Journal of Physical Mathematics & its Applications
Journal of Physical Mathematics & its Applications is an open access peer-reviewed journal from the publishers of Scientific Research and Community (SRC) which publishes articles in the form of written, video and power-point presentations from the arena of Physical Mathematics & its Applications. The journal focuses to build up accessing the complete content of articles freely from online for reader’s perusal. Article acceptance will be in the form of research, review, analysis, magazines, editorials, opinion, communications, mini-review, short communications, picturized, book review, video type and power point presentations (PPTS). Articles can be submitted through online from https://srcpublishers.com/physical-mathematics-its-applica/user/register or can send through an e-mail with attachment for physicalmathematics@onlinescientificresearch.info 0.35 Impact Factor 84.5% Citations 90% Acceptance
Mathematical Models to Understand the Dynamics of Water Borne Disease and its Impact on Public Health
Waterborne diseases, such as cholera, typhoid fever, and malaria, remain major global health concerns, and their transmission rates are largely driven by contaminated water sources, poor sanitation conditions, and human behavioral factors. Mathematical modelling has, therefore, become an important tool for enhancing knowledge of the complex dynamics of transmission of these diseases, as well as for the evaluation of the efficiencies of public health interventions. This review integrates current modelling approaches, emphasizing both model formulation and analytical methods. It highlights some of the gaps in the literature, such as the limited number of models that integrate multiple transmission pathways and the tendency for very few to properly account for delays in the implementation of interventions. Recent developments, including deterministic and stochastic models, sensitivity analyses, and the newer approach of fractional-order systems, are discussed in terms of their potential for closing the gap between theoretical frameworks and real-world epidemiological patterns. Overall, the review emphasizes the utility of mathematical modelling to help develop effective strategies for water purification, vaccination, and community education that will lower the burden of waterborne diseases.
Copyright (c) 2026 Journal of Physical Mathematics & its Applications
Burning-Rate Coefficients in Solid Propellants: Modeling, Estimation, Stability and Numerical Illustration
The burning-rate law is a central empirical and mathematical relation in the modelling of solid-propellant combustion. In its classical Saint Robert–Vieille form, it is written as r = aPn, where r is the linear burning rate, P is pressure, a is the burning-rate coefficient and n is the pressure exponent. This revised manuscript develops a coherent mathematical framework for estimating a and n, quantifying uncertainty, and embedding the fitted coefficients into nonstationary pressure dynamics. The model is extended to include thermal memory through discrete-delay and distributed-memory functional differential equations. Existence, uniqueness, positivity, continuous dependence, local and global stability, delay thresholds and Lyapunov–Krasovskii stability are stated as formal results. Each theorem is accompanied by a numerical illustration with clear colour graphics, including pressure-response curves, stability surfaces, kernel effects and Lyapunov decay. The numerical section is written in a MATLAB/Python style so that the simulations can be reproduced without relying on propellant formulation or manufacturing details. The paper remains mathematical and modelling-oriented; it does not provide operational design instructions.
Numerical Demonstration with 260 Significant Decimal Places Performed using the Wolfram Mathematica Program
"Aspin Bubbles", in order to demonstrate that it obtains all the forces of nature : Gravity, Electric, Nuclear, Casimir, Yannoe, etc., needs to work with more than 70 significant decimal places. Many scientists consider this not to be science because they don' t know how to work with so many decimal places. To leave no doubt about "Aspin Bubbles" mechanical theory, I propose this trigonometry exercise performed with the Wolfram. Mathematica program so that you can verify its high numerical accuracy.
Hermite Polynomials and the Quantum Harmonic Oscillator: An Algebraic Approach
This study presents a complete algebraic formulation of the quantum harmonic oscillator, demonstrating that the Hermite polynomials, Hn(x), emerge as the fundamental solution when the power-series method is applied to the Schrödinger equation. The physical requirement of square integrability enforces truncation of the series, leading precisely to these polynomials, which constitute the polynomial part of the energy eigenfunctions, ψn(x) = NnHn(αx)e−α2x2/2, where n is the quantum number. This quantum number directly determines the quantized energy levels through En = ℏω (n + 1/2). The analysis based on creation and annihilation operators further reveals that the action of the creation operator on the ground state generates the excited states along with their corresponding Hermite functions. The orthogonality of these polynomials, given by ensures the orthonormality of the complete set of eigenfunctions. Therefore, Hermite polynomials encapsulate both the mathematical structure and the physical essence of quantization in this fundamental system.
Regularization of Second-Order Functional Differential Equations in Weighted Spaces
This paper develops a regularization framework for second-order functional differential equations in weighted Banach spaces. The main object is the delayed equation u′′(t) + a(t)u′(t) + b(t)u(t) = F(t, u(t), u(t − τ )), and its regularized counterpart u′′ ε (t) + a(t)u′ε(t) + b(t)uε(t) + εLuε(t) = F(t, uε(t), uε(t − τ )). The analysis is motivated by previous work of Manuel J. Alves and Elena V. Alves on functional differential equations, weighted functional settings, operator methods and delayed or memory-dependent combustion models. The local assertions are reorganized into four principal results: existence and uniqueness, uniform a priori estimates, compactness and convergence, and spectral stability. Complete proofs are supplied. Additional computational examples and professional PGFPlots figures illustrate weighted decay, delay amplification, error rates, stability regions and non-stationary combustion profiles. The revised version also defines a concrete regularizing operator, includes reproducible Python and MATLAB simulations, demonstrates the explicit O(ε) convergence rate, makes the regularizing operator explicit in every theorem, gives a deeper numerical stability analysis, and compares the approach with five recent papers published in 2023–2026
Functional Differential Equations and Stability Analysis
This work investigates Functional Differential Equations, which model systems whose dynamics depend on past or future states. Stability results of solutions are studied. The work also includes analytical and numerical methods, with simulations in Python and MATLAB, and applications to delay models.
Mathematical Modelling and Optimal Control Strategies in Drug Addiction Dynamics
Drug abuse is a complex social and health problem that involves the individuals, their families, and societies at the international level. The tools that should be used in the understanding of its development should be based on the nonlinear and dynamic nature of addictive behavior. This is analyzed in terms of mathematical modeling. These models explain the changes in state, like susceptible, addicted, rehabilitated, and relapsed, depending on changes in the rate of contact, the efficacy of treatment, and the tendency to relapse. Through bifurcation behavior, stability analysis, and sensitivity studies, it is clear that a slight intervention or awareness change can cause dramatic changes in outcome. The basic reproduction number is another threshold measure that explains the continuation or the decline of addiction in a population. This review gives a summary of some of the key nonlinear modelling techniques, their assumptions, the way of analyzing them, and their use. It also indicates limitations, including a lack of real-time data and appropriate focus on psychological and social aspects. It is important to emphasize that synthesis presented the significance of additional data and more realistic models that can increase drug abuse prevention and rehabilitation and policy interventions.
Mathematical Models for Epidemiology: Prevention and Control of Infectious Diseases
Infectious diseases remain a major global health challenge, especially in developing countries with limited healthcare resources. Malaria, transmitted by mosquitoes, continues to cause significant harm in tropical and subtropical regions like Mozambique.
The 12 Principles of Fluid Dynamics in Closed Conduits Aerodynamic Analysis of a Golf Ball in Flight
In the real world, fluid flow in closed conduits manifests in both empty conduits, i.e., conduits devoid of solid obstacles and conduits packed with solid particles. The majority of packed beds reported in the literature over the last 100 years, approximately, contain spheroidal particles, for the most part. However, the particles within the packed conduits can vary over a very broad range of shapes and diameter values and when coupled with a wide choice of both conduits and available fluids, this ensemble of experimental variables complicates the study matter, immensely. Empirical studies covering this very broad dynamic range of choices has created many difficulties which has, unfortunately, contributed to a compromised data base of published results. In fact, it is not an exaggeration to suggest that more than 50% of published articles in this field are fraught with errors, misinformation and contradictions. For instance, the use of gases as the fluid of choice, in any experiment, always presents a problem because it is a compressible fluid which means that its’ density varies as a function of pressure drop. In addition, because the viscosity also varies with density, the turbulent tipping point of the packed conduit under study cannot be predicted a priori which means that when a compressible fluid is chosen as the fluid of choice in permeability studies, all fluid variables must be measured simultaneously, i.e., viscosity, density, pressure drop and velocity. This is a significant experimentation undertaking for any particular study which many times in the published literature is omitted from the experimental protocol. There is the added difficulty of managing pressure drop, since small particles render large pressure drops while large particles render small pressure drops. If one also considers the additional important variables of packed bed porosity and boundary layer (wall-effects), it becomes very clear that such empirical studies are anything but simple experiments. In fact, there are a total of seven discrete variables and one universal constant embedded in the validated mathematical expression for the pressure flow relationship of packed conduits. A dimpled golf ball, therefore, is a very interesting candidate as a choice for the study of packed beds because it is large in diameter, has a roughened surface, and its’ aerodynamic flight takes place through a compressible fluid medium (air), which stands at standard temperature and pressure (STP) in the path of the golf ball flight. Accordingly, a commercial golf ball represents an ideal study candidate and, therefore, the gold standard candidate, albeit of the most difficult variety from an empirical standpoint, to evaluate the performance claims of any fluid flow model which purports to describe the hydrodynamics of packed conduits. In this paper, a relatively new fluid flow model (QFFM) will be used to do exactly that. The aerodynamics of a golf ball in flight will be documented by taking measurements using water, an incompressible fluid, in a packed conduit containing many golf balls, something that has never been accomplished, as far as we know. The teaching of this unique fluid flow model will be used to demonstrate an experimental technique which circumvents/ solves many of the aforementioned issues for studies in packed beds. In short, this paper defines the gold standard of how an experimental protocol should be carried out for permeability in closed conduits, even when the particles are irregular (non- spherical) in shape, compressible (non-rigid), very small in diameter and/or have a roughened surface morphology and, even when the fluid of choice is compressible, such as air.
Quantum Gravity and the Distance-Velocity Profile of a Black Hole
Newton's law of universal gravitation implies that the circular velocity of a low-mass body at a distance r from the center is given by the relation v2=GM/r, while the velocity v remains unbounded. According to Einstein's theory of relativity, the relative velocity of motion is limited by a maximum speed, which is the speed of light. Furthermore, a massive object cannot reach the speed of light. In General Relativity, the circular velocity from a gravitational center is also determined by the same expression used to describe the gravitational field according to Newton. By using this fundamental relationship in combination with the relativistic limit of maximum velocity to derive the magnitude of binding energy—and consequently the binding mass of a bound state of two massive objects—we obtain two equations. According to the first, the value of the gravitational coupling parameter K can be determined as a function of the distance from the center; according to the second, the velocity of circular motion at that distance can be established. The derived equation for the coupling parameter, which we shall call the "vacuum equation," can be expressed as the sum of three equations. The first is the fermion equation, the second is the boson equation, and the third is the gravitation equation. Solving these equations individually, as well as their partial sums for the parameter K=2, leads to determining the volume ratios of selected regular polytopes in four-dimensional space (with properties described by L. Schläfli, which are in agreement with the mass ratios of selected elementary particles. The concept of regular polyhedra in three-dimensional space in connection with planetary orbits was previously used by J. Kepler. For the parameter K∈(0,1), the solution of the gravitation equation leads to determining the mass of the resulting so-called black hole after the collapse of two massive objects. The first set of solutions to the equation for the gravitational coupling parameter determines the velocity profile based on circular velocity at a distance from the gravitational center both outside and inside the black hole. This gravitational parameter is established by solving the gravitation equation, which depends on a parameter that distinguishes massive objects based on their composition—whether they are fermionic, bosonic, or a mixture of these basic components. The Planck black hole, which lies on the boundary between the masses of elementary particles and massive objects such as black holes, can be considered the limiting threshold. The second set of solutions for the parameter K > 2 determines the mass of a particle equivalent to the anti-gravitational effect expected in dark energy. Simultaneously, this group of solutions allows for the determination of conditions for the quantum mechanical evaporation of black holes based on particleanti particle creation with rest mass. This tri-particle creation involves a dark energy particle, a dark matter particle, and a baryonic particle, whose masses are in the expected ratio corresponding to their observed distribution in the universe.
Machine Learning Models for Predicting Length of Hospital Stay in the Gynecology Department of South-Kivu Hospitals
This study examines how machine learning can predict how long patients will stay in gynaecology departments in South Kivu, which helps hospitals manage beds and resources more efficiently. We analysed data from hospitals using some factors that influence hospital stays, such as patient age, diagnosis, and treatment. We performed a model that takes information at the admission of a patient. We did not integrate some factor changes that can occur while the patient is already admitted to the hospital. We tested different machine learning approaches, including generalised linear models, decision trees, and neural networks. The generalised linear model performed best, achieving 97% accuracy in predicting the length of stay. This demonstrates that machine learning can effectively support hospital planning and resource management in this setting.
On Perfect Mirror and the Existence of Particle-less Waves
A perfect mirror is a beam splitter that reflects a particle and transmits a fraction of its wave energy, resulting in a particle-less wave, accounting for dark energy and dark matter (as standing waves). Waves are periodic, residing in a quotient space as in clock times, in contrast to linear calendar times for particle motions. That is, we live in a diagonal 4-manifold, locally Euclidean. Dirac took the square-roots of the shell equation to account for particle/antiparticle, but in the 4 x 4 matrix form his equation does not square to the shell equation. Thus, we instead take the complex conjugates of the shell equation to arrive at particle/antiparticle, manifested as +/- sigma z of the Pauli matrices exhibiting opposite angular momenta, specifically, a left-handed positron changeable to a right-handed electron by a 180-degree rotation. The quotient space of waves renders action-at-a-distance for gravity and electromagnetism; in particular, the fine-structure-constant is dimensionless and leads to quantum entanglement with instant actions across spacetime. As a distinction, gravitational waves are caused by exogenous events and hence travel at the speed of light, which is analogous to an uplifting elevator with its clock running increasingly faster by General Relativity, causing variations in the spacetime curvatures.
Gravitational and Dark-Matter Concepts that Can Help Explain Cosmic Data
We discuss gravitational concepts and candidate specifications for dark matter that, together, can help explain known ratios of dark-matter effects to ordinary-matter effects and can help explain eras in the rate of expansion of the universe. The ratios pertain to galaxies and galaxy evolution, galaxy clusters, and densities of the universe. The candidate specifications for dark matter reuse, with variations, a set of known elementary particles. Regarding galaxy evolution and the rate of expansion of the universe, we deploy multipole-expansion methods that combine Newtonian gravity, aspects of motions of sub-objects of gravitationally interacting objects, and Lorentz invariance. One outgrowth from our work suggests relationships among some physics constants. Another outgrowth from our work suggests a basis for a candidate specification for quantum gravity
A New Confirmation of the Presence of Two Errors in Michelson’s Analysis of his MMX, on the PCR Academic Website, for Riverboats Contest Times Calculation, and a NewDemonstration, Contradicting the Solution in Physics
We elaborated New Interpretation of Michelson’s analysis (NI16) of his experiments (MMX), by discovering and correcting the Errors 1, 2 (MER12) in that Analysis (not in experiment), including in its faithful model of two swimmers contest (TSC). In NI16 was presented previous Demonstrations 1, 2, and 3, which all proved total times equality t1=t2 (no winner) in Water frame (WF). But this result contradicts current Physics textbooks (PTB) which states the inequality of times t2<t1, and transversal SW2 wining the contest. This result is also valid for MMX. Recently, we discovered that a numerical calculation, similar to our NI16, appears on the website Physics Classroom (PCR), by a numerical calculus for Example 1 (EX1) of the first crossing of a river 80 m wide, with speed of 3m/s, by a motorboat (BO2) with speed of 4m/s. This numerical calculation in PCR, faithfully follows the rules of plane vector calculus and reaches the result of the first crossing time of t2a=20,0s. This result in PCR completely corresponds to procedures and times from NI16 in WF. But corresponding time from PTB is t2a=30.20s, the total is double t2=60.4s and for BO1 results t2=60,4s<91.4s=t1 and SW2 wins TSC, and PTB must be corrected. Therefore, the credible PCR website, represents a Confirmation 4 for the correctness of our NI16. And based on a new geometric route of BO2 in BF frame, from trigonometric calculation, results t2=t1, as in NI16. So this calculus represents the very Demonstration 5, of NI16 correctness, allowing the reintroduction of the ether into Physics, with numerous consequences.
Rethinking Fundamental Algebra in Terms of Learning Architecture
With the aim of extending the applicability of fundamental theoretical mathematical concepts to educational contexts, the present study examines and analyzes relevant algebraic content that supports learners’ understanding of existing quantified relational premises and scientific learning content and algebraic concepts. The purpose of this analytical study is to investigate how algebraic content—such as the contributions of Van der Waerden (1973), Freudenthal (1973), and Vergnaud (1983; 1994)—should be developed and applied within mathematics education, with a focus on Vygotsky’s (1986; 1998) research-based learning framework. The study also examines theoretically grounded learning content, including multiplicative structures, Abelian semigroups, and inverse operations, which facilitate an optimal transition and interplay between fundamental algebra within a learning architecture and students’ mastery of scientific algebraic concepts. To provide a foundation for this learning architecture, the study presents a theoretical conceptualization of how the learning process should engage, systematize, extend, and conceptually integrate students’ understanding into a new algebraic context.
Modal Composition of a Terahertz Pulse in a Parallel-plate Waveguide Partially Filled with a Nonlinear GaSe or DAST Crystal
A numerical analysis of terahertz (THz) pulse propagation in a metal parallel-plate waveguide partially filled with a nonlinear crystal (PPNW) is presented. A femtosecond laser generating a linearly polarized optical pulse with a duration of 100 fs is employed for both pumping and detection of broadband THz pulses. Terahertz radiation is generated via optical rectification of the optical pulse in a nonlinear crystal that partially fills the waveguide cross-section. Nonlinear crystals such as GaSe and DAST are considered efficient frequency converters in the THz range. The modal composition of the THz pulse in the waveguide is analyzed numerically using MATLAB and COMSOL Multiphysics. The analysis is performed for two types of optical–THz interaction, namely eoo and eee, using both experimental and modeled refractive indices in the THz frequency range. It is shown that decreasing the crystal thickness between the waveguide plates leads to an expansion of the single-mode propagation bandwidth of the THz pulse spectral components. In particular, for a GaSe crystal with both thickness and height equal to 0.1 mm, the fundamental TE₁₀ mode spans the frequency range from 0.2 to 3 THz, while reducing the thickness to 60 μm extends this range up to 5 THz. These results provide guidelines for optimizing single-mode THz generation and propagation in hybrid waveguide structures.
Advances in Exploration of the Dark Universe (Dark Energy & Dark Matter)
The rudiments of a dark energy theory are providing that appears to have been developed in two complementary ways. On the one hand, this theory is based on physics and mathematics and, on the other hand, it is developed on the basis of available data. This correspondends to the discovery of the laws of planetary motion in elliptical planetary orbits by JOHANNES KEPLER in the past. He developed his laws from a large dataset. Later it was theoretically substantiated more thoroughly by ISAAK NEWTON. The focus is on deriving a formula for the equivalence of energy and time or equivalence of dark energy and the age of the universe. This derivation provides new theoretical insights and applications in theoreticsl terms and leads to the discovery of a new law of nature. The theoretical result is confronted with the numerical value calculated from the available data from the MAX PLANCK Institute for Radio Astronomy. Excellent matching of numerical values of dark energy resulting in three independent paths makes the approach plausible. THOMAS GÖRNITZ provides a theoretical basis for the equation of the equivalence of dark energy and the age of the universe. The empirical BALMER formula for the frequencies of the spectral lines in the arc spectrum of hydrogen was also theoretically justified by NIELS BOHR, who calculated the energy levels of the hydrogen atom and the fequencies of spectral lines. The derivation of a formula for calculation dark energy is described. Further formulas are deducted. The dark matter of the cosmos is calculated. A balance sheet is drawn up. Conclusions are drawn.
Induced Electric Dipole Red Shift – The Key to Replacing Big Bang Cosmology
This presentation re-examines the key foundations of Big Bang Cosmology, including red shift, General Relativity effects, dark matter, and cosmic microwave background radiation. It introduces the Induced Electric Dipole Red Shift (IEDRS) as a laboratory verifiable mechanism explaining red shift and light bending through photon interaction with non-uniform electric fields. The study proposes alternative explanations for major cosmological observations, emphasizing the role of electromagnetic fields in space. These findings suggest a revised cosmological framework, offering new perspectives on the structure and behavior of the universe.
Quantum Computing in Practice: Navigating Simulation and Optimization on the Pathto Quantum Advantage
The commercial and scientific momentum behind quantum computing is accelerating rapidly, yet a critical gap remains between theoretical promise and practical implementation. This talk bridges that gap through some of the hands-on projects spanning quantum simulation and combinatorial optimization, implemented on real quantum software frameworks including Intel Quantum SDK, Eviden myQLM, PennyLane, and Qiskit. We begin by motivating the two most tractable near-term application domains - simulation and optimization, through current industry trends, before examining concrete implementations: a GAN-accelerated surrogate simulator for neutrino interactions in the INOICAL particle physics detector; a quantum-accelerated Quantum Monte Carlo workflow for correlated electron systems, developed in collaboration with Intel and presented at ISC 2025; a QAOA-based solution to the flight-gate assignment problem, developed in industry collaboration with Atos-Eviden; and a hardware-aware Variational Quantum Eigensolver (VQE). The talk concludes with honest lessons from implementing these algorithms on real platforms, the theory-hardware gap, and why hybrid classical-quantum workflows remain the practical frontier. Attendees will leave with a grounded understanding of where quantum advantage is being engineered today, and what it actually takes to get there.
Otto H. Kegel’s Beloved Contributions to Sylow Theory – In Memory of My Adored Masterly Teacher
Let p be a prime. A p-group is a group all of whose elements have order a power of p and a p’-group is a group all of whose elements have order prime to p, that is, not containing any element of order p. A p-group is finite if and only if its order is a power of p. Every group theorist knows what are well-known as the three Sylow theorems: any finite group G has 1) subgroups of order |G|p, where |G|p is the highest power of p dividing the order |G| of G, which are called Sylow p-subgroups and are not only maximal with respect to (w.r.t.) order but also w.r.t. inclusion, whence 2) every p-subgroup of G is contained in at least one Sylow p-subgroup, and 3) all the Sylow p-subgroups are conjugate, that is, if S1 and S2 are Sylow p-subgroups of G, then there exists an element x of G such that x-1S1x = S21. The set of all Sylow p-subgroups of a group G is denoted by Sylp(G). The theorems and the maximal p-subgroups are named after Ludvig Sylow, the great Norwegian mathematician who discovered them and published them (fortunately) in December 1872 (see [26.] and https://en.wikipedia.org/wiki/Peter_Ludvig_Sylow). A Sylow p-subgroup of an arbitrary group is a p-subgroup, which is maximal w.r.t. inclusion. Every p-subgroup of a group is contained in at least one Sylow p-subgroup. A group satisfies the Sylow Theorem for the Prime p or the Sylow p-Theorem, if all of its Sylow p-subgroups are conjugate, and it satisfies the Strong Sylow Theorem for the Prime p, if each of its subgroups satisfies the Sylow p-Theorem. A locally finite group is a group all of whose finitely generated subgroups are finite. Sylow Theory of Locally Finite Groups studies when they satisfy the Sylow p-Theorem and when even the Strong Sylow p-Theorem and determines the structure of those groups. A central concept to that end is the p-uniqueness subgroup of a locally finite group, which is a finite p-subgroup being contained in a unique Sylow p-subgroup. 1When G is a finite group, P a p-subgroup of G and S ∈ Sylp(G), then the operation of P by conjugation on C(G,S) := { Sx | x ∈ G } has at least one fixed point, that is (∃ x ∈ G) (Px ⊆ S), and for P ∈ Sylp(G) exactly one, which means 4) |Sylp(G)| = |G : NG(S)| = |C(G, S)| ≡ 1 (mod p); hence G satisfies the Strong Sylow Theorem for the Prime p, that means, every subgroup U of G conjugates transitively on Sylp(U), and thus we have the Frattini argument for G (and p), that is, if N is a normal subgroup of G and P ∈ Sylp(N), then NG(P) covers G/N, that is, G = N · NG(P). Otto H. Kegel has since the swinging sixties of last century again and again showed interest in Sylow Theory and very especially in how to extend it from finite groups to locally finite groups. He summarised findings up to 1973 by him and by others in the book [22.], which became a standard book on locally finite groups. When the book was in press, he developed the new paper [11.] on Sylow Theory of Locally Finite Groups and presented it in lectures during 11 December 1973. This paper has two open questions until today. 13½ years later he presented on 8 June 1987 in four lectures [12.] a summary of results up to 1987 about Sylow Theory of Locally Finite Groups and extended them beautifully from locally finite and p-soluble groups for p ≠ 2, according to results by Brian Hartley and Andrew Rae, to locally finite groups in general for p ≥ 5. This paper has ten open questions until today. The paper at hand presents Otto H. Kegel’s achievements of and merits for Sylow Theory of Locally Finite Groups and communicates that it was a “Herzensangelegenheit” (matter close to one’s heart) for him. Otto H. Kegel passed away on his birthday 20 July 2025 at the age of 91. Being in deepest mourning, I miss him dreadfully and will always honour his memory. The paper presents references to articles on Sylow Theory of Locally Finite Groups by Otto H. Kegel and by the author. First it presents and summarises the twelve beloved contributions by Otto H. Kegel. Secondly it presents and summarises the ten+four enthusiastic contributions by the author, which add thorough comments to Otto H. Kegel’s admirable contributions regarding wide generalisations, solve two of his open issues and state twelve still open issues most of which, however, the author solved already in yet unpublished work. Thirdly it draws some conclusions from Kegel’s contributions and the author’s contributions thereby suggesting fascinating topics for future research based upon the twelve open issues of [11.] and [12.]. In the meantime the author published his solution of Open Issue 6: see 28. Flemisch, F.F.: Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p – Manuscript on Sylow theory in locally finite groups – Mathematics edition – Second edition. Journal of Mathematical & Computer Applications (JMCA), Volume 5, Issue 2 (April 2026), 1-40. https://srcpublishers.com/mathematical-computer-applicatio/issue/view/2041.
Physics-Informed Neural Network for Deep Learning Solution of Forward and Inverse Problems Involving PDE Physical Laws
Physics-based AI-driven techniques integrate physical laws into neural network (NN) training. In the Physics-Informed Neural Networks (PINNs) method, a loss function includes data-driven and physics-driven components. The NN is trained to minimize this loss thereby learning solutions consistent with both observed data and the governing physical laws. In machine learning, PINNs enable the solution of ODEs and PDEs by embedding physical laws directly into the training process. PINNs incorporate the governing equation of a system as a constraint in the loss function. This method regularizes the learning process and enables robust predictions under noisy and sparse data. The nonlinear pendulum [Guckenheimer & Holmes, Springer, 1983] governed by a secondorder nonlinear ODE is a canonical example of system dynamics. For an undamped nonlinear pendulum, we have [$\frac{d^2\theta}{dt^2}+\frac{g}{L}\sin(\theta)=0$], where (\theta(t)) is the angular displacement. This presentation focuses on PDEs for modeling more complex systems in physics, engineering, and applied sciences where solutions are sought. In particular, the time-dependent Schrödinger equation (TDSE) is introduced as the foundational PDE of non-relativistic quantum mechanics governing the evolution of quantum states and giving rise to both forward and inverse problems. A forward problem predicts the system’s evolution given the Hamiltonian together with initial and boundary conditions. In contrast, an inverse problem seeks to estimate unknown parameters from observed data. Inverse problems are often formulated as optimization problems like [$\underset{V}{\mathop{\min}}\,\|F(V)-\mathcal D_{\text{obs}}\|^2+\alpha R(V)$] where (R(V)) is the regularization term. The TDSE for a quantum system in d spatial dimensions is [$\mathrm{i}\hbar\frac{\partial\Psi(\mathbf{r},t)}{\partial t}=\hat{H}\Psi(\mathbf{r},t)$] where (\Psi(\mathbf{r},t) is the complex-valued wavefunction. A survey by Banerjee et al. (arXiv:240.01026, 2024) highlights growing interest in PINNs for medical image analysis. PDEs with sparse data are particulary promising for PINNs such as in hemodynamics (i.e., blood flow models) with Navier-Stokes equations, non-Newtonian blood rheology (e.g., Carreau-Yasuda, Herschel-Bulkley, Casson models), and reduced-order arterial models.
Vesicles play a vital role in many metabolic processes or also in drug delivery (liposomes). One of the various properties they have to enjoy for proper functioning is the stability in shape, more precisely in volume and surface area. In this talk, we will describe a rigorous mathematical description of such vesicles, explain how surface area and enclosed volume enter the game, present the mathematical findings and discuss the resulting implications for biology. This is joint work with Andrea Mondino from Oxford University and Tristan Rivière from ETH Zurich.
Integral Inequalities and Error-Certified Candidate Ranking in Computational Protein Design
Objective: This chapter aims to develop an applied mathematical framework for computational protein design by integrating integral inequalities, generalized convexity, and error-certified numerical scoring. The main objective is to improve the reliability of pre-experimental candidate prioritization by moving beyond raw model outputs and incorporating explicit error control into numerical evaluation. Theoretical Framework: The chapter is grounded in generalized convexity theory, continuous performance functionals, and quadrature-based numerical analysis. In particular, s-convexity, tgs-convexity, and trapezoid-, midpoint-, and perturbed-trapezoid-type inequalities provide the theoretical basis for deriving explicit error bounds in candidate scoring. Method: The method defines continuous score functionals for AI-designed therapeutic protein candidates and evaluates these functionals numerically through quadrature rules. The resulting approximation errors are bounded analytically using theorem-backed integral inequalities. A dysferlinopathy-oriented case study, supported by a Miyoshi myopathy data workbook, is incorporated to summarize public datasets, controlled-access resources, representative variants, and ongoing clinical studies. Results and Discussion: The findings indicate that candidate ranking based solely on numerical scores may be unreliable when quadrature error is ignored. By introducing a certification layer, the proposed framework yields a more robust and analytically defensible ranking strategy. The case study demonstrates the practical relevance of certified ranking for closely competing therapeutic binder candidates. Research Implications: The framework offers a transferable methodology for trustworthy AIassisted therapeutic design and may inform future studies in computational biology, numerical optimization, and intelligent decision-support systems. Originality/Value: This chapter contributes to the literature by combining computational protein design with generalized-convexity-based error certification. Its originality lies in transforming numerical candidate scoring into a mathematically certified ranking process, thereby strengthening the reliability of computational therapeutic prioritization.
Proof of the Conjecture about an Infinite Set of Twin Prime Numbers
Some classical problems in number theory serve as a reliable platform for solving slightly more complex mathematical problems that have long been, not quite deservedly, elevated to the rank of open mathematical problems. As the author sees it, this is precisely the case with the well-known unsolved problem of twin prime numbers.
Climate Forecasts Using the Hybrid SARIMA-LSTM Onacc Model: Optimizing Predictions for the Bimodal Humid Forest Zone of Cameroon
Cameroon is facing increasing climate variability, with direct consequences on agriculture, ecosystems, and rural development. The bimodal humid forest zone-covering the Centre, South, and East regions-is particularly vulnerable due to its specific climatic conditions, which make it both agriculturally productive and highly sensitive to fluctuations in precipitation and temperature. This study aims to calibrate and apply the SARIMA-LSTM ONACC hybrid model, developed under the auspices of the National Observatory on Climate Change (ONACC). Our goal is to optimize climate forecasting for this critical agroecological zone. Historical temperature and precipitation data (1980-2022) from selected meteorological stations were used to train and test the model. The methodology combines the Seasonal AutoRegressive Integrated Moving Average (SARIMA) model to capture linear and seasonal components, with Long Short-Term Memory (LSTM) networks to model nonlinear dependencies and residual dynamics. Preprocessing steps include normalization, seasonal decomposition, and model validation. Model performance was evaluated using RMSE, MAE, and NSE indicators. Results show that the SARIMA-LSTM ONACC model reliably reproduces the interannual variability of the region. These results provide a solid foundation for informed agricultural planning, water resource management, and the development of climate risk mitigation strategies in southern Cameroon.
Ulianov’s Formulation of the Maxwell’s Equations, from a Spherical-Shell Electron Model
Ulianov String Theory (UST) proposes a non–pointlike electron: instead of an infinitesimal particle orbiting the proton, the electron is modeled as a thin spherical shell of negative charge whose radius is comparable to the size of its orbital. The shell rotates at high angular velocity and naturally generates a magnetic field. The negative charge is distributed over the entire membrane in a manner that parallels the quantum mechanical wave function; however, rather than representing a probability density of finding a point particle, the wave equation is interpreted here as a real charge density over a geometrical surface. All of the rest mass of the electron is concentrated in a single point located at one pole of the shell-the polar mass. This point like mass determines a preferred direction of rotation, so that all electrons share the same intrinsic rotational sense. What quantum theory labels as ”opposite spin” corresponds, in this model, merely to the shell being inverted, with the polar mass located at the opposite pole. Thus, the electron possesses an extended charge distribution but a localized point of inertia. This microscopic structure allows for a natural and fully geometrical decomposition of macroscopic polarization P and magnetization M. From these fields, the usual conduction-current term μ0J in the Ampère–Maxwell law is not assumed but is instead recovered as an emergent quantity. As a result, Maxwell’s equations can be rewritten in a field-only form: ∇× E = -∂B/∂t, ∇ ・ B = 0, ∇ ・ D = 0, ∇ ×H =∂D/∂t. With D =ε_0 E+P and H =1/μ0 B -M. The central modification introduced by UST concerns the Ampere-Maxwell law. In its standard SI form, ∇ × B =μ_0 J+μ_0 ε_(0 ) ∂E/∂t. the magnetic field has two distinct sources: electric current and the time derivative of the electric field. This two-term structure reflects the traditional belief that magnetism generated by permanent magnets and magnetism generated by electric currents are fundamentally different. However, UST provides a unified microphysical explanation. Because every electron is a rotating spherical shell with a mass pole, both forms of magnetism arise from the same underlying mechanism: the alignment or misalignment of the rotation axes of many electrons. An electric current does not create a magnetic field; rather, the preexisting microscopic magnetic fields of the electrons become partially aligned as the shells move and the mass poles lag due to inertia. This alignment process parallels the way sailboats randomly oriented on a calm sea all turn in the same direction under a light wind. Thus, the magnetic field surrounding a current-carrying wire is not produced but is the collective ordering of microscopic magnetization fields. This perspective removes the need for the explicit sum in the original Ampere –Maxwell law and yields a more uniform and elegant field structure without altering any experimentally verified predictions of classical electromagnetism.
Mathematical Modeling Applied to Combustion Processes: Advances in the Theory of Combustion Modeling
Combustion modeling theory is an area of science that focuses on developing mathematical and computational models to describe and predict combustion processes, which occur when a fuel reacts with an oxidant, usually oxygen in the air. In this research, thermodynamic concepts, kinetic models and fluid mechanics models are applied to model an ideal combustion process in a combustion chamber of a jet engine in order predict the temperature variations in the combustor caused by the injection of hot gases into the chamber and by combustion.
This article explores the deep structural constraints governing algebraic composition and argues that the universe cannot be fundamentally commutative or associative. Starting from Hurwitz’s classical theorem, which classifies real normed division algebras and restricts them to four cases-ℝ, ℂ, ℍ, and ????-we show that multiplicative coherence is an extraordinarily rare geometric property. The Cayley-Dickson construction is then analyzed as a controlled extension of this framework, revealing a systematic trade-off between expressivity and structural stability: as dimension increases, commutativity, associativity, and eventually divisibility is necessarily lost. Beyond abstract algebra, the article connects these mathematical results to concrete everyday experiences, illustrating non-commutativity through the order of ordinary actions (socks and shoes) and non-associativity through musical composition, where grouping determines meaning. These examples are not merely pedagogical metaphors, but manifestations of the same structural logic governing algebraic systems. We argue that non-commutativity and non-associativity should not be understood as defects, but as signatures of a world rich enough to sustain meaning, interaction, and transformation. From this perspective, stable algebraic structures correspond to rare equilibrium states, while most cognitive, physical, and creative processes operate in regimes where structure is dynamic rather than fixed. The article thus proposes a unifying interpretation in which mathematics functions as a geometry of possibilities, explaining why certain forms of simplicity are forbidden and why complexity is a necessary condition for the emergence of a meaningful universe.
BIM 2.0 in Project Development Construction Execution and Supervision
Building Information Modeling (BIM) 2.0 represents a significant evolution in construction project management, integrating additional dimensions such as time (4D), costs (5D), and sustainability (6D). Its adoption has transformed collaboration between stakeholders in the AEC (Architecture, Engineering, and Construction) sector, enabling real-time data-driven decision-making and improving project energy efficiency and sustainability. Recent studies indicate that BIM 2.0 implementation can reduce construction costs by up to 20% and lead times by 15%. Regarding the Methodology, this article is based on a systematic review of Scopus-indexed literature (Q1) and representative case studies between 2019 and 2023. BIM 2.0 applications were analyzed in LEED certified projects and energy efficiency standards, using energy simulation tools and advanced 3D modeling. Performance metrics such as CO₂ emission reduction, resource optimization, and risk management were evaluated. Addressing the Results and Discussion, we can say that the implementation of BIM 2.0 demonstrated a 30% improvement in energy efficiency in net-zero buildings and a 25% reduction in carbon emissions during the construction phase. Interoperability with AI and IoT enabled real-time monitoring of energy consumption and early detection of design conflicts. However, challenges such as the lack of software standardization and the shortage of skilled labor persist, affecting 40% of the analyzed projects. In the Conclusions section, we determined that BIM 2.0 is consolidated as a key tool for sustainable construction, facilitating the integration of emerging technologies and improving project lifecycle management. Investment in professional training and protocol standardization is recommended to maximize its potential. Future research should address the integration of blockchain and digital twins to optimize data security and traceability.
Mathematical Examination of MHD Slip Oscillatory Blood Flow Through an Artery in the Presence of a Magnetic Field
In this study, we investigated the mathematical examination of MHD slip oscillatory blood flow through an artery in the presence of a magnetic field. The process involves the development of dimensional mathematical models representing the flow in the form of a partial differential equation and scaling the model using some quantities where the governing models were made dimensionless and some pertinent parameters, such as the Hartman number and Womersley number, were obtained. The dimensionless governing model was perturbed and reduced to ODE. Method: The perturbed ODE was solved using the Laplace Method (LM), where the function representing the flow was obtained, and we performed numerical simulation using Wolfram Mathematica, version 12. The results reveal that the Hartman number and Womersley numbers influence the blood flow, flow rate, and shear stress, respectively. In conclusion, we have been able to derive a mathematical model that represents the problem, solve it analytically using the Laplace method, and carry out numerical simulation.