5TH INTL. SYMP. ON SUSTAINABLE MATHEMATICS APPLICATIONS

In this talk, we explore a method to uncover fractal structure and fractional property relations in thermomechanical and electromechanical polymeric materials. The methodology is based on entropy dynamics where Shannon’s entropy is combined with fractional order constraints to obtain Bayesian posterior probability densities. This results in fat-tailed posterior densities where their maximum likelihood is evaluated using localized fractal and non-local fractional order operators. We argue that the entropy dynamic approach provides a means to identify the appropriate fractal and/or fractional order operators as a function of (multi)fractal material structure. We give examples in hyperelastic energy functions based on underlying multifractal structure and fractional order viscoelasticity. The modeling framework is compared to experiments on dielectric elastomers and auxetic foams where the uncertainty in the constitutive properties and the fractional order are quantified.

Previous work by the author has proposed a foundation for physics based on a Klein-4 symmetry between the four fundamental parameters mass, time, charge and space. These parameters and the algebras which specify their properties can then be seen as generated by a computational universal rewrite system, based on a zero totality state for the universe. The algebras, remarkably, combine to a 64-component group which is isomorphic to the gamma algebra of the Dirac equation, the equation which defines the fundamental (fermionic) state in physics. A very powerful version of relativistic quantum mechanics emerges from the application of this algebra, based on a state vector which is nilpotent or squaring to zero. In view of the various proposals made for founding physics on the behaviour of cellular automata, and the claim that long-range order in automata is only possible via the Klein-4 group (1), it is proposed to investigate possible connections between the Klein-4 group as used by the author in fundamental physics and the Klein-4 group as it becomes relevant to cellular automata, along with the computational developments with which they are each connected.

The Markov/cell-to-cell mapping technique (CCMT) is a systematic procedure to describe the dynamics of both linear and non-linear systems in discrete time and in system state space previously partitioned into computational cells in a similar manner used by finite difference or finite element methods [1]. An important feature of the Markov/CCMT is its capability to model the long term dynamics of chaotic systems in a probabilistic format. Markov/CCMT has been used for the failure modeling of different types of control systems, as well as for state/parameter estimation and diagnostics, accident management and global analysis of reactor dynamics. Some example applications are provided in [1-5]. A continuous-time, discrete state-space version of Markov/CCMT has also been developed [6] and implemented for dynamic probabilistic risk/safety assessment [7]. An overview of the Markov/CCMT is presented, including computational tools for applications.

The aim of this talk is to give an overview of stability criteria as they apply to a variety of coherent structures on infinite dimensional lattice dynamical systems. We will start with solitary waves of the discrete nonlinear Schrodinger equation (DNLS), discussing both a stability classification from the anti-continuum (uncoupled site) lattice limit and the famous Vakhitov-Kolokolov (VK) criterion. We will then extend considerations to discrete breathers primarily in nonlinear Klein-Gordon lattices, and will show how a direct analogy to the stability of their periodic orbits exists in connection to DNLS. Moreover, we will discuss a recently put forth criterion for their spectral stability which is analogous to the VK criterion and ``falls back'' on it upon reduction to the DNLS case. Lastly, we will discuss some intriguing connections of the discrete breather problem with that of traveling waves in (chiefly Fermi-Pasta-Ulam type) lattices and will devise yet another spectral stability criterion in that case too which will once again be the proper analogue of the VK one for the lattice traveling waves.

Complex phenomena in wide classes of natural, artificial and social systems can be satisfactorily handled by using mathematical concepts that emerge naturally within nonextensive statistical mechanics, based on nonadditive entropies. A brief introduction will be provided on tools such as the q-generalized Fourier transform, central limit theorem, entropy production in nonlinear dynamical systems, global optimization techniques, among others.

In China, the High-Speed Railway (HSR) stimulates the economic and financial development of the society. With HSR at the speed of 420-660 km/h, this transportation vehicle leads to the quick development of human resources and freight flow. This increased efficiency propels and improves GDP in the region and the country.
The HSR at 660 km/h which is tested successfully has a deep influence on the transportation market. At this speed comparable to the plane, the plane may lose its market share. In consideration of the boarding and luggage claiming time, the overall time spending on a similar trip with HSR may be quicker than the airplane’s if the distance between the destination and the departure is less than 9,000 kilometers. When we compare HSR with sky-flying vehicles, speaking of the travel’s time cost, only some special airlines like the 15,329 kilometers 18-hour flight between Singapore (SIN) and Newark (EWR) with Airbus A350-900ULR can compete with HSR’s. When we compare HSR with ground vehicles, if the sea tunnel for HSR is constructed, none of the other types of vehicles can be comparable to HSR in terms of efficiency [1]. In short, if top speed HSR such as Hyperloop with its 1000-4000 km/h is successfully constructed, as sustainable logistics, no other transportation will be comparable to such HSR [2].
However, although the province of Quebec is known for its vast amount of hydroelectricity and other recycling resources that can be implemented into the electrical system that empowers HSR, at the present stage, in the Great Montreal Area, there is no presence of HSR. Because of this reason, this paper discusses the suitability of the implementation of the HSR system and the cost-benefit of its implementation, with the reference of the Chinese one. Different mathematical models such as World Banks’s and their implications will be talked over [3]. Eventually, based on simulations and AI analysis, these models will be upgraded and be constructed to suit specifically the Great Montreal Area situation.
Based on upgraded mathematical models, this paper will conclude whether the HSR is suitable for the Great Montreal Area and discuss the correlation between the HSR implementation and the financial and economic development in the region.

To make the earth a better place to live in as the ultimate goal for the sustainable development, mankind has to understand different earth processes and their dynamics. Nepal is a mountainous country where landslide, debris flow, land tsunami and other gravitational mass flow hazards claim many human lives every year. For the prevention and mitigation of such hazards, systematic study of such events is needed.
We discuss our modeling and simulation techniques related to different gravitational mass flows, especially with physics-based two-phase mass flow model [1]. We discuss our attempts to model Glacial Lake Outburst Floods (GLOFs), in three different initial and boundary conditions for the lake geometry, volume, and conduits. The results reveal different interesting flow dynamics of the solid and fluid-phases, lake-emptying process and levee formation during the flow [2]. We also discuss the interaction of two-phase debris flow with obstacles of different orientations and number at different locations of the flow path. The results show many naturally inline phenomena, like flow redirection, formation of vacuum behind the obstacles and the phase-separation. The computation of the novel barycentric impact pressures computed from the separate phasic impact pressures are important for the design of the structural mitigation measures [3]. For the flow through laterally converging channels, we relate the flow obstruction and the contraction ratio.
We also discuss our full dimensional model developed for the modeling of bulk mixture of solid and fluid [4] and simulate the dry and wet snow avalanches of different water contents [5]. In the unified modeling and simulation techniques, we also discuss the dynamics of subaerial and partially submerged landslide and the interaction with a fluid reservoir downstream. The resulting short land tsunami wave generation, amplification and propagation along with the submarine mass movement [5, 6].
The different physics, mechanics and dynamics of the solid and fluid phases in the gravitational mass flows as revealed in our models and simulation results have enriched our understanding in the multiphase geophysical mass flow processes.

After decades of theoretical and experimental studies (see [1] for a general overview) on the most fundamental nuclear fusion in nature, the synthesis of the neutron from an electron and proton in the core of stars, R. M. Santilli and collaborators (including the Author) achieved the industrial synthesis o the neutron via a reactor known as Directional Neutron Source [2]. Subsequent systematic tests and independent verifications have shown that the same Directional Neutron Source can synthesize a new particle obtained from the synthesis of an electron and, this time, a neutron, by therefore resulting in a new strongly interacting particle called pseudo-proton [3] which is negatively charged, thus being attracted, rather than repelled by nuclei. In this lecture, we indicate that, subject to proper funding and development, pseudoproton irradiation is expected to be preferable over the current treatment of certain forms of cancer via proton irradiation [4] because, being positively charged, protons are repelled by the atomic nuclei of carcinogenic cells, thus requiring high irradiation flux and high energy with ensuing high invasive character. By contrast, due to the negative charge, pseudo-proton irradiation can be done via localized low energy beams that are absorbed by carcinogenic cells, with ensuing less invasive character.

The paper is devoted to the different aspects of qualitative analysis in dynamics of complex nonlinear systems, that are generated by applied problems of engineering practice, including fundamental problems of modelling in mechanics. Main aims are the problems of optimal mechanical-mathematical modelling and the regular schemes of decomposition in engineering design. Multiconnectivity, high-dimensionality, nonlinearity of original statement under good detalization of full initial system lead to the necessity of the problem narrowing. The generalization of reduction principle, well-known in stability theory of A.M.Lyapunov, is important goal for engineering practice. Besides, the investigated objects are treated for unified view point on formed basic postulates (stability and singularity) as singularly perturbed ones (in sense of A.N.Tikhonov, A.Nayfeh, S.Cambell), with Sustainable Mathematics Applications.
Uniform methodology, based on Lyapunov’s methods, in accordance with Chetayev’s stability postulate, is developed for mechanical systems with multiple time scales. The presented approach, with combination of stability theory and perturbations theory methods, allows to elaborate the general conception of the modelling, to build regular algorithm for constructing of the effective mechanical-mathematical models, to work out the simple schemes of engineering level for decomposition-reduction of original models and dynamic properties.
This approach enables to obtain the simplified models, presenting interest for applications, with rigorous substantiation of the acceptability. The conditions of qualitative equivalence between full model and simplified models are determined. In the applications to mechanics (for mechanical systems with gyroscopes, for electromechanical systems, for robotic systems,…) the obtained results enable to construct the models (known and new ones) by strict methods, with the substantiation of the correctness for problems of analysis and synthesis. The interpretation of these models leads to new approximate theories, acceptable in applications of engineering practice. It allows to optimize the modelling process, to cut down the engineering design time. As applications, the different examples of concrete physical nature are considered.
Besides the hierarchy of state variables is established by natural way automatically; the sequences of nonlinear shortened models (as comparison systems) are built in accordance with hierarchic structure of variables; the correspondence between original model and shortened one is revealed.
The obtained results are generalizing and supplementing ones, known in theory of perturbations; these results are developing interesting applications in engineering. With reference to Mechanics the rigorous theoretic justification is obtained for considered approximate models and theories, both traditional (K.Magnus, A.Andronov, D.Merkin,…) and new ones (in particular, inertialess model, precessional model, Aristotel’s model of point mass dynamics,…).
The author is grateful to Russian Foundation of Fundamental Investigations for support of this research.

The charged moving particle in a medium when its speed is faster than the speed of light in this medium produces electromagnetic radiation which is called the Cerenkov radiation. We derive the photon power spectrum, including the radiative corrections, generated by charged particle moving within 2D graphene sheet with implanted ions. Graphene with implanted ions, or, also 2D-glasses,
are dielectric media, enabling the experimental realization of the Cerenkov radiation. It is not excluded that LEDs with the 2D dielectric sheets will be the crucial components of detectors in experimental particle physics. So, the article represents the starting point of the unification of graphene physics with the physics of elementary particles.

The importance of theoretical models in Science and Engineering far outweighs that of experimental based models. The result of our lack of transparency towards the use of a more unified approach to analytical integration for solving some of the most difficult problems related to the Physical and Biological Sciences has forced us to become dependent on the use of experimental based models. In reality, this has never been a matter of choice for all of us but rather a direct consequence in our failure to fully understand exactly why the vast majority of differential equations behave the way they do by not admitting highly predictable patterns of analytical solutions for resolving them.
In this talk I will begin by extending the traditional concept of a “differential” in Calculus by introducing an entirely new algorithm capable of representing all mathematical equations consisting of only algebraic and elementary functions in complete specialized differential form. Such a universal algorithm would involve the use of multivariate polynomials and the differential of multivariate polynomials all defined in a very unique algebraic configuration.
At first glance this may not sound like a major breakthrough in the Physical Sciences but progressively throughout this entire presentation, it will become very apparent that such a specialized differential representation of all mathematical equations would lead to some form of a unified theory of integration. It is only from the general numerical application of such a universal theory in mathematics that we can expect to arrive at some form of a unified theory of Physics. This would be constructed from the development of very advanced physical models that would be built exclusively on general rather than on the local analytical solutions of many well known fundamental differential equations of the Physical and Biological Sciences.

We will be presenting a very large amount of empirical results that were gathered from the numerical application of the unified theory of integration that was introduced in the first part of my entire presentation on a number of very specific mathematical models. This would include a general first order ODE followed by a second order PDE where a detailed empirical analysis of the data collected on each of these differential equations would lead to their complete integration in terms of generalized analytical solutions involving only the algebraic and elementary functions.
We will also be presenting a series of Physical models which have been chosen very carefully just for demonstrating the applicability of our unified theory of integration into the Physical Sciences. These will include the equations for describing general linear elasticity and a very specific case of the Navier-Stokes equations corresponding to an incompressible fluid involving heat transfer and variable viscosity. For each of these physical models we will be developing a universal numerical process that would be based entirely on the general application of our specialized differential form representation of all mathematical equations for the exact integration of the corresponding set of PDEs in terms of only generalized exact analytical solutions that can satisfy a wide range of boundary conditions.

The theory of Nonlinear Dynamics refers to the study of time - evolving processes in physical systems, whose evolution, although deterministic, often leads to behaviors that are unpredictable for long times. When time is a continuous variable, these processes are described by systems of nonlinear differential equations. However, it is often mathematically convenient to view time as a discrete variable, and describe the dynamics via nonlinear difference equations, which are computationally much easier to analyze [1,2]. In this lecture, I will begin by reviewing the geometry of fractals, originally developed to study objects that are geometrically complex, and often form structures called strange attractors to which the solutions of differential or difference equations converge as time grows indefinitely. Next, I will describe results of physical importance regarding energy transmission in Hamiltonian lattices, which constitute excellent models of dynamical processes occurring in Solid State Physics. In such processes, it is highly desirable to consider, beyond the short range effects of nearest neighbors, long range phenomena in which particles are influenced by distant neighbors. In this framework, it is often mathematically relevant to replace ordinary derivatives by their fractional form, in which ordinary differentiation is replaced by an operator Dα where α is not a positive integer [3,4]. For example, regarding anomalous diffusive processes in complex media, one often employs fractional-order differential equations to account for nonlocal diffusion effects.

A new powerful hybrid numerical-analytic method for solving boundary value problems will be reviewed. This method, known as the unified transform or the Fokas Method, has its origin in the analysis of a particular class of nonlinear partial differential equations (PDEs) called integrable. However, in recent years it has found a large number of applications in the solution of linear evolution as well as elliptic PDEs (www.wikipedia.org/wiki/Fokas_method).

In this talk we outline: 1) The novel isomathematics for the representation of the size, shape and density of particles in deep mutual overlapping with ensuing non-linear, non-local and non-Hamiltonian interactions; 2) The Einstein-Podolsky-Rosen (EPR) "completion" via isomathematics of quantum mechanics and chemistry into hadronic mechanics and chemistry; 3) The ensuing new notion of EPR entanglement in which particles are in continuous and instantaneous communications via the overlapping of their wavepackets without need for superluminal communications; 4) The inapplicability of Bell's inequality under the indicated EPR entanglement with ensuing recovering of classical images; and 5) The progressive achievement of Einstein's determinism in the structure of hadrons, nuclei and stars and its full achievement at the limit of gravitational collapse. We then outline representative applications in physics, chemistry and biology, including expected new cancer treatments and the new conception of living organisms characterized by extended constituents in continuous and instantaneous EPR entanglement.

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