the q h laplace transform on discrete time scales in this paper we develop a q h laplace transform on specific time scales we show that the transform is reduced for h 0 to the q laplace transform reduce for q 1 to the h laplace transform and reduced for q h 1 to the z transform finally we employ the q h laplace transform to produce some key results

lyapunov functions for riemann liouville like fractional difference equations abstract discrete memory effects are introduced by fractional difference operators asymptotic stabilities of nonlinear fractional difference equations are investigated in this paper a linear scalar fractional difference equality is utilized lyapunov second direct method is proposed for nonlinear discrete fractional systems asymptotic stability conditions are provided and some examples are given

asymptotic stability of q h fractional difference equations abstract asymptotic stability of linear nabla riemann liouville q h fractional difference equation is investigated in this paper a liapunov functional is constructed for the fractional difference equation the sufficient condition for the asymptotic stability of considered equations is proposed the results are illustrated with the corresponding numerical examples

gronwall s inequality on discrete fractional calculus in this paper we introduce discrete fractional sum equations and inequalities we obtain the equivalence of an initial value problem for a discrete fractional equation and a discrete fractional sum equation then we give an explicit solution to the linear discrete fractional sum equation this allows us to state and prove an analogue of gronwall s inequality on discrete fractional calculus we employ a nabla or backward difference we employ the riemann liouville definition of the fractional difference as a result we obtain gronwall s inequality for discrete calculus with the nabla operator we illustrate our results with an application that gives continuous dependence of solutions of initial value problems on initial conditions

volumes of small balls on open manifolds lower bounds and examples question under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center point xx two theorems establish such assumptions and two examples sharply limit their weakening in particular we give an example of a metric on r4 extending to higher dimensions of positive ricci curvature whose sectional curvatures decay to 0 and such that the volume of balls goes uniformly to 0 as the center goes to infinity 0 introduction the following question came up in discussions of the first author with peter li under what curvature assumptions on a complete open manifold m can one find a constant c m r such that the volume of balls b x r can be bounded from below independent of the midpoint x 0 1 vol b x r c m r note that in the presence of a lower bound on the ricci curvature one needs to prove the above inequality only for some r since the bishop gromov volume comparison theorem gr p 124 then gives a bound c m r for r c n rn in dimension 2 theorem a below is a satisfyingpositive answer since the assumptions do not imply a lower bound on the injectivity radius while any weakening of the assumptions leads to immediate counterexamples theorem a if m2 is complete with curvature k o then there exist c m such that for radii r c m r2 in higher dimensions however we have example 1 there are convex hypersurfaces md in rd 1 such that for d 3 no lower bounds as in 0 1 exist for md in dimensions 2 we therefore assume an upper curvature bound but then one gets uniform control even on the injectivity radius the following theorem has been rediscovered several times received by the editors august 10 1987 1980 mathematics subject classification 1985 revision primary 53c20 secondary 53a07 research supported by nsf grant mcs 7g01780 and the sloan foundation both authors were supported by invitations to the i h e s and the m s r i r 1988 american mathematical society 0002 9947 88 1 00 25 per page

regular cartan groupoids and longitudinal representations abstract with the intent of laying the groundwork for a program that aims at explicitly describing the space of cartan i e multiplicative connections on a general proper lie groupoid we begin to investigate the space of such connections in the regular case we point out that there is a close relationship between cartan connections on a proper regular groupoid and representations of the groupoid on its own longitudinal bundle i e on the vector distribution tangent to its orbits this observation enables us to reduce the original problem to a simpler one we carry out a prospective study of the latter problem and apply the resulting analysis to produce a number of examples in rank two which serve to illustrate the diversity of the possible obstructions to the existence of multiplicative connections

the role of synaptic transmission in a hiv model with memory a fractional order model for hiv with two transmission modes and treatment is proposed global stability of the disease free equilibrium is analyzed cell to cell transmission has a strong impact in the value of the reproduction number increasing drug efficacy decreases the infectiousness of the disease we propose a mathematical model with memory for the dynamics of hiv epidemics where two transmission modes cell to cell and virus to cell and drug resistance are considered systems with memory or fractional order systems have largely been applied to the modeling of several real life phenomena here we consider a fractional model where the order of the non integer derivative takes values in the interval 0 5 1 0 we prove the local and global stability of the disease free equilibrium we study the role of the cell to cell transmission probability on the dynamics of the model and on the value of the reproduction number r0 for distinct values of the fractional order derivative moreover we show evidence of an improvement of hiv infected patients quality of life due to the increase of the drug efficacy in the end important inferences are drawn

integrability of lie brackets in this paper we present the solution to a longstanding problem of differential geometry lie s third theorem for lie algebroids we show that the integrability problem is controlled by two computable obstructions as applications we derive explain and improve the known integrability results we establish integrability by local lie groupoids we clarify the smoothness of the poisson sigma model for poisson manifolds and we describe other geometrical applications

tractor calculi for parabolic geometries parabolic geometries may be considered as curved analogues of the homogeneous spaces g p where g is a semisimple lie group and p c g a parabolic subgroup conformal geometries and cr geometries are examples of such structures we present a uniform description of a calculus called tractor calculus based on natural bundles with canonical linear connections for all parabolic geometries it is shown that from these bundles and connections one can recover the cartan bundle and the cartan connection in particular we characterize the normal cartan connection from this induced bundle connection perspective we construct explicitly a family of fundamental first order differential operators which are analogous to a covariant derivative iterable and defined on all natural vector bundles on parabolic geometries for an important subclass of parabolic geometries we explicitly and directly construct the tractor bundles their canonical linear connections and the machinery for explicitly calculating via the tractor calculus

geometric structures as deformed infinitesimal symmetries a general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries specifically this model consists of a lie algebroid equipped with an affine connection compatible with the lie algebroid structure the curvature of this connection vanishes precisely when the structure is locally symmetric this model generalizes cartan geometries a substantial class to the intransitive case simple examples are surveyed and corresponding local obstructions to symmetry are identified these examples include foliations riemannian structures infinitesimal g structures symplectic and poisson structures

on complete manifolds of nonnegative kth ricci curvature in this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature key tools are comparison techniques and morse theory of busemann and distance functions

discrete fractional bihari inequality and uniqueness theorem of solutions of nabla fractional difference equations with non lipschitz nonlinearities abstract in this paper the discrete fractional bihari inequality is developed based on this inequality uniqueness theorem of solutions of fractional difference equations with non lipschitz nonlinearities is derived in addition the effectiveness of proposed results is illustrated by a nonlinear numerical example

stability regions for linear fractional differential systems and their discretizations this paper concerns with basic stability properties of linear autonomous fractional differential and difference systems involving derivative operators of the riemann liouville type we derive stability regions for special discretizations of the studied fractional differential systems including a precise description of their asymptotics our analysis particularly shows that discretizations based on backward differences can retain the key qualitative properties of underlying fractional differential systems in addition we introduce the backward discrete laplace transform and employ some of its properties as the main proof tool

on the perelman s reduced entropy and ricci flat manifolds with maximal volume growth in this paper we study the ricci flat manifolds with maximal volume growth using perelman s reduced volume of ricci flow we show that if m n g is an noncompact complete ricci flat manifold with maximal volume growth satisfying rm x rightarrow 0 as d x d g x p rightarrow infty then m n has the quadratic curvature decay some applications to this result are also presented

the generalized additional symmetries of the two toda lattice hierarchy the generalized additional symmetries of the two toda lattice hierarchy are investigated in this paper the algebraic structure of this generalized additional symmetry is showed as w w and the actions of the generalized additional symmetries on function are also discussed by restricting the two toda lattice hierarchy to the semi infinite case

stability of difference volterra equations direct liapunov method and numerical procedure abstract a procedure to construct liapunov functionals for discrete volterra equations is proposed using this procedure stability conditions are derived for general volterra difference equations some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for volterra integro differential equations are presented

new definitions of exponential hyperbolic and trigonometric functions on time scales abstract we propose two new definitions of the exponential function on time scales the first definition is based on the cayley transformation while the second one is a natural extension of exact discretizations our exponential functions map the imaginary axis into the unit circle therefore it is possible to define hyperbolic and trigonometric functions on time scales in a standard way the resulting functions preserve most of the qualitative properties of the corresponding continuous functions in particular pythagorean trigonometric identities hold exactly on any time scale dynamic equations satisfied by cayley motivated functions have a natural similarity to the corresponding differential equations the exact discretization is less convenient as far as dynamic equations and differentiation are concerned

h analogue of newton s binomial formula in this letter the h analogue of newton s binomial formula is obtained in the h deformed quantum plane which does not have any q analogue for h 0 this is just the usual one as it should be furthermore the binomial coefficients reduce to for h 1 some properties of the h binomial coefficients are also given finally it is hoped that such results will contribute to an introduction of the h analogue of the well known functions h special functions and h deformed analysis

the h laplace and q laplace transforms starting with a general definition of the laplace transform on arbitrary time scales we specify the particular concepts of the h laplace and q laplace transforms the convolution and inversion problems for these transforms are considered in some detail

generalized q hermite polynomials we consider two operators a and a in a hilbert space of functions on the exponential lattice where 0 q 1 the operators are formal adjoints of each other and depend on a real parameter we show how these operators lead to an essentially unique symmetric ground state 0 and that a and a are ladder operators for the sequence the sequence n 0 is shown to be a family of orthogonal polynomials which we identify as symmetrized q laguerre polynomials we obtain in this way a new proof of the orthogonality for these polynomials when 0 the polynomials are the discrete q hermite polynomials of type ii studied in several papers on q quantum mechanics

examples of manifolds of positive ricci curvature on presente de nouveaux exemples de varietes de riemann simplement connexes de dimension 7 a courbure de ricci positive qui n admettent pas de metrique de courbure sectionnelle non negative dans les cas compacts et non compacts

elements of q harmonic analysis abstract measures that act as q analogs of lebesgue measure are introduced some q analogs of fourier and laplace transforms are studied last the connection between certain spaces of q exponential type and discrete q fourier transforms is discussed

open manifolds with sectional curvature bounded from below in this paper we study complete open manifolds with sectional curvature bounded from below we obtain some pinching conditions to insure that such manifolds are diffeomorphic to a euclidean space

impulsive fractional q integro difference equations with separated boundary conditions in this paper we discuss the existence of solutions for impulsive fractional q integro difference equations with separated boundary conditions existence results are proved via fixed point theorems due to krasnoselskii and o regan while the uniqueness of solutions is accomplished by means of banach s contraction mapping principle examples illustrating the obtained results are also presented

lie algebroids holonomy and characteristic classes abstract we extend the notion of connection in order to study singular geometric structures namely we consider a notion of connection on a lie algebroid which is a natural extension of the usual concept of a covariant connection it allows us to define holonomy of the orbit foliation of a lie algebroid and prove a stability theorem we also introduce secondary or exotic characteristic classes thus providing invariants which generalize the modular class of a lie algebroid

complete ricci flat kahler manifolds of infinite topological type we display an infinite dimensional family of complete ricci flat kahler manifolds of complex dimension 2 for which the second homology is infinitely generated these are obtained from the gibbons hawking ansatz 2 by using infinitely many sparsely distributed centers

complete manifolds with nonnegative ricci curvature and quadratically nonnegatively curved infinity in this paper we study complete riemannian manifolds with nonnegative ricci curvature and quadratically nonnegatively curved infinity we show that m is of finite topological type if it has either maximal volume growth or minimal volume growth

exploring delayed mittag leffler type matrix functions to study finite time stability of fractional delay differential equations abstract in this paper we introduce a concept of delayed two parameters mittag leffler type matrix function which is an extension of the classical mittag leffler matrix function with the help of the delayed two parameters mittag leffler type matrix function we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations via the variation of constants method in addition we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations thereafter we present finite time stability results of nonlinear fractional delay differential equations under mild conditions on nonlinear term finally an example is presented to illustrate the validity of the main theorems

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

multiplicative forms and spencer operators motivated by our attempt to recast cartan s work on lie pseudogroups in a more global and modern language we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem from this point of view the novelty of this paper is that we study forms with coefficients however the main contribution of this paper is conceptual the discovery of the relationship between multiplicative forms and cartan s work is explored here to provide a completely new approach to integrability theorems for multiplicative forms the multiplicative point of view shows that modulo lie s functor the cartan pfaffian system itself a multiplicative form with coefficients is the same thing as the classical spencer operator

large volume growth and the topology of open manifolds in this paper we study complete noncompact riemannian manifolds with nonnegative ricci curvature and large volume growth we find some reasonable conditions to insure that this kind of manifolds are diffeomorphic to a euclidean space or have finite topological type

the q coulomb problem there exist operator solutions of the q coulomb problem in both configuration and momentum space since the arguments of the corresponding amplitudes are noncommuting however there are problems of physical interpretation here we answer the question of physical interpretation by associating with the operator amplitude in momentum space a numerically valued amplitude lying in a hilbert space defined by the suq 2 algebra this new amplitude now depending on commuting arguments may be interpreted by the usual rules of quantum mechanics and may also be fourier transformed to yield the amplitude in configuration space

on non homogeneous singular systems of fractional nabla difference equations in this article we study the initial value problem of a class of non homogeneous singular systems of fractional nabla difference equations whose coefficients are constant matrices by taking into consideration the cases that the matrices are square with the leading coefficient singular non square and square with a matrix pencil which has an identically zero determinant we provide necessary and sufficient conditions for the existence and uniqueness of solutions more analytically we study the conditions under which the system has unique infinite and no solutions furthermore we provide a formula for the case of the unique solution finally numerical examples are given to justify our theory