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The purpose of this paper is to study the local zeta integrals of Friedberg-Jacquet at complex place and to establish similar results to the recent work [4] joint with C. Chen and D. Jiang. In this paper, we will (1) give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation π of GL 2 n ( C ) with a non-zero Shalika model; (2) construct a new twisted linear period Λ s , χ and give a different expression of the linear model for π; (3) give a necessary and sufficient condition on the character χ such that there exists a uniform cohomological test vector v ∈ V π (which we construct explicitly) for Λ s , χ . As a consequence, we obtain the non-vanishing of local Friedberg-Jacquet integral at complex place. All of the above are essential preparations for attacking a global period relation problem in the forthcoming paper ( [11] ).

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbers based on their corresponding rational. The proof relies on a relationship between Markov numbers and continuant polynomials which originates in Frobenius' 1913 paper., 16 pages

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

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

This paper is concerned with the global exponential stability of time periodic traveling fronts of reaction-advection-diffusion equations with time periodic bistable nonlinearity in infinite cylinders. It is well known that such traveling fronts exist and are asymptotically stable. In this paper, we further show that such fronts are globally exponentially stable. The main difficulty is to construct appropriate supersolutions and subsolutions.

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

Let G be a locally compact group, B ( G ) the Fourier–Stieltjes algebra of G and B 0 ( G ) = B ( G ) ∩ C 0 ( G ) . The space B 0 ( G ) is a closed ideal of B ( G ) . In this paper, we study the Banach algebra B 0 ( G ) under various aspects. The main emphasis is on regularity and the existence of various kinds of approximate identities, the question of when the quotient of B 0 ( G ) modulo the Fourier algebra A ( G ) is radical, the Bochner–Schoenberg–Eberlein property and a characterization of elements in B 0 ( G ) in terms of continuity of translation properties. The paper also contains a number of illustrating examples.

This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.

Stability in distribution, implying the existence of the invariant probability measure, is an important measure of stochastic hybrid system. However, the effect of Levy jumps on the stability in distribution is still unclear. In this paper, we consider a n-species competitive Lotka–Volterra model with Levy jumps under regime-switching. First, we prove the existence of the global positive solution, obtain the upper and lower boundedness. Then, asymptotic stability in distribution as the main result of our paper is derived under some sufficient conditions. Finally, numerical simulations are carried out to support our theoretical results and a brief discussion is given.

International audience; The aim of this paper is to bring to light a previously unknown geometrical method for extracting the square root in seventh century China. In order to achieve this goal, a seventh century commentary by the scholar Jia Gongyan, 賈公彥, on a Confucian canon, the Rites of Zhou Dynasty [Zhouli 周禮], is analysed. This is compared with the commentary by his contemporary Li Chunfeng,李淳風, which is referred to in another mathematical book , the Mathematical Procedures of the Five Canons , [ Wujing Suanshu 五經筭術 ]. Although these two scholars probably knew each other , they used very different methods to solve the same problem in relation to square root extraction. It is argued that the differences mainly lie in two aspects : firstly , Jia Gongyan mostly made use of geometry while Li Chunfeng used counting rods ; secondly , the two methods had different geometrical interpretations. Given the fact that the method of square root extraction Jia Gongyan uses is one among many other methods he employed in mathematics , and it has the same features as the others ; moreover , other commentators on the Confucian Canons use similar mathematical methods , this paper closes with a general discussion on mathematical cultures. It is suggested that there were three elements to mathematical practice in seventh century China : geometry , counting rods , and written texts. The interplay and structure between the three elements is seen to influence mathematical practices .

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such ultrapower characterization of an unstable class. In the present paper, we first establish the existence of so-called optimal ultrafilters on (suitable) Boolean algebras, which are to simple theories as Keisler's good ultrafilters [15] are to all (first-order) theories. Then, assuming a supercompact cardinal, we characterize the simple theories in terms of saturation of ultrapowers. To do so, we lay the groundwork for analyzing the global structure of simple theories, in ZFC, via complexity of certain amalgamation patterns. This brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability.

In this paper we consider the following critical nonlocal problem { − L K u = λ u + | u | 2 ⁎ − 2 u in Ω u = 0 in R n ∖ Ω , where s ∈ ( 0 , 1 ) , Ω is an open bounded subset of R n , n > 2 s , with continuous boundary, λ is a positive real parameter, 2 ⁎ : = 2 n / ( n − 2 s ) is the fractional critical Sobolev exponent, while L K is the nonlocal integrodifferential operator L K u ( x ) : = ∫ R n ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , x ∈ R n , whose model is given by the fractional Laplacian − ( − Δ ) s . Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of − L K (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.

The elastic net is a regularized least squares regression method that has been widely used in learning and variable selection. The elastic net regularization linearly combines an l 1 penalty term (like the lasso) and an l 2 penalty term (like ridge regression). The l 1 penalty term enforces sparsity of the elastic net estimator, whereas the l 2 penalty term ensures democracy among groups of correlated variables. Compressed sensing is currently an extensively studied technique for efficiently reconstructing a sparse vector from much fewer samples/observations. In this paper we study the elastic net in the setting of sparse vector recovery. For recovering sparse vectors from few observations by employing the elastic net regression, we prove in this paper that the elastic net estimator is stable provided that the underlying measurement/design matrix satisfies the commonly required restricted isometry property or the sparse approximation property. It is well known that many independent random measurement matrices satisfy the restricted isometry property while random measurement matrices generated by highly correlated Gaussian random variables satisfy the sparse approximation property. As a byproduct, we establish a uniform bound for the grouping effect of the elastic net. Some numerical experiments are provided to illustrate our theoretical results on stability and grouping effect of the elastic net estimator.

Konig's paradox, that he presented for the first time in 1905, preserved the same structure in all his papers: there was a number that at the same time was and was not finitely definable. Still, he changed the way for forming it, and both its consequences and its solutions changed as well. In the present paper we are going to follow the story of Konig's paradox, that is an intriguing mix of labelling, solving, criticising an “object” from different viewpoints and for different aims.

In this paper we report for the first time on the binary generalized synchronization, when for the certain values of the coupling strength two unidirectionally coupled dynamical systems generating the aperiodic binary sequences are in the generalized synchronization regime. The presence of the binary generalized synchronization has been revealed with the help of both the auxiliary system approach and the largest conditional Lyapunov exponent calculation. The mechanism resulting in the binary generalized synchronization has been explained. The finding discussed in this paper gives a strong potential for new applications under many relevant circumstances.

Using results in a previous paper "Non-semistable exceptional objects in hereditary categories", we focus here on studying the topology of the space of Bridgeland stability conditions on $D^b(Rep_k(Q ))$, where $Q$ is the acyclic triangular quiver (the underlying graph is the extended Dynkin diagram $\widetilde{\mathbb A}_2$). In particular, we prove that this space is contractible (in the previous paper it was shown that it is connected)., Comment: 51 pages

International audience; In this paper, we consider a vibrating system of Timoshenko-type in a one-dimensional bounded domain with complementary frictional damping and infinite memory acting on the transversal displacement. We show that the dissipation generated by these two complementary controls guarantees the stability of the system in case of the equal-speed propagation as well as in the opposite case. We establish in each case a general decay estimate of the solutions. In the particular case when the wave propagation speeds are different and the frictional damping is linear, we give a relationship between the smoothness of the initial data and the decay rate of the solutions. By the end of the paper, we discuss some applications to other Timoshenko-type systems.

This paper is concerned with a singular limit for the one-dimensional compressible radiation hydrodynamics model. The singular limit we consider corresponds to the physical problem of letting the Bouguer number infinite while keeping the Boltzmann number constant. In the case when the corresponding Euler system admits a contact discontinuity wave, Wang and Xie (2011) [12] recently verified this singular limit and proved that the solution of the compressible radiation hydrodynamics model converges to the strong contact discontinuity wave in the L∞-norm away from the discontinuity line at a rate of e 1 4 , as the reciprocal of the Bouguer number tends to zero. In this paper, Wang and Xie's convergence rate is improved to e 7 8 by introducing a new a priori assumption and some refined energy estimates. Moreover, it is shown that the radiation flux q tends to zero in the L∞-norm away from the discontinuity line, at a convergence rate as the reciprocal of the Bouguer number tends to zero.

In this paper, we investigate the factor properties and gap sequence of the Tribonacci sequence, the fixed point of the substitution σ( a , b , c ) = ( ab , ac , a ). Let ω p be the p-th occurrence of ω and G p ( ω ) be the gap between ω p and ω p +1 . We introduce a notion of kernel for each factor ω , and then give the decomposition of the factor ω with respect to its kernel. Using the kernel and the decomposition, we prove the main result of this paper: for each factor ω , the gap sequence { G p ( ω )} p ≤1 is the Tribonacci sequence over the alphabet { G 1 ( ω ), G 2 ( ω ), G 4 ( ω )}, and the expressions of gaps are determined completely. As an application, for each factor ω and p ∈ ℕ , we determine the position of ω p . Finally we introduce a notion of spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors.

In this paper, we generalize the notion of Kantorovich-type sampling operators using the Fejer-type singular integral. By means of these operators we are able to reconstruct signals (functions) which are not necessarily continuous. Moreover, our generalization allows us to take the measurement error into account. The goal of this paper is to estimate the rate of approximation by the above operators via high-order modulus of smoothness.

The recent work of George Andrews and Mircea Merca on the truncated version of Euler's pentagonal number theorem has opened up a new study on truncated theta series. Since then several papers on the topic have followed. The main purpose of this paper is to generalize the study to Hecke-Rogers type double series, which are associated with some interesting partition functions. Our proofs heavily rely on a formula from the work of Zhi-Guo Liu on the q-partial differential equations and q-series.

Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same $(a,b)$-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks $m(x,q,z)$ and $f_{a,b,c}(x,y,q)$ introduced by Hickerson and Mortenson, as well as $\overline{m}(x,q,z)$ and $\overline{f}_{a,b,c}(x,y,q)$ introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations., Comment: 87 pages, comments are welcome. We have extended the previous version

In this paper, we propose a novel method for obtaining the solution of the fractional differential equation in the class of second-grade fluids models. The technique described in this paper is called the double integral method. The method generates, in general, an approximate solution for the fractional diffusion equations, the energy equations, or the heat equations. In our study, we use the generalized fractional derivative in Caputo–Liouville’s sense. For the illustrations of our method, we propose the graphical representations of the approximates solutions obtained by using the double integral method. We propose interpretations and physical discussions of the solutions obtained with the double integral method.

In this paper, we present an application of the homotopy perturbation transform method to compute the approximate analytical solution of the nonlinear fractional order computer virus propagation (CVP) model. The fractional derivatives are used in Caputo sense. The proposed approximate method generates the numerical solution in the shape of a rapid convergent series by utilizing the provided initial conditions. The main purpose of the paper is to analyze the effect of variation of fractional order α on the meeting time of susceptible, infected and recovered computers. Moreover, the local stability analysis of the fractional order computer virus model is also presented using Routh–Hurwitz stability criterion.

Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\text{reg} (\mathcal{O}_X)$ is not even bounded above by any polynomial function of $d$ when $X$ is not smooth. For a normality bound in the smooth case, we establish that $\text{reg}(X) \leq n(d-2)+1$, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that $\text{reg}(X) \leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\text{reg}(X)$ under suitable assumptions., Comment: 23 pages. This paper has been largely rewritten after McCullough-Peeva's counterexamples to the Eisenbud-Goto regularity conjecture, which appeared in J. Amer. Math. Soc. in 2018. We also added new results on the regularity of smooth projective varieties of arbitrary dimension

We introduce and study spaces of multivariate functions of bounded variation generalizing the classical Jordan and Wiener spaces. Multivariate generalizations of the Jordan space were given by several prominent researchers. However, each of the proposed concepts preserves only few properties of Jordan variation which are designed to a selected application. In contrast, the multivariate generalization of the Jordan space presented in this paper preserves all known and reveals some previously unknown properties of the space. These, in turn, are special cases of the basic properties of the introduced spaces proved in the paper. Specifically, the first part of the paper describes structure properties of functions of bounded ( k , p ) -variation ( V p k functions). It includes assertions on discontinuity sets and pointwise differentiability of V p k functions and their Luzin type and C ∞ approximations. The second part presents results on Banach structure of V p k spaces, namely, atomic decomposition and constructive characterization of their predual spaces. As a result, we obtain the so-called two-stars theorems describing V p k spaces as second duals of their separable subspaces consisting of functions of “vanishing variation”.

In this paper, given an integral domain U, we investigate the main properties of a relation ← m o d which is based on the interrelation between subdomains of U and finitely generated unitary submodules of U. We shall characterize it in terms of a second relation ≺ ⋄ between n-tuples ( u 1 , … , u n ) of elements of U and subdomains D of U defined by the vanishing in ( u 1 , … , u n ) of some polynomial p ( Z 1 , … , Z n ) belonging to a specific subset of the polynomial ring in several variables D [ Z 1 , … , Z n ] . Such an equivalence shall be used in order to introduce three specific collections of subdomains X U , B U and P U , whose algebraic properties present a close connection with geometrical and combinatorial properties induced by ← m o d . On the other hand, the characterization of the subdomains of X U leads to the more general problem of finding a map Ψ associating with a subdomain D of U a collection Ψ ( D ) of subdomains of K U such that the intersection of some or of any member of Ψ ( D ) gives D. In this perspective, in the present paper we shall study two further collections of subdomains of U, denoted respectively by E U and L U , whose main properties are related to those of the families P U and B U . Finally, our investigation of all the aforementioned subdomain families shall be also related to the study of pairs ( e , ξ ) , where e ∈ U ∖ { 0 } and ξ is an idempotent ring endomorphism of U whose kernel agrees with the ideal of U generated by e. We shall exhibit several results concerning the membership of ξ ( U ) and of ξ ( U ) [ e ] to the above subdomain families.

This paper considers robustness for linear evolution equations with non-instantaneous impulsive effects in Banach spaces. We present sufficient conditions to guarantee robustness of nonuniform exponential contractions and exponential dichotomies with both non-instantaneous linear impulsive actions of the associated equation and linear perturbations of the associated linear evolution equation. This paper establishes a new framework to investigate robustness for linear evolution equations with non-instantaneous impulsive conditions and also generalizes results from classical impulsive equations using several nontrivial techniques.

In this paper we study the moduli spaces Simp n m of degree n + 1 morphisms A K 1 → A K 1 with “ramification length Simp n m is a Zariski open subset of the space of degree n + 1 polynomials over K up to A u t ( A K 1 ) . It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes- here we are prescribing, instead, the ramification data. Exploiting the topological properties of the poset that encodes the ramification behavior, we use a sheaf-theoretic argument to compute H ⁎ ( Simp n m ( C ) ; Q ) as well as the etale cohomology H e ´ t ⁎ ( Simp n m / K ; Q l ) for c h a r K = 0 or c h a r K > n + 1 , when n and m are such that n ≥ 3 m . As a by-product we obtain that H ⁎ ( Simp n m ( C ) ; Q ) is independent of n, thus implying rational cohomological stability. When c h a r K > 0 our methods compute H e ´ t ⁎ ( Simp n m ; Q l ) provided c h a r K > n + 1 and show that the etale cohomology groups in positive characteristics do not stabilize.

In this paper, we analyze the nonlinear dynamics of fractional order Duffing system. First, we present the fractional order Duffing system and the numerical algorithm. Second, nonlinear dynamic behaviors of Duffing system with a fixed fractional order is studied by using bifurcation diagrams, phase portraits, Poincare maps and time domain waveforms. The fractional order Duffing system shows some interesting dynamical behaviors. Third, a series of Duffing systems with different fractional orders are analyzed by using bifurcation diagrams. The impacts of fractional orders on the tendency of dynamical motion, the periodic windows in chaos, the bifurcation points and the distance between the first and the last bifurcation points are respectively studied, in which some basic laws are discovered and summarized. This paper reflects that the integer order system and the fractional order one have close relationship and an integer order system is a special case of fractional order ones.

Complexity may be one of the most important measurements for analysing time series data; it covers or is at least closely related to different data characteristics within nonlinear system theory. This paper provides a comprehensive literature review examining the complexity testing techniques for time series data. According to different features, the complexity measurements for time series data can be divided into three primary groups, i.e., fractality (mono- or multi-fractality) for self-similarity (or system memorability or long-term persistence), methods derived from nonlinear dynamics (via attractor invariants or diagram descriptions) for attractor properties in phase-space, and entropy (structural or dynamical entropy) for the disorder state of a nonlinear system. These estimations analyse time series dynamics from different perspectives but are closely related to or even dependent on each other at the same time. In particular, a weaker self-similarity, a more complex structure of attractor, and a higher-level disorder state of a system consistently indicate that the observed time series data are at a higher level of complexity. Accordingly, this paper presents a historical tour of the important measures and works for each group, as well as ground-breaking and recent applications and future research directions.

In the present paper, the authors introduce a new integral transform which yields a number of potentially useful (known or new) integral transfoms as its special cases. Many fundamental results about this new integral transform, which are established in this paper, include (for example) existence theorem, Parseval-type relationship and inversion formula. The relationship between the new integral transform with the H-function and the H-transform are characterized by means of some integral identities. The introduced transform is also used to find solution to a certain differential equation. Some illustrative examples are also given.

This paper presents an improvement of Palmer's linearization theorem in [10] . Palmer's linearization theorem extended the Hartman–Grobman theorem to the nonautonomous case. It requires two essential conditions: (i) the nonlinear term is bounded and Lipschitzian; (ii) the linear system as a whole possesses an exponential dichotomy. The main purpose of this paper is to weaken assumptions (i) and (ii). Also in this paper we prove that the topologically equivalent function H ( t , x ) in the linearization theorem is always Holder continuous (and has a Holder continuous inverse), so as a result we generalize and improve Palmer's linearization theorem.

This paper studies the problem of finding optimal parameters for a Poincare section used for determining the type of behavior of a time series: a deterministic or stochastic one. To reach that goal optimization algorithms are coupled with the Poincare & Higuchi (P&H) method, which calculates the Higuchi dimension using points obtained by performing a Poincare section of a certain attractor. The P&H method generates distinctive patterns that can be used for determining if a given attractor is produced by a deterministic or a stochastic system, but this method is sensitive to the parameters of the Poincare section. Patterns generated by the P&H method can be characterized using numerical measures which in turn can be used for finding such parameters for the Poincare section for which the patterns produced by the P&H method are the most prominent. This paper studies several approaches to parameterization of the Poincare section. Proposed approaches are tested on twelve time series, six produced by deterministic chaotic systems and six generated randomly. The obtained results show, that finding good parameters of the Poincare section is important for determining the type of behavior of a time series. Among the tested methods the evolutionary algorithm was able to find the best Poincare sections for use with the P&H method.

This paper is concerned with nonlinear general fractional differential equations. The existence and uniqueness of the solutions of above equations are obtained by using a new norm and Banach contraction principle. The results are the generalization of Theorem 1.1 of the paper (Jankowski, 2013).

In a recent series of papers, Kavitha et al. [2,3,4] solved three inhomogeneous nonlinear Schrodinger (INLS) integro-differential equation under the influence of a variety of nonlinear inhomogeneities and nonlocal damping by the modified extended tangent hyperbolic function method. They obtained several kinds of exact solitary solutions accompanied by the shape changing property. In this paper, we demonstrate that most of exact solutions derived by them do not satisfy the nonlinear equations and consequently are wrong. Furthermore, we study a generalized Hirota equation with spatially-inhomogenetiy and nonlocal nonlinearity. Its integrability is explored through Painleve analysis and N-soliton solutions are obtained based on the Hirota bilinear method. Effects of linear inhomogeneity on the profiles and dynamics of solitons are also investigated graphically.

a b s t r a c t Article history: Received 2 July 2014 Available online xxxx MSC: 35L10 65M60 93B05 93B40 93D15 Keywords: Controllability Stabilizability Fluid-particle System Burgers equation In this paper we study, from a control theoretic view point, a 1D model of fluid-particle interaction. More precisely, we consider a point mass moving in a pipe filled with a fluid. The fluid is modelled by the viscous Burgers equation whereas the point mass obeys Newton's second law. The control variable is a force acting on the mass point. The main result of the paper asserts that for any initial data there exist a time T> 0a nd a control such that, at the end of the control process, the particle reaches a point arbitrarily close to a given target, whereas the velocities of the fluid and of the point mass are driven exactly to zero. Therefore, within this simplified model, we can control simultaneously the fluid and the particle, by using inputs acting on the moving point only. Moreover, the main result holds without any smallness assumptions on the initial data. Alternatively, we can see our results as yielding controllability of the viscous Burgers equation by a moving internal boundary.

In this paper we introduce a notion of symplectic normal crossing divisor V and define the GW invariant of a symplectic manifold X relative to such a divisor. Our definition includes normal crossing divisors from algebraic geometry. The invariants we define in this paper are key ingredients in symplectic sum type formulas for GW invariants, and extend those defined in our previous joint work with T.H. Parker [16], which covered the case V was smooth. The main step is the construction of a compact moduli space of relatively stable maps into the pair (X,V)(X,V) in the case V is a symplectic normal crossing divisor in X.

In this paper, we obtain an explicit formula for the discrepancy between the limit of the second inner variations of p -Laplace Allen–Cahn energies and the second inner variation of their Γ -limit which is the area functional. Our analysis explains the mysterious discrepancy term found in our previous paper [8] in the case p = 2 . The discrepancy term turns out to be related to the convergence of certain 4-tensors which are absent in the usual Allen–Cahn functional. These (hidden) 4-tensors suggest that, in the complex-valued Ginzburg–Landau setting, we should expect a different discrepancy term which we are able to identify. Along the way, we partially answer a question of Kohn and Sternberg [6] by giving a relation between the limit of second variations of the Allen–Cahn functional and the second inner variation of the area functional at local minimizers. Moreover, our analysis reveals an interesting identity connecting second inner variation and Poincare inequality for area-minimizing surfaces with volume constraint in the work of Sternberg and Zumbrun [16] .

In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over Z b in reproducing kernel Hilbert spaces. The tent transformation (previously called baker’s transform) was originally used for lattice rules by Hickernell (2002) to achieve higher order convergence of the integration error for smooth non-periodic integrands, and later, has been successfully applied to digital nets over Z 2 by Cristea et al. (2007) and Goda (2015). The aim of this paper is to generalize the latter two results to digital nets over Z b for an arbitrary prime b . For this purpose, we introduce the b -adic tent transformation for an arbitrary positive integer b greater than 1, which is a generalization of the original (dyadic) tent transformation. Further, again for an arbitrary positive integer b greater than 1, we analyze the mean square worst-case error of QMC rules using digital nets over Z b which are randomly digitally shifted and then folded using the b -adic tent transformation in reproducing kernel Hilbert spaces. Using this result, for a prime b , we prove the existence of good higher order polynomial lattice rules over Z b among a smaller number of candidates as compared to the result by Dick and Pillichshammer (2007), which achieve almost the optimal convergence rate of the mean square worst-case error in unanchored Sobolev spaces of smoothness of arbitrary high order.

We show that there is the same number of ( n , l ) -alternating sign trapezoids as there is of column strict shifted plane partitions of class l − 1 with at most n parts in the top row, thereby proving a result that was conjectured independently by Behrend and Aigner. The first objects generalize alternating sign triangles, which have recently been introduced by Ayyer, Behrend and the author who showed that they are counted by the same formula as alternating sign matrices. Column strict shifted plane partitions of a fixed class were introduced in a slightly different form by Andrews, and they essentially generalize descending plane partitions. They also correspond to cyclically symmetric lozenge tilings of a hexagon with a triangular hole in the center. In addition, we also provide three statistics on each class of objects and show that their joint distribution is the same. We prove our result by employing a constant term approach that is based on the author's operator formula for monotone triangles. This paper complements a forthcoming paper of Behrend and the author, where the six-vertex model approach is used to show equinumeracy as well as generalizations involving statistics that are different from those considered in the present paper.

The new ingredient of this paper is that we consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research – supervised learning theory and numerical integration – can be used in sampling discretization of the square norm on different function classes.

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w ( x ) = x α e − x − t / x , x ∈ ( 0 , ∞ ) , t > 0 and α > 0 . When the matrix size n → ∞ , we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t ∈ ( 0 , d ] , d > 0 fixed. A particular Painleve III transcendent is involved in the approximation, as well as in the large- n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift–Zhou nonlinear steepest descent method.

We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra $P_k := \mathbb F_2[x_1,x_2,...,x_k]$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we study a minimal set of generators for $\mathcal A$-module $P_k$ in some so-call generic degrees and apply these results to explicitly determine the hit problem for $k=4$., Comment: 68 pages, Quy Nhon University preprint, Viet Nam, 2011. A shorter version of this paper was published in Advances in Mathematics

In this paper we prove that for the Gromov–Witten theory of P 1 orbifolds of ADE type the genus-2 G-function introduced by B. Dubrovin, S. Liu, and Y. Zhang vanishes. Together with our results in [11] , this completely solves the main conjecture in their paper [4] . In the process, we also found a sufficient condition for the vanishing of the genus-2 G-function which is weaker than the condition given in our previous paper [11] .

This paper addresses the mean square exponential stabilization problem of stochastic bidirectional associative memory (BAM) neural networks with Markovian jumping parameters and time-varying delays. By establishing a proper Lyapunov–Krasovskii functional and combining with LMIs technique, several sufficient conditions are derived for ensuring exponential stabilization in the mean square sense of such stochastic BAM neural networks. In addition, the achieved results are not difficult to verify for determining the mean square exponential stabilization of delayed BAM neural networks with Markovian jumping parameters and impose less restrictive and less conservative than the ones in previous papers. Finally, numerical results are given to show the effectiveness and applicability of the achieved results.

Growing biological evidence from various ecosystems (Berec et al., 2007) suggests that the Allee effect generated by two or more mechanisms can act simultaneously on a single population. Surprisingly, prey–predator system incorporating multiple Allee effect is relatively poorly studied in literature. In this paper, we consider a ratio dependent prey–predator system with a double Allee effect in prey population growth. We have taken into account the case of the strong and the weak Allee effect separately and study the stability and complete bifurcation analysis. It is shown that the model exhibits the bi-stability and there exists separatrix curve(s) in the phase plane implying that dynamics of the system is very sensitive to the variation of the initial conditions. Generic normal forms at double zero singularity are derived by a rigorous mathematical analysis to show the different types of bifurcation of the proposed model system including Bogdanov–Takens bifurcation, saddle–node bifurcation curve, a Hopf bifurcation curve and a homoclinic bifurcation curve. The complete analysis of possible topological structures including elliptic, parabolic, or hyperbolic orbits, and any combination of them in a neighborhood of the complicated singular point ( 0 , 0 ) in the interior of the first quadrant is explored by using a blow up transformation. These structures have important implications for the global behavior of the model. Numerical simulation results are given to support our theoretical results. Finally, the paper concludes with a discussion of the ecological implications of our analytic and numerical findings.

We consider nonlinear diffusion problems of the form ut=Δu+f(u) with Stefan type free boundary conditions, where the nonlinear term f(u) is of monostable, bistable or combustion type. Such problems are used as an alternative model (to the corresponding Cauchy problem) to describe the spreading of a biological or chemical species, where the free boundary represents the expanding front. We are interested in its long-time spreading behavior which, by recent results of Du, Matano and Wang [10], is largely determined by radially symmetric solutions. Therefore we will examine the radially symmetric case, where the equation is satisfied in |x| 0. Subsequently, sharper estimate of the spreading speed was obtained by the authors of the current paper in [11], in the form that limt→∞⁡[h(t)−c⁎t]=Hˆ∈R1. In this paper, we consider the case N≥2 and show that a logarithmic shifting occurs, namely there exists c⁎>0 independent of N such that limt→∞⁡[h(t)−c⁎t+(N−1)c⁎log⁡t]=hˆ∈R1. At the same time, we also obtain a rather clear description of the spreading profile of u(t,r). These results reveal striking differences from the spreading behavior modeled by the corresponding Cauchy problem.

In this paper we show how \emph{the metric theory of tensor products} developed by Grothendieck perfectly fits in the study of channel capacities, a central topic in \emph{Shannon's information theory}. Furthermore, in the last years Shannon's theory has been generalized to the quantum setting to let the \emph{quantum information theory} step in. In this paper we consider the classical capacity of quantum channels with restricted assisted entanglement. In particular these capacities include the classical capacity and the unlimited entanglement-assisted classical capacity of a quantum channel. To deal with the quantum case we will use the noncommutative version of $p$-summing maps. More precisely, we prove that the (product state) classical capacity of a quantum channel with restricted assisted entanglement can be expressed as the derivative of a completely $p$-summing norm., Comment: V2: Some proofs have been explained in more detail. New references added. Same results

This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This involves a notion of Poincar\'e duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control of the Vafa-Witten bound under conformal changes of metric. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated to conformal weights on noncommutative tori, and spectral triples associated to duals of torsion-free discrete cocompact subgroups satisfying the Baum-Connes conjecture., Comment: Final version. 38 pages