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Recently, Atangana and Baleanu have defined a new fractional derivative which has a nonlocal and non-singular kernel. It is called the Atangana–Baleanu derivative. In this paper we present a numerical technique to obtain solution of fractional differential equations containing Atangana–Baleanu derivative. For this purpose, we use the operational matrices based on Genocchi polynomials together with the collocation points which help us to reduce the problem to a system of algebraic equations. An error bound for the error of the operational matrix of the fractional derivative is introduced. Finally, some examples are given to illustrate the applicability and efficiency of the proposed method.

Given an exact symplectic manifold M and a support Lagrangian Λ ⊂ M , we construct an ∞-category Lag Λ ( M ) which we conjecture to be equivalent (after specialization of the coefficients) to the partially wrapped Fukaya category of M relative to Λ. Roughly speaking, the objects of Lag Λ ( M ) are Lagrangian branes inside of M × T ⁎ R n , for large n, and the morphisms are Lagrangian cobordisms that are non-characteristic with respect to Λ. The main theorem of this paper is that Lag Λ ( M ) is a stable ∞-category, and in particular its homotopy category is triangulated, with mapping cones given by an elementary construction. The shift functor is equivalent to the familiar shift of grading for Lagrangian branes.

In this paper, numerical solution of the mathematical model describing the deathly disease in pregnant women with fractional order is investigated with the help of q-homotopy analysis transform method (q-HATM). This sophisticated and important model is consisted of a system of four equations, which illustrate a deathly disease spreading pregnant women called Lassa hemorrhagic fever disease. The fixed point theorem is considered so as to demonstrate the existence and uniqueness of the obtained numerical solution for the governing fractional model. The proposed method is also included the Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. In order to illustrate and validate the efficiency of the future technique, the projected model in the sense of fractional order is also considered. Moreover, the physical behaviors of the obtained numerical results are presented in terms of simulations for diverse fractional order.

The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is defined. When the function being approximated is known at only finitely many points, the approximation is constructed as a projection based on this discrete set of points. In this paper we address the issue of estimating the absolute error in the approximation. The error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle, and these polynomials are then evaluated asymptotically using Riemann--Hilbert methods., 48 pages, 3 figures. v.2 corrects typos and adds Corollary 1.5 for clarity

An old and well-known conjecture of Frankl and Furedi states that the Lagrangian of an r-uniform hypergraph with m edges is maximised by an initial segment of colex. In this paper we disprove this conjecture by finding an infinite family of counterexamples for all r ≥ 4 . We also show that, for sufficiently large t ∈ N , the conjecture is true in the range ( t r ) ≤ m ≤ ( t + 1 r ) − ( t − 1 r − 2 ) .

The paper deals with linear problems defined on γ -weighted Hilbert spaces of functions with infinitely many variables. The spaces are endowed with zero-mean Gaussian measures which allows to define and study e -truncation and e -superposition dimensions in the average case and probabilistic settings. Roughly speaking, these e -dimensions quantify the smallest number k = k ( e ) of variables that allow to approximate the ∞ -variate functions by special ones that depend on at most k -variables with the average error bounded by e . In the probabilistic setting, given δ ∈ ( 0 , 1 ) , we want the error ≤ e with probability ≥ 1 − δ . We show that the e -dimensions are surprisingly small which, for anchored spaces, leads to very efficient algorithms, including the Multivariate Decomposition Methods.

The Routh–Hurwitz stability criterion is a useful tool for investigating the stability property of linear and nonlinear dynamical systems by analyzing the coefficients of the corresponding characteristic polynomial without calculating the eigenvalues of its Jacobian matrix. Recently some of these conditions have been generalized to fractional systems of order α ∈ [0, 1). In this paper we extend these results to fractional systems of order α ∈ [0, 2). Stability diagram and phase portraits classification in the (τ, Δ)-plane for planer fractional-order system are reported. Finally some numerical examples from population dynamics are employed to illustrate our theoretical results.

This paper generalizes the existing minimal mathematical model of a given generic touristic site by including a distributed time-delay to reflect the whole past history of the number of tourists in their influence on the environment and capital flow. A stability and bifurcation analysis is carried out on the coexisting equilibria of the model, with special emphasis on the positive equilibrium. Considering general delay kernels and choosing the average time-delay as bifurcation parameter, a Hopf bifurcation analysis is undertaken in the neighborhood of the positive equilibrium. This leads to the theoretical characterization of the critical values of the average time delay which are responsible for the occurrence of oscillatory behavior in the system. Extensive numerical simulations are also presented, where the influence of the investment rate and competition parameter on the qualitative behavior of the system in a neighborhood of the positive equilibrium is also discussed.

In this paper, we study an epidemic model describing two strains with non-monotone incidence rates. The model consists of six ordinary differential equations illustrating the interaction between the susceptible, the exposed, the infected and the removed individuals. The system of equations has four equilibrium points, disease-free equilibrium, endemic equilibrium with respect to strain 1, endemic equilibrium with respect to strain 2, and the last endemic equilibrium with respect to both strains. The global stability analysis of the equilibrium points was carried out through the use of suitable Lyapunov functions. Two basic reproduction numbers R 0 1 and R 0 2 are found; we have shown that if both are less than one, the disease dies out. It was established that the global stability of each endemic equilibrium depends on both basic reproduction numbers and also on the strain inhibitory effect reproduction number(s) Rm and/or Rk. It was also shown that any strain with highest basic reproduction number will automatically dominate the other strain. Numerical simulations were carried out to support the analytic results and to show the effect of different problem parameters on the infection spread.

Given a closed set E of Lebesgue measure zero on the unit circle T there is a continuous function f on T such that for every continuous function g on E there is a subsequence of partial Fourier sums S n + ( f , ζ ) = ∑ k = 0 n f ˆ ( k ) ζ k of f , which converges to g uniformly on E . This result completes one result in a recent paper by C. Papachristodoulos and M. Papadimitrakis (2019), see Papachristodoulos and Papadimitrakis (2019). They proved that for a classical one third Cantor set C there is no universal function in the disk algebra. They also proved that for a symmetric Cantor set C ∗ on T there is no universal continuous function for the classical symmetric Fourier sums. See also [2] .

In this paper, we prove the linear inviscid damping and voticity depletion phenomena for the linearized Euler equations around the Kolmogorov flow. These results confirm Bouchet and Morita's predictions based on numerical analysis. By using the wave operator method introduced by Li, Wei and Zhang, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called Kolmogorov flow. The same dissipation rate is proved for the Navier-Stokes equations if the initial velocity is included in a basin of attraction of the Kolmogorov flow with the size of $\nu^{\frac 23+}$, here $\nu$ is the viscosity coefficient., Comment: 92 pages

In this paper, we construct the global solutions near a local Maxwellian to the Vlasov-Poisson-Landau system with slab symmetry for the physical Coulomb interaction. The fluid quantities of this local Maxwellian are the approximate rarefaction wave solutions to the associated one-dimensional compressible Euler equations. We prove for the first time that for the Cauchy problem on this system, such a local Maxwellian is time asymptotically stable under suitably small smooth perturbations and global solutions tend in large time to the rarefaction waves of the compressible Euler system with the corresponding Riemann data. As a byproduct, the nonlinear stability of the same rarefaction waves for the pure Landau equation is also proved. This illustrates in our setting that the electric field does not affect the propagation of rarefaction waves to the Landau equation.

In this paper, we derive extensions of classical Lyapunov and Chetaev instability theorems and LaSalle’s invariance principle to the case of Atangana–Baleanu derivative in the Caputo sence. Moreover, we get some results to estimates fractional derivatives of quadratic and Volterra–type Lyapunov functions when γ ∈ (0, 1). Finally, we present rigorous proofs about the complete classification for global dynamics of an HIV/AIDS epidemic model with Atangana–Baleanu Caputo derivative (Chaos, Solitons and Fractals 126 (2019) 41–49).

Time-delays and fractional-order play a vital role in modeling biological systems with memory. In this paper, we propose a novel delay differential model with fractional-order for tumor immune system with external treatments. Non-negativity of the solution of such model has been investigated. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to two differenttumor time-delays τ1 and τ2. The occurrence of Hopf bifurcation is captured when any of the time-delay passes through critical value τ 1 * , or τ 2 * . Theoretical results are validated numerically by solving the governing system, using the modified Adams–Bashforth–Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model.

In this paper, we consider a new class of non-instantaneous impulsive partial stochastic differential systems with fractional sectorial operators in separable Hilbert spaces. Using the fractional calculus, the measure of noncompactness, properties of sectorial operators and fixed point theorems, we establish the optimal controllability to the these control systems without assuming the operators is compact. Finally, an example is provided to illustrate the obtained theory.

This paper introduces a novel version of variable-order (VO) space-time mobile-immobile advection-dispersion equation involving derivatives with non-singular kernels. An optimization scheme is proposed for solving this new class of VO fractional problems.The presented method is based on the hybrid of the generalized Lucas polynomials together with their operational matrices of VO fractional derivatives (which are obtained for the first time in the presented study), the collocation technique and the Lagrange multipliers scheme. The presented method transforms obtaining the solution of such problems into obtaining the solution of systems of algebraic equations. Two numerical examples are provided to show the validity and accuracy of the presented approach.

Infectious diseases have been ranked in the top ten causes of death by WHO in 2016 and despite of availability of various types of vaccines and antibiotics, a huge population is still dying by infectious diseases every day. This may happen due to, several reasons like, resistance of pathogens to antibiotics, improper hygiene and various types of difficulties in vaccination. It has also inspired mathematical modelers to develop dynamical systems predicting the infections in long run. During their spread in a particular population, infectious diseases show various kind of delays which essentially affects the dynamics. In this paper, a susceptible - vaccinated- exposed - infectious - removed (SVEIR) epidemic model is developed with vaccination and two discrete time delays. The first time delay has been incorporated for the time period used to cure the infectious population and another time delay denote the temporary immunity period. The existence of solution and its boundedness have been established. The local stability of disease free equilibrium in respect of both the delays have been discussed explicitly and we have found threshold values of both delay parameters for the local stability of disease free equilibrium. We have also established the local stability of interior equilibrium following the existence of Hopf-bifurcation. Theoretical result shows that considered model system undergoes a Hopf-bifurcation around the interior equilibrium when the time delay due to time period used to cure the infectious population crosses a threshold value. We have also discussed the direction and stability of delay induced Hopf bifurcation using normal form theory and centre manifold theorem. In presence of delay, by constructing a Lyapunov function, local asymptotic stability of the positive equilibrium point is discussed. The length of delay has been estimated to preserve the stability using Nyquist criterion. With the suitable choices of the parameters, some numerical simulations have been presented in the support of our analytical results.

This paper explores the idea of Bertrand duopoly in a context of bounded rationality, where two content providers (CP) endeavor to maximize their own profit. Each CP varies his price and fixes the level of his credibility of content. We suppose that one CP adjusts his price strategy according to a kind of smoothed information which average his marginal profit in the previous period and the one in a delayed period, while another CP makes its price strategy as the best response, which is a function the opponent action in the current period. Through a detailed analysis, we calculate explicitly the equilibrium points of the dynamical system, and we show the stability of its Nash equilibrium under some conditions. Numerical results reveal that while varying the model parameters (speed of adjustment and side payment), complex dynamic behaviors would occur, such as chaos. In addition, an appropriate method of chaos controlling is applied to force the system to go back to its stabilization behavior.

Viruses have different mechanisms in causing a disease in an organism, which largely depend on the viral species. The recent advancement, through coupling data analysis and mathematical modeling, has allowed the identification and characterization of the nature of the virus. In the present paper, the homotopy analysis method is applied to provide an approximate solution of the basic HIV viral dynamic model describing the viral dynamics in a susceptible population. The proposed method allows for the solution of the governing system of differential equations to be calculated in the form of an infinite series with components which can be easily calculated. The homotopy analysis method utilizes a simple method to adjust and control the convergence region of the infinite series solution by using an auxiliary parameter. By using the homotopy series solutions, firstly, several β − curves using an appropriate ratio are plotted to demonstrate the regions of convergence and the optimum value of ℏ, then the residual and absolute errors are obtained for different values of these regions. Secondly, the residual error functions are applied to show the accuracy of the applied homotopy analysis method. Also, the convergence theorem of homotopy analysis method for the HIV viral dynamic model is proved. Mathematica software is used for the calculations and numerical results. The results obtained show the effectiveness and strength of the homotopy analysis method.

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: $L y + \lambda_n D y = 0$, where $L,D$ are linear differential operators with polynomial coefficients not depending on $n$. For positive integer values of the parameters $r,c$ these polynomials are Sobolev orthogonal polynomials with some explicitly given measures. Some basic properties of these polynomials, including recurrence relations, are obtained., Comment: 18 pages

This paper investigates the global stability of discrete-time pathogen infection model with pathogen-to-cell and cell-to-cell transmissions and humoral immunity. We consider both latently and actively infected cells. The model incorporates three types of intracellular time delays. We use nonstandard finite difference method to discretize the continuous-time model. We establish by using Lyapunov method, the global stability of equilibria in terms of the basic reproduction number R 0 and the humoral immune response activation number R 1 . We have proven that if R 0 ≤ 1 , then the pathogen-free equilibrium Q0 is globally asymptotically stable, if R 1 ≤ 1 R 0 , then the persistent pathogen equilibrium without immune response Q* is globally asymptotically stable, and if R 1 > 1 , then the persistent pathogen equilibrium with immune response Q ¯ is globally asymptotically stable. We illustrate our theoretical results by using numerical simulations. The effect of time delay on the pathogen dynamics is also studied. We have shown that the time delay has similar effect as the drug therapy. This gives some impression to develop new class of treatment to increase the delay period and then suppress the pathogen replication.

This paper formulates and analyzes a virus dynamics model with impairment of B-cell functions. The model includes two modes of viral transmission, cell-free and cell-to-cell. The cell-free and cell-cell incidence rates are modeled by general nonlinear functions. We integrate both latently and actively infected cells as well as three time delays into the model. Nonnegativity and boundedness properties of the solutions are proven to show the well-posedness of the model. The model admits two equilibria which are determined by the basic reproduction number R 0 G . The global stability of each equilibrium is proven by utilizing Lyapunov function and applying LaSalle’s invariance principle. The theoretical results are illustrated by numerical simulations. The effect of impairment of B-cell functions on the virus dynamics is studied. We have shown that if the functions of B-cell are impaired, then the concentration of viruses is increased in the plasma.

This paper introduces a novel class of nonlinear optimal control problems generated by dynamical systems involved with variable-order fractional derivatives in the Atangana–Baleanu–Caputo sense. A computational method based on the Chebyshev cardinal functions and their operational matrix of variable-order fractional derivative (which is generated for the first time in the present study) is proposed for the numerical solution of this class of problems. The presented method is based on transformation of the main problem to solving system of nonlinear algebraic equations. To do this, the state and control variables are expanded in terms of the Chebyshev cardinal functions with unknown coefficients, then the cardinal property of these basis functions together with their operational matrix are employed to generate a constrained extremum problem, which is solved by the Lagrange multipliers method. The applicability and accuracy of the established method are investigated through some numerical examples. The reported results confirm that the established scheme is highly accurate in providing acceptable results.

This paper involves complex valued versions of Riemann-Liouville integral, Atangana-Baleanu integral operator and non-linear Telegraph equation. Under various suitable assumptions the results are established in the setting of complex valued double controlled metric space. Thereafter, by making consequent use of the fixed point method, short and simple proofs are obtained for solutions of Riemann-Liouville integral, complex valued Atangana-Baleanu integral operator and non-linear Telegraph equation.

In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian symmetric space of compact type into its Cartesian products.

In this presented paper, we investigate the novel numerical scheme for the non-linear reaction-diffusion equation and non-linear integro reaction-diffusion equation equipped with Atangana Baleanu derivative in Caputo sense (ABC). A difference scheme with the help of Taylor series is applied to deal with fractional differential term in the time direction of differential equation. We applied a numerical method based on quasi wavelet for discretization of unknown function and their spatial derivatives. A formulation to deal with Dirichlet boundary condition is also included. To demonstrate the effectiveness and validity of our proposed method some numerical examples are also presented. We compare our obtained numerical results with the analytical results and we conclude that quasi wavelet method achieve accurate results and this method has a distinctive local property. On the other hand the method is easy to apply on higher order fractional partial differential equation and integro differential equation.

We study EC- ( s , t ) -weak tractability of multivariate linear problems in the average case setting. This paper extends earlier work in the worst case setting. The parameters s ≥ 0 and t ≥ 0 allow us to study the information complexity n ( e , d ) of a d -variate problem with respect to different powers of ln e − 1 , corresponding to the bits of accuracy, and d . We consider the absolute and normalized error criteria. In particular, a multivariate problem is EC- ( s , t ) -weakly tractable iff lim d + e − 1 → ∞ ln n ( e , d ) ∕ [ d t + ln s e − 1 ] = 0 . We deal with general linear problems and linear tensor product problems. We show necessary and sufficient conditions for EC- ( s , t ) -weak tractability. In the case of general linear problem these conditions are matching. For linear tensor product problems, we also show matching conditions with the exception of some cases where s > 1 , in general.

In this paper, we introduce a parametric type of Bernoulli polynomials and study their basic properties in order to establish an extension of Euler-Maclaurin quadrature rules and compare them with the well-known ordinary case.

Minkowski's Theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body, and, moreover, the surface area measure determines a convex body uniquely. In this manuscript we prove an extension of Minkowski's theorem. Consider a measure $\mu$ on $\mathbb{R}^n$ with positive degree of concavity and positive degree of homogeneity. We show that a surface area measure of a convex set $K$, weighted with respect to $\mu$, determines a convex body uniquely up to $\mu$-measure zero. We also establish an existence result under natural conditions including symmetry. We apply this result to extend the solution to classical Shephard's problem, which asks the following: if one convex body in $\mathbb{R}^n$ has larger projections than another convex body in every direction, does it mean that the volume of the first convex body is also greater? The answer to this question is affirmative when $n\leq 2$ and negative when $n\geq 3.$ In this paper we introduce a new notion which relates projections of convex bodies to a given measure $\mu$, and is a direct generalization of the Lebesgue area of a projection. Using this notion we state a generalization of the Shephard problem to measures and prove that the answer is affirmative for $n\leq 2$ and negative for $n\geq 3$ for measures which have a positive degree of homogeneity and a positive degree of concavity. We also prove stability and separation results, and establish useful corollaries. Finally, we describe two types of uniqueness results which follow from the extension of Minkowski's theorem.

In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism L with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to L. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism f 0 over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation f of f 0 with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.

Let K 0 ( V k ) be the Grothendieck group of k-varieties. Campbell and Zakharevich have constructed a higher algebraic K-theory spectrum K ( V k ) such that π 0 K ( V k ) = K 0 ( V k ) . In this paper we construct non-trivial classes in the higher homotopy groups of K ( V k ) when k is finite or a subfield of C. To do this we give a recipe for lifting motivic measures K 0 ( V k ) → K 0 ( E ) to maps of spectra K ( V k ) → K ( E ) . We consider two special cases: the classical local zeta function, thought of as a homomorphism K 0 ( V F q ) → K 0 ( End ( Q l ) ) , and the compactly-supported Euler characteristic, thought of as a homomorphism K 0 ( V C ) → K 0 ( Q ) . We use lifts of these motivic measures to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map S → K ( V k ) is nontrivial in higher dimensions when k is finite or a subfield of C, and, moreover, that when k is finite this map is not surjective on higher homotopy groups.

The main observation of this paper is that some sequential weak compactness arguments in Hilbert space theory can be replaced by Heine/Borel compactness arguments (for the strong topology). Even though the latter form of compactness fails in (infinite-dimensional) Hilbert spaces, it nevertheless trivializes under the so-called bounded functional interpretation. As a consequence, the proof mining program of extracting computational bounds from ordinary proofs of mathematics can be applied to modified proofs which use these false Heine/Borel compactness arguments. Additionally, the bounded functional interpretation provides good logical guidance in formulating quantitative versions of analytical statements. We illustrate these claims with three minings. The bounded functional interpretation is here used for the first time in proof mining.

In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set C? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset F of C containing more than one point, every linear generating IFS of F must consist of similitudes with contraction ratios ± 3 − n , n ∈ N . In particular, a simple criterion is formulated to characterize self-similar subsets of C with equal contraction ratio in modulus.

In this work, we are concerned with the investigation of the qualitative behaviors of certain systems of non-linear differential equations of second order. We make a comparison between applications of the integral test and Lyapunov’s function approach on some recent stability and boundedness results in the literature. An example is furnished to illustrate the hypotheses and main results in this paper.

A rather tricky question is the construction of wavelet bases on domains for suitable function spaces (Sobolev, Besov, Triebel-Lizorkin type). In his monograph from 2008, Triebel presented an approach how to construct wavelet (Riesz) bases in function spaces of Besov and Triebel-Lizorkin type on cellular domains, in particular on the cube. However, he had to exclude essential exceptional values of the smoothness parameter s, for instance the theorems do not cover the Sobolev space W"2^1(Q) on the n-dimensional cube Q for n>=2. Triebel also gave an idea how to deal with those exceptional values for the Triebel-Lizorkin function space scale on the n-dimensional cube Q: he suggested to introduce modified function spaces for the critical values, the so-called reinforced spaces which are subsets of the classical Triebel-Lizorkin function spaces on the cube. In this paper we start examining these reinforced spaces and transfer the crucial decomposition theorems necessary for establishing a wavelet basis from the non-critical values to analogous results for the critical cases now decomposing the reinforced function spaces of Triebel-Lizorkin type.

In this paper we study the mapping properties of some Hardy-type operators with general bounds of integration. In particular, some estimates for Schatten-von Neumann norms of these Hardy-Steklov operators are derived and discussed.

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e., up to a specified combinatorial factor, the monotone Hurwitz number in genus g with ramification specified by a given partition is a polynomial indexed by g in the parts of the partition., Comment: 23 pages

A submetry is a metric analogue of a Riemannian submersion, and an e ϵ -Lipschitz and co-Lipschitz map is a metric analogue of an ϵ -Riemannian submersion. The stability of submetries from Alexandrov spaces to Riemannian manifolds in the Gromov–Hausdorff topology can be viewed as a parametrized version of Perelman’s stability theorem in Alexandrov geometry. In this paper, we will study the stability of e ϵ -Lipschitz and co-Lipschitz maps. Our approach is based on controlled homotopy theory and semi-concave functions on Alexandrov spaces. As applications of our stability results, we generalize fiber bundle finiteness results on Riemannian submersions and partially generalize the stability of submetries.

This paper will discuss, grill topological space which is not only a space for obtaining a new topology but generalized grill space also gives a new topology. This has been discussed with the help of two operators in minimal spaces.

In this paper we derive exact closed-form density functions of the generalized Verhulst process (see Mackevicius (2015), Jakubowski and Wisniewolski (2015)), and the Bessel process with a constant drift (see Coman et al (1998), Linetsky (2004)), which have applications in mathematical biology and queueing theory. We propose a generic probabilistic method for deriving exact closed-form density functions for these two diusion processes based on a novel application of the exponential measure change in Palmowski and Rolski (2002) and Hurd and Kuznetsov (2008), together with formulae in Borodin and Salminen (2015). Our study generalizes several known results in the literature: Proposition 2.1 generalizes results of Theorem 2 and Theorem 3 in Jakubowski and Wisniewolski (2015) to the case of the generalized Verhulst process. Proposition 2.3 provides the exact closed-form density for a Bessel process with a constant drift, and it provides an alternative expression to the spectral expansion given in Proposition 1 of Linetsky (2004).

In this paper we study the asymptotic phase space energy distribution of solution of the Schr\"{o}dinger equation with a time-dependent random potential. The random potential is assumed to be with slowly decaying correlations. We show that the Wigner transform of a solution of the random Schr\"{o}dinger equation converges in probability to the solution of a radiative transfer equation. Moreover, we show that this radiative transfer equation with long-range coupling has a regularizing effect on its solutions. Finally, we give an approximation of this equation in term of a fractional Laplacian. The derivations of these results are based on an asymptotic analysis using perturbed-test-functions, martingale techniques, and probabilistic representations., Comment: 33 pages

The problem of finding the zeros of the sum of two maximally monotone operators is of fundamental importance in optimization and variational analysis. In this paper, we systematically study Attouch–Théra duality for this problem. We provide new results related to Passty’s parallel sum, to Eckstein and Svaiter’s extended solution set, and to Combettes’ fixed point description of the set of primal solutions. Furthermore, paramonotonicity is revealed to be a key property because it allows for the recovery of all primal solutions given just one arbitrary dual solution. As an application, we generalize the best approximation results by Bauschke, Combettes and Luke [H.H. Bauschke, P.L. Combettes, D.R. Luke, A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space, Journal of Approximation Theory 141 (2006) 63–69] from normal cone operators to paramonotone operators. Our results are illustrated through numerous examples.

We study the homogenization of a class of actions with an underlying Lagrangian L defined on the set of absolutely continuous paths in the Wasserstein space P p ( R d ) . We introduce an appropriate topology on this set and obtain the existence of a Γ -limit of the rescaled Lagrangians. Our main goal is to provide a representation formula for these Γ -limits in terms of the effective Lagrangians. This allows us to study not only the “convexity properties” of the effective Lagrangian, but also the differentiability properties of its Legendre transform restricted to constant functions. For the case d > 1 we obtain partial results in terms of an effective Lagrangian defined on L p ( ( 0 , 1 ) d ; R d ) . Our study provides a way of computing the limit of a family of metrics on the Wasserstein space. The results of this paper can also be applied to study the homogenization of variational solutions of the one-dimensional Vlasov–Poisson system, as well as the asymptotic behavior of calibrated curves (Fathi (2003) [6] , Gangbo and Tudorascu (2010) [12] ). Whereas our study for the one-dimensional case covers a large class of Lagrangians, that for the higher dimensional case is concerned with special Lagrangians such as the ones obtained by regularizing the potential energy of the d -dimensional Vlasov–Poisson system.

In this paper, we study the problem of controllability of Schrodinger equation. We prove that the system is exactly controllable in infinite time to any position. The proof is based on an inverse mapping theorem for multivalued functions. We show also that the system is not exactly controllable in finite time in lower Sobolev spaces.

We present sampling theorems for reproducing kernel Banach spaces on Lie groups. Recent approaches to this problem rely on integrability of the kernel and its local oscillations. In this paper, we replace these integrability conditions by requirements on the derivatives of the reproducing kernel and, in particular, oscillation estimates are found using derivatives of the reproducing kernel. This provides a convenient path to sampling results on reproducing kernel Banach spaces. Finally, these results are used to obtain frames and atomic decompositions for Banach spaces of distributions stemming from a cyclic representation. It is shown that this process is particularly easy, when the cyclic vector is a Gårding vector for a square integrable representation.

In this paper, we practiced relatively new, analytical method known as the variational homotopy perturbation method for solving Klein–Gordon and sine-Gordon equations. To present the present method’s effectiveness many examples are given. In this study, we compare numerical results with the exact solutions, the Adomian decomposition method (ADM), the variational iteration method (VIM), homotopy perturbation method (HPM), modified Adomian decomposition method (MADM), and differential transform method (DTM). The results reveal that the VHPM is very effective.

In the paper, we generalize a result of P. Shiu on the number of square-full integers between successive squares. We extend, via a different approach, the result to k -full integers, and we also obtain explicit error terms in the process.

In a previous paper by the first two authors, a tube formula for fractal sprays was obtained which also applies to a certain class of self-similar fractals. The proof of this formula uses distributional techniques and requires fairly strong conditions on the geometry of the tiling (specifically, the inner tube formula for each generator of the fractal spray is required to be polynomial). Now we extend and strengthen the tube formula by removing the conditions on the geometry of the generators, and also by giving a proof which holds pointwise, rather than distributionally. Hence, our results for fractal sprays extend to higher dimensions the pointwise tube formula for (1-dimensional) fractal strings obtained earlier by Lapidus and van Frankenhuijsen. Our pointwise tube formulas are expressed as a sum of the residues of the "tubular zeta function" of the fractal spray in $\mathbb{R}^d$. This sum ranges over the complex dimensions of the spray, that is, over the poles of the geometric zeta function of the underlying fractal string and the integers $0,1,...,d$. The resulting "fractal tube formulas" are applied to the important special case of self-similar tilings, but are also illustrated in other geometrically natural situations. Our tube formulas may also be seen as fractal analogues of the classical Steiner formula., 43 pages, 13 figures. To appear: Advances in Mathematics

In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under reciprocal transformations. The main technical tool is in a suitable generalization of the classical Schouten-Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermanifolds. These constructions are used to the computation of the Lichnerowicz-Jacobi cohomologies of Jacobi structures. We also introduce the notion of bi-Jacobi structures and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure., 59 pages

In this paper we build a link between the Teichmuller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincare uniformization. In the case of a one-sheeted hyperboloid with n orbifold points we show that the Poisson algebra D n of geodesic length functions is the semiclassical limit of the twisted q -Yangian Y q ′ ( o n ) for the orthogonal Lie algebra o n defined by Molev, Ragoucy and Sorba. We give a representation of the braid-group action on D n in terms of an adjoint matrix action. We characterize two types of finite-dimensional Poissonian reductions and give an explicit expression for the generating function of their central elements. Finally, we interpret the algebra D n as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semi-simple point.