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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.

In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

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.

We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.

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 date of publication of Helmholtz's first paper on the foundations of geometry is discussed.

That Euler was quite aware of the subtleties of assigning a sum to a divergent series is amply demonstrated in his paper De seriebus divergentibus which appeared in Novi commentarii academiae scientiarum Petropolitanae 5 (1754/55), 205–237 (= Opera Omnia (1) 14, 585–617) in the year 1760. The first half of this paper contains a detailed exposition of Euler's views which should be more readily accessible to the mathematical community.The authors present here a translation from Latin of the summary and first twelve sections of Euler's paper with some explanatory comments. The remainder of the paper, treating Wallis' hypergeometric series and other technical matter, is described briefly. Appended is a short bibliography of works concerning Euler which are available to the English-speaking reader.

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.

This paper is dedicated to L p bounds on eigenfunctions of a Schrodinger-type operator ( − Δ g ) α / 2 + V on closed Riemannian manifolds for critically singular potentials V. The operator ( − Δ g ) α / 2 is defined spectrally in terms of the eigenfunctions of − Δ g . We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation ( ∂ t 2 + ( − Δ g ) α / 2 + V ) u = 0 . The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao [4] and Shao-Yao [27] play a key role in this paper. We construct a new reproducing operator with several local operators and some good error terms. Moreover, we shall prove that these local operators satisfy certain variable coefficient versions of the “uniform Sobolev estimates” by Kenig-Ruiz-Sogge [18] . These enable us to handle the critically singular potentials V and prove the quasimode estimates.

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.

International audience; In this paper, we consider a semi-linear fourth order parabolic equation in a bounded smooth domain Ω with homogeneous Dirichlet and Neumann boundary conditions. The main result of this paper is the null controllability and the exact controllability to the trajectories at any time T > 0 for the associated control system with a control function acting at the interior.; Dans ce papier, on considère uneéquation parabolique semi-linéaire de quatrième ordre dans un domaine borné régulier Ω avec des conditions aux limites de type Dirichlet et Neumann homogènes. Le résultat principal de ce papier concerne la contrôlabilitéà zéro et la contrôlabilité exacte pour tout T > 0 du système de contrôle associé avec un contrôle agissantà l'interieur.

The purpose of this paper is to show well-posedness results in L 2 -based Sobolev spaces for transmission, Dirichlet, Neumann, and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on a compact Riemannian manifold of dimension m ≥ 2 . The Dirichlet, transmission, and mixed problems for the nonlinear Darcy-Forchheimer-Brinkman system with L ∞ coefficients are also analyzed. First, we focus on the well-posedness of linear transmission, Dirichlet and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using a variational approach that reduces such a boundary value problem to a mixed variational formulation defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. Further, we show the equivalence between each boundary value problem for the Brinkman system with L ∞ coefficients and its mixed variational counterpart, and then the well posedness in L 2 -based Sobolev spaces by using the Necas-Babuska-Brezzi technique. The second goal of this paper is the construction of the Newtonian and layer potential operators for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using the well-posedness results for the analyzed linear transmission problems. Various mapping properties of these operators are also obtained and used to describe the weak solutions of the Poisson problems with Dirichlet and Neumann conditions for the nonsmooth Brinkman system in terms of such potentials. Finally, we combine the well-posedness results of the Poisson problems of Dirichlet, transmission, and mixed type for the nonsmooth Brinkman system with a fixed point theorem in order to show the existence of a weak solution of the Poisson problem of Dirichlet, transmission, or mixed type for the (nonlinear) Darcy-Forchheimer-Brinkman system with L ∞ coefficients in L 2 -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds of dimension m ∈ { 2 , 3 } .

Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63] ), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by e ≥ 0 . Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with e = 0 ) plus the inner layer as e → 0 , where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63] , and open a new window in the future theoretical study of chemotaxis models.

Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8] . These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x → ∞ are established.

The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G 1 , G 2 for every morphism G 1 L → L G 2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula. The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”. In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for SL 2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to GL 2 ). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for GL 2 , and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.

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.

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.

From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization, and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings.

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.

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.

Focusing on the unpredictability of person-to-person contacts and the complexity of random variations in nature, this paper will formulate a stochastic SIR epidemic model with nonlinear incidence rate and general stochastic noises. First, we derive a stochastic critical value R 0 S related to the basic reproduction number R 0 . Via our new method in constructing suitable Lyapunov function types, we obtain the existence and uniqueness of an ergodic stationary distribution of the stochastic system if R 0 S > 1 . Next, via solving the corresponding Fokker-Planck equation, it is theoretically proved that the stochastic model has a log-normal probability density function when another critical value R 0 H > 1 . Then the exact expression of the density function is obtained. Moreover, we establish the sufficient condition R 0 C 1 for disease extinction. Finally, several numerical simulations are provided to verify our analytical results. By comparison with other existing results, our developed theories and methods will be highlighted to end this paper.

Semi-Markov processes established a rich framework for many real-world problems. Semi-Markov processes include Markov processes, Markov chains, renewal processes, Markov renewal processes, Poisson processes, birth and death processes, and etc. Since semi-Markov processes describe more complex mathematical models of various objects, in many cases the obtained mathematical models are investigated numerically. But in some cases it is possible to study such mathematical models using analytical methods. In the presented paper, the authors chose to do the second way. In this paper we study the semi-Markov random walk processes with negative drift and positive jumps. The random variable or the moment, at which the process for the time reaches in zero level is introduced. An integral equation for the Laplace transform of conditional distribution of this random variable is obtained. In this paper length of jump is given by the gamma distribution with parameters α and β resulting in a fractional order integral equation. In the class of gamma distributions, the resulting general integral equation of convolution type is reduced to a fractional order differential equation with constant coefficients. And also, the exact solution of the resulting fractional differential equation with constant coefficients has been found. Finally, using form of Laplace transform the expectation and variance of the random variable are found.

A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms., Comment: v3: Final version; to appear in Advances in Mathematics

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry., Comment: 25 pages

The present work is devoted to giving new insights into the extended Malkus-Robbins (EMR) dynamo. Firstly, based on differential geometry method, i.e. Kosambi-Cartan-Chern (KCC) theory, the paper investigates the Jacobi stability of the equilibrium and periodic orbit by the eigenvalues of the deviation curvature tensor. The deviation vector is applied to analyze the trajectory behaviors near the equilibrium and periodic orbit. Secondly, the zero-zero-Hopf bifurcation is investigated. The paper obtains the conditions that two periodic solutions appear at the bifurcation point and discusses their stability. Finally, on the global dynamics, the ultimate bound sets of the system are estimated. Numerical simulations are given to verify and visualize the corresponding theoretical results.

Completely multiplicative functions whose sum is zero ($CMO$). The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors. The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem~3). The same idea applies also in a more general context (theorem~4). After general statements of that sort, including examples of $CMO$ (theorem~5), the paper is devoted to ``small'' functions, that is, functions of the form $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem~8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ and $f(2)\not=-2$ (theorem~7). A third result gives only a sufficient condition (theorem~9). The three results apply to the particular case $f(p)=-1$, the historical example of Euler. Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Hal\'asz, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Ska\l ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems~10~and~11).

Randomness is a common phenomenon in nonlinear systems. And conditions to reach synchronization are more complex and difficult when chaotic systems have random parameters. So in this paper, an adaptive scheme for synchronization of chaotic system with random parameters by using the fuzzy impulsive method and combining the properties of Wiener process and Ito differential is investigated. The main concepts of this paper are applying fuzzy method to approximate the nonlinear part of system, then using Ito differential to study the Wiener process of random parameters of chaotic system, and realizing synchronization under fuzzy impulsive method. The stability is analyzed by Lyapunov stability theorem. At the end of the paper, numerical simulation is presented to illustrate the effectiveness of the results obtained in this paper.

A periodic predator-prey model has been introduced in [5] to study the effect of water level on persistence or extinction of fish populations living in an artificial lake. By using the continuation theorem of Mawhin’s coincidence degree theory, the authors give sufficient conditions for the existence of at least one positive periodic solution. In this paper we study the problem in the general case. We begin by analyzing the invariance, permanence, non-persistence and the globally asymptotic stability for the system. Most interestingly, under additional conditions, we find that the periodic solution obtained in [5] is unique. Finally, in order to make the model system more realistic, we consider the special case when the periodicity in [5] is replaced by almost periodicity. We obtain conditions for existence, uniqueness and stability of a positive almost periodic solution. The methods used in this paper will be comparison theorems and Lyapunov functions. An example is employed to illustrate our result.

This paper presents a new chaotic system that exhibits a two wing (2W) chaotic attractor in its integer order dynamics, three-wing (3W) and four-wing (4W) chaotic attractors in its fractional-order (FO) dynamics, and an eight-wing (8W) attractor in its nonautonomous fractional dynamics. An interesting feature of the proposed system is that two distinct periodic orbits coexist with a strange attractor that gradually evolves into a 4W attractor. The asymmetry, dissimilarity and topological structure of this proposed system with respect to those available in literature, manifest increased irregularity, which in turn indicate more chaos. Besides, the authors have drawn its comparison with various well-known fractional-order chaotic systems (FOCS)s to prove its enhanced features in terms of higher Lyapunov Exponent, fractional order orbital velocities, bandwidth, density, range of dynamical behaviour, etc. A control scheme is proposed to enable switching hybrid synchronisation between the 8W nonautonomous FOCS and the 4W autonomous FOCS, using the former as master and the latter as slave. This work throws light on the potential practical applicability of the proposed system by designing a circuit using minimum circuit components possible, thus signifying the objectives of the paper are finally achieved.

Motivated by the paper Gerard-Varet and Dormy (2010) [6] [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in [6] to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.

This paper is a continue work of [4,5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term - μ ( 1 + t ) λ ρ u , where λ≥0 and μ > 0 are constants. We have showed that, for all λ≥0 and μ>0, the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initial-boundary value problem in the half space ℝ d + with space dimension d = 2,3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤ λ

Yilmaz et al. (Fluct. Noise Lett. 17, 1830002, 2018) investigated the stochastic phenomenological bifurcations of a generalized Chua circuit driven by Skew-Gaussian distributed noise. They proved it is possible to decrease the number of scrolls by properly choosing the stochastic excitation, manipulating the skewness and noise intensity parameters. Based on the latter, this paper proposes an extension of skew-gaussian noise based on the family of Scale Mixtures of Skew-normal (SMSN) distributions, which includes the skew- t , the skew-gaussian, and the gaussian noises as particular cases. The Lorenz, Generalized Lorenz, Proto–Lorenzand Rossler attractors driven by skew- t distributed noise are considered. Results show that the chaotic regime’s behavior is influenced by the freedom parameter degrees of skew- t noise, increasing the noise variance. This paper concludes that noise intensity increases by rescaling the skew- t distribution at zero mean, rather than by increasing the asymmetry parameter.

In a published paper of Journal of Chaos, Solitons and Fractals, some miss prints were found. One is about the calculation of the solution and also about the expression of the solution using Green’s function of the fractional differential equation studied in this paper; The others are for the properties for Green’s function, these miss prints affected the deriving of the main results in Khan’s paper, some corrections for Khan’s paper should be needed. In this paper, we make some corrections and give the correct proof proceeding for the results, a new example is given to validate part of the proven results.

We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an e -approximation for the d -variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on e − 1 . The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on e − 1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential ( s , t ) -weak tractability with max ( s , t ) ≥ 1 . For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential ( s , t ) -weak tractability with max ( s , t ) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).

Reduction theory has played a major role in the study of Hamiltonian systems. Whilst the Hamilton–Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators relies on approximations of a complete solution of the Hamilton–Jacobi equation. The natural question that we address in this paper is how these two topics (reduction and Hamilton–Jacobi theory) fit together. We obtain a reduction and reconstruction procedure for the Hamilton–Jacobi equation with symmetries, even in a generalized sense to be clarified below. Several applications and relations to other reduction of the Hamilton–Jacobi theory are shown in the last section of the paper. It is remarkable that as by-product we obtain a generalization of the Ge–Marsden reduction procedure [18] and the results in [17] . Quite surprisingly, the classical ansatze available in the literature to solve the Hamilton–Jacobi equation (see [2] , [19] ) are also particular instances of our framework.

In this paper, we construct a new two-dimensional convergent scheme to solve Cauchy problem of following two-dimensional scalar conservation law { ∂ t u + ∂ x f ( u ) + ∂ y g ( u ) = 0 , u ( x , y , 0 ) = u 0 ( x , y ) . In which initial data can be unbounded. Although the existence and uniqueness of the weak entropy solution are obtained, little is known about how to investigate two-dimensional or higher dimensional conservation law by the schemes based on wave interaction of 2D Riemann solutions and their estimation. So we construct such scheme in our paper and get some new results.

The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong – each is vitiated by divergent series. We concentrate on only one of the two incorrect “identities,” which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings. The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ ( 1 2 ) , and in 1925, S. Wigert obtained a generalization giving a formula for ζ ( 1 k ) for any even integer k ≥ 2 . We extend the work of Ramanujan and Wigert in this paper. The Voronoi summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function σ s ( n ) . The modified Bessel functions K s ( x ) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σ s ( n ) and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.

In the first part of the paper we introduce some geometric tools needed to describe slow–fast Hamiltonian systems on smooth manifolds. We start with a smooth bundle p : M → B where ( M , ω ) is a C ∞ -smooth presymplectic manifold with a closed constant rank 2-form ω and ( B , λ ) is a smooth symplectic manifold. The 2-form ω is supposed to be compatible with the structure of the bundle, that is the bundle fibers are symplectic manifolds with respect to the 2-form ω and the distribution on M generated by kernels of ω is transverse to the tangent spaces of the leaves and the dimensions of the kernels and of the leaves are supplementary. This allows one to define a symplectic structure Ω e = ω + e − 1 p ∗ λ on M for any positive small e , where p ∗ λ is the lift of the 2-form λ to M . Given a smooth Hamiltonian H on M one gets a slow–fast Hamiltonian system with respect to Ω e . We define a slow manifold S M for this system. Assuming S M is a smooth submanifold, we define a slow Hamiltonian flow on S M . The second part of the paper deals with singularities of the restriction of p to S M . We show that if dim M = 4 , dim B = 2 and Hamilton function H is generic, then the behavior of the system near a singularity of fold type is described, to the main order, by the equation Painleve-I, and if this singularity is a cusp, then the related equation is Painleve-II.

This review paper aims at answering a basic question on the sign of Lyapunov exponents. A few recent papers reported hyperchaotic system having the sign of Lyapunov exponents as (+, +, −, −). Such (+, +, −, −) sign of Lyapunov exponents is in contradiction with the well known (+, +, 0, −) sign of Lyapunov exponents for a 4-D hyperchaotic system. This paper thus discusses various issues related to Lyapunov exponents and proves that the reported sign of (+, +, −, −) Lyapunov exponents is actually (+, +, 0, −) or (+, 0, −, −). This clarification is very important and essential since Lyapunov exponents are the only quantitative measure for the existence of hyperchaos. Three different algorithms are used for calculating the Laypunov exponents to prove the actual sign of Lyapunov exponents for a hyperchaotic system.

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincar\'e metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the punctured unit disc endowed with the standard Poincar\'e metric. As a consequence, we obtain an optimal uniform estimate of the supremum norm of the Bergman kernel, involving a fractional growth order of the tensor power., Comment: 42 pages, 2 figures; v.2 is a final update to agree with the published paper

The paper explores an eco-epidemiological model with weak Allee in predator, and the disease in the prey population. We consider a predator-prey model with type II functional response. The curiosity of this paper is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We perform the local and global stability analysis of the equilibrium points and the Hopf bifurcation analysis around the endemic equilibrium point. Further we pay attention to the chaotic dynamics which is produced by disease. Our numerical simulations reveal that the three species eco-epidemiological system without weak-Allee induced chaos from stable focus for increasing the force of infection, whereas in the presence of the weak-Allee effect, it exhibits stable solution. We conclude that chaotic dynamics can be controlled by the Allee parameter as well as the competition coefficients. We apply basic tools of non-linear dynamics such as Poincare section and maximum Lyapunov exponent to identify chaotic behavior of the system.

The paper addresses the mathematical analysis of the fishery model in the context of the fractional derivative operator. We use the Caputo–Fabrizio derivative in the investigations. We first prove the fishery model is biologically well definite by proposing the existence and the uniqueness of its solution. The main objective of this paper is to study the dynamics of the predator and the prey in the fishery model when the fractional-order derivative is used. Notably, we analyze the impact of the fractional-order derivative on the dynamics of the fishery model explicitly. To answer this issue, we introduce a new numerical scheme based on the discretization of the fractional integral associated with the Caputo–Fabrizio derivative. The numerical simulations of the solutions of the fractional model are intended to illustrate the numerical scheme presented in our paper. We finish by analyzing the local and global asymptotic stability of the equilibrium points using the Jacobian matrix and the Lyapunov direct method. The Lyapunov function is constructed using standard construction. We notice here that the solutions of the fractional fishery model with different values of the orders describes a cycle when we depict them in three dimensional spaces in time, and furthermore the marine reserves ensure the sustainability of fractional system.

In this paper, we consider a time fractional-order derivative for a diffusive mussel–algae model. The existence of pattern formation was the subject of interest of many previous research works in the case of the diffusive mussel–algae model. Examples include the Turing instability, Hopf bifurcation, Turing-Hopf bifurcation, and others. The presence of the time–fractional–order derivative never been investigated in this model. Next to it ecological relevant, it can generate some important patterns. One of these patterns is produced by the presence of the Turing-Hopf bifurcation. Therefore, our main interest is to analyze the effect of the time fractional–order derivative on the spatiotemporal behavior of the solution, which never been achieved for the mussel-algae model. Besides, Turing–Hopf was studied exclusively on the classical reaction-diffusion systems, where it was also considered for the diffusive mussel-algae model. Thus, our paper puts the fist steps on proving the existence of this type of codimension bifurcation on the diffusive systems with time fractional–order–derivative systems. Further, a suitable numerical simulations are used for confirming the theoretical obtained results.

This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because of a singular integral in this expression. Therefore approximation methods are usually applied to calculate the spectral factor. It is shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor for all densities in this set. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. The paper will also show that the same results hold for sampling-based algorithms for the calculation of the outer function in the theory of Hardy spaces.

Though much progress has been made with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W 1 , n ( R n ) and higher order Adams inequalities on finite domain Ω ⊂ R n , whether there exists an extremal function for the critical higher order Adams inequalities on the entire space R n still remains open. The current paper represents the first attempt in this direction by considering the critical second order Adams inequality in the entire space R 4 . The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Polya-Szego type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [31] ), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R 4 of the form S ( α ) = sup ‖ u ‖ H 2 = 1 ⁡ ∫ R 4 ( exp ⁡ ( 32 π 2 | u | 2 ) − 1 − α | u | 2 ) d x , where α ∈ ( − ∞ , 32 π 2 ) . We establish the existence of the threshold α ⁎ , where α ⁎ ≥ ( 32 π 2 ) 2 B 2 2 and B 2 ≥ 1 24 π 2 , such that S ( α ) is attained if 32 π 2 − α α ⁎ , and is not attained if 32 π 2 − α > α ⁎ . This phenomenon has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R 2 . Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.

In this paper, we investigate the dynamics and the global exponential stability of a new class of Hopfield neural network with time-varying and distributed delays. In fact, the properties of norms and the contraction principle are adjusted to ensure the existence as well as the uniqueness of the pseudo almost periodic solution, which is also its derivative pseudo almost periodic. This results are without resorting to the theory of exponential dichotomy. Furthermore, by employing the suitable Lyapunov function, some delay-independent sufficient conditions are derived for exponential convergence. The main originality lies in the fact that spaces considered in this paper generalize the notion of periodicity and almost periodicity. Lastly, two examples are given to demonstrate the validity of the proposed theoretical results.

We consider models given by Hamiltonians of the form H ( I , φ , p , q , t ; e ) = h ( I ) + ∑ j = 1 n ± ( 1 2 p j 2 + V j ( q j ) ) + e Q ( I , φ , p , q , t ; e ) where I ∈ I ⊂ R d , φ ∈ T d , p , q ∈ R n , t ∈ T 1 . These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in [28] , [29] , [43] and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0 e ≪ 1 , under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O ( 1 ) . This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of [28] , [29] . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I ∈ I ⊂ R d . We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from [28] , [29] . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31] —notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [28] , [29] .