gamow vectors and borel summability in a class of quantum systems we analyze the detailed time dependence of the wave function x t for one dimensional hamiltonians h partial x 2 v x where v for example modeling barriers or wells and x 0 are compactly supported

arithmetic of normal rees algebras two reasons drive this study of the arithmetical properties of blow up like algebras first one still lacks effective criteria for the rees algebra of an ideal to be normal and second there are several new phenomena related to torsionfree symmetric algebras of modules notably bundlesthat need clarification as yet another related question there is the description of the canonical modules of such algebras a necessary step in determining their gorenstein loci the main thread for these questions is the understanding of the divisor class group essentially defined by a pair made up of a morphism and a free subgroup of finite rank the behaviour of the mapping has puzzling

the qualitative behavior of solutions of a nonlinear difference equation abstract this paper is concerned with the qualitative behavior of solutions to the difference equation x n 1 p qx n k 1 x n n 0 1 2 where the initial conditions x k x 1 x0 are non negative k 1 2 3 and the parameters p q are non negative we start by establishing the periodicity the character of semicycles the global stability and the boundedness of the above mentioned equation we also present solutions that have unbounded behavior it is worth to mention that this difference equation is a special case of an open problem was introduced by m r s kulenovic and g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002 several computational examples are given to support our theoretical discussions the presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation

the study of a class of rational difference equations abstract in this paper we study the global behavior of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we investigate the character of semicycles periodicity global stability and convergence to 2 k cycle solutions of mentioned equation

on the higher order rational recursive sequence xn axn k bx n 3k the main purpose of the current paper is to prove that every positive solution of the delay difference equationx n ax n k bx n 3 k where a b 0 the initial conditions x 3 k 1 x 3 k 2 x 0 0 and k 1 2 3 converges eventually to a period k solution we also give the results of computational examples to support our theoretical discussion

on the arithmetic and homology of algebras of linear type three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras e g cohen macaulayness factoriality the main tools are certain fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing up rings

exponential asymptotics and boundary value problems keeping both sides happy at all orders we introduce templates for exponential asymptotic expansions that in contrast to matched asymptotic approaches enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter graphic 1 and have a single boundary layer at one end of the interval for linear equations the template is a transseries that takes the form of a sliding ladder of exponential scales for nonlinear equations the transseries template is a two dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed an exponential asymptotic approach also reveals how boundary value problems force the surprising presence of transseries in the linear case and negative powers of terms in the series beyond all orders in the nonlinear case we also demonstrate how these transseries can be resummed to generate multiple scales type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter finally we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion 1 embed inline graphic 1 gif

a rational difference equation abstract we investigate the boundedness character the periodic nature and the global asymptotic stability of all positive solutions of the equation in the title with positive parameters and nonnegative initial conditions

analytic methods for obstruction to integrability in discrete dynamical systems a unique analytic continuation result is proven for solutions of a relatively general class of difference equations by using techniques of generalized borel summability this continuation allows for painleve property methods to be extended to difference equations it is shown that the painleve property pp induces under relatively general assumptions a dichotomy within first order difference equations all equations with pp can be solved in closed form on the contrary absence of pp implies under some further assumptions that the local conserved quantities are strictly local in the sense that they develop singularity barriers on the boundary of some compact set the technique produces analytic formulas to describe fractal sets originating in polynomial iterations 2004 wiley periodicals inc

dynamics of a higher order rational difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 p qy n 1 y n k n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as the global stability the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002

rees algebras of modules we study rees algebras of modules within a fairly general framework we introduce an approach through the notion of bourbaki ideals that allows the use of deformation theory one can talk about the essentially unique generic bourbaki ideal i e of a module e which in many situations allows one to reduce the nature of the rees algebra of e to that of its bourbaki ideal i e properties such as cohen macaulayness normality and being of linear type are viewed from this perspective the known numerical invariants such as the analytic spread the reduction number and the analytic deviation of an ideal and its associated algebras are considered in the case of modules corresponding notions of complete intersection almost complete intersection and equimultiple modules are examined in some detail special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way for these modules one is able to estimate the reduction number and other invariants in terms of the buchsbaum rim multiplicity

from the taylor series of analytic functions to their global analysis abstract we analyze the conditions on the taylor coefficients of an analytic function to admit global analytic continuation complementing a recent paper of breuer and simon on general conditions for natural boundaries to form a new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals the integral representations yield global information such as analytic continuability position of singularities asymptotics for large values of the variable and asymptotic location of zeros

on linear complete intersections or some of its localizations becomes a complete intersection is a major factor in section 1 we describe a jacobian formulation of serre s normality criterion for symmetric algebras it is used to provide several classes of examples of normal and non normal rees algebras of prime ideals in regular domains adding to the catalog of cohen macaulay domains that are symmetric algebras we examine almost complete intersections these bear a close relationship to certain height three ideals of

on the recursive sequence yn 1 p yn 1 qyn yn 1 abstract we investigate the periodic character and the global stability of solutions of the equation y n 1 p y n 1 qy n y n 1 with positive parameters and positive initial conditions

the oscillatory character of the recursive sequence xn 1 α βxn k 1a bxn 2k 1 abstract the aim of this work is to generalize techniques of difference equations in order to apply them to a wider class we start by establishing the boundedness the oscillatory character and the global stability of the equation x n 1 x n k 1 a bx n 2 k 1 n 0 1 2 with positive parameters k 1 2 and nonnegative initial conditions

the dimension and components of symmetric algebras let r be a commutative noetherian ring and let a4 be any finitely generated r module in this paper we compute the dimension of s m the symmetric algebra of m and identify many of the components of s m while the computation of the dimension of s m is not difficult the answer is intriguing and in view of the recent interest in symmetric algebras we hope the computation will be useful the dimension is given by the for mula dim s m max dim r p p m

study of a system of non linear difference equations arising in a deterministic model for hiv infection a deterministic model for hiv infection in the presence of combination therapy is introduced the model is then approximated by an implicit finite difference method a system of difference equations to solve the ivp system we initially purpose the discrete analogue of the continuous model of hiv infection then the boundedness the local and the global asymptotic stability of the equilibria of the system of difference equations the bistability phenomenon and the bifurcation phenomenon are also investigated in addition we will show that the associated approximational method is very robust since the equilibria and the stability conditions are independent on the time step finally numerical simulations of the system are presented when the system has meaningless solutions in view point of biology or our theory is incapable

dynamics of a rational difference equation using both theoretical and computational approaches this paper is concerned with the qualitative behavior of solutions to the difference equationx n 1 x n px n qx n k n 0 1 2 where the initial conditions x k x 1 x 0 are non negative email protected 1 2 3 and the parameters p q are non negative our concentration is on invariant intervals the character of semicycles the global stability and the boundedness of the above mentioned equation it is worth to mention that this difference equation was an open problem introduced by kulenovic and ladas in m r s kulenovic and g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002 our final comments are about informative examples

dynamics of a higher order rational difference equation abstract in this paper we will investigate a nonlinear rational difference equation of higher order our concentration is on invariant intervals periodic character the character of semicycles and global asymptotic stability of all positive solutions of x n 1 x n x n k bx n cx n k n 0 1 it is worth to mention that our results solve the open problem proposed by kulenvic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2002

rigorous bounds of stokes constants for some nonlinear ordinary differential equations at rank one irregular singularities a rigorous way of obtaining sharp bounds for stokes constants is introduced and illustrated on a concrete problem arising in applications

asymptotics beyond all orders asymptotology and borel summation asymptotics superasymptotics hyperasymptotics m berry the geometric model of crystal growth numerical analysis of the geometric model of crystal growth j m hammersley dendritic crystal growth an experimental assessment of continuum models of dendritic growth j p gollub directional solidification of solids directional growth of dilute mixtures and lamellar eutectics k kassner et al flow in a hele shaw cell viscous displacement in a hele shaw cell s tanveer the rapidly forced pendulum exponentially small estimate for separatrix splittings j scheurle et al ordinary differential equations singular perturbations of solitons y pomeau solitary water waves in the presence of small surface tension solitary waves with ripples beyond all orders j t beale problems in optics potpourri 23 additional articles index

geometric stability conditions for higher order difference equations is a higher order or delay autonomous difference equation if the order m is a fixed positive integer larger than 1 the real valued function f is assumed continuous on r and x x g r are the initial values in 1ym 0 addition to the fact that discretizations of scalar differential equations result in equations of type 1 direct applications of difference equations of order greater than 1 date back to more than half a century ago and nowadays these equations are increasingly encountered in both social and w x w x natural sciences see 4 and 6 for some applications and additional references a solution x of f x x s x is a fixed point or equilibrium of eq 1 like its differential analogs 1 can be and often is translated into a first order vector equation the instability or the asymptotic stability of x may then be determined by linearizing the associated vector function

existence uniqueness analyticity and borel summability for boussinesq equations through borel summation methods we analyze the boussinesq equations for coupled fluid velocity and temperature fields 1 ut u p u u ae2 f t u we prove that an equivalent system of integral equations in the borel variable p r dual to 1 t has a unique solution in a class of exponentially bounded functions implying the existence of a classical solution to 1 in a complex t region that includes a real positive time axis segment for analytic initial data and forcing it is shown that the solution is borel summable implying that formal series in powers of t is gevrey 1 asymptotic and within the time interval of existence the solution remains analytic with the same analyticity strip width as the initial data and forcing we also determine conditions on the integral equation solution that improve the estimate for existence time

bounds for solutions of a six point partial difference scheme abstract this paper is concerned with a class of partial difference equations which arise from discretising the one dimensional heat and the first order wave equations explicit bounds are found for their solutions and subexponential solutions are considered using the green s function we derive a stability criteria for arbitrary solutions of this equation

strongly cohen macaulay schemes and residual intersections this paper studies the local properties of closed subschemes y in cohen macaulay schemes x such that locally the defining ideal of y in x has the property that its koszul homology is cohen macaulay whenever this occurs y is said to be strongly cohen macaulay in x this paper proves several facts about such embeddings chiefly with reference to the residual intersections of y in x the main result states that any residual intersection of y in x is again cohen macaulay introduction our purpose in this paper is to investigate a property of a closed subscheme y in a noetherian cohen macaulay scheme x which we call strong cohen macaulayness this is a local property which we first describe for a local ring let x spec r where r is a cohen macaulay local ring and let y spec r i

effective normality criteria for algebras of linear type abstract the algebras studied here are subalgebras of rings of polynomials generated by 1 forms so called rees algebras with coefficients in a noetherian ring given a normal domain r and a torsionfree module e with a free resolution f 2 f 1 f 0 e 0 we study the role of the matrices of syzygies in the normality of the rees algebra of e when the rees algebra r e and the symmetric algebra s e coincide the main results characterize normality in terms of the ideal ic s e and of the completeness of the first s symmetric powers of e where c rank and s rankf0 ranke it requires that r be a regular domain special results under broader conditions on r are still more effective

dynamics of xn 1 xn 2k 1xn 2k 1 αxn 2l abstract we investigate the global stability of all positive solutions of the difference equation x n 1 x n 2 k 1 x n 2 k 1 x n 2 l where is a positive real number k 1 2 l 0 1 and the initial conditions are positive real numbers when x 1 1 is globally asymptotically stable also when 1 we show that this equation possesses solutions that converge to the two cycle 0 1 0 1

on the recursive sequence xn 1 α βxn k 1 γxn 2k 1bxn k 1 cxn 2k 1 abstract our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients our concentration is on invariant intervals the character of semicycles and the boundedness of the above mentioned equation our final comments are about informative examples

on rational recursive sequences our goal in this chapter is to gain some understanding of the dynamics of rational recursive sequences of the form x n 1 frac a sum limits i 0 k a i x n i b sum limits i 0 k b i x n i n 0 1 3 0 1 where k is a positive integer and the coefficients a b a 0 a k b 0 b k are nonnegative real numbers

behavior of lacunary series at the natural boundary abstract we develop a local theory of lacunary dirichlet series of the form k 1 c k exp z g k re z 0 as z approaches the boundary i r under the assumption g and further assumptions on c k these series occur in many applications in fourier analysis infinite order differential operators number theory and holomorphic dynamics among others for relatively general series with c k 1 the case we primarily focus on we obtain blow up rates in measure along the imaginary line and asymptotic information at z 0 when sufficient analyticity information on g exists we obtain borel summable expansions at points on the boundary giving exact local description borel summability of the expansions provides property preserving extensions beyond the barrier the singular behavior has remarkable universality and self similarity features if g k k b c k 1 b n or b n 1 n n n behavior near the boundary is roughly of the standard form re z b q x where q x 1 q if x p q q and zero otherwise the botcher map at infinity of polynomial iterations of the form x n 1 p x n 0 p turns out to have uniformly convergent fourier expansions in terms of simple lacunary series for the quadratic map p x x x 2 0 1 and the julia set is the graph of this fourier expansion in the main cardioid of the mandelbrot set

on the dimension and integrality of symmetric algebras the symmetric algebra of an r module e sym e has a relatively simple description since it is given as a quotient of a polynomial ring over r modulo an ideal j generated by linear homogeneous polynomials our first result relates the grade of j to certain determinantal ideals associated to such presentation in particular at least when r is cohen macaulay the krull dimension of sym e can be easily estimated from such formulation the expression of the grade of j is also useful in setting up conditions for the integrality and or factoriality of sym e it turns out that the integrality of sym e for r regular is related to phenomena of cohen macaulayness of the symmetric algebra there are however considerable difficulties in translating the details of a projective resolution of e into properties of the ideal 3 this analysis is carried out for some modules finally we consider for an ideal i the symmetric algebra over the ring

dynamics of the difference equation xn 1 xn pxn kxn q we study the invariant interval the character of semicycles the global stability and the boundedness of the difference equation x n 1 x n px n kx n q n 0 where the initial conditions x k x 1 x 0 are nonnegative and p q 0

the characteristics of a higher order rational difference equation in this paper we study the global behavior of the higher order nonlinear difference equationx n email protected email protected n k email protected n 2 k 1a bx n k 1 n 0 1 where a b and a b c are positive email protected 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we investigate the character of semicycles periodicity global stability and boundedness of above mentioned equation

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give

global stability of yn 1 p qyn ryn k1 yn in this paper we investigate the global stability of difference equation y n 1 p qy n ry n k1 y n n 0 1 2 where the initial conditions y k y 1 y 0 are non negative email protected 1 2 3 and the parameters p q and r are non negative we show that when rq 1 and k is odd then this equation possesses unbounded solutions

on the difference equation xn p xn sqxn t xn s abstract we investigate the persistence and global attractivity of the difference equation x n p x n s qx n t x n s n 1 2 3 with positive parameters and positive initial conditions our results not only include some previously known results but apply to some difference equations that have not been investigated so far

oscillation and asymptotic behavior of a class of higher order nonlinear recursive sequences abstract we investigate the global attractivity the periodic behavior and the invariant interval with the semicycles character of solutions of the equation x n 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 for all admissible non negative values of the parameters b c and the initial conditions x 2 k 1 x 1 x 0 we show that the solutions exhibit a periodic attitude depending on the parameters of the family of the above mentioned difference equation in comparison with each other

dynamics of a rational difference equation in this paper we investigate the periodic character invariant intervals oscillation and global stability of all positive solutions of the equationx n 1 px n x n kq x n k n 0 1 where the parameters p and q and the initial conditions x k x 0 are nonnegative real numbers as might be expected the two cases p q give rise to different dynamic behaviors in particular our results solve the open problem proposed by kulenvic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2001

global stability of a higher order rational recursive sequence abstract in this paper we study the qualitative behavior of solutions of the class of delay difference equation x n 1 x n k 1 x n 2 k 1 a bx n k 1 n 0 1 2 where the initial conditions x 2 k 1 x 1 x 0 are positive k 1 2 and the parameters a b are positive our concentration is on invariant intervals and the global stability of the above mentioned equation we obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients furthermore the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed it is also illustrated that for some special case of parameters the solution will be either a period 2 solution or will converge to the equilibrium point or will have unbounded solutions

on the recursive sequence xn 1 α βxn k 1a bxn k 1 cxn 2k 1 abstract in this paper we study the global asymptotic stability invariant intervals the character of semicycles and the boundedness of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 a bx n k 1 cx n 2 k 1 n 0 1 where a b c and are positive k 1 2 3 and the initial conditions x 2k 1 x 1 x0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable finally we present some informative examples

dynamics of a non linear difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 y n py n k y n q n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as periodicity invariant interval the character of semicycles the global stability and the boundedness of positive solutions of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002