter, screw dislocation parameter, and spacelike dislocation parameter. To accomplish this, we employ a derivation technique that centers on the observation that photon rays (light-like geodesics) and/or time-like geodesics follow paths parallel to the z-axis while maintaining a constant radius along the cylinder.

In physics, symmetry is mostly about the symmetry between elementary particles, between matter and antimatter. In the universe, we have two fundamental types of energy: energy of matter and energy of space. These two energies have a symmetry that is not geometry type, not electric charge type, it is energy type. Every physical object with mass m symmetrically diminishes the energy density of space in its center exactly for the amount of its mass. This symmetry of matter and space energies is the physical source of gravity that is also valid in the centre of black holes.

This article is the second part of a scientific project under the general name "Geometrized vacuum physics". On the basis of the Algebra of Stignatures presented in the previous article [1], this article develops the main provisions of the Algebra of Signatures. Both of the above algebras are aimed at studying the properties of an ideal vacuum, but at the same time they are universal and can be applied in various branches of knowledge. It is shown that the signature of a quadratic form is related to the topology of the metric space for which the given quadratic form is a metric. Conditions are given under which an additive imposition of metric spaces with different topologies (or signatures) leads to a total Ricci flat space similar to a Calabi-Yau manifold. A spin-tensor representation of metrics with different signatures is considered and a Dirac bundle of quadratic forms is presented. This article does not contain physical applications of the Algebra of Signatures, but the potential power of this mathematical apparatus will be demonstrated in subsequent articles of this project.

According to Quantum Mechanics a narrow wave-package of the center of mass of macroscopic objects, due to their heavy mass, remains stable over astronomical times. However Quantum Mechanics allows and energetically even prefers largely smeared out c.m. states that never been seen. Why is this the real state of the world we know? Does it suggest a micro-macro threshold with different theoretical descriptions?

Using arguments originally introduced by Einstein we show that the reductionist philosophy of quantum mechanics reverses the natural order. In quantum mechanics emission is followed by observation and interpreted classically with the collapse of the wave function, whereas in nature classical absorptions of energy precede emission. Also momentum transfers leading to emissions are asymmetric with respect to spatial orientation and time, but are disregarded in quantum mechanics because molecular impulses cannot be observed. Thus despite irrefutable physical evidence that microscopic processes evolve asymmetrically the wave function is assumed to apply symmetrically in time. The deficiencies are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. The calculus of variations is then applied to the wave function to show that it is an incomplete equation of motion because it yields twice the allowable action minimum thereby allowing the concept of reality to be redefined.

The correspondence between classical and quaternion quantum equations, permits considering the universe (vacuum) as an ideal elastic solid. Elementary particles would have to be standing or soliton-like waves. Tension induced by the compression and twisting of the elastic medium would increase the energy density, consequently generate a gravity forcing and affect the wave speed. Consequently the gravity could be described by an index of refraction. Theory was created by combining the Cauchy model of the elastic continuum with the Planck-Kleinert crystal hypothesis. The quaternion-imaginary Lagrange’an, the quaternion motion equation and the quaternionic oscillator allowed deriving: - The Schrödinger equation from the functional integral, which identifies the quaternion-imaginary quantum Hamiltonian. - The 2nd order wave equation system describing both the bosons and the gravity in terms of quaternionic Poisson equation. - The first order quaternionic wave equation system. - The family of the second order wave equation systems describing both the particles and the generated quaternionic force-fields (four-potentials). - The fundamental Planck and gravity constants. - The quaternionic continuity equation in an ideal elastic solid.

In a series of recent papers we developed a formulation of general relativity in which spacetime and the dynamics of matter evolve with a Poincaré invariant parameter τ. In this paper we apply the formalism to derive the metric induced by a `static’ event evolving uniformly along its t-axis at the spatial origin x=0. The metric is shown to vary with t and τ, as well as spatial distance r, taking its maximum value for a test particle at the retarded time τ=t−r/c. In the resulting picture, an event localized is space and time produces a metric field similarly localized, where both evolve in τ. We first derive this metric as a solution to the wave equation in linearized field theory, and discuss its limitations by studying the geodesic motion it produces for an evolving event. By then examining this solution in the 4+1 formalism, which poses an initial value problem for the metric under τ-evolution, we clarify these limitations and indicate how they may be overcome in a solution to the full nonlinear field equations.

Starting with a particle in the cube system, we investigate the exotic possibility that the quantum numbers (n_x,n_y,n_z) satisfy the Markov equation. We find that the proposed particle in the Markov-cube system follows from a non-classical information entropic space by the quantum potential Q. We study the classical correspondents of the proposed particle in the Markov-cube system. According to our proposition, this particle identified by detailed measurements of the confinement particle-energy levels is an artifact or occurs in nature.

The primary concept of time in special and general relativity (SR, GR) and quantum mechanics (QM) is coordinate time t. Here I show: SR and GR are mathematically correct, but physically t has an issue. It takes an observer as the center of time, just as the geocentric model takes Earth as the center of space. In Euclidean relativity (ER), the roles of t and proper time τ have switched. Time dilation is interpreted differently: In ER, an observed clock is slow with respect to an observer in his proper flow of time (not in its proper flow of time as in SR/GR). All energy is moving through 4D Euclidean space (ES) at the speed c. All four dimensions are distance, and “cosmic time” t is the total distance covered in ES divided by c. Unlike in previous ER models, an observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time. The Lorentz factor and gravitational time dilation are recovered in ER. So, ER predicts the same relativistic effects as SR and GR. Yet ER outperforms SR in solving time’s arrow and the c² in mc². ER also outperforms a GR-based cosmology in explaining the data from high-redshift supernovae while declaring cosmic inflation, expansion of space, dark energy, and quantum gravity redundant. ER even improves our understanding of QM: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant. I conclude: The true pillars of physics are ER and QM.

Information is physically measurable as a selection from a set of possibilities, the domain of information. This defines the term "information". The domain of the information must be known together reproducibly beforehand. As a practical consequence, digital information exchange can be made globally efficient, interoperable and searchable to a large extent by online definition of application-optimized domains of information. There are even more far-reaching consequences for physics. The purpose of this article is to present prerequisites and possibilities for a physical approach that is consistent with the precise definition of information. This concerns not only the discretization of the sets of possible experimental results, but also the order of their definition along time. The access to or comparison with the domain of information is the more frequent, the earlier it was defined. The geometrical appearance of our space is apparently a delayed statistical consequence of a very frequent connection with the common primary domain of information.

When generalized noncommutative Heisenberg algebra accommodating impacts of finite gravitational fields as specified by loop quantum gravity, doubly–special relativity, and string theory, for instance, is thoughtfully applied to the eight-dimensional manifold (M8), the generalization of the Riemannian manifold becomes imminent. By constructing the deformed affine connections on a four-dimensional Riemannian manifold, we have determined the minimal length deformation of Riemann curvature tensor and its contractions, the Ricci curvature tensor, and Ricci scalar. Consequently, we have been able to construct the deformed Einstein tensor. As in Einstein’s classical theory of general relativity, we have proved that the covariant derivative of the deformed Einstein tensor vanishes, as well. We conclude that the minimal length correction suggests a correction to the spacetime curvature which is manifest in the additional curvature terms of the corrected Riemann tensor and its contractions. Accordingly, the spacetime curvature endows additional curvature and geometrical structure.

When the minimal length approach emerging from noncommutative Heisenberg algebra, generalized uncertainty principle (GUP), and thereby integrating gravitational fields to this fundamental theory of quantum mechanics (QM) is thoughtfully extended to Einstein field equations, the possible deformation of the metric tensor could be suggested. This is a complementary term combining the effects of QM and general relativity (GR) and comprising noncommutative algebra together with maximal spacelike four–acceleration. This deformation compiles with GR as curvature in relativistic eight–dimensional spacetime tangent bundle, generalization of Riemannian spacetime, is the recipe applied to derive the deformed metric tensor. This dictates how the affine connection on Riemannian manifold is straightforwardly deformed. We have discussed the symmetric property of deformed metric tensor and affine connection. Also, we have evaluated the dependence of a parallel transported tangent vector on the spacelike four–acceleration given in units of L, where L = rℏcG3 is a universal constant, c is speed of light, and ℏ is Planck constant, and G is Newton’s gravitational constant.

Epsilon-near-zero nanoantennas can be used to tune the far-field radiation pattern due to their exceptionally large intensity-dependent refractive index. Here, we propose hybrid optical antenna based on indium tin oxide (ITO) to optically tune the deflection of radiation. In particular, a hybrid structure antenna of ITO and dielectric material, which makes the deflection angle changes 17 ∘ as incident intensities increases. Finally, the array of ITO or hybrid nanodisks can enhance the unidirectionality to be needle-like, with the angular beam width α&lt; 8∘ of main lobe. The deflection angle of radiation pattern response with needle-like lobe pave the way for further studies and applications in beam steering and optical modulation where dynamic control of the nanoantennas is highly desirable.

The classical Einstein’s Field Equation has been derived previously by demanding that the Clausius relation dS = δQ/T holds. In this letter, we show that the semiclassical Einstein’s Field Equation can be recovered using the generalized entropy Sgen by using the non-equilibrium thermodynamics where the Clausius relation is replaced by the entropy balance relation dS = δQ/T +diS, where diS is some internal entropy. Moreover, since the quantum fields violate the classical focusing, we, therefore, have to use quantum focusing in this case. Using then Jacobson’s idea of thermodynamics of spacetime we recover the semiclassical equation of motion. We, therefore, in a sense also show the validity of the semiclassical approximation, a crucial approach for establishing a number of important ideas such as the Hawking effect.

We consider a specific holographic model in the limit where the gauge field does not couple to the rest of of the holographic fields (gravity and scalar sector) and investigate a phase of matter at zero charge density, a realistic feature that may have implications for disordered strange metals. We then pick a specific form of the gauge coupling $Z(\alpha)$ with a certain disorder realization and argue that this provides a hard-gapped insulator with exponentially-suppressed conductivity by holographic methods. The limit is non-trivial: as there is backreaction at zero density amount, surviving in the coupled case.

In this paper, inspired by quantum field theory, or more specifically QED, we propose a dynamical model for relativity. By adopting the approach provided by this dynamical model, we provide a dynamical explanation for relativistic phenomena such as length contraction, time dilation and clock desynchronization. The result of the dynamical approach for clock desynchronization is especially interesting. Our dynamical approach exhausts the freedom for different clock synchronizations on frames accelerated from an initial reference frame, which is allowed by Einstein’s principle approach. Furthermore, our dynamical approach arrives at the interesting conclusion that Einstein’s equivalence principle and gravitational redshift are necessary to explain special relativistic clock desynchronization. This result shows that special relativity by itself is an incomplete theory. If our dynamical model assumptions are assumed to be valid at a fundamental level, it implies some modifications to quantum field theory and general relativity via the equivalence principle. These modifications have been discussed.

The theory of special relativity has reexamined and reconstructed the concepts of space and time, which are indispensable for describing physical laws, consequently instigating a major revolution in physics in general. Although the correctness of the special theory of relativity has been con-firmed by many observations and experiments, it is unclear why the "principle of relativity" and the "principle of the constant speed of light," which form the basis of the idea, are valid. Moreo-ver, although special relativity has advanced our understanding of time, our understanding of the fundamental question, "What is time?" is still limited. The theory of special relativity regards space as a four-dimensional construct consisting of one-dimensional time and three-dimensional space. This paper proposes an alternative framework, wherein real space is uniform and isotropic four-dimensional space, and elementary particles are waves that propagate through this four-dimensional space at a constant speed. This new perspective provides evidence to support two of Einstein’s key principles—the "principle of relativity" and the "principle of the constant speed of light." Furthermore, this model clarifies what time is and why time flows at a constant rate from the past to the future.

Within the framework of the previously proposed formulation of the quantum theory of gravity in terms of world histories, it was suggested that the universe has its own mass. This quantity is analogous to the mass of a particle in relativistic mechanics. The mass of the universe is a distribution of non-zero values of gravitational constraints, which arises and changes in time as a consequence of the initial conditions for fundamental dynamic variables. A formulation of the Euclidean quantum theory of gravity is also proposed to determine the initial state, which can be the source of the universe’s own mass. Being unrelated to ordinary matter, the distribution of its own mass affects the geometry of space and forms a dedicated frame of reference. The existence of selected reference systems is taken into account by the corresponding modification of the system of quantum gravitational constraints. A variant of such a modification of the Wheeler-De Witt equation is the operator representation of gravitational constraints, which, together with the state of the universe, determines the parameters of the reference system in the form of a distribution of the spinor field on a spatial section.

The Macroscopic Mechanical Equilibrium Condition can be shown to be the physical origin of the Equation of State of any material system, then can describe all thermodynamic quasi-equilibrium processes caused by the change of the external temperature and/or mechanical environment. If it cannot be satisfied, the system would be broken.

In a recent article we showed that the analog of the cosmological constant in two spacetime dimensions for a wide variety of integrable quantum field theories has the form ρvac = −m2/2g where m is a physical mass and g is a generalized coupling, where in the free field limit g → 0, ρvac diverges. We speculated that in four spacetime dimensions ρvac takes a similar form ρvac = −m4/2g, but did not support this idea in any specific model. In this article we study this problem for λφ4theory in d spacetime dimensions. We show how to obtain the exact ρvac for the sinh-Gordon theory in the weak coupling limit by using a saddle point approximation. This calculation indicates that the cosmological constant can be well-defined, positive or negative, without spontaneous symmetry breaking. We also show that ρvac satisfies a Callan-Symanzik type of renormalization group equation. For the most interesting case physically, ρvac is positive and can arise from a marginally relevant negative coupling g and the cosmological constant flows to zero at low energies.

The phenomena of electrical conductivity and electromigration in metallic systems are related, since in both cases the basic physical process is the scattering of conduction electrons by metal ions. Numerous searches have been made for equations connecting the conductivity with electromigration. In the case of a liquid metal, when using the Drude-Sommerfeld (DS) conduc-tivity equation, it was not possible to obtain a quantitative relationship between these phenomena, which would be correct. Attempts to find such a relationship when taking into ac-count the N. Mott correction (g - factor) in the DS equation were unsuccessful. This article proposes a different correction (b - factor) to the DS equation, which takes into account the pos-sibility of varying the momentum transferred by the conduction electron to a metal ion during the scattering. This correction allows to establish a quantitative relationship between conductivity and electromigration, as well as between electromigration in various binary systems with common components, in agreement with experiment. The proposed theory describes well, in particular, two- and multi-component metal systems of any concentration (the consistency rule for triangles A-B, B-C, C-A). The value of b - factor smoothly changes depending on the heat of vaporization of the metal, per unit volume.

In network analysis, links depict the connections between each pair of network nodes. However, such pairwise connections fail to consider the interactions among more agents, which may be indirectly connected. Such non-pairwise or higher-order connections can be signified by involving simplicial complexes. The higher-order connections become even more noteworthy when it comes to neuronal network synchronization, an emerging phenomenon responsible for the many biological processes in real-world phenomena. However, involving higher-order interactions may considerably increase the computational costs. To confound this issue, map-based models are more suitable since they are faster, simpler, more flexible, and computationally more optimal. Therefore, this paper addresses the impact of pairwise and non-pairwise neuronal interactions on the synchronization state of 10 coupled memristive Hindmarsh-Rose neuron maps. To this aim, electrical, inner linking, and chemical synaptic functions are considered as 2- and 3-body interactions in three homogenous and two non-homogenous cases. The results show that through chemical pairwise and non-pairwise synapses, the neurons achieve synchrony with the weakest coupling strengths.

Coupling the Maxwell tensor to the Riemann-Christoffel curvature tensor is shown to lead to a geometricized theory of electrodynamics. While this geometricized theory leads directly to the classical Maxwell equations, it also extends their interpretation by giving charge density and mass density, and the four-velocity that describes their motion geometric definitions. These geometric definitions are reminiscent of General Relativity’s interpretation of mass in terms of the scalar curvature R and hint at the emergence of gravity. The gravitational theory that does emerge is shown to be equivalent to Einstein’s General Relativity augmented by an energy-momentum tensor term that mimics the properties of dark matter and/or dark energy. In summary, the proposed geometrization of the Maxwell tensor puts both electromagnetic and gravitational phenomena on an equal footing with both being tied to the curvature of space-time. Using specific solutions to the proposed theory, the unification brought to electromagnetic and gravitational phenomena, as well as the relationship of those solutions to the corresponding solutions of the classical Maxwell and Einstein field equations are compared.

Abstract The value of the mass of the particles is one more mystery that current theories cannot predict, so they have to be determined experimentally. In this paper we will see how the quantization of 4D space gives rise to the quantization of mass and, in turn, the quantization of mass implies that space-time is discrete.

We propose a promising model of discrete spacetime based on nonassociative geometry and complex networks. Our approach treats space as a simplicial 3-complex (or complex network), built from "atoms" of spacetime and entangled states forming n-dimensional simplices (n=1,2,3). At large scales, a highly connected network is a coarse, discrete representation of a smooth spacetime. We show that for high temperatures, the network describes disconnected discrete space. At the Planck temperature, the system experiences the phase transition, and for low temperatures, the space becomes a triangulated discrete space. We show that the cosmological constant depends on the universe’s topology. The "foamy" structure, analogous to Wheeler’s "spacetime foam," significantly contributes to the effective cosmological constant, which is determined by the Euler characteristic of the universe.

A substantial body of phenomenological and theoretical work over the last few years strengthens the possibility that the vacuum energy density (VED) of the universe is dynamical, and in particular that it adopts the `running vacuum model' (RVM) form, in which the VED evolves mildly as $\delta \rho_{\rm vac}(H)\sim \nu_{\rm eff} m_{\rm Pl}^2{\cal O}\left(H^2\right)$, where $H$ is the Hubble rate and $\nu_{\rm eff}$ is a (small) free parameter. This dynamical scenario is grounded on recent studies of quantum field theory (QFT) in curved spacetime and also on string theory. It turns out that what we call the `cosmological constant', $\Lambda$, is no longer a rigid parameter but the nearly sustained value of $8\pi G(H)\rho_{\rm vac}(H)$ around (any) given epoch $H(t)$, where $G(H)$ is the gravitational coupling, which can also be very mildly running (logarithmically). Of particular interest is the possibility suggested in past works that such a running may help to cure the cosmological tensions afflicting the $\Lambda$CDM. In the current study, we reanalyze it in full and we find it becomes further buttressed. Using the modern cosmological data, namely a compilation of the latest SNIa+BAO+$H(z)$+LSS+CMB observations, we probe to which extent the RVM provides a quality fit better than the concordance $\Lambda$CDM model, paying particular emphasis on its impact on the $\sigma_8$ and $H_0$ tensions. We utilize the Einstein-Boltzmann system solver \texttt{CLASS} and the Monte Carlo sampler \texttt{MontePython} for the statistical analysis, as well as the statistical DIC criterion to compare the running vacuum against the rigid vacuum ($\nu_{\rm eff} = 0$). On fundamental grounds, $\nu_{\rm eff}$ receives contributions from all the quantized matter fields in FLRW spacetime. We show that with a tiny amount of vacuum dynamics ($|\nu_{\rm eff}|\ll 1$) the global fit can improve significantly with respect to the $\CC$CDM and the mentioned tensions may subside to inconspicuous levels.

This paper is another applied article based on the author's "Unified Theory of Force". The main work of this paper is as follows: (1) Reinterpretation of the Lorentz contraction effect and rigorous proof that the Lorentz contraction is a physical phenomenon based on the conservation of energy and momentum under non-completely symmetrical relative motion effects. This paper proves that the Lorentz contraction effect not only occurs in the direction of motion but also simultaneously in any direction. (2) For the first time, we attempted to explain the physical meaning of the Planck energy formula in theory and associate the Planck energy formula with the Lorentz contraction effect, which can help us to have a deeper understanding of the true physical meaning behind the Planck energy formula. At the same time, we also made reasonable theoretical predictions about the basic structure of photons.

The author introduces a novel theoretical framework that suggests matter and energy are both converted from curved space. In Space Theory, gravitation is generated by the flow of space, instead of being transmitted by the graviton as in String Theory. This theory also suggests that Newton's gravitational constant, denoted as G, may not be truly constant but could vary over time. The equivalent equation of space is S=Ec²=mc⁴, and for the gravitation of hollow sphere space, the equation is S_μν=4πGm=(4/3)π((r+a)³-r³).

Affine quantization is a relatively new procedure, and it can solve many new problems. This essay reviews this new, and novel, procedure for particle problems, as well as those of fields and gravity. New quantization tools, which are extremely close to, and even constructed from, the tools of canonical quantization, are able to fully solve selected problems that using the standard canonical quantization would fail. In particular, improvements can even be found with an affine quantization of fields, as well as gravity.

In this work, we propose a transition wave equation from quantum to classical regime in the framework of the von Neumann formalism for ensembles and then obtain an equivalent scaled equation. This leads us to develop a scaled statistical theory following the well-known Wigner-Moyal approach of quantum mechanics. This scaled nonequilibrium statistical mechanics has in it all the ingredients of the classical and quantum theory described in terms of a continuous parameter displaying all the dynamical regimes in-between the two extreme cases. Finally, a simple application of our scaled formalism consisting of reflection from a mirror by computing various quantities including probability density plots, scaled trajectories and arrival times are analyzed.

With simple but rigorous arguments we prove that the ordinary second quantization of bosonic and fermionc fields is formally equivalent to constraining the elementary degrees of freedom of the classical fields to have intrinsically periodic dynamics in time. This result confirms and extends the formal equivalence obtained in previous papers for general Hamiltonian systems in terms of canonical quantization, for elementary relativistic particles also in terms of the Feynman Path Integrals, as well as other remarkable correspondences of both phenomenological and theoretical fundamental interest.

The evolution of even and odd Schrödinger cat states of the inverted oscillator is obtained in the center-of-mass tomographic probability description of the two-mode oscillator.The notion of entangled probability distribution is reviewed. Evolution equations describing the time-dependence of probability distributions identified with quantum system states are discussed.The connection with the Schrödinger equation and the von Neumann equation is clarified.

The many problems faced by the theory of general relativity (GR) have always motivated us to explore the modified theory of GR. Considering the importance of studying the black hole (BH) entropy and its correction in gravity physics, we study the correction of thermodynamic entropy for a kind of spherically symmetric black holes under the generalized Brans-Dicke (GBD) theory of modified gravity. We derive and calculate the entropy and heat capacity. It is found that when the value of event horizon radius~$r_+$~is small, the effect of the entropy correction term on the entropy is very obvious, while for larger values~$r_+$~, the contribution of the correction term on entropy can be almost ignored. In addition, we can observe that as the radius of the event horizon increases, the heat capacity of BH in GBD theory will change from a negative value to a positive value, indicating that there is a phase transition in black holes. Given that studying the structure of geodesic lines is important for exploring the physical characteristics of a strong gravitational field, we also investigate the stability of the geodesic motion of particles in static spherically symmetric BHs within the framework of GBD theory. Concretely, we analyzes the dependence of the innermost stable circular orbit on model parameters. Also, the geodesic deviation equation is also applied to investigate the stable circular orbit of particles in GBD theory. The conditions for the stability of BH solution and the limited range of radial coordinates required to achieve stable circular orbit motion are given. Finally, we show the locations of stable circular orbits, and obtain the angular velocity, the specific energy and angular momentum of the particles which move in circular orbits.

We study the Kelvinons: monopole ring solutions to the Euler equations, regularized as the Burgers vortex in the viscous core. There is finite anomalous dissipation in the inviscid limit. However, in the anomalous Hamiltonian, some terms are growing as logarithms of Reynolds number; these terms come from the core of the Burgers vortex. In our theory, the turbulent multifractal phenomenon is similar to asymptotic freedom in QCD, with these logarithmic terms summed up by an RG equation. The small effective coupling does not imply small velocity; on the contrary, velocity is large compared to its fluctuations, which opens the way for a quantitative theory. In the leading order in the perturbation theory in this effective coupling constant, we compute running multifractal dimensions for high moments of velocity circulation, in good agreement with the data for quantum Turbulence and available data for classical Turbulence. The logarithmic dependence of fractal dimensions on the loop size comes from the running coupling in anomalous dimensions. This slow logarithmic drift of fractal dimensions would be barely observable at Reynolds numbers achievable at modern DNS.

Quantum squeezing, an intriguing phenomenon that amplifies the uncertainty of one variable while diminishing that of its conjugate, may be studied as a time-dependent process, with exact solutions frequently derived from frameworks grounded in adiabatic invariants. Remarkably, we reveal that exact solutions can be ascertained in the presence of time-variant elastic forces, eschewing dependence on invariants or frozen eigenstates formalism. Delving into these solutions as an inverse problem unveils their direct connection to the design of elastic fields, responsible for inducing squeezing transformations onto canonical variables. Of particular note, the dynamic transformations under investigation belong to a class of gentle quantum operations, distinguished by their delicate manipulation of particles, thereby circumventing the abrupt energy surges commonplace in conventional control protocols.

We explore several aspects of replica synchronization with the goas of retrieving the value of parameters for the Lorenz system. The idea is that of having a computer replica (slave) of a natural system (master, simulated in this paper), and exploit the fact that slave synchronizes with the master only if they evolve with the same parameters. As a byproduct, in the synchronized phase the state variables of the slave and that of the master are the same, thus allowing to perform measurements impossible on the real system. We review some aspects of master-slave synchronization using a subset of variables, with intermittent coupling. We show how synchronization can be achieved when some of the state variables are available for direct measurement using a simulated annealing approach, and also when they are accessible only through a scalar function, using a pruned-enriching ensemble approach, similar to genetic algorithms without cross-over.

It is often claimed that Einstein is wrong about quantum mechanics. However, when comparisons are made with respect to theoretical foundations rather than experimental results Einstein’s theories are found to be superior. Although quantum mechanics correctly predicts what it is possible to observe (the emissions) it ignores the other half of natural phenomena, what cannot be observed (the absorptions), thereby violating the conservation laws. By describing only one-half of radiation processes conceptual difficulties such as wave function collapse, infinite paths, and inscrutable mathematics seem to be insoluble mysteries. The deficiencies are corrected by introducing Hamilton’s principle S = ∫ Ldt and applying the methods of quantum field theory to derive relativistic equations of motion. The calculus of variations is used to show that the wave function represents an incomplete equation of motion because it equals twice the allowable action minimum.

We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, $Q_8$, afforded by multiplying one of the $\gamma$-matrices by the imaginary number $i$. The reason for doing this is to introduce a bivector into the spin algebra. The resulting spin equation separates into distinct and complementary spaces, one describing polarization and the other coherence. The former describes a 2D structured spin and the latter its helicity, generated by a unit quaternion. We discuss some consequences.

We still do not have satisfactory answers about the inflationary and accelerating expansions of the universe, the missing mass problem, and the origins of the four fundamental forces and the quantum phenomena. Here, we introduce a new model of matter and space and a postulate on a constant of motion, and then show that an equation of motion derived from the constant of motion could provide new understandings of the above problems. For example, we show that there is a natural inflation scenario and that the co-moving distance to the closest accelerating supernova is clearly given as $c/(\sqrt{3} H)$ ($\sim$8.1 billion ly). In addition, we show that, in a system including a proton and an electron, the equation of motion has eight acceleration terms, some of which are suspected to be responsible for the four fundamental forces and the galaxy rotation anomaly. From further discussion, we conjecture that in order to understand the gravitational and strong interactions and the quantum phenomena, it is necessary to consider the expansion of the universe. The lensing mass distribution is explained, based on another postulate on a method of representing superposition.

The exceptional Jordan algebra is the algebra of $3 \times 3$ Hermitian matrices with octonionic entries. It is the only one from Jordan’s algebraic formulation of quantum mechanics which is not equivalent to the conventional formulation of quantum theory. It has often been suggested that this exceptional algebra could explain physical phenomena not currently explained by the conventional approach, such as values of the fundamental constants of the standard model of particle physics, and their relation to gravitation. We show that this is indeed the case; and this also unravels the connection between general relativity and the weak force. The exceptional Jordan algebra also predicts a new U(1) gravitational interaction which modifies general relativity, and which provides a theoretical basis for understanding the Modified Newtonian Dynamics (MOND).

The gravitational constant is an unchanging quantity included in the law of universal gravitation. The value of this constant is known much less accurately than other fundamental physical constants, which is due to the experimental difficulties of measuring it. Here we report a simple method for the theoretical calculation of the gravitational constant using the elementary charge, speed of light, Z boson mass, and relativistic mechanics.

The time interval set approach depends on results for radiation propagation from star sources, ie spherically symmetric, where properties relate to propagation sphere surfaces, finite for any realistic event and measurement. In contrast the usual vector and one moment time set approach from Newton’s laws, applies to stationary parallel radiation. A time interval only description necessarily has to start from it's own definitions, and properties for time intervals have to be defined with time intervals. This paper is devoted to deriving time interval only set properties commutation, addition and multiplication and derivative, and to define how these can correspond consistently to one moment time set properties. Time development within the time interval description equilibrium depends on the well-known ‘mean velocity theorem’ (MVT) and the time interval version of the Legendre transform, defined in earlier papers. Radiation propagation ‘away from’ and gravitation ‘towards’ the star source can be part of the description. In the discussion parts these properties are applied within astrophysics. Included is a integral derivation of star source radiation energy for both zero and non zero wave particle mass. The time interval set is not one of the usual sets encountered. However it’s non-linearity or non-commutativity property can be related to Noether charges and structure constants, that equal the radiation wave group-velocity, for photons the light velocity c, and, different from the usual derivative to one moment time, to specifics of the time interval only set derivative to time intervals.

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Consequently, what is ruled out by the Bell-test experiments is not local realism but the additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with.

Application of the Ramsey graph theory to the analysis of physical systems is reported. Physical interactions may be very generally classified as attractive and repulsive. This classification creates the premises for the application of the Ramsey theory to the analysis of physical systems built of electrical charges, electric and magnetic dipoles. The notions of mathematical logic, such as transitivity and intransitivity relations, become crucial for understanding of the behavior of physical systems. The Ramsey theory explains why nature prefers cubic lattices over hexagonal ones for systems built of electric or magnetic dipoles. The Ramsey approach may be applied to the analysis of mechanical systems when actual and virtual paths between the states in the configurational space are considered. Irreversible mechanical and thermodynamic processes are seen within the reported approach as directed graphs. Chains of irreversible processes appear as transitive tournaments. These tournaments are acyclic; the transitive tournaments necessarily contain the Hamiltonian path. The set of states in the phase space of the physical system, between which irreversible processes are possible, is considered. The Hamiltonian path of the tournament emerging from the graph uniting these states is a relativistic invariant. Applications of the Ramsey theory to the general relativity become possible when the discrete changes in the metric tensor are assumed. Reconsideration of the concept of “simultaneity” within the Ramsey approach is reported.

In previous works we have developed the concept of a relativistic engine taking into account the phenomena of retardation of electromagnetic fields. We have shown that Newton’n third law cannot strictly hold in a distributed system, where the different parts are at a finite distance from each other and thus force imbalance is created at the system’s center of mass. As the system is affected by a total force for a finite period, mechanical momentum and energy are acquired by the system. In previous works we relied on the fact that the bodies were macroscopically natural. Lately we relaxed this assumption and studied charged bodies, thus analyzing the consequences on a possible electric relativistic engine. On the first paper on this subject we investigated this phenomena in general but gave an example of a system only at the stage of reaching a stationary state, a second paper was devoted to the same but on the atomic scale, here we shall analyze the charge relativistic engine in a general time dependent setting.

Light travels in a moving medium through densities that differ from the density of that medium at rest. This study derives a formula for light velocity through a moving medium for any direction of light propagation in that medium that, applied to the Fizeau experiment, gives a fringe shift of 0.23541. The Fresnel drag coefficient applied to the Fizeau experiment offers a fringe shift of 0.20466. The mean of the fringe shift observations in the Fizeau experiment of 0.23016 confirms both expectations.

On a rotating disk, two particles bounce at the vertices of a regular polygon in opposite directions. On the return to the entry point, a clock measures the time difference, called Sagnac effect. Due to Coriolis effects, the counterclockwise and clockwise paths are different. The particular case of the slow disk where the two trajectories are very close and almost polygonal is studied. The existence of a transition between a classical and a relativistic regime is proved. An experimental verification is proposed. Although the two Sagnac effects seem analogous, in detail their behavior is quite different.

We study an air-fluidized granular monolayer, composed in this case of plastic spheres, which roll on a metallic grid. The air current is adjusted so that the spheres never loose contact with the grid, so that the dynamics may be regarded as pseudo two-dimensional (or two-dimensional, if the effects of sphere rolling are not taken into account). We find two surprising continuous transitions, both of them displaying two coexisting phases. Moreover, in all cases, we found the coexisting phases display strong energy non-equipartition. In the first transition, at weak fludization, a glassy phase coexists with a disordered fluid-like phase. In the second transition, a hexagonal crystal coexists with the fluid phase. We analyze, for these two-phase systems, the specific diffusive properties of each phase, as well as the velocity correlations. Surprisingly, we find a glass phase at very low packing fraction and for a wide range of granular temperatures. Both phases are characterized also by a strong anti-correlated velocities upon collision. Thus, the dynamics observed for this quasi two-dimensional system unveils phase transitions with peculiar properties, very different from the predicted behavior in well know theories for their equilibrium counterparts.

This paper is mainly based on a stricter premise of the twin paradox and the assumption of inertial frame, discusses the properties of time and space under the premise of complete symmetry, and draws an interesting conclusion: the simultaneity of different reference frames is possible realized, and the space is relatively independent. And based on this, the twin paradox, cosmic inflation, ultra-distance action of quantum entanglement, microscopic space motion of particles, measurement problems and other phenomena are tentatively explained from a new angle. This interpretation is exploratory and new. At the same time, the author also proposes an experimental way to test the relative independence of space.At the same time, this paper attempts to strictly prove that Einstein's definition of simultaneity and spatial absoluteness in special relativity may be problematic.

Although the gravitational force is well known for its extremely weak value in the framework of ordinary macroscopic objects at our scale, the generally discredited framework of pushing gravity might lead us to discover a huge source of energy which is mostly lost during the action of the gravity force due to the large distances between nucleons compared with their diameters. In case this energy is identified and partially diverted from its role in gravity, we would benefit from a gigantic reservoir of energy reminiscent of Nikola Tesla's dream of a free unlimited energy source

The objective of this study was to evaluate the drying kinetics of the mass of jambu leaves and mass of jambu leaves in foam mat at different temperatures (50, 60 and 70 °C and thickness of 1.0 cm). The physicochemical characterization of the materials was performed before and after drying. Twelve different drying models were fitted to the drying kinetics data and the thermodynamic properties were calculated. The physicochemical parameters for the mass of leaves and foam after drying were moisture content from 2 to 7%, ash from 13 to 17%, protein content from 22 to 30%, lipids from 0.6 to 4% and total titratable acidity from 0.20 to 0.28% of tartaric acid. Wang & Singh and Midilli models were recommended for describing the drying kinetics. The use of foam mat promoted higher values of effective diffusion coefficient and activation energy and lower values of enthalpy and entropy, thus enhancing the reduction of drying time.

The dynamics of many complex systems can be classified as ordered, chaotic, or critical. Order offers stability and robustness, while chaos allows for change and adaptability. Criticality, then, is often seen as a balance required by living systems at different scales. In classical models, however, criticality is only found near phase transitions, restricting the parameter space (and thus the likelihood) of critical dynamics, as most parameters yield ``undesirable'' solutions. Here we show that this limitation is due to the homogeneity built-in these models, i.e., all elements sharing parameter values. By exploring heterogeneous versions of archetypal models in physics and computer science, we observe critical dynamics in a broader range of parameters, and thus could be more common than previously thought.

Abstract: A new method for constructing exact solutions of the General Relativity equations for a dusty matter with fractal property is proposed. This method allows to find the solution of the GR-equations in terms of matter velocities : the connection coefficients and the Ricci tensor of space-time are expressed in terms of matter velocities; the metric tensor and the matter density are found as functions of velocity from the GR-equations. The connection coefficients and the Ricci tensor are invariant with respect to the discrete scaling transformation of velocity , where is constant. Therefore, the found solution can be used to simulate the fractal properties of the cosmic web in terms of matter velocities. This solution includes isotropic and anisotropic distributions of matter density. In an isotropic case, there is a class of exact solutions including both the well-known Friedmann’s solution and a solution with a periodic distribution of the matter density in space. This last solution may be used to simulate the quasi-periodic distribution of matter in the cosmic web. It is possible that, the cosmic web and its fractal properties are the space-time primary properties. These properties are described with a deformation tensor of the space-time.

Many complex networks have a connectivity that might be only partially detected or that tends to grow over time, hence the prediction of non-observed links is a fundamental problem in network science. The aim of topological link prediction is to forecast these non-observed links by only exploiting features intrinsic to the network topology. It has a wide range of real applications, like suggesting friendships in social networks or predicting interactions in biological networks.The Cannistraci-Hebb theory is a recent achievement in network science that includes a theoretical framework to understand local-based link prediction on paths of length n. In this study we introduce two innovations: theory of modelling (science) and theory of realization (engineering). For the theory of modelling we first recall a definition of network automata as a general framework for modelling the growth of connectivity in complex networks. We then show that several deterministic models previously developed fall within this framework and we introduce novel network automata following the Cannistraci-Hebb rule. For the theory of realization, we present how to build adaptive network automata for link prediction, which incorporate multiple deterministic models of self-organization and automatically choose the rule that better explains the patterns of connectivity in the network under investigation. We compare Cannistraci-Hebb adaptive (CHA) network automaton against state-of-the-art link prediction methods such as structural perturbation method (SPM), stochastic block models (SBM) and artificial intelligence algorithms for graph embedding. CHA displays an overall higher link prediction performance across different evaluation frameworks on 1386 networks. Finally, we highlight that CHA offers the key advantage to explicitly explain the mechanistic rule of self-organization which leads to the link prediction performance, whereas SPM and graph embedding not. In comparison to CHA, SBM unfortunately shows irrelevant and unsatisfactory performance demonstrating that SBM modelling is not adequate for link prediction in real networks.

An exhaustive list of thirteen instances of reports confirming experimentally the relativistic Doppler relation, are examined. For those involving longitudinal Doppler, the non-relativistic relation is seen to be confirmed, within the reported experimental accuracies, to the same degree as the standard relativistic relation. Higher values of the speed of the emitter would be required to examine further the claimed confirmations. For those reports involving saturation spectroscopy, there is much confusion over the appropriate Doppler relation to be used, together with some serious analytical flaws. For the two cases that involve transverse Doppler, there are seen to be either serious faults in the theoretical part, or intrusion from the first order effect. Therefore, the reported conclusions - that the results for the experiments confirm the relativistic SR relation - cannot be justified by any of the experimental works.

A gravitational lens is mass-curved spacetime. Light that propagates freely, locally always on the fastest way, follows curved paths in it. The same applies to all effects propagating with the speed of light, also to gravity itself. They all have to follow the curvature of spacetime. A gravitational lens not only enhances the light from background objects but concurrently also their gravitational effect ! The "Gravitationally Lensed Gravity" (GLG) is a still scarcely noticed peculiar feature of general relativity. In [1] I presented a thought experiment to make the existence of this effect plausible. Here these considerations are described in terms of future and past light cones or null cones commonly used in relativity theory to illustrate the causal structure of spacetime. In the framework of general relativity, the GLG effect can be understood as a consequence of the folding of the past null cone by a gravitational lens. Starting from the basic ideas of the theory, the terms are introduced and applied. The extensive description on textbook level should be understandable also for beginners. In order not to obstruct the view on the essential physical relations, I avoid a mathematical formalism and use instead diagrams and figures, based on realistic numerical calculations, to reinforce my logical argumentation. Experts will recognize many long-known facts, but may still be surprised by the resulting conclusions. First, I present some particular aspects of the historical evolution of the concept of gravitational lensing and give a brief overview of the work of other authors on the subject GLG. I then explain the limitations of the applicability of the "Newtonian Limit" and move on to the key role of the speed of gravity in understanding GLG. Finally, I use a simple example to illustrate the possible significance of the GLG effect. With this work I intend to elicit greater interest in "Gravitationally Lensed Gravity" and to initiate a broader discussion about it.

Potential dependent special relativity and Gauss's law for the electric field have been used for gravity by assuming vacuum energy to be generated when the energy is a minimum useful expression for vacuum energy has been found. This expression shows that elementary particles are generated by gravity vacuum filling their hollow balls; using Schrödinger equation for spherically symmetric particles vacuum energy is quantized. Treating mass as vibrating spheres thus solving Schrödinger, coordinate harmonic oscillator energy relation has been found.

Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of first and second kind. In physics however, commonly the only acceptable members of this set are the Legendre polynomials. Quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone, in-depth argument for rejecting all solutions of Legendre’s equation, but the polynomial ones, in physics. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation is quintessential to the argument. We demonstrate that the evenness or oddness of the Legendre polynomials is a consequence of the same ingredients.

It is well known that the photon trajectories follow a curved path in a gravitational field. We explore here the gravitational bending of electric field lines. It seems that the electric field lines of a charge, supported in a gravitational field, follow exactly the trajectories of photons emitted isotropically from a source situated at the charge location. From a detailed examination of the electrostatic field of a charge accelerated uniformly in the instantaneous rest frame, exploiting the strong principle of equivalence, one can determine the bending of the electric field lines of a charge in a gravitational field. The fraction of electric field lines crossing a surface, stationary above or below the charge in the gravitational field, is shown to be exactly similar to the fraction of photon trajectories, emitted from a source placed at the charge location, intersecting that surface. On the other hand, for a freely falling charge in the gravitational field there is no such bending of electric field lines. The field lines continue to extend in radial straight lines from the instantaneous 'present' position of the charge, as do the trajectories of photons streaming away from the instantaneous position of a freely falling source in the gravitational field. The electric field configuration of a freely falling charge in the gravitational field is shown to be exactly the same as that of a charge moving uniformly in an inertial frame with velocity equal to the instantaneous ``present'' velocity of the freely falling charge.

Physicists have employed various approaches to solve the “paradox” and offered a consistent conclusion that the traveller twin is younger than the homebody twin. However, some authors attribute the cause of differential aging to acceleration, while others believe that acceleration should not be regarded as the source of differential aging, but frame switch is. So far there is no agreement on this issue. Actually, acceleration and frame switch should not be regarded as the source of differential aging. The author will introduce a thought experiment in which two very long spacecraft are moving in opposite directions at a speed close to the speed of light to discuss the real cause of differential aging. The author found that the real cause of differential aging in the twin “paradox” is because of the asymmetry of destination choice. People always choose their destinations in the universe rather than in spacecraft. The destinations are always stationary relative to the homebody twin and moving relative to the traveller twin. Once we choose a destination in a reference frame, the distance is the proper length in this reference frame, but the distance will contract in another moving reference frame, thus the recording time will be shorter.

We all have in mind Einstein's famous thought experiment in the elevator where we observe the free fall of a body, and then the trajectory of a light ray. Simply here, in addition to the qualitative aspect, we carry out the exact calculation, and for the first time the worldlines equations are given. We consider a uniformly accelerated reference frame in rectilinear translation, and we show that the trajectories of the particles are semi-ellipses with the center on the event horizon. The frame of reference is non-inertial, the space-time is flat, and the computations are performed within the framework of special relativity. Some experimental consequences are discussed, especially the experiment with the accelerated Michelson-Morley interferometer is solved, and we described an experiment where a new relativistic paradox appears --- a particle of matter seems to go faster than light. The differences, compared to the classical case, are important at large scale and close to the horizon, but they are small in the lift where the interest is above all theoretical. The concepts of metric, coordinated velocity and horizon are discussed, and the analogy with the black hole is made.

After proposing the Principle of Minimum Gravitational Potential, in a pursuit to find the explanation behind the correction to Newton's gravitational potential that accounts for Mercury's orbit, by finding all the higher-order corrections it is shown that the consequences of the existence of speed of light for gravity are not yet fully explored.

An axiomatic theory is proposed that reconciles the existence of an absolute scale for time (Planck time) and special relativity. According to this theory speed of light c becomes a variable which is proposed to be taken as the fifth dimension.

We study the effect of the generation of the mechanical transverse wave (MTW)　travelling in the opposite direction (OD) to a moving medium (MM) on the relativistic energy conservation law (RECL). From the viewpoint of the relativity of simultaneity (RS), the time on the coordinate coinciding with the advance end of the wave (AEW) travelling toward the rear of the MM passes faster than that on the coordinate coinciding with the wave source (WS). Then the AEW in the MM travels forward compared to that in the rest frame of reference (RFR) which is stationary relative to the medium when the time on the coordinate coinciding with the WS is same for each inertial frame of reference (IFR). Hence, the coordinate interval (CI) between the AEW and WS in the MM is observed to be larger than that between them in the RFR. We show that this difference holds true for the CI of any portion having transverse velocities mutually converted by the Lorentz transformation (LT). This difference in the CI leads to that in the rest mass (RM). We demonstrate that the RM included in wave motion (WM) in the MM is larger than one included in WM in the RFR when comparing the portions having transverse velocities mutually converted by the LT. This relation holds true for all portions in WM. Therefore, the total coordinate interval of the portion (CIP) and total RM (TRM) included in WM in the MM (WMMM) are large compared to them included in WM in the RFR. Furthermore, we compare the relativistic kinetic energy (RKE) of the MTW travelling in the OD to the MM (ODMM) with that of the MTW propagating in the direction vertical to the moving direction of the medium. We prove that the CIP and RM included in the former MTW are larger than them included in the latter MTW when comparing each portion with the same transverse velocity (TV). Moreover, the total CIP and TRM included in the former MTW are also large compared to them included in the latter MTW. The reason for these is that the latter CIP and RM are equal to them in the RFR when comparing the portions having transverse velocities mutually converted by the LT. On the other hand, the energy supplied to generate each MTW is the same. From these, we demonstrate that the RKE of the MTW travelling in the ODMM can be larger than the total relativistic energy (TRE) of the MTW propagating in the direction vertical to the moving direction of the medium. Consequently, we propose a violation of the RECL and Einstein’s principle of relativity (EPR) because the TRE is not necessarily conserved in the IFR in which the medium is moving.

In light of the self-simulation hypothesis, a simple form implementation of the principle of efficient language is discussed in a self-referential geometric quasicrystalline state sum model in three dimensions. Emergence is discussed in context of geometric state sum models.

The purpose of this paper is to demonstrate the existence of an artificial magnetic monopole and to introduce new electromagnetic equations by altering an electric field and a magnetic field vectors.As a principle device, a cylindrical condenser is prepared, and a superconducting loop is inserted into it. By this conduction, radial electric fields take a role as the centripetal force and both counterclockwise and clockwise motions are induced. As a result, a stationery wave is formed in which the nodes take a part in creating a monopole as follows.First, employing the Lorentz conservations and because node of the stationary wave has no phases, the momentum k and the vector potential A vanish and instead a magnetic potential appears in order to maintain the Lorentz conservation. This magnetic potential has relationship with an electric potential, and thus consequently, a dependent relationship is obtained between an electric field and a magnetic field vectors. Using this conclusive dependent relationship, we can derive new Maxwell equation assembly which are created by altering the electric field and the magnetic field vectors. In this process, we derive a divergent equation of magnetic fields which is not zero, i.e., the existence of a magnetic monopole. Employing these newly derived Maxwell equation, an electromagnetic wave is derived whose speed is the same as one the existing Maxwell equations provide. As a monopole configuration, this paper discusses the energy gap of the vacuum, which is a result of the Dirac equation and describes a monopole as pairs between two Cooper pairs (i.e. four electrons) whose interaction is a photon. As mentioned, because the total momenta and phases are zero, this paper defines the wave function as the Dirac function and demonstrate the condensation, employing the Bloch’s theorem. Moreover, using the macroscopic basic equations, we retrace the creation of the divergent magnetic field in view of macroscopic phenomenon., which provides results in this paper.In Result section in this paper, we succeeded in demonstrating the distribution of the divergent magnetic field of monopole in terms of both microscopic and macroscopic scales. Furthermore, Discussion section describes properties a magnetic monopole should follow.

An equation for the Fermi level of impurity isotropic semiconductors is considered when the bottom of the conduction band and the top of the valence band are at the center of the first Brillouin zone. The energy of charge carriers in the valence and conduction bands of semiconductors is considered taking into account the real positions of the top of the valence band and the bottom of the conduction band in the first Brillouin zone. A universal equation for determining the Fermi levels of impurity isotropic semiconductors is presented.

The relativistic quantum dynamics of a spin-0 scalar particle under the effects of the violation of Lorentz symmetry in the presence of a non-electromagnetic potential is analyzed. The central potential induced by the Lorentz symmetry violation is a linear electric and constant magnetic field and, analyze the effects on the eigenvalues and the wave function. We see there is a dependence of the linear charge density on the quantum numbers of the system

A given problem in physics can be solved if it is well formulated. Well formulated means that it has a bijective correspondence to physical reality. Mass Gap Problem has no bijective correspondence with the physical reality and is that’s why not solvable mathematically. It can be solved physically by the formulation of the photon’s mass accordingly to the Planck-Einstein relation.

Maxwell equation for the electric field has been solved for any medium by suggesting the wavenumber and angular frequency be complex quantities. This accounts for the field decay by interaction with the medium.This expression for the time and special decay of the electric field in a medium is used to construct a new wave function sensitive to the medium physical properties. This new wave function, unlike the conventional one, differentiates between a beam of particles in a vacuum and that enters a medium which is an attenuated due to the scattering effect.Another expression for time decaying electric field was obtained using Newton’s laws for frictional medium. This expression shows that the electric field diminishes due to friction.Fortunately, this time decaying part of the electric field is typical to that derived from Maxwell’s equations. Finally, a new Schrodinger equation sensitive to the medium properties was derived. This equation, fortunately, describes some scattering processes for Protein scattering, scattering of x-rays, opto-acoustic phonons, and Raman scattering for some materials successfully .

Density Functional Theory (DFT) full potential linearized augmented plane wave (FP-LAPW) method with the Modified Becke-Johnson (mBJ) approximation is used to perform spin polarised calculations of the transition metal perovskites MoScO₃ and WScO₃. Both depict half metallic behaviour with semiconducting and metallic in the minority and majority spins respectively. MoScO₃ and WScO₃ have indirect R− Γ band gaps in the minority spin channels of 3.61 and 3.82eV respectively. Moreover, they both show substantial magnetic moments of 2.99μ_B. In addition, we calculate the dielectric function, optical conductivity and the optical constants, namely, the refractive index, the reflectivity, the extinction and absorption coefficients.

The energy E takes the place of the gravitational mass m_G to produce gravity [1]. We wonder whether the inertial mass m_i in the inertial force is also replaced by E. The experiments to measure the inertial force and weight of a carrier are presented. Now we can test the principle of equivalence and study foundations of physics in condensed matter.

This paper discovers that current variational principle and Noether theorem for different physics systems with (in)finite freedom systems have missed the double extremum processes of the general extremum functional that both is deduced by variational principle and is necessarily taken in deducing all the physics laws, but these have not been corrected for over a century since Noether's proposing her famous theorem, which result in the crisis deducing all the physics laws. This paper discovers there is the hidden logic cycle that one assumes Euler-Lagrange equations, and then he finally deduces Euler-Lagrange equations via the equivalent relation in the whole processes in all relevant current references. This paper corrects the current key mistakes that when physics systems choose the variational extreme values, the appearing processes of the physics systems are real physics processes, otherwise, are virtual processes in all current articles, reviews and (text)books. The real physics should be what after choosing the variational extreme values of physics systems, the general extremum functional of the physics systems needs to further choose the minimum absolute extremum zero of the general extremum functional, otherwise, the appearing processes of physics systems are still virtual processes. Using the double extremum processes of the general extremum functionals, the crisis and the hidden logic cycle problem in current variational principle and current Noether theorem are solved. Furthermore, the new mathematical and physical double extremum processes and their new mathematical and physical pictures for (in)finite freedom systems are discovered. The improved variational principle and improved Noether theorem are given, which would rewrite all relevant current different branches of science, as key tools of studying and processing them.