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Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.

The report entitled "L. Liu, D. Li, F. Bai, A relative Lempel-Ziv complexity: Application to comparing biological sequences, Chem. Phys. Lett. 530 (2012) 107-112" mentions on the powerful construction of phylogenetic trees based on Lempel-Ziv algorithm. On the other hand, the method explained in the paper does not give promising result on the data set on invasive Caulerpa taxifolia in the Mediterranean Sea. The phylogenetic trees are obtained by the proposed method of the aforementioned paper in this short note. (C) 2018 Elsevier B.V. All rights reserved.

The purpose of this comment paper is to show that the results of the paper (Cui, 2021) are incorrect.

Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.

In this paper, we introduce some notions of invariancy for the Ricci tensor on real hypersurfaces in complex two-plane Grassmannians G 2 ( C m + 2 ) , namely, F -invariant and invariant Ricci tensor. Using these notions, we give non-existence theorem and characterization for the special case among the real hypersurfaces of Type ( A ) in G 2 ( C m + 2 ) , respectively. Here the distribution F is defined by F = [ ξ ] ∪ D ⊥ where D ⊥ = Span { ξ 1 , ξ 2 , ξ 3 } and [ ξ ] = Span { ξ } .

Recently, Alaca et al. [Alaca et al., Chaos, Solitons & Fractals 2006;29:1073–9] proved some fixed point theorems in intuitionistic fuzzy metric spaces by a strong definition of Cauchy sequence (see [George and Veeramani, Fuzzy Sets Syst 1994;64:395–9] and [Veeramani and Vasuki, Fuzzy Sets Syst 2003;135:409–13]), also the intuitionistic fuzzy metric space has extra conditions (see [Gregori et al., Chaos, Solitons & Fractals, 2006;28:902–5]). In this paper, we consider generalized intuitionistic fuzzy metric spaces i.e., L -fuzzy metric spaces and prove the fuzzy version of Banach and Edelstein contraction theorems in these spaces for modified definition of Cauchy sequence.

In this paper, it is shown that von Neumann continuous geometry may be regarded as the first attempt towards formulating a general quantum spacetime geometry akin to that of Cantorian spacetime E(∞) and noncommutative geometry.

The equations governing the motion of the paper sheet in a press nip and the anisotropic percolation of the water in the sheet, are derived in invariant form. In order to avoid restrictive assumptions on the configuration of the paper sheet, the system of coordinates is generated by the flow of fibrous material through the nip region. With the corresponding constitutive equations, the formulation consists of a system of partial differential equations for the metric tensor of the coordinate system, and for the water velocity. For practical use, the solution is then mapped back into a cartesian frame of reference. Quantities of industrial interest, such as the residual water content and stresses, as well as the press-induced anisotropy, can be calculated in principle. A Galerkin finite-element approximation is implemented using rectangular linear elements. Two case studies are presented, for large and small permeabilities, and the corresponding differences in water pressure and velocity are in general qualitative agreement with the observations. Finally, the predictive value of the model is demonstrated by the dependence of the solution on the imposed shear stress and its gradient across the sheet.

Gaemer's technique for recovering a function ⨍(x) from its power moments μ n = ∝ 1 0 ⨍(x)x n dx specified at n = 0, 1, 2,…N, is rederived using an expansion in Chebyshev polynomials of the second kind. Advantages of the present derivation over the original one are pointed out.

The purpose of this short paper is to present a new form of the so called early exercise premium for the American type derivatives. The decomposition we derived consists of the corresponding European derivative and a derivative with a stochastic maturity. In different particular cases we reach to the well known form for the American put option where the underlying asset is driven by a Brownian motion or a Levy process.

This paper examines the thoughts and early career of the astrophysicist and cosmologist E. A. Milne. Although Milne only turned to cosmology in 1932, many of the ideas that characterised his heterodox system of world physics can be traced back to his works from the 1920s. Contrary to what has been stated in the literature, we argue that Milne was familiar with and interested in cosmology even before 1932. The relationship between mathematics and physics, an important topic in Milne's cosmophysics, was the subject of an unpublished paper he gave in 1922. We place the paper in its historical context and comment on its relevance for Milne's later work. An annotated transcript of the address is included as an accompanying paper under the title ‘The Relations of Mathematics to Science’.

This short paper outlines the basic topological features of two new fuzzy Kahler manifolds in duality. Relation to E-infinity theory of the compactified Klein modular space in connection with ’t Hooft holographic principles is briefly discussed.

This paper based on the essay [1], studies in case that replicated observations are available in some experimental points, the parameters estimation of one dimensional linear errors-in-variables (EV) models. Asymptotic normality is established.

This paper engages with the following closely related questions that have recently received some attention in the literature: (a) what is the status of the equivalence principle in general relativity (GR)?; (b) how does the metric field obtain its property of being able to act as a metric?; and (c) is the metric of GR derivative on the dynamics of the matter fields? The paper attempts to complement these debates by studying the spin-2 approach to (quantum) gravity. In particular, the paper argues that three lessons can be drawn from the spin-2 approach: (1) different from what is sometimes claimed in the literature, central aspects of the non-linear theory of GR are already derivable in classical spin-2 theory; in particular, ‘universal coupling’ can be considered a derived ‘theorem’ in both the classical and the quantum spin-2 approach; this provides new insights for the investigation of the equivalence principle; (2) the ‘second miracle’ that Read et al. argue characterises GR is explained in the classical as well as in the quantum version of the spin-2 approach; (3) the spin-2 approach allows for an ontological reduction of the metrical part of spacetime to the dynamics of matter fields.

Finding exact and analytic solutions of nonlinear system in high dimensions is a difficult but meaningful work. In this paper, by means of the symbolic package Maple, we investigate a Painleve integrable (3+1)-dimensional generalized Burgers (gBurgers) equation. Starting from the Cole-Hopf transformation with different seed solutions, abundant localized solutions are provided, including new variable separation solutions, lumps, lump-“multiple-soliton” solutions and other interaction solutions. Specifically, the lump-two soliton solution and lump-soliton-periodic solution are depicted by the three-dimensional plots and contour plots at different times, respectively. The methods and all the resulting solutions presented in this paper can be generalized to the Painleve integrable ( N + 1 )-dimensional Burgers equation.

In a 1919 article for the Times of London, Einstein declared the relativity theory to be a ‘principle theory,’ like thermodynamics, rather than a ‘constructive theory,’ like the kinetic theory of gases. The present paper attempts to trace back the prehistory of this famous distinction through a systematic overview of Einstein’s repeated use of the relativity theory/thermodynamics analysis after 1905. Einstein initially used the comparison to address a specific objection. In his 1905 relativity paper he had determined the velocity-dependence of the electron’s mass by adapting Newton’s particle dynamics to the relativity principle. However, according to many, this result was not admissible without making some assumption about the structure of the electron. Einstein replied that the relativity theory is similar to thermodynamics. Unlike the usual physical theories, it does not directly try to construct models of specific physical systems; it provides empirically motivated and mathematically formulated criteria for the acceptability of such theories. New theories can be obtained by modifying existing theories valid in limiting case so that they comply with such criteria. Einstein progressively transformed this line of the defense into a positive heuristics. Instead of directly searching for new theories, it is often more effective to search for conditions which constraint the number of possible theories. The paper argues that the latter was the strategy that led Einstein to most of his major successes. The constructive/principle theories opposition should be considered not only as abstract classification of theories, but also as Einstein’s attempt to formulate a sort of ‘logic of discovery.’ The paper argues that most of Einstein’s scientific successes were obtained by following the principle strategy. Most of his failures happened when he was forced to fall back to the constructive strategy.

To our knowledge the article in hand is first ever study for existence of Si’lnikov type of chaos for Nuclear Spin Generator (NSG) system. This paper deals with the existence of heteroclinic Si’lnikov-type orbits within an NSG chaotic system involving three parameters and having three equilibrium points. Method of undetermined coefficient is applied for analytical analysis of heteroclinic orbits. As an outcome the Si’lnikov criteria guarantees that the NSG system has a Smale horseshoe type chaos. Uniform convergence of series expansion for heteroclinic orbit is verified in this paper. Lyapunov exponents spectrum and sensitivity dependence are discussed. Numerical simulations are compiled for the verification of analytical results and bifurcation diagrams are displayed. Though examination of Si’lnikov criterion for NSG system is an exhaustive study but we have succeeded to explore an other aspect of the system by this criterion.

The Lie symmetry and invariants for a generalized Birkhoffian system on time scales are studied, which include exact invariants and adiabatic invariants. First, the generalized Pfaff-Birkhoff principle on time scales is established, and by using Dubois-Reymond lemma the generalized Birkhoff’s equations on time scale are derived. Secondly, the determining equations of Lie symmetry for the generalized Birkhoffian system on time scales are established. We prove that if the Lie symmetry satisfies the structural equation, it leads to a conserved quantity, which is an exact invariant of the system. Again, the perturbation of Lie symmetry under the action of small disturbance is considered, the determining equations and the structural equations of disturbed system are established, and the adiabatic invariants led by the Lie symmetry perturbation for the generalized Birkhoffian system on time scales are given. Because of the arbitrariness of selecting time scales and the generality of the generalized Birkhoffian system, the results of this paper are of universal significance. The results of this paper contain the corresponding results for Birkhoffian system on time scales and classical generalized Birkhoffian system as its special cases. At the end of the paper, an example is given to illustrate the validity of the method and the results.

The economic system is an irreversible entropy increase process which is constructed by many elements and is far away from the equilibrium point; and affected by various parameters change, it is quite common that its motion state appears chaotic phenomenon due to instability. The extremely complex and not completely random aperiodic motion form of chaotic phenomenon is strongly sensitive to initial conditions. The development of nonlinear science, especially the emergence and development of chaos and fractal theory, has gradually become a powerful tool for economists to study the complexity, uncertainty and nonlinearity of social economic systems; and some visionary economists began to apply the research results of nonlinear science to economics, which has produced nonlinear economics. On the basis of summarizing and analyzing previous research works, this paper first obtains the non-degenerate equilibrium point of some typical fractional-order chaotic economic systems and transforms the equilibrium points of those systems to the origin through coordinate transformation, and then analyzes the Jacobi matrixes of new systems obtained through coordinate translation, and the parameter conditions of bifurcation in the economic systems are finally given and the numerical simulation of the fractional-order chaotic economic system evolution is carried out through bifurcation diagram, phase diagram and time series diagram. The study results of this paper provide a reference for the further study of the evolution of fractional-order chaotic economic systems with non-degenerate equilibrium points.

In this paper, we study the existence and the numerical estimates of solutions for a specific types of fractional differential equations. The nonlinear part of the problem, however, presupposes certain hypotheses. Particularly, for exact localization of the parameter, the existence of a non-zero solution is established, which requires the sublinearity of the nonlinear part at the origin and infinity. We also take into consideration several theoretical and numerical examples. One of the main novelties of this paper is to use the variational method to investigate the properties of solutions of boundary values problems involving Atangana–Baleanu fractional derivative for the first time.

This paper is concerned with the moving least squares (MLS) meshless approach for the numerical solution of two-dimensional (2D) variable-order time fractional nonlinear diffusion-wave equation (V-OTFND-WE) on arbitrary domains. The variable-order fractional derivative of this equation is expressed in the Caputo type. The proposed method is based on the MLS approximation in conjunction with the finite difference technique. For spatial derivatives , the MLS approximation and for temporal derivative, the finite difference technique are utilized to discretize the equation. The aim of the present paper is to show that the meshfree method based on the MLS approximation and collocation scheme is suitable for solving the variable-order fractional partial differential equations (PDEs). The proposed method is validated, using some different nonlinear numerical examples with complex geometries . The results obtained demonstrate the accuracy and easy implementation of the method for nonlinear variable-order time fractional PDEs.

A fractional compartmental mathematical model for the spread of the COVID-19 disease is proposed. Special focus has been done on the transmissibility of super-spreaders individuals. Numerical simulations are shown for data of Galicia, Spain, and Portugal. For each region, the order of the Caputo derivative takes a different value, that is not close to one, showing the relevance of considering fractional models., This is a preprint of a paper whose final and definite form is published by 'Chaos Solitons Fractals' (ISSN: 0960-0779). Paper Submitted 04/June/2020; Revised 22/July/2020 and 02/Aug/2020; Accepted 04/Jan/2021

Research findings have shown that evolution equations containing non-integer order derivatives can lead to some useful dynamical systems which can be used to describe important physical scenarios. This paper deals with numerical simulations of multicomponent symbiosis systems, such as the parasitic predator-prey model, the commensalism system, and the mutualism case. In such models, we replace the classical time derivative with either the Caputo fractional derivative or the Atangana-Baleanu fractional derivative in the sense of Caputo. To guide in the correct choice of parameters, we report the models linear stability analysis. Numerical examples and results obtained for different instances of fractional power α are provided for non-spatial models as well as the spatial case in one and two dimensions in other to justify our theoretical findings which include the chaotic phenomena, spatiotemporal and oscillatory patterns, multiple steady states and other spatial pattern processes. This paper further suggest an alternative approach to incessant killing of wildlife animals for pattern generation and decorative purposes.

Assume that Ω = Ω 1 ∖ Ω ¯ 2 is an annulus between two smooth bounded domains Ω 1 and Ω 2 in ℂ n (or a Stein manifold) such that Ω ¯ 2 ⋐ Ω 1 , Ω 1 is weakly q -convex, Ω 2 is weakly ( n − q − 1 ) -convex and let 1 ⩽ q ⩽ n − 2 with n ⩾ 3 . This paper is devoted to studying existence, boundedness and compactness of the weighted ∂ ¯ -Neumann operator N p , q t with weights e − t λ on the space L p , q 2 ( Ω ) (or W p , q s ( Ω ) ) for sufficiently large t and for λ ∈ C ∞ ( Ω ¯ ) . Moreover, the closedness of the ranges of the operators ∂ ¯ and ∂ ¯ ∗ are proved. Next as an application, we study the regularity of operators associated with N p , q t . Consequently, the global boundary regularity of the ∂ ¯ -equation, ∂ ¯ u = f , is studied on Ω . In the end of the paper, we study compactness of the commutator [ B p , q t , f ] of the weighted Bergman projection B p , q t and the multiplication operator by a function f ∈ C ( Ω ¯ ) on the space L p , q 2 ( Ω ) .

As the predominant factor influencing the service performance and life cycle of finished parts, surface quality is always improved by polishing technology in the machining process of aero-engine blade. The reasonable selection of polishing process parameter intervals have significant impact on surface quality, which is critically bound up with the performances of finished blade. In this paper, two different characteristic values of surface quality, including morphology feature (surface roughness) and stress property (residual stress), were analyzed respectively. To obtain the global optimal parameter intervals for surface quality improvement in Ti-6Al-4V blade polishing process, the comprehensive analysis methods were presented in this paper. In order to develop the prediction models of surface roughness and residual stress, regression analysis was conducted based on the results of Central-Composite designed polishing experiments. The significance and adequacy of the developed model was verified by using the analysis of variance (ANOVA). The sensitivity analysis was carried out to evaluate the influence of polishing process parameters on surface quality. Subsequently, stability interval and instability interval were divided according to sensitivity curves. Finally, the optimization method of process parameter intervals was proposed and the comprehensive optimal intervals were obtained. The experiments were carried out to verify the developed prediction models and the obtained global optimal parameter intervals. The results of polishing confirmation experiments indicated that the surface-quality-oriented optimal intervals of polishing process parameters were dependable. Keywords: Ti-6Al-4V blade, Surface quality, Polishing parameters, Surface roughness, Residual stress, Sensitivity analysis

In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner. The SSGFEM employs enrichments of 2 types: the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution, while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface. Thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions. Both types of enrichments have been modified by a simple local procedure of “subtracting the interpolant.” A simple local orthogonalization technique (LOT) also has been used at the nodes enriched with both enrichment functions. We prove that the SSGFEM yields the optimal order of convergence. The numerical experiments presented in this paper indicate that the conditioning of the SSGFEM is not worse than that of the standard finite element method, and the conditioning is robust with respect to the position of the mesh relative to the interface.

In a recent paper (Merida-Angulo and Thas, 2017), the authors introduced a map F which associates a Deitmar constructible set (which is defined over the field with one element, denoted by F 1 ) with any given loose graph Γ . By base extension, a constructible set X k = F ( Γ ) ⊗ F 1 k over any field k arises. In the present paper, we will show that all these mappings are functors, and we will use this fact to study automorphism groups of the constructible sets X k . Several automorphism groups are considered: combinatorial, topological and also groups induced by automorphisms of the ambient projective space. When Γ is a finite tree, we will give a precise description of the combinatorial and projective groups, amongst other results.

This paper is concerned with the spatiotemporal behaviors of a Gierer–Meinhardt system in discrete time and space form. Through the linear stability analysis, the parametric conditions are gained to ensure the stability of the homogeneous steady state of the system. Based on the bifurcation theory, as well as center manifold theorem, we derive the critical parameter values of the flip, Neimark–Sacker and Turing bifurcation respectively. Besides, the specific parameter expression to form patterns are also determined. In order to identify chaos among regular behaviors, we calculate the Maximum Lyapunov exponents. The results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamics, such as period doubling cascade, invariant cycles, periodic windows, chaotic behaviors, and some striking Turing patterns, e.g. circle, mosaic, spiral, spatiotemporal chaotic patterns and so on, which can be produced by flip-Turing instability, Neimark–Sacker–Turing instability and chaos.

This paper demonstrates how the solutions to conventional radioactive decay equations can be derived using results from matrix analysis . In particular, we draw on results for the matrix exponential function when the matrix is triangular. By applying key theorems, the paper explains how the solutions to these equations can be presented in algorithmic form and in terms of divided differences of the exponential function. Furthermore, with little additional effort, the approach yields solutions to more general variations of these equations.

In the present paper we develop a method of the vector fields Z i [9] in the theory of separation of variables. For an integrable case of the complex Kirchhoff's problem on e ⁎ ( 3 ) , which has been never considered before, we construct—with the help of this method—two types of separation of variables (SoV): symmetric and asymmetric ones. Our asymmetric SoV is unusual: it is characterized by the quadratures containing differentials defined on two different curves of separation. It is a direct analogue of asymmetric SoV for the Clebsch model [17] . In the case of symmetric SoV both curves of separation are the same. This case has an additional bonus: on zero level set of one of the Casimir functions it yields a direct analogue of the famous Weber-Neumann separated coordinates. They are also considered in the present paper in some details.

This paper presents a new subgrid multiscale stabilized formulation for non-Newtonian Casson fluid flow model tightly coupled with variable diffusion coefficients Advection–Diffusion–Reaction equation ( V A D R ). Both the subscales are chosen to be time dependent and the stabilized formulation has been reached through the complete elimination of the unresolvable scales in terms of the coarse scale solution. Hence the resultant formulation emerges as a set of equations involving only the coarse scale solution instead of multiple scales and makes the formulation simpler to handle. In this study the shear-rate dependent Casson viscosity coefficient is assumed to be a function of the solute mass concentration, resulting in a two way coupling. This paper investigates the stability and the convergence properties of the stabilized finite element solution , without imposing any additional regularity condition other than the admissible space requirement on it. The proposed expressions of the stabilization parameters play a significant role in obtaining optimal order of convergences. During various theoretical derivations the estimations of the non-linear apparent viscosity coefficient have been carefully carried out assuming sufficiently regular exact solution. The performance of the scheme has been numerically validated with lid driven cavity problem. The theoretically established rate of convergence results are appropriately verified through numerical studies. In addition the transient Casson fluid flow behavior in a flow past square cylinder has been analyzed thoroughly.

In order to understand the microstructure of pellets and improve the metallurgical properties of pellets, fractal theory is introduced to extract the edge features of pellets. Firstly, the original mineralogical phase is obtained by experiment, which is preprocessed by the histogram equalization to enhance the overall contrast. Based on the discrete Fractional Brownian Random Field Model, the algorithm is redesigned to calculate the dimension of each pixel, map the gray space of image into the dimension space, select the appropriate window size, transform and edge extraction. Comparing the algorithm in this paper with Canny operator and Laplace–Gauss operator, it is concluded that the algorithm in this paper has certain advantages in mineralogical phase edge extraction. Then, Gauss noise is added to the original gray image, and Canny operator and this algorithm are used to extract the edges of the noisy image. The numerical results of peak signal to noise ratio and root mean square error are obtained. Finally, the comparison proves that the algorithm can extract more complete edges, and has a stronger noise immunity.

The purpose of the present paper is the development of meshless diffuse approximate method in connection with the phase field formulation for solution of two-phase flow problems. The numerical approach enables single-domain, fixed-node treatment of the involved moving boundaries. The problem is formulated for Newtonian fluids based on a physical model that involves a coupled set of Navier–Stokes and Cahn–Hilliard equations. Diffuse approximate method is structured with a second order polynomial basis, Gaussian weight function, adaptive upwind scheme and eleven-node local subdomain. The pressure-velocity coupling is performed by the incremental fractional step method and explicit time discretization is used to solve the governing equations. The method is demonstrated in axisymmetry for a problem of two co-flowing immiscible fluids with different material properties that yield dripping or jetting of the core fluid. An assessment of the novel method is carried out based on comparison of the present results with the finite volume method outcome. A sensitivity study of various process parameters is carried out and verified against the previously published results. The paper represents a pioneering development in simulation of two-phase flows by a meshless method.

In earlier papers, we introduced spherical T-duality, which relates pairs of the form $(P,H)$ consisting of an oriented $S^3$-bundle $P\rightarrow M$ and a 7-cocycle $H$ on $P$ called the 7-flux. Intuitively, the spherical T-dual is another such pair $(\hat P, \hat H)$ and spherical T-duality exchanges the 7-flux with the Euler class, upon fixing the Pontryagin class and the second Stiefel-Whitney class. Unless $\mathrm{dim}(M)\leq 4$, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. In this paper, we define a canonical Poincar\'e virtual line bundle $\mathcal{P}$ on $S^3 \times S^3$ (actually also for $S^n\times S^n$) and the spherical Fourier-Mukai transform, which implements a degree shifting isomorphism in K-theory on the trivial $S^3$-bundle. This is then used to prove that all spherical T-dualities induce natural degree-shifting isomorphisms between the 7-twisted K-theories of the pairs $(P,H)$ and $(\hat P, \hat H)$ when $\mathrm{dim}(M)\leq 4$, improving our earlier results., Comment: 19 pages, clarifications added, typos corrected

In this paper Stokes’ hypothesis receives the physical justification and alternative mathematical formulation. An alternative formulation is more general and easily checked. As well as in linear elasticity , the pressure in Navier–Stokes equations can be eliminated as unknown function. To do this transformation it is necessary to use the well known derivation analogies of Navier–Cauchy (also called Lame’s) equations. In this case the well known problem of “second viscosity” disappears. Thus the fundamental defining relation for pressure is no longer necessary and the Navier–Stokes and Lame’s equations exactly coincide. The Navier–Stokes equations are not physically exact because the traditional assumption of const bulk viscosity is not correct. The analysis presented here corrects this error and more exact Navier–Stokes equations have been proposed. This paper written in a way that gives insight to mathematicians, physicists, engineers and students who may not be experts in this topic.

The present paper shows the dynamics of Hepatitis E with optimal control. The paper is analyzed by two different aspects: first, we explore the dynamics of Hepatitis E model and then applying the optimal control analysis. Secondly, we use the most appropriate and recent fractional order derivative called the Atangana–Baleanu derivative for the dynamical analysis of Hepatitis E model. The proposed model considered is locally asymptotically stable when the threshold quantity less than one. Further, we explore the stability analysis of the model when R 0 > 1 . Then, we choose some appropriate control to formulate the optimality system. The results associated to the optimal control are obtained and discussed with different strategies. Moreover, we apply Atangana–Baleanu derivative to the proposed model and obtain the required results necessary for the fractional order model. Numerical results for the optimal control problem and Atangana–Baleanu derivative are obtained and discussed in detail. The results suggest that control variables chosen should be properly applied to get rid of the infection of Hepatitis E. The Atangana–Baleanu derivative results suggest that at any time t we can check the disease status and make a useful strategy for the early elimination of Hepatitis E from the community.

In this paper, we present and analyze a prey-predator system, in which prey species can be infected with some disease. The model presented in this paper is motivated from D. Mukherjee’s model in which he has considered an SI model for the prey species. There are substantial evidences that infected individuals have the ability to recover from the disease if vaccinated/ treated properly. In this regard, Mukherjee’s model is modified by considering SIS model for prey species. Theoretical and numerical simulations show that the recovery of infected prey species plays a crucial role in eliminating the limit cycle oscillations and thus making the interior equilibrium point stable. The possibility of Hopf bifurcation around non zero equilibrium point using the recovery rate as a bifurcation parameter, is discussed. Further, the model is extended by incorporating the harvesting of predator population. A monitory agency has been introduced which monitors the exploitation of resources by implementing certain taxes for each unit biomass of the predator population. The main purpose of the present research is to explore the effect of recovery rate of prey on the dynamics of the system and to optimize the total economical net profits from harvesting of predator species, taking taxation as control parameter.

This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier-Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective--diffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective--diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259--280]. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection--diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov-Galerkin (SUPG) and the Galerkin/least-squares method (GLS). The presented method is a Galerkin/least-squares formulation with dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large- and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method., Comment: Update to postprint version

This paper is a next step in the project of systematic description of colored knot and link invariants started in previous papers. In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels $R_1\otimes R_2\otimes R_3\longrightarrow Q$ with all possible $Q$, for $|R|\leq 3$. The calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org. Using the obtained inclusive Racah matrices, we also calculated the exclusive Racah matrices with the help of trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method., 29 pages, 3 figures

This paper deals with the implementation of thermomechanically coupled gradient-enhanced elastoplasticity at finite strains. The presented algorithmic formulation heavily relies on the variational structure of the considered initial boundary value problem. Consequently, such a variational structure is elaborated. While variational formulations are well-known in the case of isothermal plasticity theory, thermomechanically coupled gradient-enhanced plasticity theory has not been considered before. The resulting time-continuous variational principle allows the computation of all unknown variables jointly and naturally from the stationarity condition of an incremental potential. By discretization of this time-continuous potential in time, a discrete approximation is obtained which represents the foundation of the algorithmic formulation. As a matter of fact, stationarity of the respective potential again defines all unknown variables — now in a time-discrete fashion. Within this paper, the necessary condition associated with stationarity – a vanishing first gradient – is solved numerically by means of Newton’s method. Hence, the first as well as the second derivatives of the incremental potential are required. They are computed by numerical differentiation based on hyper-dual numbers. By considering a perturbation with respect to hyper-dual numbers, the first as well as the second derivatives are computed in an exact manner without introducing any numerical errors. Although numerical differentiation based on hyper-dual numbers is numerically more extensive than real-valued perturbations, it is shown that the scalability of the resulting parallel finite element implementation is not dominated by this effect for sufficiently large problems.

This paper develops the analytical conditions for the existence of periodic solutions of a second-order,time-delayed, discontinuous dynamical system. A sample model consists of two linear delayed sub-systems with a switching boundary. The defined G-functions for the delayed, discontinuous systems are introduced, and sufficient and necessary conditions for a flow crossing, sliding and grazing along the switch boundary are developed for such a delayed, discontinuous system. Furthermore, nine (9) regular basic mappings in phase plane and thirty-three (33) delay-related mappings for the second-order, time-delayed, discontinuous systems are classified. Constraint equations are predicted analytically for two periodic orbits with initial functions provided posteriorly. Finally, three numerical examples are illustrated to verify the existence of generalized slowly oscillating periodic orbits without and with sliding portions. This paper improves and extends motion switchability conditions at the boundary in discontinuous dynamical systems without delay.

A three factor variance model introduced by Gatheral in 2008, called the double mean reverting (DMR) model, is well-known to reflect the empirical dynamics of the variance and prices of options on both SPX and VIX consistently with the market. One drawback of the DMR model is that calibration may not be easy as no closed form solution for European options exists, not like the Heston model. In this paper, we still use the double mean reverting nature to extend the Heston model and study the pricing of variance swaps given by simple returns in discrete sampling times. The constant mean level of Heston’s stochastic volatility is extended to a slowly varying process which is specified in two different ways in terms of the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. So, two types of double mean reversion are considered and the corresponding models are called the double mean reverting Heston-OU model and the double mean reverting Heston-CIR models. We solve Riccati type nonlinear equations and derive closed form exact solutions or closed form approximations of the fair strike prices of the variance swaps depending on the correlation structure of the three factors. We verify the accuracy of our analytic solutions by comparing with values computed by Monte Carlo simulation. The impact of the double mean reverting formulation on the fair strike prices of the variance swaps are also scrutinized in the paper.

Recently fractional differential operators with non-index law properties have being recognized to have brought new weapons to accurately model real world problems particularly those with non-Markovian processes. This present paper has two double aims, the first was to prove the inadequacy and failure of index law fractional calculus and secondly to show the application of fractional differential operators with no index law properties to statistic and dynamical systems. To achieve this, we presented the historical construction of the concept of fractional differential operators from Leibniz to date. Using a matrix based on the fractional differential operators, we proved that, fractional operators obeying index law cannot model real world problems taking place in two states, more precisely they cannot describe phenomena taking place beyond their boundaries, as they are scaling invariant, more precisely our results show that, mathematical models based on these differential operators are not able to describe the inverse memory, meaning the full history of a physical problem cannot be described accurately using these derivatives with index law properties. On the other hand, we proved that, differential operators with no index-law properties are scaling variant, thus can describe situations taking place in different states and are able to localize the frontiers between two states. We present the renewal process properties included in differential equation build out of the Atangana–Baleanu fractional derivative and counting process, which is connected to its inter-arrival time distribution Mittag–Leffler distribution which is the kernel of these derivatives. We presented the connection of each derivative to a statistical family, for instance Riemann–Liouville–Caputo derivatives are connected to the Pareto statistic, which has no well-defined average when alpha is less than 1 corresponding to the interval where fractional operators mostly defined. We established new properties and theorem for the Atangana–Baleanu derivative of an analytic function, in particular we proved that, they are convolution of the Mittag–Leffler function with the Riemann–Liouville–Caputo derivatives. To see the accuracy of the non-index law derivative to in modeling real chaotic problems, 4 examples were considered, including the nine-term 3-D novel chaotic system, King Cobra chaotic system, the Ikeda delay system and chaotic chameleon system. The numerical simulations show very interesting and novel attractors. The king cobra system with the Atangana–Baleanu presented a very novel attractor where at the earlier time we observed a random walk and latter time we observed the real sharp of the cobra. The Ikeda model with Atangana–Baleanu presented different attractors for each value of fractional order, in particular we obtain a square and circular explosions. The results obtained in this paper show that, the future of modeling real world problem relies on fractional differential operators with non-index law property. Our numerical results showed that, to not model physical problems with fractional differential operators with non-singular kernel and imposing index law in fractional calculus is rightfully living with closed eyes without ever taking a risk to open them.

It has been shown that natural interval extensions (NIE) can be used to calculate the largest positive Lyapunov exponent (LLE). However, the elaboration of NIE are not always possible for some dynamical systems, such as those modelled by simple equations or by Simulink-type blocks. In this paper, we use rounding mode of floating-point numbers to compute the LLE. We have exhibited how to produce two pseudo-orbits by means of different rounding modes; these pseudo-orbits are used to calculate the Lower Bound Error (LBE). The LLE is the slope of the line gotten from the logarithm of the LBE, which is estimated by means of a recursive least square algorithm (RLS). The main contribution of this paper is to develop a procedure to compute the LLE based on the LBE without using the NIE. Additionally, with the aid of RLS the number of required points has been decreased. Eight numerical examples are given to show the effectiveness of the proposed technique.

The local truncation error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high-order finite elements in the 1-D, 2-D and 3-D cases on uniform and non-uniform meshes. Several applications of the local truncation error are considered in the paper. It is proven that for the 1-D wave equation with a piece-wise constant wave velocity, the local truncation error is zero if the linear finite elements with the element size proportional to the wave velocity, the lumped mass matrix and the central-difference method with the time increments equal to the stability limit are used. It is shown in the 1-D and multidimensional cases that the optimal lumped mass matrix can be calculated by the minimization of the order of the local truncation error and yields the maximum possible order of accuracy. The minimization of the order of the local truncation error allows us to develop the linear finite elements and the isogeometric high-order elements with improved accuracy; i.e., accuracy is improved from order 2 p (the conventional elements) to order 4 p (the new elements) where p is the order of the polynomial approximations. New high-order boundary conditions are developed in order to keep a high-order accuracy of the developed technique. The new elements can be equally applied to linear wave propagation and heat transfer problems. It is also shown that non-uniform meshes may lead to inaccurate results due to the increase in the local truncation error. The difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach. The presented numerical examples are in good agreement with the theoretical results. The approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches.

In a recent paper [R. Qesmi, Dynamics of an opinion model with threshold-type delay, Chaos, Solitons & Fractals 98 (2020). (https://doi.org/10.1016/j.chaos.2020.110379)], we proposed a mathematical model of threshold-type delay differential equations describing the relationship between two subpopulations with opposite opinions and the opinion spread dynamics. The study there showed the possibility of a transcritical forward and backward bifurcations of positive equilibria. In the present paper, we show that the opinion model undergoes a Hopf bifurcation through which one of the bifurcation branches loses the stability and periodic solutions appear. One of the important consequences of the obtained dynamics is that the consensus of the both opinions could be lost by maintaining the balance between the time taken for an individual to become convinced of the outsider opinion, which need be short, and the number of individuals converted to the local opinion which need be low. Finally, we provide numerical simulations to illustrate and support our theoretical results. results.

This paper investigates the stability and stabilization problem of variable-order fractional nonlinear dynamic systems with impulsive effects (VO-IFNDS) via a linear feedback controller. New inequalities on the VO Caputo fractional derivatives are established in this paper, which plays an essential role in the study of the stability theory of VO-IFNDS. Based on utilizing S-procedure and analytical technique, several sufficient criteria on Mittag-Leffler stability and asymptotical stability of VO-IFNDS are presented by means of the extension of the Lyapunov direct method. Finally, numerical examples are given to show the efficiency of the proposed method.

In the present paper, first, we characterize the standard static spacetimes satisfying certain Ricci-Hessian class type equations, such as generalized quasi Einstein manifolds, m-quasi Einstein manifolds, (m, rho)-quasi Einstein manifolds, and gradient Ricci solitons. Then, we obtain some necessary and sufficient conditions about the existence of certain vector fields on the standard static spacetime. We also characterize the m-quasi Einstein standard static spacetimes admitting concurrent or parallel vector fields and get several physical properties of such spacetimes. At last, we construct non-trivial examples of m-quasi Einstein spacetimes verifying our results. (C) 2021 Elsevier B.V. All rights reserved. Research Fund of the Istanbul Technical UniversityIstanbul Technical University [TGA-2018-41211] This work is supported by the Research Fund of the Istanbul Technical University (GAP project reference TGA-2018-41211) . The authors would like to thank the referee for reviewing the paper carefully and her or his valuable comments which provided insights to improve the quality of the results.

Static spaces (with perfect fluids) appeared in a natural way both in physics (cf. Hawking and Ellis, 1975) and mathematics (cf. A. Fischer and J. Marsden, Duke Math. J. 42 (3) (1975), 519-547). These arise as solutions of the perfect static fluid equation and play a key role in general relativity, providing patterns for some celestial objects. In this paper, we investigate compact and simply connected trans-Sasakian spaces having type ( α , β ) , whose defining functions α and β satisfy the static perfect fluid equation. In particular, we derive some conditions that ensure these spaces are isometric to a 3 -sphere. First result of this work shows that the function α satisfying static perfect fluid equation and the scalar curvature τ satisfying certain inequality are not only necessary, but also sufficient conditions for a 3-dimensional compact and simply connected trans-Sasakian manifold to be isometric to a 3-sphere. In the second result of this paper, we prove that the function β satisfying the static perfect fluid equation, scalar curvature τ satisfying certain inequality and the Ricci operator satisfying a Coddazi-type equation are also requirements ensuring that a trans-Sasakian space (again compact and also simply connected) is isometric with a 3-sphere.