on a solvable system of difference equations of kth order it is shown that the next system of k difference equationsx n j a n j x n k j b n j i 1 kx n i s j i 1 c n j n n 0 j 1 k where a n j b n j c n j

application of bogomolny s transfer operator to a circular harmonic oscillator plus potential bogomolny s transfer operator has been used to find an analytical solution for the semiclassical energy eigenvalues of a simple two dimensional integrable system the system studied consists of a particle moving in an isotropic harmonic oscillator potential plus a potential the classical trajectories are used to construct the transfer matrix and an expression is derived for the eigenvalues of this matrix as a function of the energy these eigenvalue curves yield the semiclassical energy eigenvalues for the quantum system which turn out to be exactly the same as the results obtained by solving the schrodinger equation some insight into this unexpected agreement is provided by considering an exact transfer operator we show that when this operator is expanded in powers of planck s constant the leading term in the expansion is bogomolny s transfer operator

on the system of high order rational difference equations xn ayn p yn byn pxn qyn q in this paper we study the behavior of positive solutions of the system of high order rational difference equationsx n ay n p y n by n px n qy n q n 1 2 where p and q are positive integers with p

on the fuzzy difference equation x n 1 a x n x n m in this paper we study the existence the boundedness and the asymptotic behavior of the positive solutions of the fuzzy equation xn 1 a xn xn m n 0 1 where xn is a sequence of fuzzy numbers a is a fuzzy number and m 1 2

on a system of two nonlinear difference equations abstract in this paper we study the oscillatory behavior the boundedness of the solutions and the global asymptotic stability of the positive equilibrium of the system of two nonlinear difference equationsxn 1 a yn xn p yn 1 a xn yn q n 0 1 p qare positive integers

on some solvable systems of difference equations abstract we show that the following systems of difference equations x n 1 u n 1 v n y n 1 w n 1 s n n n 0 where u n v n w n s n are some of the sequences x n or y n with real initial values x 0 and y 0 are solvable in fourteen out of sixteen possible cases two cases are left unsolved probably the most interesting is the result in the case u n x n v n x n w n x n s n y n where a fascinating formula is obtained in an elegant way by using some ad hoc ideas

boundedness and asymptotic behavior of the solutions of a fuzzy difference equation in this paper we study the existence the uniqueness the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation xn 1 i 0kai xn ipi where k 1 2 ai i 0 1 k are positive fuzzy numbers pi i 0 1 k are positive constants and xi i k k 1 0 are positive fuzzy numbers

a scattering approach to the quantization of hamiltonians in two dimensions application to the wedge billiard we extend the scattering approach to quantize general hamiltonian systems in two dimensions and demonstrate it for the wedge billiard the resulting energy levels are given by the zeros of a secular equation the corresponding wavefunctions may also be constructed we verify analytically that this quantization condition is exact in the case of the integrable wedges this method has been applied successfully to computing numerically a few thousand levels for various values of the angle using these data we can check the gutzwiller trace formula and other semiclassical relations involving the scattering matrix and periodic orbits for which we find excellent agreement

the difference equation xn 1 α xn k i 0k 1cixn i has solutions converging to zero abstract the aim of this note is to show that the following difference equation x n 1 x n k i 0 k 1 c i x n i n 0 1 where k n c i 0 i 0 k 1 i 0 k 1 c i 1 and 1 has solutions which monotonically converge to zero this result shows the existence of such solutions which was not shown in the recently accepted paper a e hamza on the recursive sequence x n 1 x n 1 x n j math anal appl in press

on a nonlinear second order system of difference equations we prove that all positive solutions of the next system of difference equationsx n 1 a y n px n 1 q y n 1 a x n py n 1 q email protected n 0 where parameters a p and q are positive numbers are bounded if one of the following conditions is satisfied p 2 4 or p 1 q q 1 completely describing the boundedness character of the system in this case

domains of undefinable solutions of some equations and systems of difference equations for some classes of difference equations and systems of difference equations we find the sets of initial values for which solutions of these equations and systems are not well defined so called domains of undefinable solutions

on the system of rational difference equations xn a yn 1 xn pyn q yn a xn 1 xn ryn s abstract in this paper we study the behavior of positive solutions of the system of difference equations x n a y n 1 x n p y n q y n a x n 1 x n r y n s n 1 2 where p 2 q 2 r 2 s 2 a is a positive constant and x 1 max p r x 2 max p r x 0 y 1 max q s y 2 max q s y 0 are positive real numbers

on the positive solutions of the system of rational difference equations xn 1 1 yn k yn 1 yn xn myn m k abstract our aim in this paper is to investigate the periodic nature of solutions of the system of rational difference equations x n 1 1 y n k y n 1 y n x n m y n m k n 0 1 where k is a nonnegative integer m is a positive integer and the initial values x m x m 1 x 0 y m k y m k 1 y 0 are positive real numbers

bogomolny s semiclassical transfer operator for rotationally invariant integrable systems the transfer operator due to bogomolny provides a convenient method for obtaining a semiclassical approximation to the energy eigenvalues of a quantum system no matter what the nature of the analogous classical system in this paper the method is applied to integrable systems which are rotationally invariant in two and three dimensions in two dimensions the transfer operator is expanded in a fourier series in the angle variable while in three dimensions it is expanded in spherical harmonics in both cases when the fourier coefficients are evaluated using the stationary phase approximation we arrive at the einstein brillouin keller quantization conditions the associated maslov indices are shown to agree with the results calculated by well known simple rules the theory is applied to several rotationally invariant systems including the hydrogen atom and the isotropic harmonic oscillator in two and three dimensions the circle billiard a billiard inside a spherical cavity and a harmonic potential with a singular magnetic flux line

global behavior of a higher order nonlinear difference equation abstract in this paper we consider a higher order difference equation of the form x n 1 f x n x n k n 1 2 3 under some certain assumptions we obtain that the length of its finite semicycle is less than or equal to k moreover we concluded that the equation is permanent under some special conditions and a sufficient condition for its global asymptotical stability is given when these results are applied to the difference equation x n 1 x n k x n n 1 2 3 some sufficient and necessary conditions of global asymptotical stability of the equation are obtained we should state out that the main theorem in j math anal appl 233 1999 790 798 is included in one of our results

on a system of difference equations of odd order solvable in closed form it is shown that the next system of difference equationsx n c ny n 2 k 1 a n b ny n 2 k 1 i 1 k 1y n 2 i 1 x n 2 i y email protected nx n 2 k 1 a email protected nx n 2 k 1 i 1 k 1x n 2 i 1 y n 2 email protected n 0 where a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 2k 1 are real numbers is solved in closed form and the behavior of its well defined solutions when all the coefficients in the system are constant is described

on some systems of difference equations abstract in this note we show that the following systems of difference equations u n 1 v n 1 v n v n 1 u n 1 u n u n 1 v n 1 u n v n 1 u n 1 v n u n 1 u n 1 v n v n 1 v n 1 u n with n n 0 and complex initial values u 0 and v 0 are solvable explicitly by means of riccati equations and based on the formulae for the general solutions we present the behaviour of their solutions as n while the result is natural for the first system it is a bit surprising for the other two systems

unbounded increasing solutions of a system of difference equations with delays we consider a homogeneous system of difference equations with deviating arguments in the form y n k 1 q k n y n p k y n r k where n n 0 n 0 z p k r k are integers r k p k 0 q is a positive integer y y 1 y s t y n 0 r n 0 r 1 r s is an unknown discrete vector function s 1 is an integer r max r 1 r q y n y n 1 y n and k n ij k n i j 1 s are real matrices such that ij k n 0 n 0 1 0 and k 1 q j 1 s ij k n 0 for each admissible i and all n n 0 the behavior of solutions of this system is discussed for n the existence of unbounded increasing solutions in an exponential form is proved and estimates of solutions are given the scalar case is discussed as well

quantum section method for the soft stadium abstract the soft stadium is defined by a monomial potential with exponent as a parameter such that corresponds to the billiard the practical use of the quantum section method depends only on the partial separability of the system on both sides of the section which holds for all s in particular for 1 0 the system becomes globally separable allowing for a general test of the method for various values of the parameter we also tested the use of the asymptotic wkb type approximation in the construction of green s functions and asymptotic overlap integrals to obtain higher energy eigenvalues we show these approximations to be reliable

exact bogomolny sections for separable systems a good poincare section of a classical dynamical system defines a mapping that provides essential qualitative information about its orbits bogomolny s sections in quantum mechanics lead to an eigenvalue condition that depends on the poincare map in the semi classical limit the conditions for the construction of purely quantum mechanical bogomolny green functions are discussed here with reference to separable systems for appropriately chosen green functions the ensuing eigenvalue condition is shown to be exact this result is valid even if not a single classical orbit crosses the surface of section

global attractivity in a nonlinear difference equation abstract we consider the nonlinear difference equation x n a k 1 m b k x n k n 0 1 2 where a and bk k 1 2 m are nonnegative numbers with b mk 1bk 0 and we are interested in whether all positive solutions are attracted by the positive equilibrium l a 2 a 2 2 b

on the new solutions to the fuzzy difference equation xn 1 a bxn abstract in this paper using a generalization of division for fuzzy numbers we study the existence and global behavior of the fuzzy difference equation x n 1 a b x n n 0 1 where a and b are positive fuzzy numbers some examples are presented to illustrate the applicability of our results

pricing asian option by the fft with higher order error convergence rate under levy processes pricing asian options is a long standing hard problem there is no analytical formula for the probability density of its payoff even when the process of the underlying asset follows the simple lognormal diffusion process it is known that the density function of a discretely sampled asian option s payoff can be efficiently approximated by the fast fourier transform fft as a result we can accurately price the option under more general levy processes this paper shows that the pricing error of this approach called the fft approach can be decomposed into the truncation error the integration error and the interpolation error we prove that previous algorithms that follow the fft approach converge quadratically to improve the error convergence rate our proposed algorithms reduce the integration error by the higher order newton cotes formulas and new integration rules derived from the lagrange interpolating polynomial the interpolation error is reduced by the higher order newton divided difference interpolation formula consequently our algorithms can be sped up by the fft to achieve the same time complexity as previous algorithms but with a faster error convergence rate numerical results are given to verify the efficiency and the fast convergence of our algorithms

on first order fuzzy ricatti difference equation this paper deals with the existence the boundedness and the asymptotic behavior of the positive solutions to a first order fuzzy ricatti difference equation xn 1 a xnb xn n 0 1 where xn is a sequence of positive fuzzy numbers a b and the initial value x0 are positive fuzzy numbers moreover an example is given to demonstrate the effectiveness of the results obtained

asymptotic behavior of the positive solutions of an exponential type system of difference equations in this paper we study the existence of a unique positive equilibrium the boundedness persistence and global attractivity of the positive solutions of a system of the following two difference equations x n 1 ax n by n 1 e x n y n 1 cy n dx n 1 e y n n 0 1 where a b c d are positive constants and the initial values x 1 x 0 y 1 y 0 are positive numbers

on a solvable system of difference equations of fourth order it is shown that the next system of k difference equationsx n j a n j x n k j b n j i 1 kx n i s j i 1 c n j n n 0 j 1 k where a n j b n j c n j n n 0 j 1 k and initial values x i j i j 1 k are real numbers and where s n 1 k is defined by s km j j 1 j 1 k 1 s km k 1 m n 0 can be solved in closed form in an elegant way some applications of obtained formulas for the case when the sequences a n j b n j and c n j are k periodic are also given

the periodic character of positive solutions of the difference equation x n 1 x n x n k in this note we consider the nonlinear difference equation x n 1 f x n x n k n 0 1 where k 1 2 and the initial values gk g k 1 g 0 0 we give sufficient conditions under which every positive solution of this equation converges to a not necessarily prime period two solution if k is odd and the unique positive equilibrium if k is even

global asymptotic stability in a rational recursive sequence in this paper we study the global stability of the difference equationx n a bx n 1 cx n 1 2d x n 2 n 1 2 where a b 0 and c d 0 we show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters and every positive solution of the equation in the basin exponentially converges to the attractor

existence of bounded solutions of a class of neutral systems of functional differential equations some results on the existence of bounded solutions together with their first derivatives of a class of neutral systems of functional differential equations with complicated deviations which extend and unify numerous results in the literature are proved

pricing exotic derivatives exploiting structure in this paper we introduce a new fast and accurate numerical method for pricing exotic derivatives when discrete monitoring occurs and the underlying evolves according to a markov one dimensional stochastic processes the approach exploits the structure of the matrix arising from the numerical quadrature of the pricing backward formulas to devise a convenient factorization that helps greatly in the speed up of the recursion the algorithm is general and is examined in detail with reference to the cev constant elasticity of variance process for pricing different exotic derivatives such as asian barrier bermudan lookback and step options for which up to date no efficient procedures are available extensive numerical experiments confirm the theoretical results the matlab code used to perform the computation is available online at http www1 mate polimi it marazzina bp htm 2013 elsevier b v all rights reserved

on the system of rational difference equations abstract in this paper we study the behavior of the positive solutions of the system of two difference equations x n a 1 y n p y n a y n 1 x n r y n s n 1 2 where p 1 r 1 s 1 a 0 and x1 r x2 r x0 y1 max p s y2 max p s y0 are positive real numbers

an efficient convergent lattice algorithm for european asian options financial options whose payoff depends critically on historical prices are called path dependent options their prices are usually harder to calculate than options whose prices do not depend on past histories asian options are popular path dependent derivatives and it has been a long standing problem to price them efficiently and accurately no known exact pricing formulas are available to price them under the continuous time black scholes model although approximate pricing formulas exist they lack accuracy guarantees asian options can be priced numerically on the lattice a lattice divides the time to maturity into n equal length time steps the option price computed by the lattice converges to the option value under the black scholes model as n unfortunately only subexponential time algorithms are available if asian options are to be priced on the lattice without approximations efficient approximation algorithms are available for the lattice the fastest lattice algorithm published in the literature runs in o n 3 5 time whereas for the related pde method the fastest one runs in o n 3 time this paper presents a new lattice algorithm that runs in o n 2 5 time the best in the literature for such methods our algorithm exploits the method of lagrange multipliers to minimize the approximation error numerical results verify its accuracy and the excellent performance

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

on a symmetric system of max type difference equations we study behavior of positive solutions of the following max type system of difference equationsx n 1 maxc y n px n 1 p y n 1 maxc x n py n 1 p email protected n 0 where p email protected 0 extending some results in the literature among other results we prove that if p email protected 0 1 then every positive solution of the system converges to 1 1

on the system of rational difference equations xn a yn 3 yn byn 3 xn qyn q abstract in this paper we investigate the behaviour of the positive solutions of the system of rational difference equation x n a y n 3 y n by n 3 x n q y n q n 1 2 where q 3 is a positive integer with 3 q a and b are positive constants and the initial values x q 1 x q 2 x 0 y q 1 y q 2 y 0 are positive real numbers

accurate and efficient lattice algorithms for american style asian options with range bounds abstract asian options are popular path dependent options and it has been a long standing problem to price them efficiently and accurately since there is no known exact pricing formula for asian options numerical pricing formulas like lattice models must be employed a lattice divides a certain time interval into n time steps and the pricing results generated by the lattice called desired option values for convenience converge to the theoretical option value as n since a brute force lattice pricing algorithm runs in subexponential time in n some heuristics like interpolation method are used to strike the balance between the efficiency and the accuracy but the pricing results might not converge due to the accumulation of interpolation errors for pricing european style asian options the evaluation on the major part of the lattice can be done by a simple formula and the interpolation method is only required on the minor part of the lattice thus polynomial time algorithms with convergence guarantee for european style asian options can be derived however such a simple formula does not exist for american style asian options this paper suggests an efficient range bound algorithm that bracket the desired option value by taking advantages of the early exercise property of american style options we show that part of the lattice can be evaluated by a simple formula the interpolation method is required on the remaining part of the lattice and the upper and the lower bounds option values produced by the proposed algorithm are essentially numerically identical thus the theoretical option value is said to be obtained practically when the range bound algorithm runs on a lattice with large number of time steps

existence of solutions with a single semicycle for a general second order rational difference equation by making use of inclusion theorem we show in this paper the existence of solutions with a single semicycle for a general second order rational difference equation as a corollary our results positively confirm conjectures 4 8 3 and 5 4 6 in m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc 2002

on a system of difference equations which can be solved in closed form we present an elegant method for solving the system of difference equationsx n 1 1x email protected n y n email protected n 1y n email protected n 0 where parameters a and b and the initial values x 0 and y 0 are positive numbers and by using obtained formulae we show that every solution of the system converges to a two periodic solution

global asymptotic stability for two recursive difference equations two sufficient conditions are obtained for the global asymptotic stability of the following two recursive difference equationsx n 1 x nx n 1 x n 2 ax n x n 1x n 2 a n 0 1 2 andx n 1 x n 1 x nx n 2 ax nx n 1 x n 2 a n 0 1 2 where a 0 and the initial values x 2 x 1 x 0 0 some known results are included

on positive solutions of some classes of max type systems of difference equations global attractivity results on positive solutions of some special cases of the next max type system of difference equations xn i max1 xn l1 i i p1 i j1 1m1 i xn j1 i1 sj1 i1 qj1 i1 xn lri i i pri i jri 1mri i xn jri iri sjri iri qjri iri n n0 where ri n i 1 k k n sjt it 1 k qjt it 0 i 1 k t 1 ri and pj i 0 1 mj i n i 1 k j 1 ri are presented we also present some results on the behavior of positive solutions of the difference equation xn max p1xn 1 pkxn k xn max p1xn 1 pkxn k where pj j 1 k are nonnegative numbers

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

on a third order system of difference equations abstract we show that the system of difference equations x n 1 a 1 x n 2 b 1 y n z n 1 x n 2 c 1 y n 1 a 2 y n 2 b 2 z n x n 1 y n 2 c 2 z n 1 a 3 z n 2 b 3 x n y n 1 z n 2 c 3 n n 0 where the parameters a i b i c i i 1 2 3 and initial values x j y j z j j 0 1 2 are real numbers can be solved developing further the results in the literature

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

trichotomy of a system of two difference equations abstract in this paper we study the boundedness and the asymptotic behavior of the positive solutions of the system of difference equations x n 1 a i 1 k a i x n p i j 1 m b j y n q j y n 1 b i 1 k c i y n p i j 1 m d j x n q j where k m 1 2 a b ai ci bj dj i 1 k j 1 m are positive constants pi qj i 1 k j 1 m are positive integers such that p1

on a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results in the literature

on the fuzzy difference equations of finance in this paper our intention is to demonstrate the applicability of fuzzy difference equations in the field of finance in order to achieve that we follow the work done by deeba et al and papashinopoulos et al and we study the existence of positive solutions to fuzzy financial difference equations initially we appose a method realized by buckley that deals with fuzzy analogues of the elementary compound interest problem afterwards we approach this case via the fuzzy difference equations method after making a comparison between the two methods we notice that they end up to the same general formula the main contribution of this paper is the simplicity of the method presented which gives us the opportunity to create formulas for cases which are specialized and also more realistic than the very basic one of the compound interest moreover by using difference equations we can study the factors which cause the change in the value of money in different time periods as a source for these cases of finance we use the work of kwapisz and we study a well known problem of financial mathematics this problem deals with the time value of money and as a direct application of it we study the balance of bank deposits finally a numerical example is presented for better understanding

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied

on boundedness of the difference equation xn 1 pn xn 3s 1xn s 1 with period k coefficients in this note we study the following non autonomous equation xn 1 pn xn 3s 1xn s 1 n 0 1 where pn 0 is a period k sequence and s 1 2 with gcd 3s k 1 and the initial conditions x 3s 1 x0 0 we show that every positive solution of this equation is bounded