The Thermodynamic Formalism provides a rigorous mathematical framework to study quantitative and qualitative aspects of dynamical systems. At its core there is a variational principle and corresponding, in its simplest form, to the Maximum Entropy principle, used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of scienThe Thermodynamic Formalism provides a rigorous mathematical framework to study quantitative and qualitative aspects of dynamical systems. At its core there is a variational principle and corresponding, in its simplest form, to the Maximum Entropy principle, used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of science, in particular, has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.ce, in particular, has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.

In this manuscript a new approach into analyzing the local stability of equilibrium points of non-linear Caputo fractional planar systems is shown. It is shown that the equilibrium points of such systems can be a stable focus or unstable focus. In addition, it is proposed that previous results regarding the stability of equilibrium points have been incorrect, the results here attempt to correct such results. Lastly, it is proposed that a Caputo fractional planar system cannot undergo a Hopf bifurcation, contrary to previous results prior. Though, it is shown that such systems can undergo a Hopf bifurcation (topologically).

This paper aims to investigate the global asymptotic stability of following system of higher order difference equations with quadratic terms: x_{n+1}=A+B((y_{n})/(y_{n-m}²)),y_{n+1}=A+B((x_{n})/(x_{n-m}²)) where A and B are positive numbers and the initial values are positive numbers. We also study the rate of convergence and oscillation behaviour of the solutions of related system.

We review a modern differential geometric description of the fluid isotropic motion and featuring it the diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. There is analyzed the adiabatic liquid dynamics, within which, following the general approach, there is explained in detail, the nature of the related Poissonian structure on the fluid motion phase space, as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product. We also present a modification of the Hamiltonian analysis in case of the isotermal liquid dynamics. We study the differential-geometric structure of the adiabatic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and invariant theory. In particular, we construct an infinite hierarchies of different kinds of integral magneto-hydrodynamic invariants, generalizing those before constructed in the literature, and analyze their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, some generalization of the canonical Lie-Poisson type bracket is obtained.

We use a system biology approach to translate the interaction of Bacillus Calmette-Gurin (BCG) + interleukin 2 (IL-2) for the treatment of bladder cancer into a mathematical model. The model is presented as a system of differential equations with the following variables: number of tumor cells, bacterial cells, immune cells, and cytokines involved in the tumor-immune response. This work investigates the delay effect induced by the proliferation of tumor antigen-specific effector cells after the immune system destroys BCG-infected urothelium cells following BCG and IL-2 immunotherapy in the treatment of bladder cancer. For the proposed model, three equilibrium states are found analytically. The stability of all equilibria is analyzed using the method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs).

In real financial market, the delayed market feedback and the delayed effect of government macro-control are inevitable. And both the delay of market feedback and the delay of macro-control effect bring about the mathematical difficulties in studying stabilization and synchronization of the hyper-chaotic financial system. However, employing Lyapunov function method, differential mean value theorem, suitable bounded hypotheses and pulse control technology results in the globally asymptotical stabilization and synchronization criteria. It is the first paper to drive the stabilization and synchronization criteria under the assumptions of the double delays. Finally, numerical examples illuminate the effectiveness of the proposed methods.

In this paper, the existence of two nontrivial stationary solutions for the nonlinear Gilpin Ayala two species competition model is given by using the mountain pass lemma, and the local stability criterion of the trivial solution is given by using Lyapunov function method. Based on the local stability criterion, we give some suggestions on how to avoid the population extinction. This is, when the population is on the verge of extinction, we should try our best to avoid the diffusion behavior and reduce the diffusion coefficient, otherwise the species are easy to go extinct. Numerical example shows the effectiveness of the proposed method.

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.

We give an operator solution to an advanced-retarded differential equation. The application of the operators involved produces a solution in terms of Bessel functions.

The symmetry properties of an ornament contain information about its civilisation and its interactions with other cultural sources. Two-dimensional periodic ornaments can be strictly classified into a limited set of 17 mathematical symmetry groups, also known as wallpaper groups. The collection of ornaments thus classified for a civilisation is characteristic of the cultural group and serves as a fingerprint to identify that group. If the distribution of wallpaper groups is available for several societies, mathematical methods can be applied to determine similarities and differences between the art practices of these communities. This method allows a systematic approach to the general ornamental practices within a culture and their interactions in the form of similarity of fingerprints. We test the feasibility of the method on examples of medieval Armenians, Byzantium, Seljuks first in Persia and then in Anatolia and among Arabs in the Middle East. For this purpose we present the distribution of the planar ornaments and calculate the Euclidean distances in pairs. We tested to what extend geographical and religious factors could account for the observed similarity of ornamental groups between cultures. The results suggest an intensive interaction between the Seljuk Turks and Arab craftsmen who produced the ornaments. Therefore the cultural interactions are religiously motivated.

In this paper, the authors employ Mountain Pass Lemma, the method of weak solution regularization and Lyapunov function method to derive the unique existence of globally exponential stable positive stationary solution of a single-species model with diffusion and delayed feedback. The obtained stability criterion illuminates that under some suitable conditions, a certain internal competition is conducive to the overall stability of the population, and a certain amount of family planning is conducive to the overall stability of the population. A numerical example and three tables show the effectiveness of the proposed methods.

This work probes the combined effects of magnetic field and viscous dissipation on heat field and examine the second law analysis (entropy generation) in an electrically conducting fluid under the effect of wall mass transfer over continuous stretched non-isothermal surface with variable viscosity. The viscosity of the fluid is assumed to be an inverse linear function of temperature. The governing equation for the problem are changed to dimensionless ordinary differential equations by using similarity transformation and solved numerically by using Rung Kutta and Shooting technique. Velocity, concentration and temperature distribution are obtained and used to compute the entropy generation and the Bejan number in the flow field. The effect of variable viscosity, Schmidt number, Hartman and Reynolds number on the velocity, concentration, temperature, entropy generation and Bejan number are studied and discussed.

The purpose of this paper is threefold: (i) to introduce and study the Euler-Lagrange prolongations of flatness PDEs solutions (best approximation of flatness) via associated least squares Lagrangian densities and integral functionals on Riemannian manifolds; (ii) to analyze some decomposable multivariate dynamics represented by Euler-Lagrange PDEs of least squares Lagrangians generated by flatness PDEs and Riemannian metrics; (iii) to give examples of explicit flat extremals and non-flat approximations.

In this paper, our goal is to establish the relationship between solutions of local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, also Minty local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, under generalized approximate $\eta$-convexity conditions for locally Lipschitzian functions

Financial system is essentially chaotic and unstable if there is not any external inputs. By means of Lyapunov function method, design of switching law, novel fuzzy assumption, $L^p$ estimation technique and Laplace semigroup theory, the author presents the boundedness and LMI-based (globally) asymptotical input-to-state stability criteria of financial systems. Particularly, the globally asymptotical stability in the meaning of switching implies that when the time $t$ is big enough, the dynamic of any subsystem must approach its unique equilibrium point. Besides, the global financial crisis often erupts periodically, which illuminates that the global stability in the classical sense is actually meaningless. So the stability in the meaning of switching proposed in this paper is suitable and appropriate. Numerical examples illuminate the effectiveness of the obtained results.

Increased data acquisition by uncalibrated, heterogeneous digital sensor systems such as smartphones present new challenges. Binary metrics are proposed for the quantification of cyber-physical signal characteristics and features, and a standardized constant-Q variation of the Gabor atom is developed for use with wavelet transforms. Two different continuous wavelet transform (CWT) reconstruction formulas are presented and tested under different signal to noise ratio (SNR) conditions. A sparse superposition of Nth order Gabor atoms worked well against a synthetic blast transient using the wavelet entropy and an entropy-like parametrization of the SNR as the CWT coefficient-weighting functions. The proposed methods should be well suited for sparse feature extraction and dictionary-based machine learning across multiple sensor modalities.

By partly generalizing the Lipschitz condition of existing results to the generalized Lipschitz one, the author utilizes a fixed point theorem, variational method and Lyapunov function method to derive the unique existence of globally asymptotical input-to-state stability of positive stationary solution for Gilpin-Ayala competition model with diffusion and delayed feedback under Dirichlet zero boundary value. Remarkably, it is the first paper to derive the unique existence of the stationary solution of reaction-diffusion Gilpin-Ayala competition model, which is globally asymptotical input-to-state stability. And numerical examples illuminate the effectiveness and feasibility of the proposed methods.

This article presents a new method of segmenting images with gray levels. The method is based on determining several thresholds for separation of gray levels. The determination of these thresholds is done using the certainty of the neutrosophic information. The concept of this method can be stated simply: to choose the local maximums for the neutrosophic certainty.

Data acquisition by uncalibrated, heterogeneous digital sensor systems such as smartphones present emerging signal processing challenges. Binary metrics are proposed for the quantification of cyber-physical signal characteristics and features, and a highly standardized constant-Q variation of the Gabor atom is developed for use with wavelet transforms. Two different CWT reconstruction schemas are presented and tested under different SNR conditions. A sparse representation of the Nth order Gabor atoms worked well against a test blast synthetic using the wavelet entropy and a comparable entropy-like parametrization of the SNR as the CWT coefficient-weighting functions. The proposed methods should be well suited for dictionary-based machine learning.

Data acquisition by uncalibrated, heterogeneous digital sensor systems such as smartphones present emerging signal processing challenges. Binary metrics are proposed for the quantification of cyber-physical signal characteristics and features, and a highly standardized constant-Q variation of the Gabor atom is developed for use with wavelet transforms. Two different CWT reconstruction schemas are presented and tested under different SNR conditions. A sparse representation of the Nth order Gabor atoms worked well against a test blast synthetic using the wavelet entropy and a comparable entropy-like parametrization of the SNR as the CWT coefficient-weighting functions. The proposed methods should be well suited for dictionary-based machine learning.

We compute explicitly traces of one-dimensional diffusion processes. The obtained trace forms can be regarded as Dirichlet forms on graphs. Then we discuss conditions ensuring the trace forms to be conservative. Finally, the obtained results are applied to the one-dimensional diffusion related to the Bessel’s process of order $\nu$.

Data acquisition by uncalibrated, heterogeneous digital sensor systems such as smartphones present emerging signal processing challenges. Binary metrics are proposed for the quantification of cyber-physical signal characteristics and features, and a highly standardized constant-Q variation of the Gabor atom is developed for use with wavelet transforms. Two different CWT reconstruction schemas are presented and tested under different SNR conditions. A sparse representation of the Nth order Gabor atoms worked well against a test blast synthetic using the wavelet entropy and a comparable entropy-like parametrization of the SNR as the CWT coefficient-weighting functions. The proposed methods should be well suited for dictionary-based machine learning.

Data acquisition by uncalibrated, heterogeneous digital sensor systems such as smartphones present emerging signal processing challenges. Binary metrics are proposed for the quantification of cyber-physical signal characteristics and features, and a highly standardized constant-Q variation of the Gabor atom is developed for use with wavelet transforms. Two different CWT reconstruction schemas are presented and tested under different SNR conditions. A sparse representation of the Nth order Gabor atoms worked well against a test blast synthetic using the wavelet entropy and a comparable entropy-like parametrization of the SNR as the CWT coefficient-weighting functions. The proposed methods should be well suited for dictionary-based machine learning.

In this paper it is shown that mathematical description of strain, constitutive law and dynamics obtained by direct replacement of integer order derivatives with Caputo or Riemann-Liouville fractional order partial derivatives, having integral representation on finite interval, in case of a guitar string, is nonobjective. The basic idea is that different observers, using this type of descriptions, obtain different results which cannot be reconciled, i.e. transformed into each other using only formulas that link the coordinates of the same point in two fixed orthogonal reference frames and formulas that link the numbers representing the same moment of time in two different choices of the origin of time measuring. This is not an academic curiosity! It is rather a problem: which one of the obtained results is correct?

Often, when solving forward, inverse or data assimilation problems, only a part of the solution is needed. As a model, we consider the stationary diffusion problem. We demonstrate an algorithm that can compute only a part or a functional of the solution, without calculating the full inversion operator and the complete solution. It is a well-known fact about partial differential equations that the solution at each discretisation point depends on the solutions at all other discretisation points. Therefore, it is impossible to compute the solution only at one point, without calculating the solution at all other points. The standard numerical methods like a conjugate gradient or Gauss elimination compute the whole solution and/or the complete inverse operator. We suggest a method which can compute the solution of the given partial differential equation 1) at a point; 2) at few points; 3) on an interface; or a functional of the solution, without computing the solution at all points. The required storage cost and computational resources will be lower as in the standard approach. With this new method, we can speed up, for instance, computation of the innovation in filtering or the likelihood distribution, which measures the data misfit (mismatch). Further, we can speed up the solution of the regression, Bayesian inversion, data assimilation, and Kalman filter update problems. Applying additionally the hierarchical matrix approximation, we reduce the cubic computational cost to almost linear $\mathcal{O}(k^2n \log^2 n)$, where $k\ll n$ and $n$ is the number of degrees of freedom. Up to the hierarchical matrix approximation error, the computed solution is exact. One of the disadvantages of this method is the need to modify the existing deterministic solver.

In this paper, we consider a make-to-order supply chain which satisfies demand that is dependent on both price and quoted lead -time. The manufacturer chooses the lead -time and the order quantity, and the retailer sets the revenue shares. The interactions between the manufacturer and the retailer are modelled as a Nash Game, and the existence and uniqueness of pure strategy equilibrium are demonstrated. A mechanism that enables the supply chain to coordinate the decisions of the members is developed. Lastly, we also analyze how the supply chain system parameters impact the optimal supply chain decisions and the supply chain performance.

With the revolution of computing and biology technology, data sets containing information could be huge and complex that sometimes are difficult to handle. Dynamic computing is an efficient approach to solve some of the problems. Since neighborhood multigranulation rough sets(NMGRS) were proposed, few papers focused on how to calculate approximations in NMGRS and how to update them dynamically. Here we propose approaches for computing approximations in NMGRS and updating them dynamically. First, static approaches for computing approximations in NMGRS are proposed. Second, search region in data set for updating approximations in NMGRS is shrunk. Third, matrix-based approaches for updating approximations in NMGRS while decreasing or increasing neighborhood classes are proposed. Fourth, incremental algorithms for updating approximations in NMGRS while decreasing or increasing neighborhood classes are designed. Finally, the efficiency and validity of the designed algorithms are verified by experiments.

This work promotes new methods in Mathematical Modeling, consisting in the use of the methods of Non-standard Analysis in Topoi, having as its main purpose, the mathematical definitions of the pseudoparticles from the title, with arguments from Biology/Physiology (Mathematical Neuroscience), Physics (String Theory and Emergent Quantum Mechanics), and Cybernetics (Global Brain, including Natural and Artificial Intelligence). The connections between brain and mind will be scketched via the genetic/epigenetic interplay. The topoi model the intuitionistic logic (multi-valued) and have been used in Quantum Physics while Non-Standard Analysis in SET (= the Category of sets) has been applied in Mathematical Economics; topics from Theory of Categories were also used in the study of Consciousness; however, the combination topoi - non-standard analysis was never used until now in Applied Mathematics. Another important objective is to produce progress in this aria of Pure Mathematics also - the build of non-standard analysis in classes of topoi, already used in Physics. We propose the logic of non-standard extensions in topoi as a model of the human thinking (based on infons/receptons), these theories representing new top and very difficult results in Abstract Mathematics either.

The symmetry properties of an ornament contain information about its civilisation and its interactions with other cultural sources. Two-dimensional periodic ornaments can be strictly classified into mathematical wallpaper groups. The collection of ornaments thus classified for a civilisation is characteristic of the cultural group and serves as a fingerprint to identify that group. If the distribution of wallpaper groups is available for several societies, multi-dimensional scaling algorithms can be applied to determine similarities and differences between the art practices of these communities. This method allows a systematic approach to the general ornamental practices within a culture and their interactions in the form of similarity of fingerprints. We test the feasibility of the method on examples of medieval Armenians, Byzantium, Seljuks first in Persia and then in Anatolia and among Arabs in the Middle East. For this purpose we present the distribution of the planar ornaments and calculate the corresponding Pearson correlation coefficients in pairs. The results suggest an intense interaction between the Seljuk Turks and Arab craftsmen, as well as between Armenian and Byzantine artisans who made the ornaments.

In the present paper, the effect of the non-linear thermal radiation on the neutral gas mixture in the unsteady state is investigated for the first time. The unsteady BGK technique of the Boltzmann kinetic equations for a neutral non-homogenous gas is solved. The solution of the unsteady case makes the problem more general significance than the stationary one. For this purpose, the moments' method, together with the traveling wave method, is applied. The temperature and concentration are calculated for each gas component and mixture for the first time.Furthermore, the study is held for aboard range of temperatures ratio parameter and a wide range of the molar fraction. The distribution functions are calculated for each gas component and the gas mixture. The significant non-equilibrium irreversible thermodynamic characteristics the entire system is acquired analytically. That technic allows us to investigate the consistency of Boltzmann's H-theorem, Le Chatelier principle, and thermodynamics laws. Moreover, the ratios among the different participation of the internal energy alteration are evaluated via the Gibbs formula of total energy. The final results are utilized to the argon-helium non-homogenous gas at different magnitudes of radiation force strength and molar fraction parameters. 3D-graphics are presented to predict the behavior of the calculated variables, and the obtained results are theoretically discussed.

Data acquisition by uncalibrated, heterogeneous digital sensor systems such as smartphones present emerging signal processing challenges. Binary metrics are proposed for the quantification of cyber-physical signal characteristics and features, and a highly standardized constant-Q variation of the Gabor atom is developed for use with wavelet transforms. Two different CWT reconstruction schemas are presented and tested under different SNR conditions. A sparse representation of the Nth order Gabor atoms worked well against a test blast synthetic using the wavelet entropy and a comparable entropy-like parametrization of the SNR as the CWT coefficient-weighting functions. The proposed methods should be well suited for dictionary-based machine learning.

We present a new mathematical model of disease spread reflecting specialties of covid-19 epidemic by elevating the role social clustering of population. The model can be used to explain slower approaching herd immunity in Sweden, than it was predicted by a variety of other mathematical models; see graphs Fig. \ref{GROWTH2}. The hierarchic structure of social clusters is mathematically modeled with ultrametric spaces having treelike geometry. To simplify mathematics, we consider homogeneous trees with $p$-branches leaving each vertex. Such trees are endowed with algebraic structure, the $p$-adic number fields. We apply theory of the $p$-adic diffusion equation to describe coronavirus' spread in hierarchically clustered population. This equation has applications to statistical physics and microbiology for modeling {\it dynamics on energy landscapes.} To move from one social cluster (valley) to another, the virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy's levels composing this barrier. As the most appropriate for the recent situation in Sweden, we consider {\it linearly increasing barriers.} This structure matches with mild regulations in Sweden. The virus spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This behavior matches with the covid-19 epidemic, with its cluster spreading structure. Our model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model. We present socio-medical specialties of the covid-19 epidemic supporting our purely diffusional model.

Energy efficiency of Data Center (DC) operations heavily relies on IT and cooling systems performance. A reliable and efficient cooling system is necessary to produce a persistent flow of cold air to cool servers that are subjected to constantly increasing computational load due to the advent of IoT- enabled smart systems. Consequently, increased demand for computing power will bring about increased waste heat dissipation in data centers. In order to bring about a DC energy efficiency, it is imperative to explore the thermal characteristics analysis of an IT room (due to waste heat). This work encompasses the employment of an unsupervised machine learning modelling technique for uncovering weaknesses of the DC cooling system based on real DC monitoring thermal data. The findings of the analysis result in the identification of areas for energy efficiency improvement that will feed into DC recommendations. The methodology employed for this research includes statistical analysis of IT room thermal characteristics, and the identification of individual servers that frequently occur in the hotspot zones. A critical analysis has been conducted on available big dataset of ambient air temperature in the hot aisle of ENEA Portici CRESCO6 computing cluster. Clustering techniques have been used for hotspots localization as well as categorization of nodes based on surrounding air temperature ranges. The principles and approaches covered in this work are replicable for energy efficiency evaluation of any DC and thus, foster transferability. This work showcases applicability of best practices and guidelines in the context of a real commercial DC that transcends the set of existing metrics for DC energy efficiency assessment.

In the upper charged layers of the atmosphere, the plasma is very rarefied. The collisions between its molecules are almost non-existent, and the driving forces behind them are the Lorentz forces resulting from the electric and magnetic fields. For this reason, we are interested in studying the behavior of non-collision plasmas because of its essential applications, such as the movement of satellites in the charged atmosphere. In this paper, the flow problem of collisionless gaseous plasma is examined. For that propose, we solve the unsteady Vlasov-Maxwell system of non-linear partial differential equations analytically. Methods of moments and traveling wave parameters are used to acquire an exact solution. Specific macroscopic properties of collisionless gaseous plasma are calculated along with electrical and magnetic fields. Further, thermodynamic estimation, such as entropy and entropy production, is presented. Those calculations allow us to measure the consistency with the laws of non-equilibrium thermodynamics. Relations between internal energy modification participations are predicted using Gibbs' equation for collisionless plasma. The modification effect of internal energies due to electro-magnetic fields is found to be small compared with the internal energy change due to the effect of entropy. That is because these fields are self-induced by plasma particles due to the sudden movement of the rigid plane plate. The results are accomplished according to the typical argon gaseous plasma model. Three-dimensional diagrams showing the measured variables are drawn to investigate and discuss their behavior. The problem has many commercial applications for the movement of objects in the charged atmosphere.

Data acquisition by uncalibrated, heterogeneous digital sensor systems such as smartphones present emerging signal processing challenges. Binary metrics are proposed for the quantification of cyber-physical signal characteristics and features, and a highly standardized constant-Q variation of the Gabor atom is developed for use with wavelet transforms. Two different CWT reconstruction schemas are presented and tested under different SNR conditions. A sparse representation of the Nth order Gabor atoms worked well against a test blast synthetic using the wavelet entropy and a comparable entropy-like parametrization of the SNR as the CWT coefficient-weighting functions. The proposed methods should be well suited for dictionary-based machine learning.

In this study, we mainly focus on the behavior of an electron gas produced by argon plasma that is bounded by a moving rigid flat plate. The effects of an external magnetic field on the electrons collected with each other, with positive ions, and with neutral atoms in the plasma fluid are investigated. We used the BGK model of the Boltzmann kinetic equation, which is fundamental in the study of the gas dynamics from the continuum flow to the free-molecular regimes with Maxwellian velocity distribution functions of the various species. An exact analytical solution of the model equations for the unsteady flow is given using the moment and the traveling wave methods. The behavior of the mean velocity of electrons is illustrated, which compatible with the variation of the shear stress, viscosity coefficient, and the initial and boundary conditions. Besides, the thermodynamic prediction is investigated by applying the principles of irreversible thermodynamics and Gibbs formula. It is found that the thermodynamic variables verify the Boltzmann’s H–theorem, the principle of Le Chatelier, and the Thermodynamics Second Law for the non-equilibrium processes of the system. Qualitative agreements with previous related papers are introducing. 3-Dimensional graphics for the calculating variables are offering, and their behavior is deeply discussed.

In this paper we provide an ``expository overview" of classic epidemic projection models. Starting with the simple case of an epidemic that grows exponentially we then investigate ``compartmental" models. These assume that the growth of an infected population is limited endogenously by the size of the underlying pool of susceptibles. We then describe a new family of so-called "Exo-r" statistical models, which hinge on an exogenously driven growth rate of the infected population. This family, which can be used to model both infections and deaths, captures parsimoniously both the depletion of susceptibles and the effect of interventions such as lockdowns and ``social distancing". The model is used to fit numbers of Covid-19 infections in China. It is also used to model and project deaths in the United States. Results are used to inform a discussion on i) the challenges at hand and ii) the extent to which epidemic projection models may be useful despite being wrong.

In the present time, the biggest problem of the world is the outbreak of novel coronavirus. Novel coronavirus (COVID-19), this one name has become a part of our daily lives over the past few months. Beyond the boundaries of medical science, coronavirus is now the main subject of research in all fields like Applied Mathematics, Economy, Philosophy, Sociology, Politics upto living room. The epidemic has brought unimaginable changes in our traditional habits and daily routines. Thousands of people in our country are fighting with the rest of the world to survive in various new situations. There are different kinds of coronavirus appeared in different times. In this time, Severe Acute Respiratory Syndrome Coronavirus-2 (SARS-CoV-2) is responsible for the coronavirus disease of 2019 (COVID-19). This virus was first identified towards the end of 2019 in the city of Wuhan in the province of Hubei in China. Within very short duration of time and very fast, it has spread throughout a large part of the world. In this study, the main aim is to investigate the spreading rate, death rate, recovery rate due to corona virus infection and to study the future of the coronavirus in India by using mathematical modeling based on the previous data. Mathematical models, in this situation, are the important tools in recruiting effective strategies to fight this epidemic. India is at high risk of spreading the disease and is facing many losses in socio-economic aspects. With current infection rates and existing levels of personal alertness, the number of infected people in India will increase at least in the next three months. Proper social awareness, maintain of social distance, large rate of testing and separation may break the chain of the Coronavirus-2.

The outburst of the CoVid-19 pandemic has raised several questions leading to a complex system in terms of modeling. Indeed, the modeling of the epidemic, at the level of a country, needs considering each of the different sources of contamination as well as the public health authorities' strategy, in a specific way. With this in mind, in the present paper, we develop a mathematical model of the CoVid-19 epidemic in Senegal. In the model, the population is subdivided into five compartments: susceptible, infected but asymptomatic, symptomatic, quarantined, and recovered immune people. In addition, due to its important impact on the propagation of the disease, we add one more variable: the number of infected objects. %\vspace{1mm} Therefore, the model corresponds to a system of six non-linear ordinary differential equations we submit to an analytical study to prove the relevancy of the model, simulate the evolution of the epidemic, and retrieve epidemiological parameters, namely the infection rate and the basic reproduction number, ${\cal R}_0$. Based on the senegalese territory CoVid-19 data, we simulate various scenarios as for the evolution of the epidemic in the country, in order to predict the peak and its magnitude with regard to the application of barrier measures. We also explore the option of collective immunity with special protection for vulnerable people. In doing so, non-available parameters are identified using some mathematical identification technics.

The COVID-19 epidemic is an emerging infectious disease of the viral zoonosis type caused by the corona-virus strain SARS-CoV-2, is classified as a human-to-human communicable disease and is currently a pandemic worldwide. In this paper, we propose conceptual mathematical models the epidemic dynamics of four compartments. We have collected data from the Djibouti health ministry. We define the positivity, boundedness of solutions and basic reproduction number. Then, we study local and global stability and bifurcation analysis of equilibrium to examine its epidemiological relevance. Finally, we analyze the fit of the data in comparison with the result of our mathematical results, to validate the model and estimating the important model parameters and prediction about the disease, we consider the real cases of Djibouti from 23th March to 10th June, 2020.

In this paper we provide an ``expository overview" of classic epidemic projection models. Starting with the simple case of an epidemic that grows exponentially we then investigate ``compartmental" models. These assume that the growth of an infected population is limited endogenously by the size of the underlying pool of susceptibles. We then describe a new family of so-called "Exo-r" statistical models, which hinge on an exogenously driven growth rate of the infected population. This family, which can be used to model both infections and deaths, captures parsimoniously both the depletion of susceptibles and the effect of interventions such as lockdowns and ``social distancing". The model is used to fit numbers of Covid-19 infections in China. It is also used to model and project deaths in the United States. Results are used to inform a discussion on i) the challenges at hand and ii) the extent to which epidemic projection models may be useful despite being wrong.

Boolean functions and networks are commonly used in the modeling and analysis of complex biological systems, and this paradigm is highly relevant in other important areas in data science and decision making, such as in the medical field and in the finance industry. In a Boolean model, the truth state of a variable is either 0 or 1 at a given time. Despite its apparent simplicity, Boolean networks are surprisingly relevant in many areas of application such as in bioinformatics to model gene expressions and interactions. In the latter case, a gene is either ``on" or ``off" depending on its expression level. Despite the promising utility of Boolean modeling, in most practical applications the Boolean network is not known. Automated learning of a Boolean network and Boolean functions, from data, is a challenging task due in part to the large number of unknowns (including both the structure of the network and the functions) to be estimated, for which a brute force approach would be exponentially complex. In this paper we develop a new information theoretic methodology that we show to be significantly more efficient than previous approaches. Building on the recently developed optimal causation entropy principle (oCSE), that we proved can correctly infer networks distinguishing between direct versus indirect connections, we develop here an efficient algorithm that furthermore infers a Boolean network (including both its structure and function) based on data observed from the evolving states at nodes. We call this new inference method, Boolean optimal causation entropy (BoCSE), which we will show that our method is both computationally efficient and also resilient to noise. Furthermore, it allows for selection of a set of features that best explains the process, a statement that can be described as a networked Boolean function reduced order model. We highlight our method to the feature selection in several real-world examples: (1) diagnosis of urinary diseases, (2) Cardiac SPECT diagnosis, (3) informative positions in the game Tic-Tac-Toe, and (4) risk causality analysis of loans in default status. Our proposed method is effective and efficient in all examples.

We compute explicitly trace of one-dimensional diffusion process which can be regarded as a Dirichlet form on graphs and we study its conservativeness property. Finally, we give an example of a discretization of one dimensional diffusion of Bessel's process with order $\nu$.

The famous Toms effect (1948) consists of a substantial increase of the critical Reynolds number when a small amount of soluble polymer is introduced into water. The most noticeable influence of polymer additives is manifested in the boundary layer near solid surfaces. The goal of the present paper is a group analysis of the boundary layer equations in two mathematical models of the flow of aqueous polymer solutions: the second grade fluid (Rivlin and Ericksen, 1955) and the model derived by Pavlovskii (1971). The equations of the unsteady two-dimensional boundary layer in the Pavlovskii and Rivlin-Ericksen models are analyzed for the first time here. These equations have no definite type so that finding their exact solutions is very important in order to understand the mathematical nature of the above mentioned models. The problem of group classification with respect to the arbitrary function of the longitudinal coordinate and time present in the equations, which sets the pressure gradient of the external flow, arises. All functions for which an extension of the admitted Lie group occurs are found. The task includes the ratio of two characteristic length scales. One of them is the Prandtl scale, and another is defined as the square root of the normalized coefficient of relaxation viscosity (Frolovskaya and Pukhnachev, 2018) and does not depend on the characteristics of the motion. The paper contains a number of exact solutions in the Pavlovskii model including a solution describing the flow near a critical point. Among the solutions of the new model of the boundary layer, a special place is taken by the solution of the stationary problem of flow around a rectilinear plate. Within the framework of the Prandtl theory of the boundary layer, such a solution was constructed by Blasius (1908). As is well-known, this solution has a non-removable defect: the transverse velocity near the edge of the plate increases without bound. The introduction of a relaxation term into the model makes it possible to eliminate this singularity.

Local binary pattern is one of the visual descriptors and can be used as a powerful feature extractor for texture classification. In this paper, a novel representation for face recognition is proposed, called it Bilateral Line Local Binary Patterns (BL-LBP). This scheme is an extension of Line Local Binary Patterns descriptors in the statistical learning subspace. The present bilateral descriptors are fused with an ensemble learning of calibrated SVM models. The performance of this scheme is evaluated using 5 standard face databases. It is found that it is robust against illumination variation, diverse facial expressions and head pose variations and its recognition accuracy reaches 98 percent, running on a mobile device with a processing speed of 63 ms per face. Results suggest that our proposed method can be very useful for the vision systems that have limited resources where the computational cost is critical.

Fractional differential mathematical model unfolding the dynamics of the COVID-19 pandemic in India is presented and explored in this paper. The purpose of this study is to estimate the future outbreak of disease and potential control strategies using mathematical models in India as a whole country as well as in some of the states of the country. This model is calibrated based on reported cases of infections over the month of April 2020 in India. We have used iterative fractional complex transform method to find approximate solutions of the model having modified Riemann Liouville fractional differential operator. We have also carried out a comparative analysis between actual and estimated cumulative cases graphically, moreover, most sensitive parameters for basic reproduction number$(R_0)$ are computed and their effect on transmission dynamics of COVID-19 pandemic is investigated in detail.

In the wake of the novel coronavirus, SARS-CoV-19, the world has undergone a critical situation in which grave threats to global public health emerged. Among human populations across the planet, travel restraints, border enforcement measures, quarantine, and isolation provisions were implemented in an attempt to control and limit the spread of the contagion. Decisions on how to implement and enforce various control policies should be determined based on available real-world evidence and theoretical prediction. In this study, we propose a novel exportation- importation epidemic model associated with the quarantine and hospitalized-isolation policies by considering the two-body system: a source country of a contagious disease and a neighboring country that is initially disease-free. The model is first applied to the original COVID-19 data in China, Italy, and the Republic of Korea (ROK) and observed through consistent fitting results with equivalent goodness-of-fit. Then, the data are estimated per the fitting parameters. Driven by these parametric settings and considering the normalized population, the numerical analysis, and epidemiological exploration, this work further elucidates the substantial impact of quarantine policies, healthcare facilities, and the public counter-compliance effect.

A mathematical model is proposed for quantitative description of the outbreak of novel coronavirus COVID-19 Although the model is relatively simple, the comparison with the public data shows that an exact solution solution of the model (with the correctly-specified parameters) leads to the results, which are in good agreement with the measured data in China and Austria. Prediction of the total number of the COVID-19 cases is discussed and examples are presented using the measured data in Austria, France and Poland.

Background: Outbreak of the Covid-19 is now an ongoing global health emergency. At the end of December 2019, the first infection was reported in Wuhan and the world did not pay attention to this extremely contaminated disease and plucked to react rapidly. The World is in an vulnerable state in disease spreading, facing a great loss of lives and socio-economic aspects also. That is why we have proposed a potential mathematical model with data analysis to predict and control the outcome of this pandemic. Methods: The model presented the epidemic dynamics of multiple compartments. We collected available online data of Spain. In primary step, we estimated the parameters using either the data analysis or reference papers. Then we did the data fitting analysis in comparison with the outcome of our mathematical results. The results of the system depended not only the parameters also on social consciousness. Results: It is found that disease progression in this model is determined by the basic reproductive ratio, $R_0$, the actual epidemic of $R_0$ and effective $R(t)$ of each day. If $R_0>1$, the number of latently infected individuals grows exponentially; endemic solution is stable while infection rate decays if $R_0<1$. The optimal control theory stated that vaccination and treatment strategies are highly effective for reducing both susceptible and infected population and to increase the recover rate high. In Spain, after state of alarm (quarantine) on 14 March 2020, reported cases increasing for 13 days only and from the 14th day, daily reported cases started to decline albeit with small fluctuation. Our proposed model approximates that the disease in Spain could be fully under control by after July 2020. Conclusion: Outbreak will be in control of health care system, reduce the death rate and will ensure social-economic stability.

This article presents a method of segmenting images with gray levels that uses Onicescu's information energy calculated in the context of the neutrosophic theory. Starting from the information energy calculation for complete neutrosophic information, it is shown how to extend its calculation for incomplete and inconsistent neutrosophic information. The segmentation method is based on calculation of thresholds for separating the gray levels using the local maximum points of the Onicescu information energy.

Prior to lockdown the spread of COVID-19 in UK is found to be exponential, with an exponent α=0.207 In case of COVID-19 this spreading patterns is quantitatively better described with mobility-driven SIR-SEIR model [2] rather than the homogenous mixing models Lockdown has dramatically slowed down the spread of COVID-19 in UK, and even more significantly has changed the growth in the total number of infected from exponential to quadratic. This significant change is due a transition from a mobility-driven epidemic spreading to a spatial epidemic which is dominated by slow growth of spatially isolated clusters of infected population. Our results strongly indicated that, to avoid a return to exponential growth of COVID-19 (also known as “second wave”) mobility restrictions should not be prematurely lifted. Instead mobility should be kept restricted while new measures, such as wearing mask and contact tracing, get implemented in order to allow a safe exit from lockdown.

In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid-ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “No-Arbitrage” principle in financial models with the bid-ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can build a trading strategy that provides one with an arbitrage opportunity. We will generalize this structure to any two-price finite-period financial model.

An oil and vinegar scheme is a signature scheme based on multivariate quadratic polynomials over finite fields. The system of polynomials contains $n$ variables, divided into two groups: $v$ vinegar variables and $o$ oil variables. The scheme is called balanced (OV) or unbalanced (UOV), depending on whether $v = 0$ or not, respectively. These schemes are very fast and require modest computational resources, which make them ideal for low-cost devices such as smart cards. However, the OV scheme has been already proven to be insecure and the UOV scheme has been proven to be very vulnerable for many parameter choices. In this paper, we propose a new multivariate public key signature whose central map consists of a set of polynomials obtained from the multiplication of block matrices. Our construction is motivated by the design of the Simple Matrix Scheme for Encryption and the UOV scheme. We show that it is secure against the Separation Method, which can be used to attack the UOV scheme, and against the Rank Attack, which is one of the deadliest attacks against multivariate public-key cryptosystems. Some theoretical results on matrices with polynomial entries are also given, to support the construction of the scheme.

Rapidly spreading disease, COVID-19 is classified as the human-to-human transmission-able disease and currently is a pandemic in the Globe. In this paper, we propose conceptual mathematical models for COVID-19 outbreak and it's control measurement; quarantine, hospitalization and the effect of panic and anxiety. In this situation, mathematical models are a important tool to employ an effective strategy in order to fight against this pandemic. We establish the positivity and boundedness of solutions, local and global stability analysis of equilibria to examine its epidemiological relevance. To validate the model and estimating the important model parameters and prediction about the disease, we consider the real cases of Italy from $15^{th}$ Feb to $13^{th} $ April 2020. In a series of graphical map, we have presented the comparative study to estimate the current scenarios and to predict the control measurement time boundary of the outbreak.

The asymptotic dynamics of the classes of rational difference equations (RDEs) of third order defined over the positive real-line as $$\displaystyle{x_{n+1}=\frac{x_{n}}{ax_n+bx_{n-1}+cx_{n-2}}}, \displaystyle{x_{n+1}=\frac{x_{n-1}}{ax_n+bx_{n-1}+cx_{n-2}}}, \displaystyle{x_{n+1}=\frac{x_{n-2}}{ax_n+bx_{n-1}+cx_{n-2}}}$$ and $$\displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n}}}, \displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n-1}}}, \displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n-2}}}$$ is investigated computationally with theoretical discussions and examples. It is noted that all the parameters $a, b, c$ and the initial values $x_{-2}, x_{-1}$ and $x_0$ are all positive real numbers such that the denominator is always positive. Several periodic solutions with high periods of the RDEs as well as their inter-intra dynamical behaviours are studied.

Nanofluids are a novel class of heat transfer fluid that plays a vital role in industries. In mathematical investigations, these fluids are modeled in terms of traditional integer-order partial differential equations (PDEs). It is recognized that traditional PDEs cannot decode the complex behavior of physical flow parameters and memory effects. Therefore, this article intends to study the mixed convection heat transfer in nanofluid over an inclined vertical plate via fractional derivatives approach. The problem in hand is modeled in connection with Atangana-Baleanu fractional derivatives without singular and local kernel having strong memory. The human blood is considered as base fluid dispersing carbon nanotube (CNTs) (single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes(MWCNTs )) into it to form blood-CNTs nanofluid. The nanofluids are considered to flow in a saturated porous medium under the influence of an applied magnetic field. The exact analytical expressions for velocity and temperature profiles are acquired using the Laplace transform technique and plotted in various graphs. The empirical results indicate that the memory effect decreases with increasing fractional parameters in the case of both temperature and velocity profiles. Moreover, the temperature profile is higher for blood-SWCNTs by reason of higher thermal conductivity whereas, this trend is opposite in case of velocity profile due densities difference.

The world is now undergoing through a global emergency due to COVID-19 which needs immediate remedies in order to strengthen the healthcare facility to save the nations. Looking towards to the remedies, research on different aspects including the genomic and proteomic level characterizations of the SARS-CoV2 are necessarily important. In this present study, the spatial representation/composition of twenty amino acids across the primary protein sequences of SARS-CoV2 have been looked into through different parameters viz. Shannon entropy, Hurst exponent in order to fetch the autocorrelation and amount of information over the spatial representations. Also frequency distribution of each of the amino acids over the protein sequences have been chalked out.

This paper analyzes the robust dissipativity of uncertain stochastic generalized neural networks (USGNNs) with Markovian jumping parameters and time-varying delays. In practical applications most of the systems refer to uncertainties, hence, the norm-bounded parameter uncertainties and stochastic disturbance are considered. Then, by constructing an appropriate Lyapunov-Krasovskii functional (LKF) and by employing integral inequalities LMI-based sufficient conditions of the considered systems are established. Numerical simulations are given to show the merit of the presented results.

The world is now undergoing through a global emergency due to COVID-19 which needs immediate remedies in order to strengthen the healthcare facility to save the nations. Looking towards to the remedies, research on different aspects including the genomic and proteomic level characterizations of the SARS-CoV2 are necessarily important. In this present study, the spatial representation/composition of twenty amino acids across the primary protein sequences of SARS-CoV2 have been looked into through different parameters viz. Shannon entropy, Hurst exponent in order to fetch the autocorrelation and amount of information over the spatial representations. Also frequency distribution of each of the amino acids over the protein sequences have been chalked out.

The purpose of this paper is to bring to light a method through which the global in time existence for arbitrary large in H¹ initial data of a strong solution to 3D periodic Navier-Stokes equations follows. The method consists of subdividing the time interval of existence into smaller sub-intervals carefully chosen. These sub-intervals are chosen based on the hypothesis that for any wavenumber m, one can find an interval of time on which the energy quantized in low-frequency components (up to m) of the solution u is lesser than the energy quantized in high-frequency components (down to m) or otherwise the opposite. We associate then a suitable number m to each one of the intervals and we prove that the norm ||u(t)||_H1 is bounded in both mentioned cases. The process can be continued until reaching the maximal time of existence T_max which yields the global in time existence of strong solution.

Comments are exploited by product vendors to measure satisfaction of consumers. With the advent of Natural Language Processing (NLP), comments on Google Play can be processed to extract knowledge on applications such as their reputation. Proposals in that direction are either informal or interested merely on functionality. Unlike, this work aims to determine reputation of Android applications in terms of confidentiality, integrity, availability and authentication (CIAA). This work proposes a model of assessing app reputation relying on sentiment analysis and text analysis of comments. While assuming that comments are reliable, we collect Google Play applications subject to comments which include security keywords. An in-depth analysis of keywords based on Naive Bayes classification is made to provide polarity of any comment. Based on comment polarity, reputation is evaluated for the whole application. Experiments made on real applications including dozens to billions of comments, reveal that developers lack to make efforts to guarantee CIAA services. A fine-grained analysis shows that not security reputed applications can be reputed in specific CIAA services. Results also show that applications with negative security polarities display in general positive functional polarities. This result suggests that security checking should include careful comment analysis to improve security of applications.

The purpose of this paper is to bring to light a method through which the global in time existence for arbitrary large in H¹ initial data of a strong solution to 3D periodic Navier-Stokes equations follows. The method consists of subdividing the time interval of existence into smaller sub-intervals carefully chosen. These sub-intervals are chosen based on the hypothesis that for any wavenumber m, one can find an interval of time on which the energy quantized in low-frequency components (up to m) of the solution u is lesser than the energy quantized in high-frequency components (down to m) or otherwise the opposite. We associate then a suitable number m to each one of the intervals and we prove that the norm ||u(t)||_H1 is bounded in both mentioned cases. The process can be continued until reaching the maximal time of existence T_max which yields the global in time existence of strong solution.

In this work, the problem is considered for eliminating a non-uniform rectilinear smearing of an image by mathematical and computer-based way, for example, a picture of several cars taken with a fixed camera and moving with different speeds. The problem is described by a set of 1-dimensional Fredholm integral equations (IEs) of the first kind of convolution type with a 1-dimensional point spread function (PSF) at uniform smearing and by a set of new 1-dimensional IEs of a general type (not convolution type) with a 2-dimensional PSF at non-uniform smearing. The problem is also described by one 2-dimensional IE of convolution type with a 2-dimensional PSF at uniform smearing and by a new 2-dimensional IE of a general type with a 4-dimensional PSF at non-uniform smearing. The problem for solving Fredholm IE of the first kind is ill-posed (unstable). Therefore, IEs of convolution type are solved by the Fourier transform (FT) method and Tikhonov's regularization (TR), and IEs of general type are solved by the quadrature/cubature and TR methods. Moreover, the magnitude of the image smear Δ is determined by the original “spectral method”, which increases the accuracy of image restoration. It is shown that the use of a set of 1-dimensional IEs is preferable to one 2-dimensional IE in the case of non-uniform smearing. In the inverse problem (image restoration), the Gibbs effect (the effect of false waves) in the image may occur. It can be edge or inner. The edge effect is well suppressed by the proposed technique “diffusing the edges". In the case of an inner effect, it is eliminated with difficulty, but the image smearing itself plays the role of diffusing and suppresses the inner Gibbs effect to a large extent. It is shown (in the presence of impulse noise in an image) that the well-known Tukey median filter can distort the image itself, and the Gonzalez adaptive filter also distorts the image (but to a lesser extent). We propose a modified adaptive filter. A software package was developed in MatLab and illustrative calculations are performed.

Periodic travelling waves are observed in various brain activities including visual, motor, language, sleep, and so on. There are several neural field models describing periodic waves assuming nonlocal interaction and, possibly, inhibition, time delay, or some other properties. In this work we study the influence of asymmetric connectivity functions and of time delay on the emergence of periodic waves and on their properties. Nonlinear wave dynamics is studied, including modulated and aperiodic waves. Multiplicity of waves for the same values of parameters is observed. External stimulation in order to restore wave propagation in a damaged tissue is discussed.

A regularization approach to model selection, within a generalized HJM framework, is introduced which learns the closest arbitrage-free model to a prespecified factor model. This optimization problem is represented as the limit of a one-parameter family of computationally tractable penalized model selection tasks. General theoretical results are derived and then specialized to affine term-structure models where new types of arbitrage-free machine learning models for the forward-rate curve are estimated numerically and compared to classical short-rate and the dynamic Nelson-Siegel factor models.

In this study, we first present a time-fractional L$\hat{e}$vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L$\hat{e}$vy-time fractional diffusion equation of European-style options. Introduce a more general model from the models based on the L$\hat{e}$vy-time fractional diffusion equation and review some recent findings regarding of the Europe option pricing of risk-neutral free.

The ongoing outbreak of the novel coronavirus pneumonia (also known as COVID-19) has triggered a series of stringent control measures in China, such as city closure, traffic restrictions, contact tracing and household quarantine. These containment efforts often lead to changes in the contact pattern among individuals of the population. Many existing compartmental epidemic models fail to account for the effects of contact structure. In this paper, we devised a pairwise epidemic model to analyze the COVID-19 outbreak in China based on confirmed cases reported during the period February 3rd--17th, 2020. By explicitly incorporating the effects of family clusters and contact tracing followed by household quarantine and isolation, our model provides a good fit to the trajectory of COVID-19 infections and is useful to predict the epidemic trend. We obtained the average of the reproduction number $R=1.494$ ($95\%$ CI: $1.483-1.507$) for Hubei province and $R=1.178$ ($95\%$ CI: $1.145-1.158$) for China (except Hubei), suggesting that some existing studies may have overestimated the reproduction number by neglecting the dynamical correlations and clustering effects. We forecasted that the COVID-19 epidemic would peak on February 13th ($95\%$ CI: February $9-17$th) in Hubei and 6 days eariler in the regions outside Hubei. Moreover the epidemic was expected to last until the middle of March in China (except Hubei) and late April in Hubei. The sensitivity analysis shows that ongoing exposure for the susceptible and population clustering play an important role in the disease propagation. With the enforcement of household quarantine measures, the reproduction number $R$ effectively reduces and epidemic quantities decrease accordingly. Furthermore, we gave an answer to the public concern on how long the stringent containment strategies should maintain. Through numerical analysis, we suggested that the time for the resumption of work and production in China (except Hubei) and Hubei would be the middle of March and the end of April, 2020, respectively. These constructive suggestions may bring some immeasurable social-economic benefits in the long run.

In this study, by using Least Square Method, the dataset for the Gross Domestic Product per capita is modeled as a function satisfying the fractional differential equation. The function itself is assumed to be the finite summation of its previous values and the derivatives with unknown coefficients. Then, the prediction for the upcoming years is done by having an approach dividing the dataset into 4 regions corresponding to four different tasks. The mathematical model of the Gross Domestic Product (GDP) per capita of the countries (and union) which are Brazil, China, European Union (EU), India, Italy, Japan, UK, the USA, Spain, and Turkey is constructed with a new methodology called as the deep assessment method which comes from the expressing an arbitrary function modeling the dataset as the finite summation of its previous values and the derivatives with unknown coefficient. The method uses the fractional calculus properties combining with Least Square Method and is compared to Long short-term memory (LSTM) algorithm which is a special type of neural network used for time sequences in general.

A good recommender system can infer customers’ preferences based on their historical purchase records, and recommend products that the customers may be interested in, saving them a lot of time and energy. For enterprises, it is difficult to recommend accurately to each customer, and the bad recommendation may be counterproductive. Customer loyalty is an indicator that measures the preference relationship between customers and products in the field of marketing. A hypothesis is proposed in this study: if companies can divide customers into different groups based on customer loyalty, the recommendation effect on certain groups is better than that on overall customers. In this study, customer loyalty is measured by four features of the RFML model. All customers are viewed as points on a four-dimensional space, which are clustered by the k-means model. Two recommendation algorithms based on complex networks are tested: recommendation algorithm based on bipartite graph and PersonalRank (BGPR), and recommendation algorithm based on a single vertex set network and DeepWalk (SVDW). The experimental results show that customer loyalty has improved the effectiveness of the two algorithms over 14%, and the recommendation effect is the best on customer groups with a loyalty level of 4 (the highest level is 5). The recommendation algorithms with customer loyalty are better than using them alone.

In this paper, we study the existence of the random -pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg equation with multiplicative noise in stratonovich sense. It is shown that a random -pullback attractor exists in when its external force has exponential growth. Due to the stochastic term, the estimate are delicate, we overcome this difficulty by using the Ornstein-Uhlenbeck(O-U) transformation and its properties.

Rough volatility models are popularized by \cite{gatheral2018volatility}, where they have shown that the empirical volatility in the financial market is extremely consistent with rough volatility. Fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form as of now and therefore, we must rely on numerical methods to obtain a solution. In this paper, we give a short introduction to option pricing theory and an overview of the current advancements on the rough Heston model.

Primary energy consumption is one of the key drivers of global CO₂ emissions that, in turn, heavily depend on the efficiency of involved technologies. Either the improvement in technology efficiency or the expansion of non-fossil fuel consumption require large investments. The planning and financing of such investments, by policy makers or global energy firms, require, in turn, reliable measures of associated global spreads and their evolution in time. In this paper, our main contribution is the introduction of index measures for accessing global spreads (that is, measures of inequality or inhomogeneity in the statistical distribution of a related quantity of interest) of technology efficiency and CO₂ emission in primary energy consumption. These indexes are based on the Gini index, as used in economical sciences, and generalised entropy measures. Regarding primary energy sources, we consider petroleum, coal, natural gas and non-fossil fuels. Between our findings, we attest some stable relations in the evolution of global spreads of technology efficiency and CO₂ emission, and a positive relation between changes in global spreads of technology efficiency and use of non-fossil fuel.

In this paper, initially a mathematical model is formulated for transient frequency of power system considering time delays which occur while transmitting the control signals in open communication infrastructure. Time delay negligence in a power system leads to improper measurement of frequency variation in power system. The study of impact of time delays on the stability of power system is performed by estimating the decay rate of frequency wave form using Kalman Filter (KF). In power system, there is a possibility of multiple time delays. This paper also focusses on developing Interacting Multiple Model (IMM) Algorithm with multiple model space using Kalman Filter (KF) as state estimator tool. The multiple time delays in power system is considered as multiple model space. The result shows that KF provides better estimate of correct model for a particular input-set. The qualitative properties of Riccati difference equation (RDE) in terms of state error covariance of IMMKF are also analyzed and presented.

We introduce a new, and quite general variational model for opinion dynamics based on pairwise interaction potentials and a range of opinion evolution protocols ranging from random interactions to global synchronous flows in the opinion state space. The model supports the concept of topic “coupling”, allowing opinions held by individuals to be changed via indirect interaction with others on different subjects. Interaction topology is governed by a graph that determines interactions. Our model, which is really a family of variational models, has, as special cases, many of the previously established models for the opinion dynamics. After introducing the model, we study the dynamics of the special case in which the potential is either a tent function or a constructed bell-like curve. We find that even in these relatively simple potential function examples there emerges interesting behavior. We also present results of preliminary numerical explorations of the behavior of the model to motivate questions that can be explored analytically

In this age of Facebook, Instagram, and Twitter, there is rapidly growing interest in understanding network-enabled opinion dynamics in large groups of autonomous agents. The phenomena of opinion polarization, the spread of propaganda and fake news, and the manipulation of sentiment is of interest to large numbers of organizations and people. Whether it is the more nefarious players such as foreign governments that are attempting to sway elections or it is more open and above board, such as researchers who want to make large groups of people aware of helpful innovations, what is at stake is often significant. In this paper, we review opinion dynamics including the extensions of many classical models as well as some new models that deepen understanding. For example, we look at models that track the evolution of an individual’s power, that include noise, and that feature sequentially dependent topics, to name a few. While the first papers studying opinion dynamics appeared over 60 years ago, there is still a great deal of room for innovation and exploration. We believe that the political climate and the extraordinary (even unprecedented) events in the sphere of politics in the last few years will inspire new interest and new ideas. It is our aim to help those interested researchers understand what has already been explored in a significant portion of the field of opinion dynamics. We believe that in doing this, it will become clear that there is still much to be done.

We develop a mathematical model to provide epidemic predictions for the 2019-nCov epidemic in Wuhan, China. We use reported case data up to January 31, 2020 from the Chinese Center for Disease Control and Prevention and the Wuhan Municipal Health Commission to parameterize the model. From the parameterized model we identify the number of unreported cases. We then use the model to project the epidemic forward with varying level of public health interventions. The model predictions emphasize the importance of major public health interventions in controlling 2019-nCov epidemics.

Deep coal beds have been suggested as possible usable underground geological locations for carbon dioxide storage. Furthermore, injecting carbon dioxide into coal beds can improve the methane recovery. Due to importance of this issue, a novel investigation has been done on adsorption of carbon dioxide on various types of coal seam. This study has proposed four types of Gaussian Process Regression (GPR) approaches with different kernel functions to estimate excess adsorption of carbon dioxide in terms of temperature, pressure and composition of coal seams. The comparison of GPR outputs and actual excess adsorption expresses that proposed models have interesting accuracy and also the Exponential GPR approach has better performance than other ones. For this structure, R2=1, MRE=0.01542, MSE=0, RMSE=0.00019 and STD=0.00014 have been determined. Additionally, the impacts of effective parameters on excess adsorption capacity have been studied for the first time in literature. According to these results, the present work has valuable and useful tools for petroleum and chemical engineers who dealing with enhancement of recovery and environment protection.

In this article, we investigate two types of double dispersion equations in two dierent dimensions. Double dispersion equation were derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Conservation laws are obtained for these equations by the application of the multiplier method. Finally, travelling waves and line travelling waves are respectively considered for these two equations.

In the present study, a mathematical model of non-steady partial differential equation from the process of oxygen mass transport in the human pulmonary circulation is proposed. Mathematical modelling of this kind of problems lead to a non-steady partial differential equation and for its numerical simulation, we have used finite differences. The aim of the process is the exact numerical analysis of the study, wherein consistency, stability and convergence is proposed. The necessity of doing the process is that, we would like to increase the order of numerical solution to a higher order scheme. An increment in the order of numerical solution makes the numerical simulation more accurate, also makes the numerical simulation being more complicated. In addition, the process of numerical analysis of the study in this order of solution needs more research work.

Pressure fluctuations beneath hydraulic jumps downstream of Ogee spillways potentially damage stilling basin beds. This paper deals with the extreme pressures underneath free hydraulic jumps along a smooth stilling basin. The experiments were conducted in a laboratory flume. From the probability distribution of measured instantaneous pressures, the pressures with different non-exceedance probabilities (P*a%) could be determined. It was verified that the maximum pressure fluctuations, as well as the negative pressures, are located at the positions closest to the spillway toe. The minimum pressure fluctuations are located at the downstream of hydraulic jumps. It was possible to assess the cumulative curves of P*a% related to the characteristic points along the basin, and different Froude numbers. To benchmark, the results, the dimensionless forms of mean pressures, standard deviations, and pressures with different non-exceedance probabilities were assessed. It was found that an existing methodology can be used to interpret the present data, and pressure distribution in similar conditions, by using a new third-order polynomial relationship for the standard deviation (σ*X) with the determination coefficient (R2) equal to 0.717. It was verified that the new optimized adjustment gives more accurate results for the estimation of the maximum extreme pressures than the minimum extreme pressures.

A mathematical model given by a two - dimensional differential system is introduced in order to understand the transition process from the normal hematopoiesis to the chronic and accelerated acute stages in chronic myeloid leukemia. A previous model of Dingli and Michor is refined by introducing a new parameter in order to differentiate the bone marrow microenvironment sensitivities of normal and mutant stem cells. In the light of the new parameter, the system now has three distinct equilibria corresponding to the normal hematopoietic state, to the chronic state, and to the accelerated acute phase of the disease. A characterization of the three hematopoietic states is obtained based on the stability analysis. Numerical simulations are included to illustrate the theoretical results.

This work - in two parts - tries to promote a new method in Mathematical Modeling, not ever used before, namely the use of the methods of Non-standard Analysis in Topoi. For the general mathematical theory of topoi (Topoi Theory) we refer to the book [10] . For non-standard analysis in the particular Boolean topos SET (the category of sets) we refer to the book [11]. This first part of the work is related, from an applicative point of view, mainly with Mathematical Neuroscience, more precisely a model of the human thinking, consciousness and subconsciousness. The connections between brain and mind will be also scketched, but a closer study also need some further topics to be studied in Part II: Artificial Intelligence, namely (Quantum) neural networks and (Quantum) Turing machine. However both parts (I. and II.) should be considered together, because they complement each other. The topoi model the intuitionistic logic (multi-valued) and have been used in Quantum Physics (see [9] and its references) while Non-Standard Analysis in SET (introduced by Abraham Robinson[31]) has been applied in Mathematical Economics (see [2] and its references); however, the combination was never used until now in Applied Mathematics. Even more, the Non-standard Analysis in Topoi is not too much studied from the point of view of Pure Mathematics either. One of the main objective of this work is to produce progress in this aria of Mathematics also, mainly for the SET-type topoi and exponential topoi, those topoi used in Quantum Physics and in which we need Non-stantard Analysis for our purposes. Based on the paper [13] we propose the logic of non-standard extensions in topoi as a model of the human thinking (based on infons), these theories representing top and very difficult results in Abstract Mathematics. We will analise the relation between brain and mind via the genetic-epigenetic interplay. More details will be provided in the Introduction, followed by a short presentation of the general theory, later connected with the (Quantum) Yang-Baxter equations in topoi. The connections between these equations and braid groups and knots, with intended applications in Neuroscience and Artificial Intelligence will be further analysed in Part II. Besides promoting a new idea of modeling, the present paper aims to promote the building of a Research Proposal for the European $''$Human Brain Project$''$ (https://www.humanbrainproject.eu/en/), and the NSF-Europe Program $''$Collaborative Research in Computational Neuroscience$''$ (https://www.nsf.gov/funding/pgm\_summ.jsp?pims\_id=5147), jointly with the new founded $''$National Centre for the Research of the Brain$''$ of the Romanian Academy of Science (https://acad.ro/centreAR/CNCC/CNCC.pdf, in Romanian). $''$Blue Brain Project$''$ (https://www.epfl.ch/research/domains/bluebrain/) could also be of big interest. We hope that the journals edited by MDPI (Axioms, in particular) and by the Simion Stoilow Institute of the Romanian Academy - Revue Roumaine de Mathematique Pures et Appliqu (http://imar.ro/journals/Revue\_Mathematique/home\_page.html) - among others - will host special issues related to the research subjects described in both parts of this work, starting with this one. For further details and some of the progress on this research subject, see www.ovidiufpasarescu.com.}

It is well known that the Continuous Galerkin Finite Element (CGFE) method is globally consistent with respect to the first law of thermodynamics. This means that, for any mesh, all obtained discrete solutions will conserve total energy. One might expect, that the method is, also, globally consistent with respect to the second law of thermodynamics. In this paper, we formally study if such conjecture is true. The heat conduction equation is used as the physical model for this analysis. In the present study it is proved that the conjecture is false: at least, for standard piecewise linear (1D and 2D) elements, the CGFE method is not always globally consistent with respect to the second law of thermodynamics. In other words, some obtained discrete solutions can violate the global postulate of the second law, which asserts that total entropy can never decrease.

In this paper, initially a mathematical model is formulated for transient frequency of power system considering time delays which occur while transmitting the control signals in open communication infrastructure. Time delay negligence in a power system leads to improper measurement of frequency variation in power system. The study of impact of time delays on the stability of power system is performed by estimating the decay rate of frequency wave form using Kalman Filter (KF). In power system, there is a possibility of multiple time delays. This paper also focuses on developing Interacting Multiple Model(IMM) Algorithm with multiple model space using Kalman Filter(KF) as state estimator tool. The multiple time delays in power system is considered as multiple model space The result shows that KF provides better estimate of correct model for a particular inputset. The qualitative properties of Riccati difference equation(RDE) in terms of state error covariance of IMMKF are also analyzed and presented.

This article aims to discuss, the stability and boundedness character of the solutions of the rational equation of the form \begin{equation}\label{eql21.1} y_{t+1}=\frac{\nu\epsilon^{-y_t}+\delta\epsilon^{-y_{t-1}}}{\mu+\nu y_t+\delta y_{t-1}},\quad t\in N(0). \end{equation} Here, $\epsilon>1, \nu,\delta,\mu\in (0,\infty)$ and $y_0, y_1$ are taken as arbitrary non-negative reals and $N(a)=\{a,a+1,a+2,\cdots \}$. Relevant examples are provided to validate our results. The exactness is tested using MATLAB.

In this study, we investigate spectral structure of conformable Sturm-Liouville problems and with this end, we obtain representation of solutions under different initial conditions and asymptotic formulas for eigenfunctions, eigenvalues, norming constants and normalized eigenfunctions. Consequently, we prove the existence of infinitely many eigenvalues. Also, we compare the solutions with graphics with different orders, different eigenvalues, different potentials and so, we observe the behaviors of eigenfunctions. We give an application to the α-orthogonality of eigenfunctions and reality of eigenvalues for conformable Sturm-Liouville problems defined by [15] in the last section.

The known three-component reaction-diffusion system modeling competition and co-existence of different language speakers is under study. A modification of this system is proposed, which is examined by Lie symmetry method; furthermore exact solutions in the form of traveling fronts are constructed and their properties are identified. Plots of the traveling fronts are presented and the relevant interpretation describing the language shift occurred in Ukraine during the Soviet times is suggested

This article presents a new method of segmenting images with gray levels. The method is based on determining several thresholds for separation of gray levels. The determination of these thresholds is done using the certainty of the neutrosophic information. The concept of this method can be stated simply: to choose the local maximums for the neutrosophic certainty.

The work is devoted to the investigation of virus quasispecies evolution and diversification due to mutations, competition for host cells, and cross-reactive immune responses. The model consists of a nonlocal reaction-diffusion equation for the virus density depending on the genotype considered as a continuous variable and on time. This equation contains two integral terms corresponding to the nonlocal effects of virus interaction with host cells and with immune cells. In the model, a virus strain is represented by a localized solution concentrated around some given genotype. Emergence of new strains corresponds to a periodic wave propagating in the space of genotypes. The conditions of appearance of such waves and their dynamics are described.

In this paper, we investigate the equilibrium points, stability of two equilibrium points, convergences of negative equilibrium point, periodic solutions, and existence of bounded or unbounded solutions of a system of nonlinear difference equations x_n+1 =x_n-1y_n - 1, y_n+1 = y_n-1x_n - 1 n = 0,1,..., where the initial values are real numbers. Additionally we present some numerical examples to verify our theoretical results.

We consider a model of predator-prey interaction at fractional-order where the predation obeys the ratio-dependent functional response and the prey is linearly harvested. For the proposed model, we show the existence, uniqueness, non-negativity as well as the boundedness of the solutions. Conditions for the existence of all possible equilibrium points and their stability criteria, both locally and globally, are also investigated. The local stability conditions are derived using the Magtinon's theorem, while the global stability is proven by formulating an appropriate Lyapunov function. The occurance of Hopf bifurcation around the interior point is also shown analytically. At the end, we implement the Predictor-Corrector scheme to perform some numerical simulations.

In this paper, the fixed-time stabilization problem for a class of uncertain chained system is addressed by using a novel nonsingular recursive terminal sliding mode control approach. A fixed-time controller and an adaptive law are designed to guarantee the uncertain chained form system both Lyapunov stable and fixed-time convergent within the settling time. The advantage of the controller based on the sliding mode is that the settling time does not depend on the system initial state. Furthermore, we use RBF neural network to estimate the uncertainty of the system. Finally, the simulation results demonstrate the performance of the control laws.

We address the dynamics of two-dimensional Navier-Stokes models with infinite delay and hereditary memory, whose kernels are a much larger class of functions than the one considered in the literature, on a bounded domain. We prove the existence and uniqueness of weak solutions by means of Faedo-Galerkin method. Moreover, we establish the existence of global attractor for the system with the existence of a bounded absorbing set and asymptotic compact property.

In light of Liu et al.'s original works, this research article revisits the solution of Burgers's nonlinear equation. The researcher found two exact and explicit solutions for groups $G_4$ and $G_6$, as well as a general solution. Their applications were conducted by using numerical calculations.

In light of Liu's original works, this paper revisits the solution of general Burgers's nonlinear equation. We obtain two exact and explicit solutions for group $G_4$ and $G_6$, and a most general solution as well. As applications, a numerical example is carried out.

This paper presents an extension of the resuts obtained in previous work by the author concerning the application of global optimization techniques to the design of finite strategic games with mixed strategies. In that publication the Fuzzy ASA global optimization method was applied to many examples of synthesis of strategic games with one previously specified Nash equilibrium, evidencing its ability in finding payoff functions whose respective games present those equilibria, possibly among others. That is to say, it was shown it is possible to establish in advance a Nash equilibrium for a generic finite state strategic game and to compute payoff functions that will make it feasible to reach the chosen equilibrium, allowing players to converge to the desired profile, considering that it is an equilibrium of the game as well. Going beyond this state of affairs, the present article shows that it is possible to "impose" multiple Nash equilibria to finite strategic games by following the same reasoning as before, but with a slight change: using the same fundamental theorem of Richard D. McKelvey, modifying the original prescribed objective function and globally minimizing it. The proposed method, in principle, is able to find payoff functions that result in games featuring an arbitrary number of Nash equiibria, paving the way to a substantial number of potential applications.

This paper investigates an inverse problem of option pricing in the extended Black--Scholes model. We identify the model coefficients from the measured data and attempt to find arbitrage opportunities in financial markets using a Bayesian inference approach. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by Markov Chain Monte Carlo (MCMC), which explores the posterior state space. The efficient sampling strategy of MCMC enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown drift and volatility coefficients from the measured data.

In this study, based on our previous study, we examined the mathematical properties, especially the stability of the equilibrium for our proposed mathematical model. By means of the results of the stability in this study, we also used actual data representing transient booms and resurgent booms, and conducted parameter estimation for our proposed model using Bayesian inference. In addition, we conducted a model fitting to five actual data. By this study, we reconfirmed that we can express the resurgences or minute vibrations of actual data by means of our proposed model.

This paper investigates the inverse option problems (IOP) in the extended Black--Scholes model arising in financial markets. We identify the volatility and the drift coefficient from the measured data in financial markets using a Bayesian inference approach, which is presented as an IOP solution. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by a Markov Chain Monte Carlo (MCMC) algorithm, which exploits the posterior state space. The efficient sampling strategy of the MCMC algorithm enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown trend and volatility coefficients from the measured data.

In this paper, the author obtains an analytic exact form of Heaviside function, which is also known as Unit Step function and constitutes a fundamental concept of the Operational Calculus.In particulat, this function is explicitly expressed in a very simple manner by the aid of purely algebraic representations. The novelty of this work is that the proposed explicit formula is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Hence, it may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.