Here we will electrically-quantum model the neutron and proton as a three-phase alternating current electric generator, for this, we will use the theory of three-phase electric generators and vector or phasor diagrams. We will use this model to hypothesize how the quarks that form neutrons and protons with a combined mass of 10 MeV/c², can generate, through quarks-antiquarks-gluon interactions, a mass of 939.56 MeV/c² for the neutron and a mass of 938.27 MeV/c² for the proton. Using quantum mechanics and the theory of the generalization of the Boltzmann constant in curved space-time we will calculate the number of quarks-antiquarks-gluons and gravitons inside a neutron. We will also analyse βˉ decay using vector diagrams, we will calculate the tension of the string T₀ and the length of the string ℓs between quarks-antiquark and finally, we will model a black hole using a neutron equivalent model. In version 2 of this article, we will update the vector diagrams of the βˉ decay and theoretically calculate the energy of the W⁻ boson, the Z⁰ boson, and the Θ angle, Θ = arc cos (W⁻/Z⁰). We will demonstrate, through vector diagrams, that quarks-antiquarks-gluon interactions are the cause of the origin of the W⁻ boson and the Z⁰ boson. We will also demonstrate how the fine structure constant α originates. It is important to make clear, in future updates, we will describe the relationship that exists between the theory described in this paper and the Isospin theory, flavour and colour theory

Nuclear technologies have a strong potential and a unique role to play in delivering reliable low carbon energy for a future net-zero society. However, to assure the sustainability required for the long-term success, nuclear will need to deliver innovative solutions as proposed in iMAGINE. One of the most attractive features, but also a key challenge for the envisaged highly integrated nuclear energy system is the need for a demand driven salt clean-up system. The work described provides an insight into the interplay between a potential salt clean-up system and the reactor operation in a plutonium started core in a dynamic approach. The results presented will help to optimize the parameters for the salt clean-up process as well as to understand the differences which appear between a core started with enriched Uranium and Plutonium as the fissile material. The integrated model is used to investigate the effects of the initial fissile material on core size, achievable burnup, and long term operation. Different approaches are tested to achieve a higher burnup in the significantly smaller Pu driven core. The effects of different clean-up system throughputs on the concentration of fission products in the reactor salt and its consequences are discussed for general molten salt reactor design. Finally, an investigation of how a plutonium loaded core could be used to provide fuel for future reactors through fuel salt splitting is presented with the outcome that one Pu started reactor of the same size as a uranium started core could deliver fuel for 1.5 new cores due to enhanced breeding. The results provide an essential understanding for the progress of iMAGINE as well as the basis for inter-disciplinary work required for optimizing iMAGINE.

Different aspects in applying the nucleation theorem to the description of crystallization of liquids are analyzed. It is shown that, by employing the classical Gibbs' approach in the thermodynamic description of heterogeneous systems and assuming that the basic assumptions of classical nucleation theory commonly employed in application to crystallization hold, a general form of the nucleation theorem can be formulated valid not only for one-component but generally for multi-component systems. This result is taken then as the starting point for the derivation of particular forms of this theorem for the cases that the deviation from equilibrium is caused by variations of either composition of the liquid phase, temperature, or pressure. In this procedure, recently developed by us expressions for the curvature dependence of the surface tension, respectively, the dependence of the surface tension on pressure and/or temperature are employed. It is shown that the formulation of the nucleation theorem as proposed by Kashchiev [J. Chem. Phys. 76, 5098-5102 (1982)] holds also for multi-component systems as far as mentioned above assumptions are fulfilled. In the application of classical nucleation theory to crystallization processes it is assumed as one of its basic ingredients that the bulk properties of the critical clusters are widely identical to the properties of the newly evolving crystal phase. This assumption is, however, in general, it is not true. This limitation of the theoretical description can be overcome by the application of the generalized Gibbs approach for the specification of the dependence of the properties of critical crystal clusters on the degree of metastability of the liquid phase. Applying this method, it is demonstrated that a similar formulation of the nucleation theorem as derived based on classical nucleation theory holds true also in cases when a dependence of the state parameters of the critical clusters on the degree of deviation from equilibrium is appropriately accounted for.

This study proposes a machine-learning-based framework for detecting mechanical damage in pipelines, utilizing physics-informed datasets collected from simulations for mechanical damage. The framework provides an effective workflow from dataset generation to damage detection and identification for three types of pipeline events: welds, clamps, and corrosion defects. While the study initially focused on optimizing the CNN structure using various advanced optimizers, it also investigated the impact of sensing systems on data classification and the effect of noise on classification performance. The study's analysis highlights the importance of selecting the appropriate sensing system for the specific application. The authors also found that the proposed framework is robust to experimentally relevant levels of noise, suggesting its applicability in real-world scenarios where noise is present. Overall, this study contributes to the development of a more reliable and effective method for detecting mechanical damage in pipelines. The proposed framework provides an effective workflow for damage detection and identification, and the findings on the impact of sensing systems and noise on classification performance add to its robustness and reliability.

This article will continue ansatz gage matrix of Iyer Markoulakis Helmholtz Hamiltonian mechanics points’ fields gage to Pauli Dirac monopole particle fields ansatz gage general formalism at Planck level, by constructing a Pauli Dirac Planck circuit matrix field gradient of particle monopole flow loop. This circuit assembly gage (PDPcag) that maybe operating at the quantum level, demonstrates the power of point fields matrix theoretical quantum general formalism of Iyer Markoulakis Helmholtz Hamiltonian mechanics transformed to Coulomb gage metrics, to form eigenvector fields of magnetic monopoles as well as electron positron particle gage metrics fields. Eigenvector calculations performed based on Iyer Markoulakis quantum general formalism are substituted for gage values of typical eigenvectors of dipolar magnetically biased monopoles with their conjugate eigenvectors, as well as eigenvector fields that are of the electron and positron particles. Then they are compiled to form combinatorial eigenvector matrix bundle of the monopole particle circuit field constructs assembly. Evaluation of this monopole particle fields matrix provided eigenvector fields results like SUSY, having Hermitian quantum matrix with electron-positron annihilation alongside north and south monopoles collapsing to dipolar “stable” magnetism, representing electromagnetic gaging typical metrics fields. Applying experimental observations on magnetic poles with measuring magnetic forces John Hodge’s results were showing asymmetrical pole forces; author has mathematically constructed asymmetric\strings\gage\metrics to characterize electromagnetic gravity, putting together while integrating with stringmetrics gravity that author has been reporting in earlier published articles. Physical Analysis with generalization of mass-charge and charge-fields gage metrics to quantum relativity gage metrics fields are proposed based on author’s proof formalism paper providing derivational algorithmic steps, to determine gage parametric values within the equation of Coulomb gage. Vortex fields’ wavefunctions and the scalar potential characterized by a function and a coupling constant having quantum density matrix together define the gage metrics quantifiable observable measurement physics of electron-positron cross-diagonal fields; contrastingly, diagonal terms of PDPcag matrix characterizes electron-positron particle eigenvector fields, while Hilbert Higgs mass metrics characterizes eigen-matter. Author is already working with Christopher O’Neill about magic square symmetry configurations to quantitatively understand symmetry, structure, and the real space geometry that are expected to form out of vacuum quanta point fields’ quantitative quantum general formalism theory of Iyer Markoulakis. In addition, author is currently collaborating with Manuel Malaver’s astrophysical Einstein Minkowski modified space time metrics evaluations of the sense-time-space relativistic general metrics to have means to account for curving or shaping of spacetime topology of a five-dimensional sense-time-space. Manuel Malaver’s specialization with modified Einstein Maxwell equations for modeling galaxies and stars cosmological physics, utilizing Einstein-Maxwell-Tolman- Schwarzschild and Reissner-Nordström spacetime and black holes theoretical formalisms have author of this paper collaboratively model quantum astrophysics of dark energy Star’s theory with Einstein-Gauss-Bonnet gravity equations.

proposal for a fundamental theory is described in which classical and quantum physics as a representation of the universe as a gigantic dendrogram are unified. The latter is the explicate order structure corresponding to the purely number-theoretical implicate order structure given by p-adic numbers. This number field was zero-dimensional, totally disconnected, and disordered. Physical systems (such as electrons, photons) are sub-dendrograms of the universal dendrogram. Measurement process is described as interactions among dendrograms; in particular, quantum measurement problems can be resolved using this process. The theory is realistic, but realism is expressed via the the Leibnitz principle of the Identity of Indiscernible. The classical-quantum interplay is based on the degree of indistinguishability between dendrograms (in which the ergodicity assumption is removed). Depending on this degree, some physical quantities behave more or less in a quantum manner (versus classic manner). Ideologically, our theory is very close to Smolin’s dynamics of difference and Rovelli’s relational quantum mechanics. The presence of classical behavior in nature implies a finiteness of the Universe-dendrogram. (Infinite Universe is considered to be purely quantum.) Reconstruction of events in a four-dimensional space type is based on the holographic principle. Our model reproduces Bell-type correlations in the dendrogramic framework. By adjusting dendrogram complexity, violation of the Bell inequality can be made larger or smaller.

The usage of Quantum Similarity through the equation Z = {∀θ ∈ Z → ∃s ∈ S ∧ ∃t ∈ T : θ = (s, t)}, represents a way to analyze the way communication works in our DNA. Being able to create the object set reference for z being (s, t) in our DNA strands, we are able to set logical tags and representations of our DNA in a completely computational form. This will allow us to have a better understanding of the sequences that happen in our DNA. With this approach, we can also utilize mathematical formulas such as the Euler–Mascheroni constant, regression analysis, and computational proofs to answer important questions on Quantum biology, Quantum similarity, and Theoretical Physics.

We propose the Hamiltonian formalism of Bianchi type 1 cosmological model for perfect fluid. We have considered both the equation of state parameter ω and the cosmological constant Λ as the function of volume V(t) which is defined by the product of three scale factors of the Bianchi type 1 line element. We propose a Lagrangian for the anisotropic Bianchi type-1 model in view of a variable mass moving in a variable potential . We can decompose the anisotropic expansion in terms of expansion and shearing motion by Lagrangian mechanism. We have considered a canonical transformation from expanding scale factor to scalar field ø which helps us to give the proper classical definition of that scalar field in terms of scale factors of the mentioned model. This definition helps us to explain the cosmological inflation. We have used large anisotropy(as in the cases of Bianchi models) and proved that cosmic inflation is not possible in such large anisotropy. Therefore we can conclude that the extent of anisotropy is less in case of our universe. Otherwise the inflation theory which explained the limitations of Big Bang cannot be resolved.Part II is contained with some analysis of the lagrangian ; derived in Part I ; on the quintessence model.

We try to solve three decades-old physics challenges. List all elementary particles. Describe dark matter. Describe mechanisms that govern the rate of expansion of the universe. We propose new modeling. The modeling uses extensions to harmonic oscillator mathematics. The modeling points to all known elementary particles. The modeling suggests new particles. Based on those results, we do the following. We explain observed ratios of dark matter amounts to ordinary matter amounts. We suggest details about galaxy formation. We suggest details about inflation. We suggest aspects regarding changes in the rate of expansion of the universe. We interrelate the masses of some elementary particles. We interrelate the strengths of electromagnetism and gravity. Our work seems to offer new insight regarding applications of harmonic oscillator mathematics. Our work seems to offer new insight regarding three branches of physics. The branches are elementary particles, astrophysics, and cosmology.

We develop and apply new physics theory. The theory suggests specific unfound elementary particles. The theory suggests specific constituents of dark matter. We apply those results. We explain ratios of dark matter amounts to ordinary matter amounts. We suggest details about galaxy formation. We suggest details about inflation. We suggest aspects regarding changes in the rate of expansion of the universe. The theory points to relationships between masses of elementary particles. We show a relationship between the strength of electromagnetism and the strength of gravity. The mathematics basis for matching known and suggesting new elementary particles extends mathematics for harmonic oscillators.

Here, applying the theory of the generalization of the Boltzmann constant in curved space-time, we will model the mechanism by which elementary particles are formed. We will determine the relationship between gravity and each of the elementary particles that form the standard model following analogies to that of stellar bodies. In order to achieve our goal, we will propose new models for photons, quarks and gluons; with this, we will demonstrate why there are stable and unstable elementary particles; why the first family of elementary particles forms hadrons and why the second and third families cannot form hadrons; why the fermions comply with the Pauli exclusion principle and why the bosons do not comply with the Pauli exclusion principle, etc. Finally, we will analyse the generalization of the ADS/CFT correspondence and propose a theory of everything (T.O.E.).

Computationally modelling a nuclear reactor startup core for a benchmark against the existing models is highly desirable for an independent assessment informing nuclear engineers and energy policymakers. This work presents a startup core model of the UK’s first Evolutionary Pressurised Water Reactor (EPR) based on Monte Carlo simulations of particle collisions using Serpent 2, a continuous-energy Monte Carlo reactor physics burnup code. Coupling between neutronics and thermal-hydraulic conditions with the fuel depletion is incorporated into the multi-dimensional branches, obtaining the thermal flux and fission rate (power) distributions radially and axially from the three dimensional (3D) single assembly level to a 3D full core. Shannon entropy is employed to characterise the convergence of the fission source distribution, with 3 billion neutron histories tracked by parallel computing. Source biasing is applied for the variance reduction. Benchmarking the proposed Monte Carlo 3D full-core model against the traditional deterministic transport computation suite used by the UK Office for Nuclear Regulation (ONR), a reasonably good agreement within statistics is demonstrated for the safety-related reactivity coefficients, which creates trust in the EPR safety report.

The safety of lithium-ion batteries is critical to the safety of battery electric vehicles (BEVs). The purpose of this work is to develop a method to predict battery thermal runaway in full electric vehicle crash simulation. The thermal-electrical-mechanical coupled finite element analysis is used to model an individual lithium-ion battery cell, a battery module, a battery pack, and a battery electric vehicle with 24 battery modules in a live circuit connection. The lithium-ion battery is modeled using a representative approach, with each battery internal component individually modeled to represent its geometric shape and realistic thermal, mechanical, and electrical properties. A resistance heating solver and Randles circuit model built with generalized voltage source are used to simulate the battery electrical behavior. The thermal simulation of the battery accounts for the heat capacity and thermal conductivity of various cell component materials, as well as heat conduction, radiation, and convection at their interfaces. The mechanical property of battery cell and battery module models is validated using spherical punch tests. The electrical property of the battery cell and battery module models is verified against CircuitLab simulation in an external short circuit test. The simulation result of battery module’s internal resistance agrees with the experimental data and the literature value. The multi-physics coupling phenomenon is demonstrated with a cylindrical compression simulation on the battery module. The multi-physics BEV model with 24 live battery modules is used to simulate the external short-circuit test and the side pole impact test. The simulations run time is less than 24 hours. The results demonstrated the feasibility of using representative battery model and multi-physics analysis to predict battery thermal runaway in full electric vehicle crash analysis.

The sPHENIX experiment is a state-of-the-art jet and heavy flavor physics detector which successfully recorded its first Au + Au collision data at 200 GeV at RHIC. sPHENIX will provide heavy flavor physics measurements covering an unexplored kinematic region and unprecedented precision at RHIC to probe the parton energy loss mechanism, parton transport coefficients in quark-gluon plasma, and the hadronization process under various medium conditions. At the center of sPHENIX, the Monolithic-Active-Pixel-Sensor (MAPS) based VerTeX detector (MVTX) is a high-precision silicon pixel detector. The MVTX provides excellent position resolution and the capability of operating in continuous streaming readout mode, allowing precise vertex determination and recording a large data sample, both of which are particularly crucial for heavy flavor physics measurements. In this work, we will show the general performance of heavy-flavor hadron reconstruction. In addition, we will discuss the commissioning experience with sPHENIX. Finally, we will provide the projection of b-hadron and jet observables and discuss the estimated constraints on theoretical models.

The Hyper-Torus Universe Model (HTUM) proposes a novel framework to unify quantum mechanics, cosmology, and consciousness. This paper introduces HTUM's core concept: a higher-dimensional hyper-torus encompassing all possible states of existence. HTUM suggests that the universe functions as a quantum system where all outcomes are inherently connected, with consciousness playing a significant role in actualizing reality. The model introduces several key innovations, including the universe's 4-dimensional toroidal structure (4DTS), providing a new geometric interpretation of spacetime. HTUM presents dark matter and dark energy as nonlinear probabilistic phenomena within this toroidal framework, offering a unique perspective on their nature and interactions. The model proposes a mechanism for wave function collapse through universal self-observation, bridging quantum mechanics and gravity. It introduces a Topological Vacuum Energy Modulator (TVEM) function, offering a potential resolution to the cosmological constant problem. Furthermore, HTUM integrates consciousness as a fundamental aspect of the universe, influencing quantum state actualization and potentially resolving the hard problem of consciousness. The model explores the implications of a timeless singularity and the emergence of time as a product of causal relationships within the toroidal structure. This framework addresses critical challenges in modern physics, including the nature of quantum entanglement, the origin and fate of the universe, and the relationship between mind and matter. The paper discusses the mathematical formulation of HTUM, its implications for quantum mechanics and cosmology, and its potential to bridge the gap between science and philosophy. We present detailed predictions for the cosmic microwave background (CMB) power spectrum, gravitational wave signatures, and large-scale structure formation, demonstrating how HTUM differs from the standard ΛCDM model. Our analysis reveals distinctive oscillatory patterns in the relative difference between HTUM and ΛCDM predictions, with deviations reaching up to ±20% at certain scales. The paper also explores the unification of fundamental forces within the HTUM framework, proposing novel approaches to particle physics and quantum field theory. It introduces the concept of unified mathematical operations, suggesting a more holistic approach to mathematical thinking that aligns with the model's interconnected view of the universe. Additional testable predictions and proposed experimental approaches are discussed, including advanced space-based tests and laboratory experiments probing quantum gravity effects. These specific, quantifiable predictions offer multiple avenues for empirical validation, highlighting HTUM's potential to be distinguished from ΛCDM and other alternative cosmological models. HTUM represents a significant new perspective in our understanding of the universe, inviting further research and exploration into the nature of reality, consciousness, and the fundamental structure of the cosmos. By providing a comprehensive framework that unifies various aspects of physics, cosmology, and consciousness studies, HTUM offers a promising path toward a more complete understanding of our universe.

Physics theory has yet to settle on specific descriptions for new elementary particles, for dark matter, and for dark energy forces. Our work extrapolates from the known elementary particles. The work suggests well-specified candidate descriptions for new elementary particles, dark matter, and dark energy forces. This part of the work does not depend on theories of motion. This work embraces symmetries that correlate with motion-centric conservation laws. The candidate descriptions seem to explain data that prior physics theory seems not to explain. Some of that data pertains to elementary particles. Our theory suggests relationships between masses of elementary particles. Our theory suggests a relationship between the strengths of electromagnetism and gravity. Some of that data pertains to astrophysics. Our theory seems to explain ratios of dark matter effects to ordinary matter effects. Our theory seems to explain aspects of galaxy formation. Some of that data pertains to cosmology. Our theory suggests bases for inflation and for changes in the rate of expansion of the universe. Generally, our work proposes extensions to theory in three fields. The fields are elementary particles, astrophysics, and cosmology. Our work suggests new elementary particles and seems to explain otherwise unexplained data.

Here we will explain the correlation between the Kerr-Newman diagram and the theory: RLC electrical modelling of a black hole and the early universe. We will develop a black hole model that describes and explains both theories. In the development of our analysis, we will emphasize the following conditions: M² > Q² + a², M² = Q² + a² and M² < Q² + a² and we will also analyse the condition in which the mass of a black hole reaches its critical mass M = Mc. We will demonstrate that the interpretation of the Kerr-Newman diagram is equivalent to the interpretation of the theory: RLC electrical modelling of a black hole and the early universe. Finally, we will generalize the proposed model for a black hole and propose a black hole model formed by negative particles, neutroniumd, which after reaching critical conditions of mass, pressure, volume, temperature, density, etc.; decays into a positively charged black hole, protoniu. Let us remember that both models are formed by charged particles of matter (they do not contain antimatter), but their vector configuration means that the net charges of both proposed black holes are zero.

The demand for improving the nuclear waste management has since long been identified as one of the major hurdles for widespread use of nuclear energy. Nuclear waste management, through partitioning and transmutation (P&T) has been researched since the 1990ies with partitioning being a pre-requisite for the process. Recently, an innovative approach of reactors directly operating on spent nuclear fuel, iMAGINE, has been proposed which could deliver on the aims of P&T as a side effect to more efficient nuclear energy production in the future. The results shown here confirm that it is possible to achieve the aims of P&T without prior partitioning, but it is certainly less efficient since the high concentration of MAs, required for efficient burning, are impossible to obtain in a short operational time. The proposed nuclear waste management approach will be a long-term effort when it is accomplished without partitioning/separation technologies. However, none of the analyses contradicts this effort – the key points are a) when the technology for treating the waste is possible and reliable, the time horizon will not be a major concern b) the waste management is now intrinsically linked with energy production instead of requiring dedicated costly facilities, delivering a promising economic basis c) the waste management is now associated with long-term energy production and massively improved resource utilization. In summary, the objectives of P&T are achievable without prior partitioning – this is a ground-breaking result. However, the approach will require continued nuclear energy production for a very long time. In case of a nuclear phase-out decision, a much more efficient transmutation scenario would have to be envisaged as already discussed several years ago for Germany.

The study presented hereafter shows a new methodology to reveal traces of polyethylene (the most common microplastic particles, known as a structure of C2H4) in a sample of ocean water by the irradiation of a 50 keV, 1 µA electron beam. This is performed by analyzing the photon (produced by the electrons in water ) fluxes and spectra (i.e. fluxes as a function of photon energy) at different types of contaminated water with an adequate device and in particular looking at the peculiar interactions of electrons/photons with the potential abnormal atomic hydrogen (H), oxygen (O), carbon (C), phosphorus (P) compositions present in the water, as a function of living and not living organic organisms with a PO4 group RNA/DNA strands in a cluster configuration through a volumetric cells grid.

Creating self-learning algorithms, developing deep neural networks and improving other methods that "learn" for various areas of human activity is the main goal of the theory of machine learning. It helps to replace the human with a machine, aiming to increase the quality of production. The theory of artificial neural networks, which already have replaced the humans in problems of detection of moving objects, recognition of images or sounds, time series prediction, big data analysis and numerical methods remains the most dispersed branch of the theory of machine learning. Certainly, for each area of human activity it is necessary to select appropriate neural network architectures, methods of data processing and some novel tools from applied mathematics. But the universal problem for all these neural networks with specific data is the achieving the highest accuracy in short time. Such problem can be resolved by increasing sizes of architectures and improving data preprocessing, where the accuracy rises with the training time. But there is a possibility to increase the accuracy without time growing, applying optimization methods. In this survey we demonstrate existing optimization algorithms of all types, which can be used in neural networks. There are presented modifications of basic optimization algorithms, such as stochastic gradient descent, adaptive moment estimation, Newton and quasi-Newton optimization methods. But the most recent optimization algorithms are related to information geometry, for Fisher-Rao and Bregman metrics. This approach in optimization extended the theory of classic neural networks to quantum and complex-valued neural networks, due to geometric and probabilistic tools. There are provided applications of all introduced optimization algorithms, what delighted many kinds of neural networks, which can be improved by including any advanced approaches in minimization of the loss function. Afterwards, we demonstrated ways of developing optimization algorithms in further researches, engaging neural networks with progressive architectures. Classical gradient based optimizers can be replaced by fractional order, bilevel and, even, gradient free optimization methods. There is a possibility to add such analogues in graph, spiking, complex-valued, quantum and wavelet neural networks. Besides the usual problems of image recognition, time series prediction, object detection, there are many are other tasks for modern theory of machine learning, such as solving problem of quantum computations, partial differential and integro-differential equations, stochastic processes and Brownian motion, making decisions and computer algebra.

Spinors are used for the computation of probability in quantum mechanics. They are treated as elements of a complex vector space in contrast to a real vector space. Spinor theory is abstract mathematics with an ambiguous interpretation. The overall phase of spinors does not affect the computation of probability in quantum mechanics. If two spinors have an overall phase of imaginary number i, then they can be treated like vectors in a real vector space. The square of the magnitude of the number, which is the sum of dot product and the term having basis vector i of cross product of two such vectors, is equal to the probability computed using the corresponding spinors. Therefore, the geometry of such spinors can be easily depicted in a three-dimensional space like vectors in a real vector space. Spinors are not isotropic vectors in Hilbert space. The sum of dot product and cross product of two complex numbers is equivalent to the quotient of the division of one complex number by another. Similarly, the sum of dot product and cross product of two vectors is the quotient of the division of one vector by another. Using the rules of division of vectors, we can find the rules of multiplication of vectors. The rules of multiplication and division of basis vector i match the rules of multiplication and division of imaginary number i. Therefore, basis vector i is nothing but imaginary number i. Multiplication of dot and cross product of two vectors to the second vector will give us the components of the first vector that are parallel and orthogonal to the second vector. Vectors are also made up of complex numbers like spinors. The reason for finding dot product of complex numbers in the process of computing the probability is to ignore the overall phase in contrast to the phase difference. This is misconstrued as complex vector space in quantum mechanics. 3-D number which is nothing but a spinor with new notation is an extension of vector algebra. It can have a real number as a term in addition to the terms i, j and k. In all other respects, it is a vector. 3-D numbers are part of the number system like real numbers and complex numbers. The real numbers are one dimensional numbers, the complex numbers are two-dimensional numbers and the 3-D numbers as well as vectors and spinors are three-dimensional numbers. As polarisation and spin are one and the same, we can straight away apply 3-D numbers to polarisation of light. 3-D numbers will greatly simplify the study of areas of science where three-dimensional space is involved.

This study delves into the intricate interplay between cosmology and gravity in dilaton-based models, enhancing our understanding. Exploring boundary theories coupled to gravity, we derive a four-dimensional effective theory from the cosmological constant. Dilaton gravity is then scrutinized, revealing elegant solutions and insights into brane embedding. The analysis extends to probe scalar fields, employing the WKB approximation in both conformal and non-conformal cases. In summary, this work contributes to a nuanced understanding of the complex relationship between cosmology and gravity in dilaton-based frameworks.