Search

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 However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show.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.

The translation into English of Aleksandr Yakovlevich Khinchin's (1894–1959) 1926 paper entitled ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ is made available for the first time. Here, Khinchin presented the famous foundational debate between L.E.J. Brouwer and David Hilbert of the 1920s in terms of a search for a mathematics with content. His main aim seems to have been to make intuitionism ideologically acceptable to his audience at the Communist Academy by means of the claim that insofar as Brouwer's intuitionism had a clear ‘subject matter’ and Hilbert's new program was a concession to intuitionism, the alleged victory of intuitionism not only implied the defeat of ‘empty’ formalism, but also showed the compatibility and affinity of Marxism with the newest developments in modern mathematics. This introduction provides a tentative exploration of the issue of what was tactical (or due to ideological pressure) and what was real scientific interest (or due to ignorance) (or what was both) in Khinchin's 1926 paper in the form of a detailed commentary, especially, on the tactical side of his presentation of the positions of Brouwer and Hilbert.

It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.

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.

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

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.

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.

The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2] . One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on R n , because of the lack of bounds. We give conditions so that the results of [2] can be extended to R n . The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Zitkovic [8] . They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in R n and not on a bounded domain. Xing and Zitkovic developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result.

In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).

In this paper, we study the Cauchy problem for systems of 3-D quasilinear wave equations satisfying the null condition with initial data of low regularity. In the radially symmetric case, we prove the global existence for every small data in H 3 × H 2 with a low weight. To achieve this goal, we will show how to extend the global iteration method first suggested by Li and Chen (1988) [32] to the low regularity case, which is also another purpose of this paper. Finally, we apply our result to 3-D nonlinear elastic waves.

The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ⁡ ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).

In this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space and may open new avenues of research in hypercomplex analysis.

We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2 + 1 Levin–Wen model to obtain 1 + 1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

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.

We investigate a steady planar flow of an ideal fluid in a bounded simply connected domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. In this paper, we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vortex patch problem has a unique solution.

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 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 } .

This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem { u t = u x x + f ( t , u ) , x ∈ R , t > 0 , u ( x , 0 ) = u 0 , x ∈ R , where u 0 is a nonnegative bounded function with compact support and f is a rather general nonlinearity that is periodic in t and satisfies f ( ⋅ , 0 ) = 0 . In the autonomous case where f = f ( u ) , the convergence of every bounded solution to an equilibrium has been established by Du and Matano (2010). However, the presence of periodic forcing makes the problem significantly more difficult, partly because the structure of time periodic solutions is much less understood than that of steady states. In this paper, we first prove that any ω-limit solution is either spatially constant or symmetrically decreasing. Furthermore, we show that the set of ω-limit solutions either consists of a single time-periodic solution or it consists of multiple time-periodic solutions and heteroclinic connections among them. Next, under a mild non-degenerate assumption on the corresponding ODE, we prove that the ω-limit set is a singleton, which implies the solution converges to a time-periodic solution. Lastly, we apply these results to equations with bistable nonlinearity and combustion nonlinearity, and specify more precisely which time-periodic solutions can possibly be selected as the limit.

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.

The aim of this paper is to present some results concerning the problem of minimal projections and extensions. Let X be a reflexive Banach space and let Y be a closed subspace of X of codimension one. Let W be a finite-dimensional Banach space. We present a new sufficient condition under which any minimal extension of an operator A ∈ L ( Y , W ) is strongly unique. In this paper we show (in some circumstances) that if 1 λ ( Y , X ) , then a minimal projection from X onto Y is a strongly unique minimal projection. Moreover, we introduce and study a new geometric property of normed spaces. In this paper we also present a result concerning the strong unicity of best approximation.

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.

Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space R n , including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in [25] , combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.

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.

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.

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .

In this paper we study the so called optimal polynomial meshes for domains in K ⊂ R d , d ≥ 2 . These meshes are discrete point sets Y n of cardinality c n d which have the property that ‖ p ‖ K ≤ A ‖ p ‖ Y n for every polynomial p of degree at most n with a constant A > 1 independent of n . It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C 2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d = 2 .

This is the first in a series of three papers where we study the integral manifolds of the charged three-body problem. The integral manifolds are the fibers of the map of integrals. Their topological type may change at critical values of the map of integrals. Due to the non-compactness of the integral manifolds one has to take into account besides ‘ordinary’ critical points also critical points at infinity. In the present paper we concentrate on ‘ordinary’ critical points and in particular elucidate their connection to central configurations. In a second paper we will study critical points at infinity. The implications for the Hill regions, i.e. the projections of the integral manifolds to configuration space, are the subject of a third paper.

This paper is a follow-up to the recent paper of Pillichshammer and Sonnleitner (2020) [12] . We show that the isotropic discrepancy of a lattice point set is at most d 2 2 ( d + 1 ) times its spectral test, thereby correcting the dependence on the dimension d and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test – and with it the isotropic discrepancy – is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.

The paper concerns operators in ideal topological spaces. Some characterizations of Hayashi–Samuel spaces, Ψ - C sets and semi-open sets of ∗ -topology are investigated. Continuity and decomposition are also part of this paper.

Jleli and Samet (2018) introduced a new metric space and named it as F -space. In this paper we consider the notion of α - ψ -contraction in the setting of F -metric spaces. We present some fixed point and coupled fixed point results in the generalized setting. Moreover, our purpose in this paper is to concerned with the solution of nonlinear neutral differential equation x ′ ( t ) = − a ( t ) x ( t ) + b ( t ) g ( x ( t − r ( t ) ) ) + c ( t ) x ′ ( t − r ( t ) ) with unbounded delay using fixed point theory in F -metric space.

We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors., Comment: 65 pages, 4 figures; a number of typos fixed; the exposition has been significantly revised, fixing a technical error in the non-compact case in the process; this paper is now restricted to the simple normal crossings case; the arbitrary normal crossings case will be detailed in a followup paper

This paper is the second part of our comprehensive study on the structure of the solutions for the following Chern–Simons–Higgs equation: (0.1) { Δ u + 1 e 2 e u ( 1 − e u ) = 4 π ∑ j = 1 N δ p j , in Ω , u is doubly periodic on ∂ Ω , where Ω is a parallelogram in R 2 and e > 0 is a small parameter. In part 1 [29] , we proved the non-coexistence of different bubbles in the bubbling solutions and obtained an existence result for the Chern–Simons type bubbling solutions under some nearly necessary conditions. Mean field type bubbling solutions for (0.1) have been constructed in [27] . In this paper, we shall study two other important issues for the mean field type bubbling solutions: the necessary conditions for the existence and the local uniqueness. The results in this paper lay the foundation to find the exact number of solutions for (0.1) .

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.

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.

In this paper we extend the idea of weighted density of Balcerzak et al. (2015) by using a modulus function and introduce the idea f -density of weight g of subsets of ω ≔ { 0 , 1 , … } (at the same time extending the notion of f -density (Aizpuru et al., 2014)), which we name d g f where g : ω → [ 0 , ∞ ) satisfies g ( n ) → ∞ and n ∕ g ( n ) ↛ 0 and f is a modulus function. The aim of this paper is to show that we can get new ideals Z g ( f ) consisting of sets A ⊂ ω for which d g f ( A ) = 0 different from all the previously constructed ideals Z g of Balcerzak et al. (2015) and moreover they retain all the nice properties of the ideals Z g .

The first part of this paper considers the controllability for a functional semilinear differential inclusion governed by a family of operators { A ( t ) : t ∈ [ 0 , b ] } generating an evolution operator in a Banach space in the presence of noninstantaneous impulse effects. In the second part of this paper we study the controllability for a fractional noninstantaneous impulsive semilinear differential inclusion with delay, where the linear part is an infinitesimal generator of a C 0 − semigroup. Using a weakly convergent criterion in the space of piecewise continuous functions and weak topology theory (for weak sequentially closed graph operators) we establish sufficient conditions to guarantee controllability results. Examples are given to illustrate the abstract results.

Quasi-elliptic cohomology is a variant of elliptic cohomology theories. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. Thus, the constructions on it can be made in a neat way. This theory reflects the geometric nature of the Tate curve. In this paper we provide a systematic introduction of its construction and definition., Comment: Final Version. 26 pages. To appear in Advances in Mathematics. In this paper we generalize the construction in arXiv:1612.00930. The subtle point of this generalization is explained in Section 2

This paper deals with the existence of multiple solutions for the quasilinear equation − div A ( x , ∇ u ) + V ( x ) | u | α ( x ) − 2 u = f ( x , u ) in R N , which involves a general variable exponent elliptic operator in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like | ξ | q ( x ) − 2 ξ for small | ξ | and like | ξ | p ( x ) − 2 ξ for large | ξ | , where 1 α ( ⋅ ) ≤ p ( ⋅ ) q ( ⋅ ) N . Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works A. Azzollini et al. (2014) [4] and N. Chorfi and V. Radulescu (2016) [11] from cases where the exponents p and q are constant, to the case where p ( ⋅ ) and q ( ⋅ ) are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the Cerami compactness condition.

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.

Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of M\"obius decomposition space, a far-reaching generalisation of the notion of M\"obius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of M\"obius intervals, which contains the universal M\"obius function (but is not induced by a M\"obius category), can be realised as the homotopy cardinality of a M\"obius decomposition space $U$ of all M\"obius intervals, and that in a certain sense $U$ is universal for M\"obius decomposition spaces and CULF functors., Comment: 35 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v3: minor expository improvements. Final version to appear in Adv. Math

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.

This is a chiefly expository paper on the subject of the title which, in our view, has not received a detailed treatment in the literature which is commensurate with its importance. We expect the results presented here to be useful in a number of contexts. For example, several of them will be applied in a forthcoming paper by the authors., Comment: 25 pages

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

In this paper, we replace the classical differential operators with the fractal-fractional differential operators corresponding to the power law, exponential decay, and the generalized Mittag-Leffler kernels. These operators have two parameters created: the first is a fractal dimension and the second is a fractional order. The numerical schemes are combination of the Lagrange interpolating polynomial and theory of fractional calculus. In the case of δ = k = 1 the numerical solutions for the proposed models are found to be in an excellent agreement with the finite difference methods. We investigate the effects of the fractal-fractional order on the oscillations in the Fractal-Fractional Brusselator Chemical Reaction (FFBCR). All calculations in this paper were done using the mathematica package.

This paper considers the problem of establishing L p -improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension n of the ambient space and the dimension k of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang [28] and extended by Zorin-Kranich [29] , and on recent results for geometric nonconcentration inequalities [11] . As an initial application of this new approach, this paper establishes sharp restricted strong type L p -improving inequalities for certain model quadratic submanifolds in the range k n ≤ 2 k .

Most recently, some new double sequence spaces B ( M u ) , B ( C ϑ ) where ϑ = { b , b p , r , f , f 0 } and B ( L q ) for 0 q ∞ have been introduced as the domain of four-dimensional generalized difference matrix B ( r , s , t , u ) in the double sequence spaces M u , C ϑ where ϑ = { b , b p , r , f , f 0 } and L q for 0 q ∞ , and some topological properties, dual spaces, some new four-dimensional matrix classes and matrix transformations related to these spaces have also been studied by Tug and Basar and Tug (see [1] , [2] , [3] , [4] ). In this paper, we introduce new strongly almost convergent double sequence spaces B [ C f ] and B [ C f 0 ] whose B ( r , s , t , u ) -transforms are in the spaces [ C f ] and [ C f 0 ] , respectively. The main body of this paper is designed by the investigation of the following hypothesis. Firstly, we examine some topological properties and inclusion relations including the new double sequence spaces B [ C f ] and B [ C f 0 ] . Also, we determine the α-dual, β ( b p ) -dual and γ-dual of the space B [ C f ] . Finally, we give the necessary and sufficient conditions on an infinite matrix transforming from [ C f ] over C f , and we also characterize the classes ( [ C f ] ; M u ) , ( B [ C f ] ; C f ) and ( B [ C f ] ; M u ) .