In Honor of Nobel Laureate Prof. M Stanley Whittingham
Using the category theory approach, we start defining a class of objects that is the class of bodies in a state of equilibrium. Interaction is the set of morphisms between objects in the category. The action $I_{01}$ (a morphism) of an external body $varPhi_1$ on the body $varPhi_0$ generate internal process $I_{0}$ (an automorphism). A set of automorphisms are related with the ``natural'' tendency of the body to evolve to a new equilibrium state that came's the measure of some property in $varPhi_1$. The notion of equilibrium is central and based on a dual relationship between two opposite categories. An equilibrium state is described by a set of scalar fields related with the observation process or during a modelling process. Then the system is described by an algebra over a field $F$, an $F-albebra$. Intensive and extensive physical properties and observer algebras are studied and some applications of the theory are discussed.
Despite many attempts and money invested in the last decades, the achievement of a form of controlled nuclear fusion has been essentially prohibited to date by the 'repulsive' Coulomb force between natural, positively charged nuclei that, for the fusion of two deuterons into the helium, acquires the extremely big value of 230 Newtons at the mutual distance of 1 fm. On the other hand, in nuclear physics it has been believed for about one century that negatively charged electrons and positively charged nuclei cannot form a bound state because not allowed by quantum mechanics, despite their reciprocal Coulomb attraction. Following decades of mathematical, theoretical, experimental and industrial studies, R.M. Santilli has obtained a quantitative representation of the synthesis of the neutron from protons and electrons, and of the ensuing synthesis of the so-called pseudo-nuclei, described by the laws of Hadronic Mechanics [1] according to the Einstein-Pdolsky-Rosen argument that quantum mechanics is not a complete theory. These Pseudo-nuclei, that have a negative charge, can easily win the Coulomb barrier, being attracted instead of being repulsed by their positively charged counterparts, thus solving the above stated problem and allowing a new kind of fusion called Hyperfusion, without release of harmful radiations and fully controllable [2].
In this lecture we present some theoretical background and a review from the engineering point of view of available nuclear fusions that, even though currently limited in the amount of net energy production, are nevertheless clearly controllable and sustainable.
Keywords:Studies in the axiomatic formulation of the irreversibility of nuclear fusions with the most general possible SA and NSA forces were initiated in the 1967 Ph. D. Thesis [1] [2] via the generalization of Lie algebras into Albert’s Lie-admissible and Jordan-admissible algebras [3] and the formulation of the first known deformations of Lie algebras with product (Eq. (8) of [1])
which, twenty years later, were followed by tens of thousands of papers on q-deformations. In 1978, after joining the Department of Mathematics of Harvard University under DOE support, Santilli achieved [4] the axiomatic formulation of irreversibility via bi-modules in which motions forward (backward) in time are represented with the ordering of all products to the right (ordering of all products to the left)
The Lie-admissible branch of hadronic mechanics [5] [6] (see also [7]-[17]) is then characterized by two inequivalent iso-mathematics, one per each direction of time, resulting in the Lie-admissible generalization of Heisenberg’s equation for an observable Q in the infinitesimal and finite forms in which SA interactions are represented by the Hamiltonian H and forward NSA interactions are represented by the Santillian S [1] [2]
where e> (e<) is the exponentiation in the forward (backward) iso-envelope, with particular case for R = 1 − F/H, S = 1
We then have the time rate of increase of the energy expected for nuclear fusions, 
showing that Lagrange’s and Hamilton’s external terms F are represented by the Jordan algebra content of the Lie-admissible algebra. Note that given quantum mechanical models of nuclear fusions can be completed into hadronic models via four different simple non-unitary transformations 
, with a consistent representation of irreversibility, the extended share of nuclear constituents and their broadest possible SA and NSA interactions. Applications to sustainable and controlled nuclear fusions are presented in Refs. [18] [19]. Tutoring lecture [20] are recommendable for physicists.
We recall the experimental evidence according to which nuclei are composed by extended protons and neutrons in conditions of partial mutual penetration with ensuing interactions that are: linear, local and potential, thus variationally self-adjoint (SA) [1], as well as non-linear in the wave functions (as pioneered by W. Heisenberg), non-local because defined on volumes (as pioneered by L. de Broglie and D. Bohm) and non-derivable from a potential because of contact, thus zero range and variationally non-self-adjoint (NSA) type, as pioneered by R. M. Santilli in 1978 then at Harvard University under DOE support [1]. We then review the foundations of the time reversal invariant Lie-isotopic mathematics, also known as Santilli iso-mathematics, proposed in the volume [2] which is based on the preservation of the abstract axioms of 20th century applied mathematics and the use of their broadest possible realization, thus including the isotopy of [2]-[6]: 1) The quantum mechanical enveloping associative algebra of Hermitean operators ξ : {A, B, ...; AB = A × B, I} representing SA interactions via a Hamiltonian H(r, p) into the iso-associative enveloping algebra 
with iso-product 
, iso-unit 
and the Santillian 
for the representation of NSA interactions; 2) Lie’s theory into the Lie-Santilli iso-theory; 3) Numeric fields 
into Santilli iso-fields 
of iso-numbers 
; 4) Functional analysis into the iso-functional form; 5) Metric spaces 
over 
into iso-metric iso-spaces 
over 
and iso-metrics 
; 6) Newton-Leibnitz local differential calculus into a non-linear, non-local and NSA form with iso-differential 
; 7) Geometries on S over F into iso-geometries on isospaces 
over . Iso-mathematics characterizes the iso-mechanical branch of hadronic mechanics with Lie-isotopic generalization of Heisenberg equation for the time evolution of an observable 
in the infinitesimal and finite forms [2] [3]
where 
is the exponentiation in 
. It should be indicated that iso-mathematics and iso-mechanics can be constructed via the simple non-unitary transformation 
provided it is applied to the totality of conventional formalisms. We finally indicate that iso-mathematics and iso-mechanics have permitted the first and only known numerically exact and time invariant representation of experimental data for stable nuclei [7]-[8]. Tutoring lecture [9] may be a good introduction to iso-mathematics for physicists. A knowledge of this lecture is a necessary pre-requisite for the subsequent lecture on the broader, irreversible, Lie-admissible mathematics used for irreversible nuclear fusions.
A metal alloy is a collection of crystalline grains with an average size L, the space between which is filled with impurities.
Classical theories of metal alloys are based on the idea of a “random phase”: - atoms in a crystal lattice according to chemistry. composition can be arranged randomly.
The DFK model puts forward the idea of a “commensurate phase”, which is realized by the strong interaction of crystal sublattices: - an alloy AB of equiatomic composition is considered as a commensurate crystal with AB molecules (N = L). If the main periods of the sublattices are equal, respectively: for crystal A - a, for crystal B - b, then the period of the AB alloy crystal is equal to 
. When heated (T>V0), the sublattices become independent and return to the original periods (a, b).
Having asked the question about the atomic composition of the grain of the AxB1-x alloy, we proceed from the main hypothesis - metal alloys are described by commensurate phases of the DFK structure. Let us project an alloy of cubic symmetry AxB1-x (x≥0.5) onto a one-dimensional DFK model. Assuming that the alloy is a commensurate phase of the DFK model with CH(Ax) and CH(B1-x) sublattices and strong interchain interaction, V0~1 - we will show that x can only take discretely defined values x=x0.
To the alloy grain AxB1-x (x ≥ 0.5) we associate elastically periodic chains CH(Ax) and CH(B1-x) of N and L atoms of the same size, respectively, then if the period of the chain CH(Ax) = 1, then the period of the chain CH(B1-x) is equal to 
.
Thus, x can only take discrete values x0:
The chemical composition of the AxB1-x alloy grain has the form Ax0B1-x0. We choose in (1) the first fractional-rational values 
≥ 1, with the smallest denominators, because commensurate phases [1] with strong interaction can only be realized with them. From (1) we have: x0 = 0.50; 0.70; 0.89, i.e. very limited number of options depending on V0.
The fallen atoms, with density Δx=x-x0, are located between the grains, determining their size:
L~1/ Δx. (2)
The interaction between sublattices is carried out by the potential Vij, which is significant if the temperature T 0. At T> V0 the chains are independent.
Keywords:A dynamic theory of ordering of the DFK model hole dislocations is put forward.
The results of the developed FK and DFK models are incorporated into the AIM theory to answer the question: how does the process of ordering dislocations of the DFK model (two interacting elastically periodic chains of N atoms) proceed into a commensurate crystal?
It is known that after a strong tug at the ends of one of the chains of the DFK model, half of the atoms of the first chain leave the region of interaction with the second. As a result: the remaining half of the atoms form with the atoms of the second chain a commensurate crystal with a double period. Each element of this crystal is a Frenkel-Kontorova (FK) hole dislocation [1].
A FK is a formation that does not decay in space and has a number of properties that coincide with the characteristics of a point particle, namely: – mass M; incompressible size (equal to 2); kinetic and potential energy, etc.
A commensurate crystal after a jerk is not formed immediately, but as atoms fly out of the region of interaction of chains, during the movement of dislocations to the center of the system.
On a chain (length N), dislocations appear at its edges and are arranged in pairs symmetrically relative to the center.
It is of interest to write down the spatiotemporal equation of the FK ordering process, starting from the departure of the first two edge atoms of the stretched chain to the departure of its last L= N/2 atoms.
Emitted atoms are, in the AIM theory, countable characteristics of FK dislocations, analogues of moments in time.
Let us associate the emitted J-atom of the DFK chain with a dislocation with number J at its end, J≤J0, J0 = L/2. The remaining FK dislocations numbered i, 1 ≤ i < J, are located between the center of the chain and its edge. FK move towards the center of the chain.
We will describe the dynamics of dislocation ordering depending on the number of the ejected J-atom using the AIM* model. * (AIM- ab initio mundu (lat.) – from the beginning of the world)
The theory states:
1. DFK dislocation with number i is an analogue of the i-th moment of time, located at the moment of time J at a distance 
from the center of the system.
2. R(J) - discrete Lagrangian of the AIM model.
R(J) has the form:
3. 
determined from the equation:
4. Our goal is to find all values of ; 1≤ i 0, 
, with the final solution:
where L is the main parameter of the model.
5. Parameters MJ, V(J) are found from the system of inequalities:
where MJ is the mass of atoms in a chain of 2J dislocations, atoms are called DFK dislocations; 
- period of an elastic-periodic chain, with an elasticity coefficient equal to 1;
V(J) is the periodic potential in which the dislocation chain is located.
Thus, the AIM theory is an open FK model with an increasing number of particles and with “running” ones, i.e. J-dependent periods, masses and potential amplitudes.
As shown earlier: - in systems with periodic potentials, inhomogeneous dynamic solutions inevitably arise that are not destroyed. Consequently, in a system with potential (1) it is impossible to obtain a homogeneous solution (3) as the final result.
In this regard, it is necessary to introduce additional terms into Lagrangian (1), ensuring the destruction of nonlinear excitations.
We believe that the initial terms of the “destruction mechanism” should be the first and second harmonics V(J) with the main and doubled periods alternating on (off) depending on the parity of J.
We expect that in a system with the first and second harmonics alternately turning on (off) the previous one-dimensional solitons stop, but over time two-dimensional non-decaying dynamic excitations are formed.
After some time JK ~ J0 in (1), the following harmonics VK(J) are turned on (off). At points J = JK on the temporary dislocation chain of the AIM system, phase transitions occur with a change in the spatial dimension of dynamic excited states.
To summarize, for the AIM model we write:
AIM - model
where 
is the “destruction mechanism”, with each moment of time divided into K-instants, with the corresponding harmonics of the external potential.
From general considerations it follows that VK(J) ~ V0(J), V0(J0) = 0. The phase transition points JK are determined by inequalities (4).
The AIM model is a cosmological application.
From the Big Jerk through the Big Bang and Beyond
From a cosmological point of view, the number of moments in time is equal to the optimally round number, i.e. 
.
Let’s compare the generally accepted concepts with the concepts of the AIM model:
1. “Matter” – energy excitations on the time chain;
2. “Dark energy” - one-dimensional phase of matter (stationary);
3. “Dark matter” - two-dimensional phase of matter (stationary);
4. “Visible matter” - three-dimensional phase of matter (dynamic);
5. The next phase is four-dimensional, etc.
Let us estimate the phase composition of matter at different stages of the development of the Universe.
Let X be the dynamic weight part of the Universe, then for a state of K phases we write:
X+KX+K2X+K3X+…+KK-1X=1; those. X= (K-1)/(KK-1).
When K=3, X+3X+9X=1; X3≈8%. Based on estimates of modern cosmology, for time intervals Tk we have:
T3=30 billion years; T2=90 billion years; T1=180 billion years; T4=7.5 billion years; T5=1.5 billion years...; T = ∑Tk ≈ 310 billion years.
The AIM+ model assumes the return of the emitted atoms of the DFK –chain to the point of their departure, with the formation of two interacting subsystems in the AIM model.
Keywords:If we follow the classical theory of crystal growth, then when they reach a size L > Lc in the electronic bands, the natural energy widths of the levels should overlap, and, therefore, [1,2], super radiant states should form, but they do not exist - an experimental fact.
We expect that the idea of a “dielectric molecule” solves this problem.
To overcome the fundamental contradiction between classical theories and classical experiments, we conducted exploratory studies of multilevel systems (nanodielectrics).
In experiments [3-6], chemical methods were used to determine: - to what minimum size a diamond can be divided along “cleavage” planes [7]. In them, starting from highly dispersed nanodiamond powders, we came to a water-diamond compound. A water-diamond compound is a crystal consisting of identical nanodiamonds - carbon cubes of diamond symmetry; Water molecules (H+-(OH)-) are attached to the surfaces of the “diamond molecules” as rigid rods connecting the cubes to each other.
Let us describe the process of obtaining a new substance: - first, diamonds were dissolved with KOH alkali, which turned out to be the best solvent [3]. After dissolving Diamond powders and then washing out the alkali with water, water-diamond solutions of DiamondH2O are obtained [4].
When this solution is deposited on heated silicon surfaces, thin films [5] that are difficult to destroy are formed.
When a large amount of DiamondH2O is slowly dried in a glass, thick transparent films are formed on its surface due, as we think, to the strong wetting of the glass with “water-diamond clay”. The presence of a protective film was easily confirmed - the glass and samples stopped interacting with hydrofluoric acid (HF).
Physicochemical and X-ray structural analyzes of the films led us [6] to the chemical formula of the water-diamond compound – Diamond80%(H2O)20%.
From the formula of the compound, the size of the “diamond molecule” (Diamond) follows. It is equal to ≈4.2 nm. Thus, for the Diamond80%(H2O)20% molecule we have a cube of diamond symmetry of approximately 12000 carbon atoms, on the surface of which 2000 water molecules are attached.
If we accept the hypothesis of a “dielectric molecule” arising from our experiments, then each energy level of a dielectric crystal should be assigned a new quantum number - its coordinate in the sample (an analogue of a quasi-momentum). Then the situation with crystals is equivalent to the case of a quantum system, when L~N.
As a result: - in dielectrics, the energy widths of electronic levels with the same quantum numbers do not intersect, and therefore super radiant states are not formed.
Keywords:In order to develop the FK model, the DFK model (Developed Frenkel-Kontorova model) is put forward: - two one-dimensional sequences of N and L point atoms, masses m and M; with coordinates {xi} and {yj}, connected by elastic springs with the laws of elastic dispersion 
and 
. Chains CH1 and CH2 interact with each other by potential Vi,j.
The Hamiltonian of the DFK model has the form:
From the analysis of the ground state of the DFK model (N = L) [1], the following conclusion follows: - when one of the Hooke’s chains is stretched by force F, an abrupt transition to the incommensurate phase occurs (F>Fc), in which part of the atoms of the stretched chain CH1 leaves the interaction with CH2. The number of atoms falling out of the Vi,j interaction space 
, where V0 = max Vi,j.
With strong interaction (V0 ~ 1) and strong stretching (F>Fc ~1), the size of the dislocation is 2, and the number of precipitated atoms is N/2. In this case, the incommensurate phase will be a periodic chain of hole dislocations, i.e., commensurate crystal with doubled period.
Keywords:By constructing graphs of the function β(x1): - with K = 5, 10, 20, 40... - the structural components of the FK model are analyzed with an assessment of the exponential increase in the accuracy of calculations of the ground state, depending on the length of the chain. In [1], solutions to the ground state of the FK model (“NSU Ladders”) were constructed only for N<100, the positions of each CH atom were determined with an accuracy of 10-15. In the future, for the most complete identification of the boundary effects of the FK model, it is necessary to construct graphs with N>104 (V0<0.01). In this case, the positions of each CH atom will have to be determined with an accuracy of better than 10-1500.
Keywords:In [1], the FK model (β=1) was written in the continuum approximation, after which an exact analytical solution was found for a soliton moving at speed w (Frenkel-Kontorova dislocation).
In [1] there is no answer to the question about the presence in the system of other non-dislocation solutions that are localized in space and do not decay in time.
Such solutions were found in exact numerical calculations.
Let us write the Hamiltonian as the sum of kinetic K and potential U energies, with the law of elastic dispersion of the general form:
Let us consider not only Hooke’s law, but also a function that does not allow the intersection of atoms in space (x≠o), for example, with two local minima x≈α and x=1+β:
Assuming discrete time - t = j h, j = 1.2 ..., where h is the time step - we have solutions to the system of Newton’s equations for the k-th CH atom of the form:
System of equations (3) is an algorithm for constructing dynamic solutions of the FK model.
In [2], the “Chain” program with algorithm (2)-(3) constructs dynamic solutions of the FK model.
An example of a dynamic solution with Hooke's law of elastic dispersion is considered and in it an energy excitation that does not decay in time, moving with a non-uniform speed and with an energy lower than the rest energy of the dislocation, is found.
When considering the dynamics of the FK model [2]: - α=0.5, β=0, γ=0.044, V0=0.03, Δ=1, a phase transition from β- to α-phase was found, with a decrease in the size of the CH by almost two times.
Conclusions:
1 The dynamics of the FK model and the dynamics of its continuum approximation do not always and everywhere coincide.
2 For the original discrete model, the result of an exact solution of the string limit may turn out to be erroneous. For example, in [3] an exact expression was obtained for the partition function of the FK model in the continuum approximation. Based on the above, it can be argued that this solution is not applicable to the original FK model.
3 If we accept that in local field theories ∇φ - this is a gradient analogue of Hooke’s chain, then we assume that in the center of the black hole matter with a changed metric and with fields collapsing to the size ɑN is grouped.
Keywords:In 1938, the authors: Yakov Ilyich Frenkel and Tatyana Abramovna Kontorova (LFTI) put forward the Frenkel-Kontorova model (FK model) [1], which served as the basis for the creation of many theories of highly nonlinear processes [2].
It is of interest to develop the FK model in order to expand the scope of its natural science applications.
The FK model is an elastically periodic chain of atoms (CH) in a periodic potential.
CH - is a one-dimensional sequence of N point atoms of masses m with coordinates {xi} and period β, interconnected by elastic springs with the law of elastic dispersion Φ(x). Φ(x) - most often this is Hooke's law, 
.
The periodic potential V(x) has even symmetry and period a=1.
In most previous works [1,2], it was assumed 
that solutions would be simplified by eliminating boundary effects from consideration. As it turned out [3,4], this assumption is wrong: - in the discrete FK model there is no small parameter 1/N, so it is necessary to find exact solutions taking into account the position of its boundary atoms.
The potential energy of the FK system has the form:
where N is the number of atoms in the chain, N=2K+1; xi is the distance of the i-th atom to the center of the chain.
In [1] 
.
If β=0, then in the ground state of the FK model the point CH is at the minimum of the potential V(x=0). When β≠0 the position of the central atom does not change x0=0. The positions of the remaining atoms relative to the center are not even and are determined from a system of nonlinear equilibrium equations.
- system of N equilibrium equations. Taking into account x-i = - xi, we have:
At x0=0, all coordinates xi and β are functions of x1, x1 ∈ [0, 0.5], therefore, solutions for the ground state of the FK model can be obtained by minimizing U(x1) with respect to x1 [3,4]. Comparing the exact solutions [3,4] with their continuum approximations [5], we found that the properties of the ground state of the discrete FK model coincide with the properties of its continuum approximation only in the homogeneity region.
Keywords: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 ma = 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 di erential 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 scienti c 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 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 de ned. They are introduced iso-lines and they are deducted the main equations of isolines. They are given criteria for iso-perpendicularity and iso-parallel of iso-lines. Iso-reflections, iso-rotations, iso-translations and iso-glide iso-re
flections 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.
As it is well known, Isaac Newton had to develop the di essential calculus, (jointly with Gottfried Leibniz), with particular reference to the historical de nition of velocities as the time derivative of the coordinates, v = dr=dt, in order to write his celebrated equation ma = 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 di erential calculus solely admitted the characterization of massive points. The di erential 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 scienti c 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 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 isolines. They are given criteria for iso-perpendicularity and iso-parallel of iso-lines. Iso-reflections, iso-rotations, iso-translations and iso-glide iso-re
flections 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.
Previously [1] - [3] the following were presented: FK, DFK, AIM, AIM+ models.
The FK model is an elastic-periodic chain of atoms (CH1) in a periodic potential, which is described by commensurate and incommensurate phases.
In the DFK model, the periodic potential of the FK model is replaced by a second elastically periodic chain of atoms (CH2).
In the AIM model, the cosmological applications of the FK + DFK models are considered, through the creation of a time chain (CH1 + CH2), a type of open system.
The rubaiyat of Omar Khayyam voiced the cosmological idea of the AIM model: -
“Oh, woe! Nothingness is embodied in our flesh,
Nothingness is surrounded by a border of celestial spheres.
We tremble in horror from birth to death:
We are ripples on Time, but it is nothing.”
In the AIM+ model CH1 atoms returning to the jerk point form a cloud of gas, which condenses on the energy excitations of the time chain (CH1+ CH2), forming associated states with them.
Let us call the emerging states “living cells” (LC). LC can be two-dimensional, three-dimensional, etc. dimensions. It is possible that the first three-dimensional LC was formed at the stage of inflationary growth of the Universe long before the point of the “Big Bang”; let’s call it “inflaton”.
AIM+ - the model considers the phase of inflationary growth of the Universe as the initial stage of the development of a microbial colony.
It is known that communities of biological cells are open biological systems, which at the initial stages develop according to an exponential law (I-phase). But then they plateau very quickly. We believe that the I-phase can be extended by creating a coherent state for the LC (CR mode), when the entire community develops coherently.
In experiments [4]-[5], the problem of creating one of the possible CR modes in microbiological systems was solved: - stimulating the growth of a colony of a microbiological culture of e.coli with a physical device with spatial coherence, by resonantly matching the size of a biological cell with the coherence period of the device. As a result: - we observed CR stimulated by an external field - modes with single and multiple subcultures of the microbiological culture.
Keywords:In this overview it will be shown how the first successful example of real multiscaling for metals was achieved. Multiscale simulation in the present context comprises the involvement of all length scales from atomistics via micromechanical contributions to macroscopic materials behavior and further up to applications for components - multiscale materials modelling (MMM).
The main focus of this work will be put on new developments with special emphasis on MD-simulations as well as other modelling tools such as Monte Carlo (MC) or Finite Element (FE) methods and how they interact within the present approach. It will be shown that each method is superior on its respective length scale. The parameters which transport the relevant information from one length scale to the next one are decisive for the success of physical multiscaling – and is demonstrated by a recent international state of the art summary shown in ref. [1]. In the past, the different involved methods were combined into one simulation. However, it is nowadays obvious that the preferred way to succeed in reliably understanding the mechanical behavior of materials is to apply scale bridging techniques in sequential multiscale simulations in order to achieve physically based practical material solutions without any experimental adjustment. This opens the door to successful virtual material design strategies.
In this presentation micromechanical material modelling is not limited to metals but will be extended to other material classes. This approach can also be applied to composites, as shown in the literature overview in [2] as well as to many aspects of material problems in modern technical applications where several disciplines meet - physics, materials science as well as engineering.
A main focus will be put on the problem of fatigue of metals. Here, multiscale materials modelling will provide, e.g., S-N (or Wöhler) diagrams and can answer questions such as the influence of lattice type or material properties on fatigue behavior - without performing extensive experiments as required in the past. This will provide tremendous acceleration for industrial development of materials and components as shown by examples in [3].
Keywords:In my own work, the motion and negative pressure pulse of blast wave in the shock tube are simulated numerically. Based on the formation mechanism of blast wave, the initial operational condition of blast wave is given. A nice hologram is obtained by means of computational fluid imaging, which is useful to make comparision between the numerical results and experimental data. It is found that the negative pressure region is formed after the coefficient wave behind blast wave caught up with the wave head of shock wave. Isodense curve with oval shape is formed due to the influence of nozzle. Numerical results can reflect the operational process of blast wave very well. To this end, a further understanding of the operation and negative pressure phenomenon of blast wave can be obtained, which provides valuable hints for engineering applications.
Keywords:In the present work, wave function formula of two-phase viscoelastic EDA media is established. Then, elastic parameters of two-phase viscoelastic EDA media is obtained. Moreover, wave field simulation of two-phase viscoelastic EDA media is carried out by means of pseudo-spectra method. Apart from fast longitudinal wave and transverse wave, there is also slow longitudinal wave, which can be observed in two-phase viscoelastic EDA media. The amplitude of slow longitudinal wave is larger than those in fast longitudinal wave and transverse wave in the fluid phase whereas it is the opposite case in the solid phase. It can be observed that anisotropy of media makes the wavefront of longitudinal wave and transverse wave deviate from the circle shape in isotropic media. The wavefront shape of transverse wave is more complex than hat of longitudinal wave. In addition, there exists wavefront cusp phenomenon. Viscoelasticity of media can cause the drop of seismic wave amplitude. It can be concluded that two-phase viscoelastic EDA media can reflect multiphase, viscoelasticity and anisotropy fairly well. This work will have great significance for investigating fractured reservoirs
Keywords:The phenomenon of “cold thermonuclear fusion” (CTF) involves the occurrence of high-energy nuclear reactions under normal conditions, with energies of at least 1 MeV. Is it possible? We think so.
To understand the realism of CTF, let us consider a physical process in which great energy is present at the real and virtual levels.
It is known that during first-order phase transitions a large amount of energy is released, but this energy is volumetric - usually it is not localized in space and not synchronized in time.
While searching for the desired phase transition, we found a high-energy and surface-localized process - gas desorption from the metal matrix.
The electron levels of the absorbed gas hybridize with the electrons of the metal matrix, forming a narrow energy band with them.
When heated, a positively charged ion first flies out of the sample, to which a band electron is attached after some time. The electron recombination time is determined by the widths of the levels with a single desorption decay channel.
If we apply the DFK model to gas desorption, then this process is described by structural phase transitions with a ladder dependence of the gas concentration inside the sample on temperature, with abrupt changes in pressure at the steps of the ladder [1]. Pressure restrains the escape of gas ions, being the main reason that limits the rate of its outflow and creates an internal stress field.
Superradiant levels (SRLs) do not form in crystals under normal conditions, but when heated, gas ions escaping from the crystal matrix become part of an open quantum mechanical system. As a result, SRL appear.
From [2,3] it follows that the widths of the SRL desorption channel of decay are not limited in any way and in microcrystals can reach several MeV.
We consider the following CTF model realistic: - a matrix of a metal that adsorbs hydrogen well, for example Pd, saturated with deuterium when heated, pushes out the deuterium nucleus. A superradiant electron E- should join it, but there is a faster, nuclear desorption channel - the virtual collapse of one of the internal deuterium nuclei into two virtual neutrons with the further formation of two tritium nuclei, or tritium and a neutron:
where E- is an electron at a superradiant level; 
, 
- - virtual neutron and neutrino, thus we have:
or:
Microscopic Pd crystals in this process play the role of an electron accelerator, catalyzing the nuclear process. Under nonequilibrium conditions, the neutron channel of the CTS (3) can kinematically prevail over the tritium channel (2), which we have repeatedly observed.
Keywords:The Frenkel-Kontorova model (FK model) is a closed mechanical system with conserved total energy and number of particles.
The developed Frenkel-Kontorova model (DFK model) is an open mechanical system with non-conserved energy and numbers of interacting particles.
The quantum analogue of the FK model is the theory of a quantum mechanical particle in a periodic potential.
Quantum analogues of the DFK model are open quantum mechanical theories with periodic potentials, for example, the Kronig-Penney model with a constant friction force and the model of a quantum mechanical particle tunneling through the cut ends of a periodic potential [1].
It is known that in closed quantum mechanical theories there are no transitions between discrete energy levels, therefore their wave functions do not decay, and the energy spectra are purely real.
The energy spectra of open quantum systems are described by complex values with damped wave functions.
For us, superradiant states are of greatest interest. Preliminary quote from [2]: - “as an example of a physical phenomenon possible during the interaction of an intense electromagnetic wave with matter, we cite the phenomenon of superradiance or, to be precise, coherent spontaneous radiation predicted by R. Dicke [3]. This prediction is nontrivial, since it is known from the elementary theory of radiation that spontaneous emission is an incoherent process. However, there may be excited states of a system of 𝑁 atoms in which the radiation intensity is 𝑁2 times greater than that of an individual atom.”
Qualitatively, the resulting state of N identical atoms brought to one point can be conveniently represented as atoms with two levels, the width of the upper one increases N times compared to the initial one.
An even more interesting case is when the levels are not identical, quote from [4]: -
“The problem of the influence of energy dissipation on the properties of the unstable states themselves remains one of the main problems of modern quantum physics. The irreversible flow of energy from the observed system to another (continuum) leads to the fact that each excited level of the spectrum of the observed system has a finite width. In reality, all excited states of physical systems have a finite lifetime. Their properties are studied using external fields that excite these states, which then decay along one or another channel. The weak influence of the continuum on the spectrum of the system can be taken into account using standard perturbation theory, but as soon as the widths of the levels are compared with the distance between them, the perturbation theory stops working. New approaches are required to analyze the emerging situation.”
In [4,5], several theorems were proven based on general provisions of local field theory. Of these, we will highlight two that are of key importance for our work.
In two systems of N energy levels with identical quantum numbers, in the first the levels decay one at a time, in the second through L channels.
As shown in [4, 5], when the widths of the levels of the corresponding decay channels intersect, in the first case one, and in the second L, fast-decay levels are identified, the widths of which, depending on the degree of overlap, take up almost the entire width of the initial levels (Nγ). The widths of the remaining N–L levels decrease at a fixed total zone width W and the widths of the initial levels γ by (Nγ/W)2 times.
Despite the fact that the results of theorems [4,5] are confirmed by exact solutions of nonlocal models [1], optical experiments - the absence of superradiant states in microcrystals - cast doubt on these results.
Keywords:In earlier articles I have completed an original Lie algebra investigation of the Standard Model of the elementary particles and turned its differential geometry method in the outward direction over the periodic table and the Bohr Aufbau of the full crystal structure of the atoms therein. The present article recapitulates this and how further development leads to useful models in arbitrary scale of e.g. real structure, templates and interactive computer program of the chemical elements, compounds and composition.
Keywords:Quote from [1,2]: -“Many experiments measuring the deformation of solids under static loads have revealed sudden yielding and other deviations from normal behavior, now known as the “Portvin-Le Chatelier effect.” If we follow historical truth, then the honor of the discovery of this phenomenon should be associated with the names of Felix Savard (1837) and Antoine Philibert Masson (1841). Masson described a steep, almost vertical (σ-ε diagram) increase in stress, accompanied by very little deformation, up to a value at which there was a sudden sharp increase in deformation at constant stress. In experiments of this type with dead loads used in testing machines in the 19th century, this phenomenon took on the form that later led to the use of the term "staircase effect."
For small and large deformations, this effect has been studied by many over the past two centuries, but a satisfactory explanation has not yet been achieved.
It makes sense to compare the experimental ladders of the Portvin-Le Chatelier effect [1,2] with the already existing theoretical ladders [3,4].
In [1,2], in experiments on stretching Al with a purity of 99.99% shows several detailed graphs of the σ-ε dependence, for example, [9, p. 74] and [10, p. 288].
If you compare the staircase [1, p. 74] with the staircases [3,4], then their similarities are revealed - they almost coincide. But if you look at [1, p. 74] more carefully, especially at the initial stretching section, then qualitative differences are noticeable. First of all, this is the absence of strictly vertical segments in the experimental graphs. Consequently, the FK model is not enough to explain the EPLC, so it needs to be modified and replaced with the DFK model.
From the point of view of the DFK model, the initial stretching segment is associated with the general stretching of two CHs united by the potential Vlj. Further, at a certain critical force Fc, failure occurs with compression of CH2 and abrupt stretching of CH1 followed by interchain capture. The process is repeated until the sample breaks.
The first prediction of the new model is that when stretched, the sample becomes chemically inhomogeneous in length and composition of m and M atoms.
The most important question for the EPLC within the framework of the DFK model arises - the nature of Hooke's chains.
If stretchable CH1 is logically associated with an AL crystal, then the nature of CH2 may be associated with metal impurities. Let's follow this hypothesis.
In metals with a small amount of impurities, for example, in ALR%, the metal impurity R% is capable of being ordered into a cubic crystal at high temperatures. Impurity period CH2 – one-dimensional projection of the crystal R%,
, where % is the number of impurities in the main matrix. Suppose that in our case % = 10-6, then the period CH2
Comparing the number of steps [1, p. 74] with R=100, we find an approximate match.
As a result of stretching, the period of the R-sublattice changes from r=100 to r=1.
From the temperature graphs of the EPLC [2] it is clear that the EPLC disappears at T> Tc.
EPLC is a special case of phenomena in metal alloys AxB1-x.
Keywords:
The FK model [1] was successfully applied to the description of incommensurate phases and charge density waves [2]. Less successful in describing amorphous phases in CoPd alloys [3] and completely unsuccessful in analyzing solid-phase reactions [4] and structures with high-temperature superconductivity.
The prospects of the DFK model are primarily related to temperature effects in binary alloys:
1. At T> Tc, the size of the high-temperature grain L0 is not limited by anything, L0 >> L. Perhaps this explains the “Shape Memory Effect”, when the shape of the high-temperature phase sample is remembered.
2. c - the period of the crystal lattice of the AB alloy with the elasticity coefficients of the sublattices (λ, κ) is equal to: 
. Perhaps this formula explains the “Invar Effect”, when by heating a sample does not expand, and sometimes even contracts.
3. From the analysis of the states of “charge density waves” [2], we assume, that we are talking about waves of “exciton density”, then from the DFK model, the chemical formulas of crystallites for alloys with high temperature superconductivity should be given by the formulas AxB1-xC with disparity parameters x ≈ 0.70; 0.89, etc.
Keywords:The objective of present work is to put forward the Santilli's basic theoretical conventions, his key findings, his research drilldown and experimental confirmations of IsoRedShift (IRS), IsoBlueShift (IBS) and NoIsoShift (NIS) with the appropriate mathematical formulations renown as Santilli's Isomathematics. Prof. Santilli has carried out a step by step isotopic lifting of the physical laws of special relativity resulting in a new theory today specially known Santilli isorelativity. In his 1991 hypothesis Santilli established the requirement to realize the light as electromagnetic waves propagating within a universal substratum. Santilli's studies have presented a significant reconsideration of the special theory of relativity. Prof.Santilli accomplished an efficient measurements and established that while light traverses from Zenith to the Horizon, the entire spectrum of Sunlight experiences an IRS. In this work we have concentrated on concurrence of Santilli's IRS and IBS with the adages of special relativity beneath their appropriate mathematical interpretations. Besides we focused on the innovative experimental verifications of IRS by Santilli.
Keywords:The famous EPR argument [1] by A. Einstein, B. Podolsky and R. Rosen in the year 1935 expressed their historical view that Quantum mechanics could be completed into a form recovering classical ‘determinism’ at least under limit conditions. Santilli’s seminal contribution provided formulation of novel mathematical, physical and chemical methods to show that interior dynamical systems admit classical counterpart in full accordance with the EPR argument via representation by Isomathematics [ 2,4 ]. Moreover, Einstein’s determinism for strong interactions [3] is progressively achieved in the interior of hadrons, nuclei and stars and it is fully achieved in the interior of gravitational collapse known as isodeterminism [ 6]. In this paper we investigate the metaphysical implications of the Hindu philosophical notion of ‘determinism’ in the light of Santilli’s path breaking proposal as to ' ‘determinism’ can be established in the limiting sense at the core of a black hole [5,6].’ ‘Determinism’ is a philosophical view where all the events are completely determined by previously existing cause [7,8]. “Everything that happens has been predestined to happen by an omnipotent, omniscient divinity. We show that Santilli’s determinism (isodeterminism) is in tune with the ‘determinism’ prescribed in the Hindu philosophy.
Keywords:Tsai et al [1] employed a plate impact test configuration to study shock wave propagation in a S-2 glass fiber reinforced – polyester matrix composite (GRP). Hugoniot Elastic Limit (HEL), Hugoniot and Precursor Decays were determined from plate impact data using thin (~6.8 mm) and thick (~13.6 mm) GRP targets. The HEL like points in the VISAR (“free surface velocity profiles”) data were significantly influenced by the impact velocity or shock stress levels. The interpretations of experimentally observed HEL and nonlinearity in the data for two different GRP thicknesses at a range of impact velocities remained speculative and inconclusive.
To provide some insight into the effect of deformation and matrix microcracking on the VISAR profiles (or particle velocity history), computer simulations of both thin and thick plate impact experiments were performed using the ABAQUS finite element code [2]. Fraser [3] implemented a constitutive model [4] that is based on a Helmholtz Free Energy function and continuum damage mechanics. Fraser performed computational modeling of the thin (6.8 mm) plate impact tests and determined the parameters for strain-based damage initiation and propagation models. The damage modes were: matrix shear cracking, volume expansion under compressive loading, delamination, and fiber breaking in tension and shear. The respective model constants were calibrated through comparisons between the VISAR data and computed free surface velocity profiles. Based on the simulation results, it is suggested that the HEL point is due to elastic-elastic cracking (EEC) of the matrix materials under compressive loading. In simulations, the damage (microcracking of the matrix) emanates from the impact plane and progressively damage the GRP target plate in the plate impact experiments.
Recently, Rahim [5] has performed ABAQUS simulations of shock wave propagation in thick (~ 13.6 mm) GRP using the calibrated model constants for the thin targets in order to verify and validate the generality of the constants in predicting damage evolution as the wave travels further away from the impact plane.
The most fundamental nuclear fusion in nature, the synthesis of the neutron from the hydrogen atom in the core of stars, is a physical process of extreme importance, as it literally allows the stars to "turn on". This synthesis cannot be described by quantum mechanics, due to its impossibility to describe the "excess mass" in the neutron, bigger than the sum of the masses of the proton and of the electron, as well as for other technical insufficiencies, and this is an important confirmation of the EPR argument on the lack of completeness of quantum mechanics. For this reason, Santilli proposed in April 1978 the "completion" of quantum mechanics into an axiom-preserving but non-unitary form which he called hadronic mechanics [1]. In this lecture, we outline Santilli's achievement via hadronic mechanics of a numerically exact representation of "all" characteristics of the neutron in its synthesis from the hydrogen at the non-relativistic and relativistic levels [2], subsequent systematic experiments on the laboratory synthesis of the neutron from the hydrogen conducted by Santilli and a number of collaborators including the author [3], and the importance of this synthesis for basic advances on controlled nuclear fusions.
Keywords:The general application of Specialized Differential Forms in Science and Engineering has resulted into the creation of a very unique table called Numerically Controlled System of Analytics or (NCSA) for short by which any type of differential equation may now be completed integrated only in terms of generalized analytical solutions involving only the algebraic and elementary functions. Over time when such a new method of computational analysis is applied correctly then this would have the effect of reducing our excessive dependency on the use of many types of well known experimental based models in the Physical Sciences in favor of a more Universal Algebraic Theory.
In this talk I will begin by highlighting the importance of using a Numerically Controlled System of Analytics table in fluid dynamics and in mechanics of material for integrating the corresponding set of PDEs only in terms of generalized analytical solutions as a complete alternative to conventional methods of integration. I will be demonstrating how to correctly setup such a table that would lead to defining a very special type of database by which complete generalized analytical solutions to PDEs may be logically deduced only by computation.
I will also be revealing a very important mathematical property of Specialized Differential Forms that has led to redefining the whole concept of a composite function in terms of providing us with a very practical way of measuring its degree of composition regardless of whether or not they are defined in either explicit or in implicit form. This would make it possible while in the process of setting up our Numerically Controlled System of Analytics table for analysis on various cases of the Naiver-Stokes equations to extend the scope of new potential forms of analytical solutions to PDEs by including composite functions of various degree of compositions that can be defined in either explicit or in implicit form.
Under this new type of measure for the degree of composition for all composite functions, the simple wave equation for example that forms the basis of representation of solutions to the time dependent and independent Schrödinger equation for uni-electron and multi-electron structure may be elevated to include composite functions as well where the exact order of composition would have to be determined by computation only. This would have the potential of providing us with a much better understand on the exact physical structure of a wave when applied to many areas of the Physical and Biological Sciences as a result of solving for certain types of differential equations and systems of differential equations based on the method of Specialized Differential Form.
G. Bateson presented an influential definition of (qualitative) information as a difference that makes a difference for someone/something. The author presented a treatise in 2008 [1] which from some modification of Bateson’s definition unfolded stepwise and rather rigorously what is enfolded in this very definition of information as some ‘atom’ of reality. This represented a systematic establishment of a differentiated ontology (including epistemology as some ‘head’ of the ontological ‘body’) including novel treatments of core issues in philosophy. The category of causality was presented as implied in the very concept of (qualitative) information, branching into a specified number of different, while strictly connected types of causality, framed in said differential ontology. One result was a deeper refoundation of the field of formal logics.
A later reflection on the category of the ‘border’ inside this framework of ‘qualitative informatics’ led to the insight that mathematical number theory could become reconstituted from the Fibonacci algorithm as representing the basic bridge between the qualitative essence of information and its quantitative aspects. Such reconstitution of number theory became presented in a treatise by the author in 2011 [2]. This was established in stepwise tandem between ordinal and cardinal aspects of the Fibonacci algorithm, leading to some Fibonacci based redefinition of the four basic arithmetic operations.
These results can be regarded as basically complementary to the achievements of R.M. Santilli [3] and P. Rowlands [4] with regard to innovation of a more abstract, universal and potent mathematics, including more universal algebra. In the case of Santilli the applications towards innovative physics and related technological inventions, have showed to be rather astonishing.
The paper will revisit Spencer-Brown’s ambitious – while quite condensed and partly cryptical – Laws of Form [5], in order to make an assessment of its potential contributions to deep-logics/mathematics/science vs. its limitations in said respects. The paper will clarify to what extent his logic(s) of classes can be accommodated within the logic of sentences by translations applying ‘the calculus of indications’. Also, the paper will investigate the adequacy of giving primacy to operators vs. to variables/relata when establishing truth functions.
Keywords:A solution of the limitation of Calculus in terms of not being able to establish some form of a unified theory of analytical integration was achieved by introducing an entirely new computational algorithm called specialized differential forms or (SDF) that involve the use of Multivariate Polynomials and the differential of Multivariate Polynomials, all defined in a very unique algebraic configuration. Such a unique algorithm would lead to some form of a unified theory of analytical integration that would be driven entirely by computation. This new universal theory in mathematics would be applicable for solving those DEs whose exact solution can only be expressed in terms of the algebraic and elementary functions. Concrete "numerical evidence" of this will be shown on a number of very simple mathematical models involving a first order ODE and a second order PDE. A series of Physical models have been carefully chosen just for demonstrating the applicability of the current unified theory of integration into the Physical Sciences. These will include the PDEs used for describing a very specific case of the Navier-Stokes equations corresponding to an incompressible fluid involving heat transfer and variable viscosity. To fundamentally illustrate the complete versatility of our unique algorithm for solving PDEs, we would be describing a general process for attempting to solve the set of PDEs that govern general linear elastic boundary value problems.
Keywords:A solution of the limitation of Calculus in terms of not being able to establish some form of a unified theory of analytical integration was achieved by introducing an entirely new computational algorithm called specialized differential forms or (SDF) that involve the use of Multivariate Polynomials and the differential of Multivariate Polynomials, all defined in a very unique algebraic configuration. Such a unique algorithm would lead to some form of a unified theory of analytical integration that would be driven entirely by computation. This new universal theory in mathematics would be applicable for solving those DEs whose exact solution can only be expressed in terms of the algebraic and elementary functions. Concrete "numerical evidence" of this will be shown on a number of very simple mathematical models involving a first order ODE and a second order PDE. A series of Physical models have been carefully chosen just for demonstrating the applicability of the current unified theory of integration into the Physical Sciences. These will include the PDEs used for describing a very specific case of the Navier-Stokes equations corresponding to an incompressible fluid involving heat transfer and variable viscosity. To fundamentally illustrate the complete versatility of our unique algorithm for solving PDEs, we would be describing a general process for attempting to solve the set of PDEs that govern general linear elastic boundary value problems.
Keywords:A solution of the limitation of Calculus in terms of not being able to establish some form of a unified theory of analytical integration was achieved by introducing an entirely new computational algorithm called specialized differential forms or (SDF) that involve the use of Multivariate Polynomials and the differential of Multivariate Polynomials, all defined in a very unique algebraic configuration. Such a unique algorithm would lead to some form of a unified theory of analytical integration that would be driven entirely by computation. This new universal theory in mathematics would be applicable for solving those DEs whose exact solution can only be expressed in terms of the algebraic and elementary functions. Concrete "numerical evidence" of this will be shown on a number of very simple mathematical models involving a first order ODE and a second order PDE. A series of Physical models have been carefully chosen just for demonstrating the applicability of the current unified theory of integration into the Physical Sciences. These will include the PDEs used for describing a very specific case of the Navier-Stokes equations corresponding to an incompressible fluid involving heat transfer and variable viscosity. To fundamentally illustrate the complete versatility of our unique algorithm for solving PDEs, we would be describing a general process for attempting to solve the set of PDEs that govern general linear elastic boundary value problems.
Keywords:A solution of the limitation of Calculus in terms of not being able to establish some form of a unified theory of analytical integration was achieved by introducing an entirely new computational algorithm called specialized differential forms or (SDF) that involve the use of Multivariate Polynomials and the differential of Multivariate Polynomials, all defined in a very unique algebraic configuration. Such a unique algorithm would lead to some form of a unified theory of analytical integration that would be driven entirely by computation. This new universal theory in mathematics would be applicable for solving those DEs whose exact solution can only be expressed in terms of the algebraic and elementary functions. Concrete "numerical evidence" of this will be shown on a number of very simple mathematical models involving a first order ODE and a second order PDE. A series of Physical models have been carefully chosen just for demonstrating the applicability of the current unified theory of integration into the Physical Sciences. These will include the PDEs used for describing a very specific case of the Navier-Stokes equations corresponding to an incompressible fluid involving heat transfer and variable viscosity. To fundamentally illustrate the complete versatility of our unique algorithm for solving PDEs, we would be describing a general process for attempting to solve the set of PDEs that govern general linear elastic boundary value problems.
Keywords:A solution of the limitation of Calculus in terms of not being able to establish some form of a unified theory of analytical integration was achieved by introducing an entirely new computational algorithm called specialized differential forms or (SDF) that involve the use of Multivariate Polynomials and the differential of Multivariate Polynomials, all defined in a very unique algebraic configuration. Such a unique algorithm would lead to some form of a unified theory of analytical integration that would be driven entirely by computation. This new universal theory in mathematics would be applicable for solving those DEs whose exact solution can only be expressed in terms of the algebraic and elementary functions. Concrete "numerical evidence" of this will be shown on a number of very simple mathematical models involving a first order ODE and a second order PDE. A series of Physical models have been carefully chosen just for demonstrating the applicability of the current unified theory of integration into the Physical Sciences. These will include the PDEs used for describing a very specific case of the Navier-Stokes equations corresponding to an incompressible fluid involving heat transfer and variable viscosity. To fundamentally illustrate the complete versatility of our unique algorithm for solving PDEs, we would be describing a general process for attempting to solve the set of PDEs that govern general linear elastic boundary value problems.
Offline low-cost tangible coding games, for example: RANGERS problem-solving game [1] focuses on the key 4IR Skills, namely: Computational Thinking, Critical Thinking, Collaboration and Problem solving. Offline coding games[2] and Unplugged math and coding activities[3] are inspired by computer science but can be easily implemented without the use of a computer and used to support the learning and teaching of mathematics concepts to learners who come from any learning environment, like the African underserved communities such as the rural areas, farms, informal townships.
During the presentation, participants will be introduced to the fundamentals of coding concepts and how we can apply the concepts and results to improve learners’ problem solving skills through a series of engaging interactive activities which have a direct application to the teaching and learning of school mathematics. Following this, they will interact with the Rangers problem-solving offline coding app. Afterwards, a time of reflection will help to identify the connections between coding activities and mathematics.
The theory of hyperstructures, started in 1934, uses the multivalued operations or hyperoperations. The fundamental relations connect, by quotients and partitions, this theory with the corresponding classical one. T. Vougiouklis in 1990 was introduced the Hv-structures, the largest class of hyperstructures, by defining the weak axioms where the non-empty intersection replaces the equality. The number of Hv-structures defined on a set, is extremelly greater than the number of the ordinary structures and the classical hyperstructures defined on the same set. This fact leads the Hv-structures to admit more applications and, moreover, change the philosophy of finding the appropriate structure. Specifically, we reduce the number of special Hv-structures if we have more axioms, restrictions and properties. For this, we claim that the Hv-structures are more appropriate models to express theories as the Lie-Santilli’s admissibility.
Hv-structures have lot applications in mathematics and in other sciences. These applications range from biomathematics -conchology, inheritance- and hadronic physics or on leptons, in the Santilli’s iso-theory, to mention but a few. This theory, moreover, is closely related to fuzzy theory; consequently, can be widely applicable in linguistic, in sociology, in industry and production, too. In this presentation we focus on Lie-Santilli’s admissible theory especially on the Hypernumbers or Hv-numbers. Special Hv-fields, the e-hyperfields, can be used, as isofields or genofields, in such way they should cover additional properties and satisfy more restrictions.
Keywords:
