Offline tangible coding games, such as the RANGERS and TANKS problem-solving games [1] serve as a strong bridge connecting Computational Thinking Concepts to Mathematics. Offline coding games[2] and activities[3] are inspired by computer science but introduce Computational Thinking Concepts without using a computer. This approach enriches the teaching and learning of Mathematics, making abstract concepts more accessible and engaging[4].

In this presentation, participants will be introduced to coding and computational thinking concepts in a fun and playful manner through a series of engaging hands-on activities. The session will begin with interactive exercises designed to demystify coding, followed by hands-on interaction with the RANGERS and TANKS problem-solving offline coding applications. These applications provide a tangible way to explore computational thinking concepts, demonstrating how these skills are directly related to mathematical reasoning. 

Following these activities, participants will engage in a reflective discussion to identify and articulate the strong connections between computational thinking concepts and Mathematics. This reflection time is crucial as it helps participants solidify their understanding and see the practical implications of intergrating these concepts into their teaching practices. The session aims to highlight the value of offline coding games in making computational thinking a natural part of Mathematics education.

It has been proposed that life is a Singularity (1). The primary means of sustaining that state biologically is Symbiogenesis, the assimilation of factors in the environment that pose existential threats. But why not just destroy them? Answer: Because then there would be no history to reference in order to evolve effectively. Moreover, given that, Symbiogenesis must have evolved from a Singularity as its reference point, generating a ‘holism’. This hypothetically answers the question as to where mathematics emerged from as follows. Tegmark (“Our Mathematical Universe”) stipulates that math is inherent to the Cosmos; since Symbiogenesis  passively assimilates the math inherent to the Cosmos, incorporating it into physiology as the basis of consciousness, there is a seamless integration of math both physiologically and consciously. For example, Rowlands’ Rewrite Math (“Foundations of Physical Law”) mirrors the mechanism of Epigenetic Inheritance, Rowlands’ zero ‘attractor’ behaving like the cell does, a new datum assessed by the existing data set behaving in the same way as the egg and sperm of the parent, facilitating adaptation to change. Lou Kauffman’s ‘Knot Math’ similarly resembles embryogenesis, the twisting of the circle generating knots, the twisting of the embryo generating the stages of its development. Proof of a true ‘knot’ is the ability to unknot it to re-form the circle; homologously, the cell must undergo meiosis to form the egg or sperm in order to then reproduce. And Soft Logic Math as the family of real numbers and ‘zeroes’ reflects how physiology has evolved as a two-tiered system, the real numbers being the horizontal accumulation of data, the family of zeroes as the vertical generation of physiologic traits ontogenetically and phylogenetically. This  intimate relationship between the Singularity, Symbiogenesis and evolution can also be seen as a Fibonacci sequence, the Singularity (1) + Symbiogenesis (2) = Evolution (3), or 1 + 2 = 3, out of which emerges empathy as a product. These fundamental insights provide for sustainability of life based on the ubiquity of the Fibonacci sequence. 

A problem of the connection of cosmology with elementary particle physics is shown on the level of uncertainty relations. At the scales about 10^-2 m the contribution of one single type virtual elementary particles in the lower boundary of vacuum energy is considered. The observed value of vacuum energy or energy density on the large scale of the Universe corresponds only to this scale. This is the energy about 3.34 GeV per each one cubic meter. The minimal high energy physics scale achieved by experiments at present is considered. The lower boundary of the energy is generated by the quantum vacuum of empty space and the quantum vacuum limited by matter in the Universe mainly at scales down to 10^-15 m and more much are not in agreement with the observed value, as that is established. These lower limits for the energies of the vacuum are considered in the model of estimating where they generate by the presence of virtual particles in free space and the virtual particles which are limited by matter and exist together with matter in the Universe. The numerical values of the boundary energies are obtained using the computer algorithm.

This review paper examines the status of artificial intelligence (AI) technology in Ethiopian STEM (Science, Technology, Engineering and Mathematics) schools and the possibility of implementing AI programs in the future. Many developed and developing countries are using AI to help grow and improve their economies and to leverage their technology and services. The primary use of AI technology in schools is generally to make more creative the teenagers learning in mathematics and science. This article aims to provide alternative directions on implementing artificial intelligence programs in Ethiopian STEM schools, with an emphasis on learning from developed countries and sharing best practices. Primary and secondary data are used: Secondary data are analyzed on theory-based evidence while primary data are analyzed based on structured questionnaires. In order to achieve the goal, select journals, research and other related websites are reviewed. The findings of this review indicate that in STEM schools, there are many teenagers with specific interests and abilities in mathematics and coding. This knowledge is needed for artificial intelligence. The encouragement and reflection on the advantages of basic AI concepts for youths is necessary as it can help to engage talented students in learning. This paper thoroughly analyzes relevant research and interview data to highlight key insights, status, challenges, and future directions for AI implementation in Ethiopian STEM schools.

Mineral-, metal- and mine-rich Sweden was a leading powerhouse of Chemistry in the mid- eighteenth to mid-nineteenth centuries with, still most in the world, 20 of the then known elements discovered by Swedes (n = 12), and another telling world record presented by the mineral gadolinite found in 1882 in the village of Ytterby on an island in the Stockholm archipelago, from which eight elements were originally identified; in increasing atom number order, and first isolation in parenthesis, nr 21 Scandium (Nilson 1879), 39 Yttrium (Wöhler 1838), 64 Gadolinium (de Marignac 1880), 65 Terbium (Mosander 1843), 67 Holmium (Cleve 1878), 68 Erbium (Mosander 1843), 69 Thulium (Cleve 1879) and 70 Ytterbium (von Welsbach 1906).

The shining star in this Eldorado was Jöns Jacob Berzelius (1779-1848),  “the father of modern Chemistry”. He was initially trained and worked as a physician, which widened his scope, and from his enormous production here only will be mentioned that he discovered four elements;  determined  all  then known atomic  weights and  innumerable  other chemical properties and conditions; developed the subject of Electrochemistry; coined the terms ”allotrope”, ”catalysis”, ”polymer”, ”isomer”, ”protein” and others; and formulated the still valid distinction between ”organic” and ”inorganic” Chemistry. 

Fundamentally, he also invented the unambiguous system of chemical notation still used today, i.e. the first letter(s) of the Latin name of the atoms with a numerical suffix of the amount of them in the compound, e.g. CaCO₃, calcium carbonate, containing one Calcium, one Carbon and three Oxygen atoms. No other factors are involved, so the consequence would be that the layering between  the atoms is dependent upon their sizes and proportions, i.e. stoichiometric, which was  Berzelius’  main hypothesis. But nobody knew the fabric of an atom, and he also, somewhat prophetically, suggested that electric forces might be involved to bind the atoms together. However, in the mid-19th century, Kekulé in Germany, Frankland in Great Britain and others developed the theory of  "combining power", in which compounds were joined owing to an attraction between positive and negative poles, and from there the ‘valence’ concept has mushroomed to a plethora of localized varieties which have taken over the whole body from the real atoms in a tail-wags-the-dog  way. A “network”, a “skeletal”, a “ball-and-stick” chemical formula is a phantom figure of an artificial structure and reduces the true atom build to a point or a mere intersection. It is high time to return to the stoichiometry of the atoms themselves! 

Since many years I have been studying an alternative to the Standard Model, following the original differential Lie algebra as outlined in his Norwegian Ph.D. Thesis Over en Classe geometriske Transformationer at Christiania (now Oslo) University in 1871.1,2 I could spend days praising the geniality and innovation of this work. However, the main point here is that it has no direct connections with his continuous groups or root space classifications that the Standard Model has been tied up with to annoying near-fit, but that it is a tangential line congruence algebra that together with the Bohr Aufbau system allows a precise reproduction of the periodic table and its molecular combinations.3-5 Regrettably, though, no one understood Lie ́s thesis: it got excellent marks but soon went into oblivion in the faculty archives. 100 years later I went there and got a photo copy of it (now one can get it electronically), and 1984 I together with professor R.M. Santilli translated it into English (internet open access available at hadronicpress.com/lie.pdf ).2

The aim of this communication is to review and discuss the findings from their chemical model and formula implications. Now that we have the natural figures of all atoms the phantom diagrams of their inferred force lines should be replaced by their real structure.

In the development of the theory, an approach with strict observance of the principles of physical reality, causality, and logical understanding is used. The main goals were clarifying the space-time nature of the physical vacuum, and identifying the physical models of stable elementary particles. The proposed physical model of the physical vacuum corresponds to the etheric substance predicted by the ancient Greek physicists Plato and Aristotle and maintained until the beginning of the 20th century, but a similar model has not been suggested before. It is a specific grid with oscillation properties made up of two types of sub-elementary particles with sizes in the scale about 1x10^-20 (m) and called the Cosmic Lattice. The elements of the Cosmic Lattice are held by a Supergravitational Law (SG) which differs from Newtonian gravity in that SG forces are inversely proportional to the cube of the distance. Therefore, they are super-strong at the microscopic level.  Their experimental manifestation is the attractive and repulsive Casimir forces between two bodies with highly polished surfaces. It is inferred that the sub-particles forming the Cosmic Lattice also build elementary particles as helical structures. Experimental results from particle colliders and in particular the characteristics of the first unstable particles such as pions and kaons and their decay were used to infer initially the shape of protons and neutrons. The narrow standard deviation of mass and lifetime of the pions and kaons lead to the conclusion that the protons and neutrons from which they emerge have some toroidal shape in which they are locked. Therefore, with a single cut, they come out with a very narrow standard deviation. From the additional decay of pions, it was inferred that the elementary particles are made of helical structures left-handed and right-handed, while handedness defines the sign of charge as a specific modulation of the Cosmic Lattice. The electron is a 3-body system of helical structures whose oscillation and rotational motion interact with the oscillating properties of the Cosmic Lattice and exhibit quantum mechanical features. The proton and neutron are with the same substructure but with different shapes. The proton is a twisted toroid like a 3-D Hippoped curve, while the neutron is double-folded. At the neutron, the electrical charge is locked by the SG forces at the near field, but in motion, it exhibits a magnetic moment. In the atomic nuclei, they are held by a balance between repulsive Coulomb forces and attracted SG forces. Using their shapes and following the building trend of the nuclei a complete match to the shape of the Periodic Table of Elements is obtained. Chemical valences, bond direction, and isotope stability are apparent. In theory, this is called the Atlas of Atomic Structures. The validation of the Atlas is supported by experimental results from 16 different fields of physics. The physical dimensions of the proton, neutron, and electron are identified. 

The theory offers a hypothetical scenario for the creation of sub-elementary and elementary particles through a unique crystallization that takes place on the surface and surrounding of a superdense protomatter known as a black hole indirectly observed at the center of galaxies. This leads to a very different view of the universe and to an explanation of the serious inconsistencies with the Big Bang model. Instead of this single burst, it is concluded that galaxies have cycles of active existence and hidden unobservable phases of recycling and crystallization of elementary particles ending with the birth of a new visible galaxy with a new Cosmic Lattice. In such a case, the redshift of the galaxies appears not to be Doppler but of a cosmological nature as a result of a weak difference in their Cosmic Lattices, which depends on the individual mass of the galaxies. The theory was first published in 2001, cataloged in the National Library of Canada in 2002, and published as a book, scientific papers and reported in many scientific conferences.

Negative signed numbers can represent a variety of concepts depending on the specific context in physics. Sometimes, their meaning is ambiguous, and in quantum equations, the physical interpretation of these quantities can be uniquely challenging. For example, the Dirac equation is well-known to have both positive and negative signed solutions. Today, physicists still debate the physical significance of the negative signed solutions. One way to confer meaning to these ambiguous negative signed quantities is to express them in an expanded dimensional canvas. But how might such a canvas be conceived?

Part 1 of this paper sketches a new approach showing considerable promise. The fundamental symmetry of positivity and negativity seem to be built into the very fabric of the universe at subatomic scales. In consonance with Rowlands’ concept of totality zero, these fundamental symmetries can be shown to naturally admit an alternative explanation for conceptualising the shape and content of space at subatomic scales. The approach posited is that a recursive pseudo Riemannian cobordism (PRC) of manifolds can describe the shape of space using an expanded nD + 1 space coordinate system. The advantage of this system is that it confers physicality on abstract nD mathematical spaces that are conventionally assumed to map the structure of reality. The revised shape of space admits physical content consistent with experimental findings in an altogether different way. The physical system of two surfaces connected by a small, variable length, recursive dimension, constitutes a recursive cobordism, a modification of the one conceptualised by P. Yodzis. 

The new frame of thinking distinguishes between the content of space – light and matter – and the shape of space in which that content exists. Two elementary particles – analogous to the photon and electron - are described as embedded deterministic objects in this space. 

By adding a small, variable length +1 dimension that runs perpendicular to every direction in nD space, the emergent nD + 1 space physical system allows us to account for the ubiquitous negative signed quantities that emerge in certain quantum equations. It also admits a qualitative, deterministic expression of Maxwell’s equations, the Schrödinger equation and the Dirac equation.

This naturally leads to a deeper understanding of Rowlands' fundamental parameters of space, time, mass and charge, and delivers new insight into our understanding of condensed matter physics. Most tellingly, the approach opens the way to conceptualising subatomic particles, such as photons and electrons, as real physical objects existing in 3D + 1 space rather than appearing as statistical probabilities in abstract 3D space.

Negative signed numbers can represent a variety of concepts depending on the specific context in physics. Sometimes, their meaning is ambiguous, and in quantum equations, the physical interpretation of these quantities can be uniquely challenging. One way to confer meaning to these ambiguous negative signed quantities is to express them in an expanded dimensional canvas. 

In Part 1, a new approach showing considerable promise was introduced. Part 2 of this paper extends this approach. It was noted that positive and negative signed quantities seem to be built into the very fabric of the universe at subatomic scales. In consonance with Rowlands’ concept of totality zero, certain fundamental symmetries were shown to naturally admit an alternative explanation for conceptualising the shape and content of space at subatomic scales. The new approach posited that a pseudo Riemannian cobordism (PRC) of manifolds can describe the shape of space using an expanded nD + 1 space coordinate system. The advantage of this system is that it confers physicality on an otherwise abstract mathematical space. The revised shape of space admits physical content consistent with experimental findings in an altogether different way. Two particles – the fundamental particle of light analogous to the photon and the fundamental particle of matter, analogous to the electron, were described. 

In Part 2, the structure and dynamics of two additional particles analogous to the proton and neutron will be described in a PRC. The topology of these particles provides compelling answers to certain questions that have long eluded quantum theorists. In particular, the structure and dynamics of particles analogous to quarks and their force carrying gluons can be described deterministically, and this offers significant advantages. The elusive reasoning that gives rise to the strong interaction, the property of confinement, and the origin of fractional colour charges can all be seen as inevitable consequences of the new understanding. This completes the cobordism manifold picture of subatomic particles that constitute the atom.

This new approach opens the way to reconceptualising protons and neutrons as real physical objects existing in 3D + 1 space rather than appearing as statistical excitations associated with an infinitely extensive quantum field in 3D space. This naturally leads to a deeper understanding of subatomic reality, the fundamental parameters of space, time, mass and charge, and delivers new insight into our understanding of condensed matter physics. 

Physics, at the fundamental level, is concerned only with a single system, that of the fermion or fundamental particle, and it is possible to conceive the actions of the entire universe as those of a single fermion. We have had a quantum mechanical equation for the fermion – the Dirac equation for nearly a hundred years – but no one has imagined that that equation alone could even lead to all the developments encoded in the Standard Model of particle physics. This is partly because the equation is not normally expressed in its most significantly meaningful form, but it is also because ‘derivation’ is generally taken to mean a deductive mathematical consequence given certain conditions rather than an unfolding of the innate structure that is built into the equation as a result of more fundamental principles. The equation is not necessarily the source of these principles, rather the codification of them. In fact, given these additional considerations, it is possible to see the equation in its most physically meaningful form as the source of all current aspects of the Standard Model and even some things beyond it. In addition, the full statement of the equation is not necessary for these derivations, only the definition of the fermion creation operator that the equation requires. So, the question that we will be answering is the more restricted one of whether physics can be defined by a single operator, rather than a single equation.

As it is well known, Isaac Newton had to develop the  differential calculus, (jointly with Gottfried Leibniz), with particular reference to the historical definition of velocities as the time derivative of the coordinates, $v = dr/dt$, in order to  write his celebrated equation $m a = F(t, r, v)$, where $a = dv/dt$ is the acceleration and $F(t, r, v)$ is the Newtonian force acting on the mass $m$. Being local, the differential calculus solely admitted the characterization of  massive points. The differential calculus and the notion of massive points were adopted by Galileo Galilei and Albert Einstein for the formulation of their  relativity, thus acquiring a fundamental role in 20th century sciences.

In his  Ph. D. thesis of 1966 at the University of Turin, Italy, the Italian-American scientist Ruggero Maria Santilli pointed out that Newtonian forces  are the most widely known in dynamics, including action-at-a-distance  forces derivable derivable from a potential, thus representable with a Hamiltonian, and other forces that are  not derivable from a potential or a Hamiltonian, since they are contact dissipative and non-conservative forces caused by the motion of the mass $m$ within a physical medium. Santilli pointed out that, due to their lack of dimensions, massive points can solely experience action-at-a-distance Hamiltonian forces.

On this ground, Santilli initiated a long scientific journey for the generalization of Newton's equation into a form permitting the representation of the actual extended character of massive bodies whenever moving within physical media, as a condition to admit non-Hamiltonian forces. Being a theoretical physicist, Santilli had a number of severe physical conditions for the needed representation. One of them was the need for a representation of extended bodies and their non-Hamiltonian forces  to be invariant over time as a condition to predict the same numerical values under the same conditions but at different times.

The resulting new calculus, today known as  Santilli IsoDifferential Calculus, or IDC for short, stimulated a further layer of studies that finally signaled the achievement of mathematical and physical maturity. In particular, we note: the isotopies of Euclidean, Minkowskian, Riemannian and symplectic geometries; the  isotopies of classical Hamiltonian mechanics, today known as the  Hamilton-Santilli isomechanics, and the isotopies of quantum mechanics, today known as the  isotopic branch of Hadronic mechanics.

The main purpose in this lecture  is to represent some recent researches of Santilli iso-mathematics in the area of the plane geometry. This lecture  is devoted to the iso-plane geometry. It summarizes the most recent contributions in this area. 
Straight iso-lines are introduced. Iso-angle between two iso-vectors is defined. They are introduced iso-lines and they are deducted the main equations of iso-lines. They are given criteria for iso-perpendicularity and iso-parallel of iso-lines.  Iso-reflections, iso-rotations, iso-translations and iso-glide iso-reflections are introduced. We define iso-circles and  they are given the iso- parametric iso-representations of the iso-circles. We introduce iso-ellipse, iso-parabola and iso-hyperbola and they are given some of their basic properties. The lecture is provided with suitable examples.

The usual concept of an electron worldline in Minkowski space assumes that particles have smooth time-like curves representing the movement of a centre of mass. Events are then points on the worldline and can occur with arbitrarily small inter-event spacing. Mass by itself is not a direct kinematic feature of such curves. Instead, mass and energy are input from dynamical behaviour.  An alternative model, explored in this paper is to assume that mass represents an upper bound on the frequency of special events on the worldline. This imposes a lower bound on the measure of causal regions between such events and produces two characteristic scales associated with particle mass. The scales are classical analogs of the de Broglie and Compton scales respectively. The model provides a direct basis for Feynman's original non-relativistic path-integral approach to quantum mechanics[1] as well as his relativistic chessboard model[2] and its extension to 3+1 dimensions[3].

In this work, a nonlocal elasticity model of Klein-Gordon type, characterized by nonlocality in space and time, is developed for the investigation of wave propagation in isotropic elastic media [1, 2]. Nonlocal elasticity theory having a close link to the underlying microstructure has the advantage to capture effects at small scales [3]. Specifically, nonlocal elasticity is valid down to the Ångström-scale and it can be considered as a generalized continuum theory of Ångström-mechanics as it has been shown in [4]. For the first time in the framework of nonlocal elasticity theory, the proposed nonlocal elasticity model of Klein-Gordon type possessing one characteristic internal time scale parameter in addition to the characteristic internal length scale parameter describes spatial and temporal nonlocal effects at small scales. 

The dispersion relations of the considered isotropic nonlocal model of Klein-Gordon type are analytically determined predicting in addition to the acoustic modes (low-frequency modes), optic modes (high-frequency modes) as well as frequency band-gaps between the acoustic and optic modes. The ranges of the frequency band-gaps for longitudinal and transverse waves are determined. Moreover, the phase and group velocities are calculated for the acoustic and optic branches of longitudinal and transverse waves showing that all four modes exhibit normal dispersion with positive group velocity. 

The proposed nonlocal model of Klein-Gordon type possessing only 4 constitutive parameters (2 elastic constants, 1 length scale and 1 time scale) provides an appropriate framework for the modelling of accurate frequency band-gaps and overall physically realistic dispersive wave propagation.

Positive integer factorization has raised concern particularly in the security of communication technology, based on the fact that positive integers can be employed in the safety of information in cryptography. This study, revealed an infinite square table which is instrumental in the analysis of all positive integer regarding their prime factors. Formula or instructions came up following the orderly arrangement of the positive integers. Moreover, a formula to test primality of a number become obvious in this study. In this regard, positive integers particularly can all potentially be factorized. While it is a solution in mathematics, this can pose threat in the safety communication technology. Nevertheless, this is on one side a solution in cryptography for the fact that a new page is opened to advance the safety of information. Had it been the case that this study did not reveal this fact, the information technology could be at risk. Therefore, the study is about knowledge to communication technology stakeholders to consider safety modification in information systems.

There is an ongoing debate as to whether our biologic origin is dualistic or monistic. Theoretically, there should be unequivocal scientific evidence for one state or the other being the primary source of the evolutionary origin of serial Symbiogenesis. The essence of this article is that the process of Symbiogenesis- the assimilation of factors in the environment that have posed existential threats as the physiologic basis for consciousness provides that hypothetical evidence. Peter Rowlands’ Rewrite Mathematics is an algorithmic homolog of Epigenetic Inheritance that explains the how and why of mathematics running true to physiology. However, it only accounts for synchronic or ‘real time’ consciousness, not for diachronically emergent nonlocal consciousness. It is how and why Gödel’s Incompleteness Theorems are incomplete….they overlook Gödel’s nonlocal consciousness that was necessary for formulating the formal math.

A new gravitational theory in full symbiosis with quantum mechanics, applicable to all bound fieds is presented. We call it the Quantal Theory of Gravity (QTG). It is in no way an attempt to repair the General Theory of Relativity (GTR). It is a whole new theory. It allows us to end the long lasting dichotomy, of not being able to handle the macroscopic world with the mathematical frameworks developed for the atomic scale. And vice versa, it ends our incapability to describe the atomic world with the mathematical tools which we shaped to describe the macroscopic world. In other words, we now have the same look at both the cosmos and the atom. Unlike what we thought so far, there is strictly no boundary between the microscopic world and the macroscopic world. It is all the same Nature, from protons to galaxies. The content of this presentation, is practically a summary, along with illustrations of our recent article “Quantal Theory of Gravity: Essential Points and Implications”, by C.B. Marchal et al, published in Annals of Physics, Volume 454, in July 2023, 169346.

Peter Rowlands appears to be the only physicist who has written a book on foundational laws in physics [1]. He identifies four fundamental symmetries that are foundational to physics: space, time, mass and charge. A group relationship, a zero-totality condition and a nilpotent Dirac equation are the primary mathematical structures used to build the foundational laws. [2]. The duality between space-time and mass-charge is so exact that any reversal of role between discrete space and continuous time also produces a corresponding reversal of role between continuous mass and discrete charge [3]. One of us has argued that his ‘principle of duality’ is so ubiquitous in both mathematics and physics that it should be promoted into a ‘law’ based on the quantum mechanical law of entanglement [4]. Rowlands identifies three distinct mathematical processes: (A) conjugation, (B) complexification and (C) dimesionalization. Their corresponding physical manifestations are dualistic in nature: (a) conserved/nonconserved, conjugated/nonconjugated, + / –, (b) real/complex (the relativistic duality) and (c) the discrete/continuous, or the dimensional/nondimensional options. A classic case of the discrete/continuous representation is the well-known continuous wave/discrete particle duality. Rowlands’ principle of duality UNITES the theory of general relativity (GR), which is a theory about gravity not a theory of gravity, and quantum mechanics (QM). It does not UNIFY them [5].

The Navier-Stokes problem in ℝ³ consists of solving the equations:

where v = v(x, t) is the velocity of the incompressible viscous fluid, p = p(x, t) is the pressure, the density ρ = 1, f = f(x, t) is the force, v₀ = v₀(x) is the initial velocity.

The aim of this talk is to explain and prove the author’s result concerning the Navier-Stokes problem (NSP) in ℝ³ without boundaries.

It is proved that the NSP is contradictory in the following sense:

If one assumes that the initial data and the solution to the NSP exists for all t ≥ 0, then one proves that the solution v(x, t) to the NSP has the property v(x, 0) = 0.

This paradox (the NSP paradox) shows that:

The NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution defined on all t ≥ 0.

In the exceptional case, when the data are equal to zero, the solution v(x, t) to the NSP exists for all t ≥ 0 and is equal to zero, v(x, t) ≡ 0.

The results, mentioned above, are proved in the author’s monographs [1], [5] and paper [3].

Our results solve the millennium problem concerning the Navier-Stokes equations, see [5].

These results are based on the author’s theory of integral equations with hyper-singular kernels, see [2], [4].

In paper [6], p.472, Theorem 2, there is a statement that, for f(x, t) = 0 and u₀(x) sufficiently small, the solution to the NSP exists for all t ≥ 0 if m ≤ q, where m is the dimension of the space and the solution is in L^q. In our case m = 3 and q = 2, so the condition m ≤ q does not hold. Therefore, the claim in [6], p. 472, is not applicable in our case.

The Navier-Stokes problem in ℝ³ consists of solving the equations:

where v = v(x, t) is the velocity of the incompressible viscous fluid, p = p(x, t) is the pressure, the density ρ = 1, f = f(x, t) is the force, v₀ = v₀(x) is the initial velocity.

The aim of this talk is to explain and prove the author’s result concerning the Navier-Stokes problem (NSP) in ℝ³ without boundaries.

It is proved that the NSP is contradictory in the following sense:

If one assumes that the initial data and the solution to the NSP exists for all t ≥ 0, then one proves that the solution v(x, t) to the NSP has the property v(x, 0) = 0.

This paradox (the NSP paradox) shows that:

The NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution defined on all t ≥ 0.

In the exceptional case, when the data are equal to zero, the solution v(x, t) to the NSP exists for all t ≥ 0 and is equal to zero, v(x, t) ≡ 0.

The results, mentioned above, are proved in the author’s monographs [1], [5] and paper [3].

Our results solve the millennium problem concerning the Navier-Stokes equations, see [5].

These results are based on the author’s theory of integral equations with hyper-singular kernels, see [2], [4].

In paper [6], p.472, Theorem 2, there is a statement that, for f(x, t) = 0 and u₀(x) sufficiently small, the solution to the NSP exists for all t ≥ 0 if m ≤ q, where m is the dimension of the space and the solution is in L^q. In our case m = 3 and q = 2, so the condition m ≤ q does not hold. Therefore, the claim in [6], p. 472, is not applicable in our case.

In this talk I will be presenting the main highlights of what I consider a very important development in the Medical Sciences for acquiring a much better understanding of the human body by demonstrating how the unique mathematical properties of  SDF  would play a very significant role in the development of more advanced and reliable theoretical models  for the human body.   This would require performing a complete analysis only on those  "general"  analytical solutions   that can be obtained using the very unique computational feature of  SDF  on  the  Naiver-Stokes equations for the  "Mechanical"  aspect of the human body that is largely influenced from the general Mechanical properties of fluids    and on the Schrodinger equation for the “Chemical”  aspect of the human body.

Currently there exist no such advanced theoretical models of  the human body that would be  based entirely on general analytical solutions of  PDEs because of  the severe limitation of Calculus  which if successfully resolved by the method of  SDF would become immeasurable in terms of reducing our excessive dependency on the use of experimental models in favor of a more universal algebraic theory for the  Physical and Biological Sciences.
 

We would like to draw attention in the present paper to a curious mathematical observation concerning fractional differential equations describing physical systems, whose time evolution for integer derivatives has a time-honored conservative form. This observation, although known to the general mathematical community [1, 2, 3], has not, in our view, been satisfactorily addressed. More specifically, we follow the recent exploration of Caputo-Riesz time-space-fractional nonlinear wave equation [4], in which the authors introduced an energy-type functional and proposed a finite-difference scheme to approximate the solutions of the continuous model. The relevant Klein-Gordon equation is considered, where we explore the sine-Gordon nonlinearity with smooth initial data. For the Riesz and Caputo derivative coefficients α=β=2, we naturally retrieve the exact, analytical form of breather waves expected from the literature. 

Focusing on the Caputo temporal derivative variation within 1<β<2 values for α=2, however, we observe artificial dissipative effects, which lead to complete breather disappearance, over a time scale depending on the value of β. We compare such findings to single degree-of-freedom linear and nonlinear oscillators in the presence of Caputo temporal derivatives and also consider anti-damping mechanisms to counter the relevant effect. These findings also motivate some interesting directions for further study, e.g., regarding the consideration of topological solitary waves, such as kinks/antikinks and their dynamical evolution in this model.

This course is a continuation of the first course that was given in Panama last year at the SIPS 2023 conference. Our primary objective for this year will always remain the same by providing a more transparent solution on the current major limitation of Calculus in terms of not being able to establish some form of a unified theory of analytical integration.

The complete mathematical solution that was presented at the last SIPS Workshop was described in the form of Specialized Differential Forms or SDF for short with some major applications in the field of the Physical Sciences that would include Fluid Dynamics, Mechanics of Material, Quantum Mechanics and even in Cosmology. We will be demonstrating at this Workshop how the unique mathematical properties of SDF would play a major role in the development of more reliable theoretical models of the human body by working only with the general analytical solutions of the Navier-Stokes equations for the Mechanical aspect and the Schrödinger equations for the Chemical aspect of the human body.

Currently there exist no such theoretical models of the human body that would be based entirely on general analytical solutions of PDEs because of the severe limitation of Calculus which if successfully resolved by the method of SDF would become immeasurable in terms of reducing our excessive dependency on the use of experimental models in the Physical and Biological sciences.

This course is a continuation of the first course that was given in Panama last year at the SIPS 2023 conference. Our primary objective for this year will always remain the same by providing a more transparent solution on the current major limitation of Calculus in terms of not being able to establish some form of a unified theory of analytical integration.

The complete mathematical solution that was presented at the last SIPS Workshop was described in the form of Specialized Differential Forms or SDF for short with some major applications in the field of the Physical Sciences that would include Fluid Dynamics, Mechanics of Material, Quantum Mechanics and even in Cosmology. We will be demonstrating at this Workshop how the unique mathematical properties of SDF would play a major role in the development of more reliable theoretical models of the human body by working only with the general analytical solutions of the Navier-Stokes equations for the Mechanical aspect and the Schrödinger equations for the Chemical aspect of the human body.

Currently there exist no such theoretical models of the human body that would be based entirely on general analytical solutions of PDEs because of the severe limitation of Calculus which if successfully resolved by the method of SDF would become immeasurable in terms of reducing our excessive dependency on the use of experimental models in the Physical and Biological sciences.

This course is a continuation of the first course that was given in Panama last year at the SIPS 2023 conference. Our primary objective for this year will always remain the same by providing a more transparent solution on the current major limitation of Calculus in terms of not being able to establish some form of a unified theory of analytical integration.

The complete mathematical solution that was presented at the last SIPS Workshop was described in the form of Specialized Differential Forms or SDF for short with some major applications in the field of the Physical Sciences that would include Fluid Dynamics, Mechanics of Material, Quantum Mechanics and even in Cosmology. We will be demonstrating at this Workshop how the unique mathematical properties of SDF would play a major role in the development of more reliable theoretical models of the human body by working only with the general analytical solutions of the Navier-Stokes equations for the Mechanical aspect and the Schrödinger equations for the Chemical aspect of the human body.

Currently there exist no such theoretical models of the human body that would be based entirely on general analytical solutions of PDEs because of the severe limitation of Calculus which if successfully resolved by the method of SDF would become immeasurable in terms of reducing our excessive dependency on the use of experimental models in the Physical and Biological sciences.

Peter Rowlands's development of fundamental physics using the nilpotent concept opens up deeper questions about homological applications of the same topological relationships in other domains. I will argue that the very idea of a homological relationship between one set of phenomena conceived in terms of a Clifford algebra, and another set of phenomena, may itself be attributable to the fact that the perception of homology derives from equivalences of "nothing" at different levels. To demonstrate this, examples can be drawn from fields as diverse as AI and music. 

In [1] and [2] models of elementary particles are proposed based on combinatorial substructures for quarks.
These papers succeed in given combinatorial models for many particle interactions. In [3] a vector version of the Harari, Shupe models is 
given, in which each particle is a four-vector and particle interactions correspond to vector identities. The Lambek model can be matched directly with the Shupe model, but contains 
extra information that allows the vectorial work. In [4] a so-called Helon model is given by Bilson-Thompson that uses framed three braids and can be seen as a generalization of the Rishon models of Harari and Schupe. In fact, we find (joint work with David Chester and Xerxes Arsiwalla) that the Lambek model is a nearly perfect intermediary between the Helon model and the Rishon model. There is a direct correspondence between Lambek's four-vectors and the braids in the Helon model, up to a slight readjustment. This means that we are in possession of a dictionary that lets us discuss and compare the structures in these models and to examine possible generalizations of them. We also can use this point of view to see some of the limitations of the Helon model that arise from the non-commutativity of the Artin Braid Group. The talk will present these structures, and our speculations about generalizations and relationships with other topological work such as found in the papers of the author [5], the work of Witten [6] and alternate topological intepretations such as [7], [8], [9] and [10].

Nonlocal elasticity and strain gradient elasticity theories are challenging generalized continuum theories to model crystals at small scales like the Ångström-scale (see,e.g., [1,2]), where classical elasticity is not valid and leads to unphysical singularities. The theory of first strain gradient elasticity in its modern form dates back to Toupin [3] and Mindlin [4]. A mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin-Mindlin anisotropic first strain gradient elasticity theory is presented [2]. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. The 3+11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and 1+6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. The numerical values of the obtained quantities are computed for some representative cubic materials using an interatomic potential (MEAM) [2, 5]. Moreover, the isotropy conditions of strain gradient elasticity are given and discussed. A generalization of the Voigt average towards the sixth-rank constitutive tensor of the gradient-elastic constants is given to determine the 5 isotropic gradient-elastic constants [2].