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Metric spaces of generalized roundness zero have interesting non-embedding properties. For instance, we note that no metric space of generalized roundness zero is isometric to any metric subspace of any L p -space for which 0 p ≤ 2 . Lennard, Tonge and Weston gave an indirect proof that l ∞ ( 3 ) has generalized roundness zero by appealing to non-trivial isometric embedding theorems of Bretagnolle, Dacunha-Castelle and Krivine, and Misiewicz. In this paper we give a direct proof that l ∞ ( 3 ) has generalized roundness zero. This provides insight into the combinatorial geometry of l ∞ ( 3 ) that causes the generalized roundness inequalities to fail. We complete the paper by noting a characterization of real quasi-normed spaces of generalized roundness zero.

In this paper we study Appell polynomials by connecting them to random variables. This probabilistic approach yields, e.g., the mean value property which is fundamental in the sense that many other properties can be derived from it. We also discuss moment representations of Appell polynomials. In the latter part of the paper the presented general theory is applied to study some classical explicitly given polynomials.

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ϕ A ( T ) as the automorphism of the Grothendieck group K 0 ( A ) induced by the Auslander–Reiten translation τ in the derived category Der b ( mod A ) of the module category mod A of finite dimensional left A-modules. We say that A is of cyclotomic type if the characteristic polynomial χ A of ϕ A is a product of cyclotomic polynomials, equivalently, if the Mahler measure M ( χ A ) = 1 . In [6] we have considered many examples of algebras of cyclotomic type in the representation theory literature. In this paper we study the Mahler measure of the Coxeter polynomial of accessible algebras. In 1933, D.H. Lehmer found that the polynomial T 10 + T 9 − T 7 − T 6 − T 5 − T 4 − T 3 + T + 1 has Mahler measure μ 0 = 1.176280 . . . , and he asked whether there exist any smaller values exceeding 1. In this paper we prove that for any accessible algebra A either M ( χ A ) = 1 or M ( χ B ) ≥ μ 0 for some convex subcategory B of A. We introduce interlaced tower of algebras A m , … , A n with m ≤ n − 2 satisfying χ A s + 1 = ( T + 1 ) χ A s − T χ A s − 1 for m + 1 ≤ s ≤ n − 1 . We prove that, if Spec ϕ A n ⊂ S 1 ∪ R + and A n is not of cyclotomic type, then M ( χ A m ) M ( χ A n ) . We construct several examples.

Let C be the complex Levi-Civita field and let c 0 ( C ) or, simply, c 0 denote the space of all null sequences z = ( z n ) n ∈ N of elements of C . The natural inner product on c 0 induces the sup-norm of c 0 . In a previous paper Aguayo et al. (2013), we presented characterizations of normal projections, adjoint operators and compact operators on c 0 . In this paper, we work on some B ∗ -algebras of operators, including those mentioned above; then we define an inner product on such algebras and prove that this inner product induces the usual norm of operators. We finish the paper with a characterization of closed subspaces of the B ∗ -algebra of all adjoint and compact operators on c 0 which admit normal complements.

We introduce a covering conjecture and show that it holds below A D R + “ Θ is regular” . We then use it to show that in the presence of mild large cardinal axioms, PFA implies that there is a transitive model containing the reals and ordinals and satisfying A D R + “ Θ is regular” . The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first application of the core model induction that goes significantly beyond the region of A D + + θ 0 Θ .

In this paper we give a more precise construction of the boundary layer used to prove the strong decay of the acoustic waves in the paper [4] . Indeed, the construction in [4] is not completely correct in the case when eigenvalues are not simple and some additional conditions should be imposed on the basis of eigenvectors chosen. There is an extra interesting algebraic structure based on a sequence of symmetric operators that should be used in this choice. This construction can be extended to the case of Robin boundary condition as well as to the full acoustic operator which includes the heat conductivity effect. Moreover, this paper allows us to extend the construction of the viscous boundary layer in the hydrodynamic limit of the Boltzmann equation used in our paper [7] to the general case.

We study in this paper, a one parameter deformation of the q -Laguerre weight function. An investigation is made on the polynomials orthogonal with respect to such a weight. With the aid of the two compatibility conditions previously obtained in Chen and Ismail (1997) and the q -analog of a sum rule obtained in this paper, we derive expressions for the recurrence coefficients in terms of certain auxiliary quantities, and show that these quantities satisfy a pair of first order non linear difference equations. These difference equations are similar in form to the recognized asymmetric discrete Painleve systems such as α q -P-IV and α q -P-V.

The effect of omnivory in predator–prey system is debatable regarding its stabilizing or destabilizing characteristics. Earlier theoretical studies predict that omnivory is stabilizing or destabilizing depending on the condition of the system. The effect of omnivory in the food chain system is not yet properly understood. In the present paper, we study the effect of omnivory in a tri-trophic food chain system on the famous Hastings and Powell model. Omnivory enhances the chance of predator switching between prey and middle predator. The novelty of this paper is to study the effect of predator switching of the top predator which is omnivorous in nature. Our results suggest that in the absence of switching, an increase of omnivory stabilizes the system from chaotic dynamics, however, if we further increase the strength of omnivory, the system becomes unstable and middle predator goes to extinction. It is also observed that the predator switching enhance the stability and persistence of all populations.

This paper provides the theoretical foundation for the component-by-component (CBC) construction of randomly shifted lattice rules that are tailored to integrals over R s arising from practical applications. For an integral of the form ∫ R s f ( y ) ∏ j = 1 s ϕ ( y j ) d y with a univariate probability density ϕ , our general strategy is to first map the integral into the unit cube [ 0 , 1 ] s using the inverse of the cumulative distribution function of ϕ , and then apply quasi-Monte Carlo (QMC) methods. However, the transformed integrand in the unit cube rarely falls within the standard QMC setting of Sobolev spaces of functions with mixed first derivatives. Therefore, a non-standard function space setting for integrands over R s , previously considered by Kuo, Sloan, Wasilkowski and Waterhouse (2010), is required for the analysis. Motivated by the needs of three applications, the present paper extends the theory of the aforementioned paper in several non-trivial directions, including a new error analysis for the CBC construction of lattice rules with general non-product weights, the introduction of an unanchored variant for the setting, the use of coordinate-dependent weight functions in the norm, and the strategy for fast CBC construction with POD (“product and order dependent”) weights.

In our recent paper, we introduced the notions of relative operator ( α , β ) -entropy and Tsallis relative operator ( α , β ) -entropy as a parameter extensions of relative operator entropy and Tsallis relative operator entropy. In this paper, we give upper and lower bounds of these new notions according to operator ( α , β ) -geometric mean introduced in Nikoufar et al. (2013) [14] .

In the first part of this paper, we obtain mirror formulas for twisted genus 0 two-point Gromov–Witten (GW) invariants of projective spaces and for the genus 0 two-point GW-invariants of Fano and Calabi–Yau complete intersections. This extends previous results for projective hypersurfaces, following the same approach, but we also completely describe the structure coefficients in both cases and obtain relations between these coefficients that are vital to the applications to mirror symmetry in the rest of this paper. In the second and third parts of this paper, we confirm Walcher's mirror symmetry conjectures for the annulus and Klein bottle GW-invariants of Calabi–Yau complete intersection threefolds; these applications are the main results of this paper. In a separate paper, the genus 0 two-point formulas are used to obtain mirror formulas for the genus 1 GW-invariants of all Calabi–Yau complete intersections.

The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.

Given a compact Lie group G, in this paper we give symbolic criteria for operators to be nuclear and r-nuclear on L p ( G ) -spaces, with applications to distribution of eigenvalues and trace formulae. Since criteria in terms of kernels are often not effective in view of Carleman's example, in this paper we adopt the symbolic point of view. The criteria here are given in terms of the concept of matrix symbols defined on the noncommutative analogue of the phase space G × G ˆ , where G ˆ is the unitary dual of G. No regularity of the kernel (or of the symbol) is assumed so that several of the obtained criteria extend to the more general setting of compact topological groups.

This paper investigates the maximal singularity-free ellipse of 3-RPR planar parallel mechanisms. The paper aims at finding the optimum ellipse, by taking into account the stroke of actuators, in which the mechanism exhibits no singularity, which is a definite asset in practice. Convex optimization is adopted for the mathematical framework of this paper which requires a matrix representation for the kinematic properties of the mechanism under study in order to solve the latter optimization problem. Based on the nature of the expressions involved in the problem, two situations may arise: dealing with either convex or non-convex expressions. Both situations are treated separately with two very fast and systematic algorithms. For the first situation, an exact method is applied while for the second one, which is a general form of the first situation, convex optimization is accompanied with an iterative procedure. The computational time for the two proposed algorithms are considerably low compared with other methods proposed in the literature which opens an avenue to use the proposed algorithms for real-time purposes.

This paper addresses the problem of defining a simple End-Effector design for a robotic arm that is able to grasp a given set of planar objects. The OCOG (Objects COmmon Grasp search) algorithm proposed in this paper searches for a common grasp over the set of objects mapping all possible grasps for each object that satisfy force closure and quality criteria by taking into account the external wrenches (forces and torque) applied to the object. The mapped grasps are represented by feature vectors in a high-dimensional space. This feature vector describes the design of the gripper. A database is generated for all possible grasps for each object in the feature vector space. A search algorithm is then used for intersecting all possible grasps over all parts and finding a common grasp suitable for all objects. The search algorithm utilizes the kd-tree index structure for representing the database of the sets of feature vectors. The kd-tree structure enables an efficient and low cost nearest-neighbor search for common vectors between the sets. Each common vector found (feature vector) is the grasp configuration for a group of objects, which implies the future end-effector design. The final step classifies the grasps found to subsets of the objects, according to the common vectors found. Simulations and experiments are presented for four objects to validate the feasibility of the proposed algorithm. The algorithm will be useful for standardization of end-effector design and reducing its engineering time.

The characterization of all Darboux polynomials and rational first integrals of the generalized Lorenz system, x ˙ = a ( y − x ) , y ˙ = b x + c y − x z , z ˙ = d z + x y , was published very recently in [K. Wu, X. Zhang, Bull. Sci. Math. 136 (2012) 291–308]. In this paper we improve that work in two aspects. On the one hand, we obtain the same results in a much more straightforward way. To do that, we show the equivalence under generic conditions, c ≠ 0 , between the Lorenz system and the generalized Lorenz system by means of a linear scaling in time and coordinates. Thus, from the well-known results on Darboux polynomials and its algebraic integrability for the Lorenz system, it is direct to obtain the corresponding results for the generalized Lorenz system. On the other hand, in the case c = 0 , we find a new Darboux polynomial of the generalized Lorenz system, not detected in the above paper, which is also a first integral. In this way, we complete the list provided by the authors of all Darboux polynomials and rational first integrals of the generalized Lorenz system.

The Calkin theorem provides a one-to-one correspondence between all operator ideals A(H) over the separable infinite-dimensional Hilbert space H and all symmetric sequence ideals a(N) over the index set N≔{1,2,…}. The main idea of the present paper is to replace a(N) by the ideal z(N0) that consists of all sequences (αh) indexed by N0≔{0,1,2,…} for which (α0,α1,α1,…,αh,…,αh︷2hterms,…)∈a(N). This new kind of sequence ideals is characterized by two properties: (1) For (αh)∈z(N0) there is a non-increasing (βh)∈z(N0) such that ∣αh∣≤βh. (2) z(N0) is invariant under the operator S+:(α0,α1,α2,…)↦(0,α0,α1,…). Using this modification of the Calkin theorem, we simplify, unify, and complete earlier results of [4,5,7–9,13,14,19–21,25] The central theorem says that there are canonical isomorphisms between the linear spaces of all traces on A(H), all symmetric linear forms on a(N), and all 12S+-invariant linear forms on z(N0). In this way, the theory of linear forms on ideals of a non-commutative algebra that are invariant under the members of a non-commutative group is reduced to the theory of linear forms on ideals of a commutative algebra that are invariant under a single operator. It is hoped that the present approach deserves the rating “streamlined”. Our main objects are linear forms in the purely algebraic sense. Only at the end of this paper continuity comes into play, when the case of quasi-normed ideals is considered. We also sketch a classification of operator ideals according to the existence of various kinds of traces. Details will be discussed in a subsequent publication.

In this paper, we introduce and study an implicit iterative method to approximate a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup in real Hilbert spaces. Further, we prove that the nets generated by the implicit iterative method converge strongly to the common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. Furthermore, we justify our main result through a numerical example. The results presented in this paper extend and generalize the corresponding results given by Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Model. 48 (2008) 279–286] and Cianciaruso et al. [F. Cianciaruso, G. Marino, L. Muglia, Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert space, J. Optim. Theory Appl. 146 (2010) 491–509].

In this paper, we investigate the bounds of the coefficients of several classes of bi-univalent functions. The results presented in this paper improve or generalize the recent works of other authors.

In this paper, we are concerned with the global wellposedness of 2-D density-dependent incompressible Navier–Stokes equations (1.1) with variable viscosity, in a critical functional framework which is invariant by the scaling of the equations and under a nonlinear smallness condition on fluctuation of the initial density which has to be doubly exponential small compared with the size of the initial velocity. In the second part of the paper, we apply our methods combined with the techniques in Danchin and Mucha (2012) [10] to prove the global existence of solutions to (1.1) with constant viscosity and with piecewise constant initial density which has small jump at the interface and is away from vacuum. In particular, this latter result removes the smallness condition for the initial velocity in a corresponding theorem of Danchin and Mucha (2012) [10] .

In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of N possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–Cech compactification β N . This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–Cech compactification of N .

The purpose of this paper is to study the L2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γt(x))K(t)dt, where γt(x) is a C∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝN × ℝn, satisfying γ0(x) ≡ x, ψ is a C∞ cut-off function supported on a small neighborhood of 0 ∈ ℝn, and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝN. The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L2. The case when K is a Calderon-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderon- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of Lp boundedness, while the third paper deals with the special case when γ is real analytic.

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables., Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398, which will not be submitted to any journal

We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood--Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the (n+1)^(n-1) conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures. Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied in [CEF] to counting problems in number theory and finite group theory. Representation stability is also used in [C] to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds., Comment: 91 pages. v2: minor revisions throughout. v3: final version, to appear in Advances in Mathematics

In this paper, we first introduce the concepts and properties of the square-mean weighted pseudo almost automorphy and the square-mean bi-almost automorphy for a stochastic process. With these preliminary settings and by virtue of the theory of the semigroups of the operators, the Banach fixed point theorem and the stochastic analysis techniques, we investigate the well-posedness of the square-mean weighted pseudo almost automorphic solutions for a general class of non-autonomous stochastic evolution equations that satisfy either global or only local Lipschitz condition. Moreover, we estimate the boundedness of attractive domain for the case where the only local Lipschitz condition is taken into account. Finally, we provide two illustrative examples to show the practical usefulness of the analytical results that we establish in the paper.

This paper is devoted to the construction of a solution for the “Inhomogeneous skew Brownian motion” equation, which first appeared in a seminal paper by Sophie Weinryb, and recently, studied by Etore and Martinez. Our method is based on the use of the Balayage formula. At the end of this paper we study a limit theorem of solutions.

The famous Baker–Campbell–Hausdorff series defines a group composition on the set of the so called grouplike elements of a completed free Lie algebra as we can find in P. Cartier’s paper on the Campbell–Hausdorff formula in 1956. According to A. Baider and R.C. Churchill, we call this group the Campbell–Hausdorff group over the alphabet X of the free Lie algebra. By a certain coproduct and certain bialgebra endomorphisms of the free associative algebra over a set X which should be at most countably infinite, we get an alternative realisation of this Campbell–Hausdorff group. This realisation is much more comfortable to deal with than the classical one. The usual composition of endomorphisms gives a near-ring structure to this group. In this paper we consider a metric on the Campbell–Hausdorff near-ring over the alphabet N and the compatibility of this metric with the near-ring compositions and we make some topological remarks.

Let D ▵ ( n ) be the double Ringel–Hall algebra of the cyclic quiver △ ( n ) and let D ▵ ( n ) be the modified quantum affine algebra of D ▵ ( n ) . We will construct an integral form D ▵ ( n ) Z for D ▵ ( n ) such that the natural algebra homomorphism from D ▵ ( n ) Z to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct the integral form U Z ( g l n ) of the universal enveloping algebra U ( g l n ) of the loop algebra g l n = g l n ( Q ) ⊗ Q [ t , t − 1 ] , and prove that the natural algebra homomorphism from U Z ( g l n ) to the affine Schur algebra over Z is surjective. In a subsequent paper (Fu [10] ), we will use affine Schur algebras to give BLM realization of U Z ( g l n ) , and this enables us to give a new proof of the statements about U Z ( g l n ) given in this paper.

This paper presents the properties of topological pressure dimension, which is an extension of the entropy dimension. Specifically, this paper studies the relationships among different types of topological pressure dimension and identifies an inequality relating them. This analysis calculates analogs of many known results of topological pressure. In particular, we will show the value of the pressure dimension is always equal to or greater than 1 for any positive constant potential function.

The classical n -logarithm is a multivalued analytic function defined inductively: Li n ( z ) : = ∫ 0 z Li n − 1 ( t ) d log t , Li 1 ( z ) = − log ( 1 − z ) . In this paper we give a simple explicit construction of the Grassmannian n -logarithm, which is a multivalued analytic function on the quotient of the Grassmannian of n -dimensional subspaces in C 2 n in generic position to the coordinate hyperplanes by the natural action of the torus ( C ∗ ) 2 n . The classical n -logarithm appears at a certain one dimensional boundary stratum. We study Tate iterated integrals, which are homotopy invariant integrals of 1-forms d log f i where f i are rational functions. We give a simple explicit formula for the Tate iterated integral which describes the Grassmannian n -logarithm. Another example is the Tate iterated integrals for the multiple polylogarithms on the moduli spaces M 0 , n , calculated in Section 4.4 of Goncharov (2005) [13] using the combinatorics of plane trivalent trees decorated by the arguments of the multiple polylogarithms. Variations of mixed Hodge–Tate structures on X are described by a Hopf algebra H • ( X ) . We upgrade Tate iterated integrals on a (rational) complex variety X to elements of H • ( X ) . The coproducts of these elements are very interesting invariants of the iterated integrals. In general their calculation is a non-trivial problem. We show however, that working modulo the ideal of H • ( X ) generated by constant variations, there is a simple way to calculate the coproduct. It is a pleasure to dedicate this paper to Andrey Suslin, whose works Suslin (1984) [21] and Suslin (1991) [22] played an essential role in the development of the story.

The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equation ∑ k = 2 n ( ∑ i 1 = 2 k ∑ i 2 = i 1 + 1 k + 1 … ∑ i n − k + 1 = i n − k + 1 n ) f ( ∑ i = 1 , i ≠ i 1 , … , i n − k + 1 n x i − ∑ r = 1 n − k + 1 x i r ) + f ( ∑ i = 1 n x i ) = 2 n − 2 ∑ 1 ≤ i j ≤ n ( f ( x i + x j ) + f ( x i − x j ) ) − 2 n − 5 ( n − 2 ) ∑ i = 1 n f ( 2 x i ) ( n ∈ ℕ , n ≥ 3 ) in β-Banach modules on Banach algebras. The concept of Ulam-Hyers-Rassias stability (briefly, HUR-stability) originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

This paper reviews the theoretical basis and its application for a multiple type of Duffing oscillation. This paper uses a suitable theoretical model to examine the structural instability of a folding truss which is limited so that only vertical displacements are possible for each nodal point supported by both sides. The equilibrium path in this ideal model has been found to have a type of “hill-top bifurcation” from the theoretical work of bifurcation analysis. Dynamic analysis allows for geometrical non-linearity based upon static bifurcation theory. We have found that a simple folding structure based on Multi-Folding-Microstructures theory is more interesting when there is a strange trajectory in multiple homo/hetero-clinic orbits than a well-known ordinary homoclinic orbit, as a model of an extended multiple degrees-of-freedom Duffing oscillation. We found that there are both globally and locally dynamic behaviours for a folding multi-layered truss which corresponds to the structure of the multiple homo/hetero-clinic orbits. This means the numerical solution depends on the dynamic behaviour of the system subjected to the forced cyclic loading such as folding or expanding action. The author suggests simplified theoretical models for hill-top bifurcation that help us to understand globally and locally dynamic behaviours, which depends on the static bifurcation problem. Such models are very useful for forecasting simulations of the extended Duffing oscillation model as essential and invariant nonlinear phenomena.

The main purpose of this paper is to study the generalized Hilbert operator {equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt {equation*} acting on the weighted Bergman space $A^p_\om$, where the weight function $\om$ belongs to the class $\R$ of regular radial weights and satisfies the Muckenhoupt type condition {equation}\label{Mpconditionaabstract} \sup_{0\le r, Comment: This paper has been accepted for publication in Advances in Mathematics

There are various discrepancies that are measures of uniformity for a set of points on the unit hypercube. The discrepancies have played an important role in quasi-Monte Carlo methods. Each discrepancy has its own characteristic and some weakness. In this paper we point out some unreasonable phenomena associated with the commonly used discrepancies in the literature such as the L p -star discrepancy, the center L 2 -discrepancy (CD) and the wrap-around L 2 -discrepancy (WD). Then, a new discrepancy, called the mixture discrepancy (MD), is proposed. As shown in this paper, the mixture discrepancy is more reasonable than CD and WD for measuring the uniformity from different aspects such as the intuitive view, the uniformity of subdimension projection, the curse of dimensionality and the geometric property of the kernel function. Moreover, the relationships between MD and other design criteria such as the balance pattern and generalized wordlength pattern are also given.

In this paper, we study degenerate CR embeddings f of a strictly pseudoconvex hypersurface M ⊂ C n + 1 into a sphere S in a higher dimensional complex space C N + 1 . The degeneracy of the mapping f will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d of the second fundamental form and all of its covariant derivatives is n (here, n is the CR dimension of M ), then f ( M ) is contained in a complex plane of dimension n + d + 1 . The converse of this statement is also true, as is easy to see. When the total rank d exceeds n , it is no longer true, in general, that f ( M ) is contained in a complex plane of dimension n + d + 1 , as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n , then partial rigidity may still persist, but there is a “defect” k that arises from the ranks exceeding n such that f ( M ) is only contained in a complex plane of dimension n + d + k + 1 . Moreover, this defect occurs in general, as is illustrated by examples.

Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f ′ ′ + P ( z ) f = 0 , where P ( z ) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E. Hille’s 1969 book Lectures on Ordinary Differential Equations are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille’s book.

In this paper, weighted extra-weak and weak type inequalities have been characterized for the one-sided Hardy–Littlewood maximal function on the plane. We have addressed conditions on pair of weights for which the dyadic one-sided maximal function on higher dimension is locally integrable. In the process, we characterize weights for which the one-sided Hardy-Littlewood maximal function satisfies restricted weak type inequalities on the plane, thus extending the result of Kerman and Torchinsky to the one-sided Hardy-Littlewood maximal function.

Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these derivations act on a differential category. A differential Tannakian theory is developed. The main application is to the Galois theory of linear differential equations with parameters. Namely, we show the existence of a parameterized Picard-Vessiot extension and, therefore, the Galois correspondence for many differential fields with, possibly, non-differentially closed fields of constants, that is, fields of functions of parameters. Other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic, which will appear in a separate paper. This application is based on differential categories developed in the present paper, and not just differential algebraic groups and their representations., Comment: 90 pages, minor corrections

The subject of this paper is to give the description, up to topological conjugacy, of possible minimal sets of triangular maps of the square of type 2 ∞ . In [4] , we give a general method allowing to embed any zero-dimensional almost 1–1 extension of the dyadic odometer (in particular any dyadic Toeplitz system) as a minimal set of a triangular map of this type. In this paper we present a method (a combination of that described in [4] with one introduced in [1] ) of similarly embedding a special class of zero-dimensional almost 2–1 extensions of the odometer. We conjecture that these two embedding theorems exhaust all possibilities for nonperiodic minimal sets. The paper was inspired by the last unsolved problem in the Sharkovski classification program of triangular maps: does there exist a triangular map with positive entropy attained on the set of uniformly recurrent points but with entropy zero on the set of regularly recurrent points. The paper answers this question positively, concluding the program.

In this paper, we continue investigations concerning generalized Orlicz–Lorentz sequence spaces λ φ initiated in the papers of Foralewski et al. (2008) [16] , [17] (cf. also Foralewski (2011) [11] , [12] ). As we will show in Example 1.1 , Example 1.2 , Example 1.3 the class of generalized Orlicz–Lorentz sequence spaces is much more wider than the class of classical Orlicz–Lorentz sequence spaces. Moreover, it is shown that if a Musielak–Orlicz function φ satisfies condition δ 2 λ , then λ φ has the coordinatewise Kadec–Klee property. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space λ φ , a strong condition of δ 2 type and the notion of regularity of function φ are introduced. Finally, criteria for non-squareness of λ φ , of their subspaces of order continuous elements ( λ φ ) a as well as of finite dimensional subspaces λ φ n of λ φ are presented.

We develop a constructive completion method in general Minkowski spaces, which successfully extends a completion procedure due to Buckner in two- and three-dimensional Euclidean spaces. We prove that this generalized Buckner completion is locally Lipschitz continuous, thus solving the problem of finding a continuous selection of the diametric completion mapping in finite dimensional normed spaces. The paper also addresses the study of an elegant completion procedure due to Maehara in Euclidean spaces, the natural setting of which are the spaces with a generating unit ball. We prove that, in these spaces, the Maehara completion is also locally Lipschitz continuous, besides establishing other geometric properties of this completion. The paper contains also new estimates of the (local) Lipschitz constants for the wide spherical hull.

In our paper [10] we discussed an error in the recent literature on abstract fractional differential equations and we proposed a different approach to treat these type of problems. By noting that the results by Hernandez et al. (2010) [10] are not applicable for problems with nonlocal conditions, in this paper we study the existence of mild solutions for a class of abstract fractional differential equations with nonlocal conditions. An application involving a partial differential equation with a fractional temporal derivative and nonlocal conditions is considered.

The paper is concerned with the long-time behaviour of the travelling fronts of the damped wave equation αutt + ut = uxx − V′(u) on ℝ. The long-time asymptotics of the solutions of this equation are quite similar to those of the corresponding reaction-diffusion equation ut = uxx − V′(u). Whereas a lot is known about the local stability of travelling fronts in parabolic systems, for the hyperbolic equations it is only briefly discussed when the potential V is of bistable type. However, for the combustion or monostable type of V, the problem is much more complicated. In this paper, a local stability result for travelling fronts of this equation with combustion type of nonlinearity is established. And then, the result is extended to the damped wave equation with a case of monostable pushed front.

This paper deals with the semi-implicit scheme with respect to the t -variable combined with a finite element spatial approximation of evolutionary Hamilton–Jacobi–Bellman equations with nonlinear source terms. We establish a convergence and a quasi-optimal L ∞ -asymptotic behavior, involving a weakly coupled system of discrete parabolic quasi-variational inequalities (PQVIs), for the solution of which an iterative discrete scheme of monotone kind is introduced and analyzed. Furthermore, the simple numerical example shows that the estimates introduced in this paper are efficient.

In this paper, we introduce the new notion of g-monotone mapping and prove some fixed point theorems for multi-valued and single-valued g-increasing mappings in partially ordered metric spaces. The mappings considered in this paper are assumed to satisfy certain metric inequalities which are established by an altering distance function. The presented results extend and improve the main results of Choudhury and Metiya [B.S. Choudhury, N. Metiya, Multi-valued and single-valued fixed point results in partially ordered metric spaces, Arab J. Math. Sci. 17 (2011) 135–151].

In this paper we discuss the existence and global attractivity of k -almost automorphic sequence solution of a model of cellular neural networks. We consider the corresponding difference equation analogue of the model system using suitable discretization method and obtain certain conditions for the existence of solution. Almost automorphic function is a good generalization of almost periodic function. This is the first paper considering such solutions of the neural networks.

In this paper, we study the long-time behavior of the solution for the linearized ideal MHD around sheared velocity and magnetic field under Stern stability condition. We prove that the velocity and magnetic field will converge to sheared velocity and magnetic field as time approaches infinity. Moreover a new depletion phenomenon is proved: the horizontal velocity and magnetic field at the critical points will decay to 0 as time approaches infinity.

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

In this paper we develop a potential theory for strongly degenerate parabolic operators of the form L : = ∇ X ⋅ ( A ( X , Y , t ) ∇ X ) + X ⋅ ∇ Y − ∂ t , in unbounded domains of the form Ω = { ( X , Y , t ) = ( x , x m , y , y m , t ) ∈ R m − 1 × R × R m − 1 × R × R | x m > ψ ( x , y , y m , t ) } , where ψ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L . Concerning A = A ( X , Y , t ) we assume that A is bounded, measurable, symmetric and uniformly elliptic (as a matrix in R m ). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of A and the Lipschitz constant of ψ. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, and several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains.

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.