on the laplace transform of the frechet distribution we calculate exactly the laplace transform of the frechet distribution in the form x 1 exp x 0 0 x for arbitrary rational values of the shape parameter i e for l k with l k 1 2 the method employs the inverse mellin transform the closed form expressions are obtained in terms of meijer g functions and their graphical illustrations are provided a rescaled frechet distribution serves as a kernel of frechet integral transform it turns out that the frechet transform of one sided levy law reproduces the frechet distribution

lasalle type theorems for stochastic differential delay equations the main aim of this paper is to establish the lasalle type asymptotic convergence theorems for the solutions of stochastic differential delay equations these stochastic versions are then applied to establish sufficient criteria for the stochastically asymptotic stability of the delay equations several examples are also given for illustration

nonclassical properties of semi coherent quantum states we study the properties of semi coherent quantum states introduced by mathews and eswaran in 1973 establishing relations between them and other families of nonclassical states such as displaced number states and multiphoton states we also analyse the photon number distribution photon statistics the wigner and marginal distribution functions and show the ways for possible further generalizations

new q deformed coherent states with an explicitly known resolution of unity we construct a new family of q deformed coherent states zq where 0 q 1 these states are normalizable on the whole complex plane and continuous in their label z they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem and expressed in terms of one of the two jacksons s q exponentials they also permit exact evaluation of matrix elements of physically relevant operators we use this to show that the photon number statistics for the states is sub poissonian and that they exhibit quadrature squeezing as well as an enhanced signal to quantum noise ratio over the conventional coherent state value finally we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics

numerical solutions of stochastic differential delay equations under the generalized khasminskii type conditions the classical existence and uniqueness theorem of the solution to a stochastic differential delay equation sdde requires the local lipschitz condition and the linear growth condition see e g 11 12 and 20 the numerical solutions under these conditions have also been discussed intensively see e g 4 10 13 16 17 18 21 22 and 24 recently mao and rassias 14 and 15 established the generalized khasminskii type existence and uniqueness theorems for sddes where the linear growth condition is no longer imposed these generalized khasminskii type theorems cover a wide class of highly nonlinear sddes but these nonlinear sddes do not have explicit solutions whence numerical solutions are required in practice however there is so far little numerical theory on sddes under these generalized khasminskii type conditions the key aim of this paper is to close this gap

quantum algebraic structures compatible with the harmonic oscillator newton equation we study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its newton equation examples of such structures are given they include undeformed and q deformed oscillators as well as the su 2 and the deformed suq 2 lie algebras which appear in a variety of physical models

difference ladder operators for a harmonic schrodinger oscillator using unitary linear lattices the formalism of raising and lowering operators is developed for the difference operator analogue of a quantum harmonic oscillator which acts on functions on a discrete support the grid under consideration is a mixed version of an equidistant lattice and a q linear grid several properties of the grid are described the grids under consideration are referred to by the name unitary linear lattices the ladder difference operators are derived and compared with the continuum situation the arising spectral problems for these operators are dealt by using the theory of bilateral jacobi operators in weighted l2 z spaces eventual applications to mathematical physics and numerical schrodinger theory are briefly discussed

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

state feedback impulse elimination of linear time varying singular systems abstract linear time varying singular systems e t x a t x b t u are treated a necessary and sufficient condition for the existence of linear state feedback to eliminate impulsive behavior is derived the condition obtained is a natural extension of existing time invariant theory

convergence and stability of the compensated split step theta method for stochastic differential equations with piecewise continuous arguments driven by poisson random measure abstract this paper deals with the numerical solutions of stochastic differential equations with piecewise continuous arguments sdepcas driven by poisson random measure in which the coefficients are highly nonlinear it is shown that the compensated split step theta csst method with 0 1 is strongly convergent in p th p 2 moment under some polynomially lipschitz continuous conditions it is also obtained that the convergence order is close to 1 p in terms of the stability it is proved that the csst method with 1 2 1 reproduces the exponential mean square stability of the underlying system under the monotone condition and some restrictions on the step size without any restriction on the step size there exists 1 2 1 such that the csst method with 1 is exponentially stable in mean square moreover if the drift and jump coefficients satisfy the linear growth condition the csst method with 0 1 2 also preserves the exponential mean square stability some numerical simulations are presented to verify the conclusions

solvability conditions consistency and weak consistency for linear differential algebraic equations and time invariant singular systems the general case we present several solvability concepts for linear differential algebraic equations daes with constant coefficients on the positive time axis as well as for the associated singular systems and investigate under which conditions these concepts are met next we derive necessary and sufficient conditions for global consistency of initial conditions for the dae as well as for the system and generalize these conditions with respect to our concept of weak consistency our distributional approach enables us to generalize results in an earlier paper where singular systems are assumed to have a regular pencil in the sense of gantmacher in particular we establish that global weak consistency in the system sense is equivalent to impulse controllability

projected euler maruyama method for stochastic delay differential equations under a global monotonicity condition abstract in this paper we investigate a projected euler maruyama method for stochastic delay differential equations with variable delay under a global monotonicity condition this condition admits some equations with highly nonlinear drift and diffusion coefficients we appropriately generalize the idea of c stability and b consistency given by beyn et al 2016 to the case with delay moreover the method is proved to be convergent with order one half in a succinct way finally some numerical examples are included to support our theoretical results

canonical commutation relations on the interval on the hilbert space h l 2 0 l where 0 l is a bounded interval of r1 the domain for the canonical commutation relation ccr and the ccr quasi algebra is constructed it is shown that the bogolubov inequality for a bose gas in a box is fulfilled

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

the projected explicit ito taylor methods for stochastic differential equations under locally lipschitz conditions and polynomial growth conditions abstract although the numerical methods to stochastic differential equations with the coefficients of locally lipschitz and polynomial growth have been discussed commonly by some authors there are few works on the high strong order numerical methods in this paper the mean square convergence of the general projected explicit ito taylor methods is considered under the assumption that drift and diffusion coefficient functions of stochastic differential equations satisfy the global monotonicity condition suitable local lipschitz conditions and polynomial growth conditions our analysis follows the idea of stochastic c stability and stochastic b consistency after giving the selection strategy of optimal parameters we prove that the projected explicit ito taylor methods owning optimal parameters share the same mean square convergence orders with non projected ones finally two numerical experiments are presented to show the effectiveness of theoretical results

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

coherent states and hermite polynomials on quaternionic hilbert spaces coherent states similar to the canonical coherent states of the harmonic oscillator labeled by quaternions are established on the right and left quaternionic hilbert spaces on the left quaternionic hilbert space reproducing kernels are established as was claimed by adler and millard j math phys 1997 38 2117 26 it is proved that these coherent states cannot be realized as a group action on a quaternionic hilbert space as an extension of the complex hermite polynomials quaternionic hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions

new generalized coherent states we construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation de q x dx e q qx with 0 q 1 and e q 0 1 we use this function in place of the usual exponential to generate new coherent states q z from the vacuum which are normalized and continuous in their label z these states allow the resolution of unity and a corresponding weight function is furnished by the exact solution of the associated stieltjes moment problem they also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally ordered function of boson operators we exemplify this by showing that the photon number statistics for these states is sub poissonian for any q 1 the states q z are squeezed we obtain and discuss their signal to quantum noise ratio the function e q x allows a natural generation of multiboson coherent states of arbitrary multiplicity which is impossible for the usual coherent states for q 1 all the above resul

impulse controllability and observability of rectangular descriptor systems the presence of impulsive responses in descriptor systems and how it relates to impulse controllability and impulse observability is considered it is shown that the equivalence between impulse controllability observability and the existence of an impulse eliminating semistate feedback output injection gain although true for square descriptor systems does not hold for rectangular descriptor systems in general hence necessary and sufficient conditions for the existence of an impulse eliminating semistate feedback output injection gain are presented

general displaced su 1 1 number states revisited the most general displaced number states based on the bosonic and an irreducible representation of the lie algebra symmetry of su 1 1 and associated with the calogero sutherland model are introduced here we utilize the barut girardello displacement operator instead of the klauder perelomov counterpart to construct new kind of the displaced number states which can be classified in nonlinear coherent states regime too with special nonlinearity functions they depend on two parameters and can be converted into the well known barut girardello coherent and number states respectively depending on which of the parameters equal to zero a discussion of the statistical properties of these states is included significant are their squeezing properties and anti bunching effects which can be raised by increasing the energy quantum number depending on the particular choice of the parameters of the above scenario we are able to determine the status of compliance with flexible statistics major parts of the

an exactly solvable periodic schrodinger operator the authors explicitly determine the band spectrum of a periodically strongly singular schrodinger operator in l2 r associated with the differential expression d2 dx2 s2 1 4 cos2 x s 2 1 4 sin2 x x epsilon r x epsilon pi z 2 0 s s 1 the corresponding density of states is calculated analytically

generation of coherent states of photon added type via pathway of eigenfunctions we obtain and investigate the regular eigenfunctions of simple differential operators xr dr 1 dxr 1 r 1 2 with the eigenvalues equal to 1 with the help of these eigenfunctions we construct a non unitary analogue of a boson displacement operator which will be acting on the vacuum in this way we generate collective quantum states of the fock space which are normalized and equipped with the resolution of unity with the positive weight functions that we obtain explicitly these states are thus coherent states in the sense of klauder they span the truncated fock space without first r lowest lying basis states 0 1 r 1 these states are squeezed sub poissonian in nature and reminiscent of photon added states in agarwal and tara 1991 phys rev a 43 492

theta schemes for sddes with non globally lipschitz continuous coefficients this paper establishes the boundedness convergence and stability of the two classes of theta schemes namely split step theta sst scheme and stochastic linear theta slt scheme for stochastic differential delay equations sddes with non globally lipschitz continuous coefficients when the drift f x y satisfies one sided lipschitz condition with respect to the present state x and the diffusion g x y obeys the global lipschitz condition with respect to the present term x but the delay terms y in the drift and diffusion may be highly nonlinear this paper first examines the strong convergence rates of the theta schemes for sddes it is also proved that the two classes of theta schemes for 1 2 1 converge strongly to the exact solution with the order 1 2 but for 0 1 2 the linear growth condition on drift f x y in x is needed for the strong convergence rates the exponential mean square stability of the theta schemes with 1 2 1 is also investigated for highly nonlinear sddes

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

discrete time approximations of stochastic delay equations the milstein scheme in this paper we develop a strong milstein approximation scheme for solving stochastic delay differential equations sddes the scheme has convergence order 1 in order to establish the scheme we prove an infinite dimensional ito formula for tame functions acting on the segment process of the solution of an sdde it is interesting to note that the presence of the memory in the sdde requires the use of the malliavin calculus and the anticipating stochastic analysis of nualart and pardoux given the nonanticipating nature of the sdde the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel

nonlinear semi coherent states their nonclassical features and phase properties semi coherent states semi css were previously introduced and their nonclassical properties studied in this paper we first propose a general formalism for the construction of nonlinear semi css as an extension of semi css and then their explicit form in the number state basis is generally deduced next as some physical appearances of the formalism after choosing two well known nonlinearity functions nonclassical features and quantum statistical properties of the associated states are studied by evaluating quadrature squeezing higher order squeezing and the mandel parameter as well as their phase properties and number and phase squeezing of the introduced states finally the results are compared with the properties of semi css

on the completeness of photon added coherent states we demonstrate explicitly the completeness of photon added coherent states pacss introduced by agarwal and tara agarwal g s and tara k 1991 phys rev a 43 492 and defined up to normalization by m z m 0 1 2 where is the boson creation operator and z are conventional glauber klauder coherent states we find the analytical form of the positive weight function in their resolution of unity by solving the associated stieltjes power moment problem we furnish an example of generation of another set of pacss which are complete

temporally stable coherent states for infinite well and poschl teller potentials this article is a direct illustration of a construction of coherent states which has been recently proposed by two of us jpg and jk we have chosen the example of a particle trapped in an infinite square well and also in poschl teller potentials of the trigonometric type in the construction of the corresponding coherent states we take advantage of the simplicity of the solutions which ultimately stems from the fact they share a common su 1 1 symmetry a la barut girardello many properties of these states are then studied both from mathematical and from physical points of view

time domain solution to descriptor variable systems the time domain solution to the fast subsystem of the descriptor variable system of the form ex spl dot ax bu with e singular is given by laplace inverse transformation a new and significant feature of the solution is that it contains impulse terms excited by the values of sufficiently smooth input u and its derivatives at the initial time point based on the solution the notions of consistent initial conditions classical solutions and impulse free solutions are discussed perfectly the impulse controllability is discussed also which admits the same interpretation as the controllability at infinity in frequency domain

q graded heisenberg algebras and deformed supersymmetries the notion of q grading on the enveloping algebra generated by products of q deformed heisenberg algebras is introduced for q complex number in the unit disk within this formulation we consider the extension of the notion of supersymmetry in the enveloping algebra we recover the ordinary z2 grading or grassmann parity for associative superalgebra and a modified version of the usual supersymmetry as a specific problem we focus on the interesting limit q 1 for which the arik and coon deformation j math phys 17 524 1976 of the heisenberg algebra allows one to map fermionic modes to bosonic ones in a modified sense different algebraic consequences are discussed

nonlinear conserved charge coherent state and its relation to nonlinear entangled state by suitably averaging over u 1 group action on a two mode nonlinear coherent state we derive the nonlinear conserved charge coherent state based on it a new nonlinear entangled state f g and its deductive state are introduced their completeness relation is proved a new two mode nonlinear squeezing operator in f g representation is obtained which in turn leads to a nonlinear negative binomial state

on the completeness of coherent states generated by binomial distribution we demonstrate the completeness of boson and spin coherent states generated by binomial distribution by furnishing the exact analytical form of their respective positive weight functions in resolution of unity these weight functions are solutions of the associated truncated stieltjes power moment problem we elaborate on the non unique character of these solutions

convergence and stability of two classes of theta milstein schemes for stochastic differential equations abstract this paper examines convergence and stability of the two classes of theta milstein schemes for stochastic differential equations sdes with non global lipschitz continuous coefficients the split step theta milstein sstm scheme and the stochastic theta milstein stm scheme for 1 2 1 this paper concludes that the two classes of theta milstein schemes converge strongly to the exact solution with the order 1 for 0 1 2 under the additional linear growth condition for the drift coefficient these two classes of the theta milstein schemes are also strongly convergent with the standard order this paper also investigates exponential mean square stability of these two classes of the theta milstein schemes for 1 2 1 these two theta milstein schemes can share the exponential mean square stability of the exact solution for 0 1 2 similar to the convergence under the additional linear growth condition these two theta milstein schemes can also reproduce the exponential mean square stability of the exact solution

deformed photon added nonlinear coherent states and their non classical properties in this paper we will try to present a general formalism for the construction of deformed photon added nonlinear coherent states dpancss f m which in a special case lead to the well known photon added coherent state pacs m some algebraic structures of the introduced dpancss are studied and particularly the resolution of the identity as the most important property of generalized coherent states is investigated meanwhile it will be demonstrated that the introduced states can also be classified in the f deformed coherent states with a special nonlinearity function next we will show that these states can be produced through a simple theoretical scheme a discussion on the dpancss with negative values of m i e f m is then presented our approach has the potentiality to be used for the construction of a variety of new classes of dpancss corresponding to any nonlinear oscillator with known nonlinearity function as well as arbitrary solvable quantum system with known discrete non degenerate spectrum finally after applying the formalism to a particular physical system known as the poschl teller p t potential and the nonlinear coherent states corresponding to a specific nonlinearity function some of the non classical properties such as the mandel parameter second order correlation function in addition to first and second order squeezing of the corresponding states will be investigated numerically

strong convergence of split step backward euler method for stochastic differential equations with non smooth drift in this paper we are concerned with the numerical approximation of stochastic differential equations with discontinuous nondifferentiable drifts we show that under one sided lipschitz and general growth conditions on the drift and global lipschitz condition on the diffusion a variant of the implicit euler method known as the split step backward euler ssbe method converges with strong order of one half to the true solution our analysis relies on the framework developed in d j higham x mao and a m stuart strong convergence of euler type methods for nonlinear stochastic differential equations siam journal on numerical analysis 40 2002 1041 1063 and exploits the relationship which exists between explicit and implicit euler methods to establish the convergence rate results

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

controllability and observability of linear time varying singular systems controllability and observability of analytically solvable linear time varying singular systems are considered our analysis includes the possible impulsive zero input response based on the definitions of controllability and observability naturally extended from the ones adopted in the time invariant setting sufficient and necessary conditions for these two properties are derived furthermore according to the specific definitions of these concepts used here the authors show that the dual relationship between controllability and observability does not hold

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

convergence rate of numerical solutions to sfdes with jumps abstract in this paper we are interested in numerical solutions of stochastic functional differential equations with jumps under a global lipschitz condition we show that the p th moment convergence of euler maruyama numerical solutions to stochastic functional differential equations with jumps has order 1 p for any p 2 this is significantly different from the case of stochastic functional differential equations without jumps where the order is 1 2 for any p 2 it is therefore best to use the mean square convergence for stochastic functional differential equations with jumps moreover under a local lipschitz condition we reveal that the order of mean square convergence is close to 1 2 provided that local lipschitz constants valid on balls of radius j do not grow faster than log j

impulse analysis of linear time varying singular systems impulsive part of solution to initial value problem of linear time varying singular systems is given explicitly impulse observability and impulse controllability for the systems are redefined

photon added coherent states as nonlinear coherent states the states m defined as up to a normalization constant and where m is a non negative integer are shown to be the eigenstates of where is a nonlinear function of the number operator the explicit form of is constructed the eigenstates of this operator for negative values of m are introduced the properties of these states are discussed and compared with those of the state m the eigenstates corresponding to the positive and negative values of m are shown to be the result of nonunitarily deforming the number states m and 0 respectively

almost sure convergence rate of θ em scheme for neutral sddes abstract this paper is concerned with almost sure convergence of em scheme for neutral stochastic differential delay equations under local lipschitz conditions with 1 2 1 it is revealed that the strong convergence rate under local lipschitz condition is 1 2 while almost sure convergence rate is less than 1 2 provided the local lipschitz constant valid on balls of radius r grows not faster than log r

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

infinite horizon optimal control of forward backward stochastic differential equations with delay we consider a problem of optimal control of an infinite horizon system governed by forward backward stochastic differential equations with delay sufficient and necessary maximum principles for optimal control under partial information in infinite horizon are derived we illustrate our results by an application to a problem of optimal consumption with respect to recursive utility from a cash flow with delay

mittag leffler coherent states we create a family of boson coherent states using the functions of mittag leffler ml e z 0 and their generalizations e z 0 instead of exponentials these states are shown to satisfy the usual requirements of normalizability continuity in the label and the resolution of unity with a positive weight function this last quantity is found for arbitrary 0 as a solution of an associated stieltjes moment problem in addition for m 1 2 3 and 1 corresponding to em z we propose and analyse special q deformations 0 q1 of the functions em z which serve as a tool to define q deformed coherent states of ml type we provide the expressions for expectation values of physical quantities for all the above states we discuss physical properties of these states noting that they are squeezed the ml coherent states are sub poissonian in nature whereas the q deformed ml states can be sub and super poissonian depending on q all these states are shown to be eigenstates of deformed boson operators whose commutation relations are given

new p q μ ν f deformed states we construct a new family of deformed states these states are normalizable on the whole complex plane and continuous in their label z under fixed values of parameters they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem in addition to the mathematical characteristics the quantum statistical properties of these states are analytically and numerically discussed in detail in the context of conventional as well as deformed quantum optics we find that for non deformed photons the states exhibit quadrature squeezing and their photon number statistics is sub poissonian on the other hand for deformed photons the states are super poissonian and no quadrature squeezing occurs

coherent states for isospectral oscillator hamiltonians coherent states for a family of isospectral oscillator hamiltonians are derived from a suitable choice of annihilation and creation operators the fock bargmann representation is also considered

stochastic differential delay equation moment stability and application to hematopoietic stem cell regulation system we study the moment stability of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise the results established here are applied to examining the local stability of the hematopoietic stem cell hsc regulation system in the presence of noise the stability of the first moment for the solutions of a linear differential delay equation under stochastic perturbation is identical to that of the unperturbed system however the stability of the second moment is altered by the perturbation we obtain using laplace transform techniques necessary and sufficient conditions for the second moment to be bounded in applying the results to the hsc system we find that the system stability is sensitive to perturbations in the stem cell differentiation and death rates but insensitive to perturbations in the proliferation rate

stability of analytical and numerical solutions of nonlinear stochastic delay differential equations this paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations sddes we derive sufficient conditions for the stability contractivity and asymptotic contractivity in mean square of the solutions for nonlinear sddes the results provide a unified theoretical treatment for sddes with constant delay and variable delay including bounded and unbounded variable delays then the stability contractivity and asymptotic contractivity in mean square are investigated for the backward euler method it is shown that the backward euler method preserves the properties of the underlying sddes the main results obtained in this work are different from those of razumikhin type theorems indeed our results hold without the necessity of constructing or finding an appropriate lyapunov functional

coherent states for systems with discrete and continuous spectrum for a system with one degree of freedom coherent states that are parametrized by classical canonical action angle variables are introduced these states also possess continuity of labelling a resolution of unity and temporal stability the insistence on canonical action angle variables strongly restricts any remaining arbitrariness in the coherent state definition such states are introduced for semibounded hamiltonian operators having either a discrete or a continuous spectrum hamiltonians that have both discrete and continuous parts in their spectrum are also discussed

convergence and stability of the semi tamed euler scheme for stochastic differential equations with non lipschitz continuous coefficients recently explicit tamed schemes were proposed to approximate the sdes with the non lipschitz continuous coefficients this work proposes a semi tamed euler scheme which is also explicit to solve the sdes with the drift coefficient equipped with the lipschitz continuous part and non lipschitz continuous part it is shown that the semi tamed euler converges strongly with the standard order one half to the exact solution of the sde we also investigate the stability inheritance of the semi tamed euler schemes and reveal that this scheme does have advantage in reproducing the exponential mean square stability of the exact solution numerical experiments confirm the theoretical analysis

photon added squeezed vacuum one photon state as an even odd nonlinear coherent state we show that the photon added squeezed vacuum a m s z 0 and the photon added squeezed one photon state a m s z 1 where m is a non negative integer may be regarded as even and odd nonlinear coherent states respectively to achieve this we derive an operator valued function f n m of the number operator n a a such that a m s z i i 0 1 are eigenstates of f n m a 2 based on this and using the unified method developed by shanta et al we find that a m s z i can be equivalently expressed in exponential forms the eigenstates of f n m a 2 are also constructed and their nonclassical features are studied in detail

new physics in landau levels the most general displaced number coherent states based on the heisenberg su 2 and su 1 1 lie algebras symmetries are constructed they depend on two parameters and can be converted into the well known photon added two variable glauber coherent states and displaced number states respectively depending on which of the parameters is equal to zero the relations of the weyl heisenberg algebra guarantee a corresponding resolution of the identity conditions a discussion of the statistical properties of these states is included significant are their squeezing properties which can be raised by increasing the energy and angular momentum quantum numbers n and m the maximum squeezing is obtained for bext 0 depending on the particular choice of parameters in the above scenarios we are able to determine the status of compliance with poissonian statistics in the limiting case we obtain a major result about the non classical properties of the glauber minimum uncertainty coherent states in other words in addition to the requirement to minimize uncertainty conditions they carry non classical features too finally a theoretical framework is proposed to generate them

the construction of some important classes of generalized coherent states the nonlinear coherent states method considering some important classes of generalized coherent states known in the literature we demonstrated that all of them can be created via conventional methods i e the lowering operator eigenstate and the displacement operator techniques using the nonlinear coherent states approach as a result we obtained a unified method to construct a large class of coherent states which have already been introduced by different prescriptions

nth even order minimum uncertainty products this paper considers the problem of finding the quantum states that minimize the products of the even nth order fluctuation of two canonically conjugate operators the problem is first attacked in an abstract form and an equation derived for the desired states a consideration of the n 2 case then leads to a connection with the new concept of squeezed states of the electromagnetic field and the usual exact solution involving the coherent state the concept of higher order squeezing is touched on to motivate the further discussion the cases n 4 6 and 8 are then taken up and approximate solutions to them found via a new numerical technique during these calculations it is noted that only the first few terms in the expansion in number states figure in the solution this observation is then exploited to find an approximate closed solution to the problem valid for all n

numerical solutions of stochastic differential delay equations under local lipschitz condition in this paper a variant of the euler maruyama method is used to define the numerical solutions for stochastic differential delay equations sddes with variable delay the key aim is to show that the numerical solutions will converge to the true solutions of the sddes under the local lipschitz condition

q ν deformation of generalized basic hypergeometric states we provide with a q deformation of the generalized hypergeometric coherent states defined such that the normalization function is given by generalized basic hypergeometric functions these states are eigenstates of suitably defined deformed lowering operators we study the domain of convergence of the corresponding normalization function on the basis of these states we investigate generalized basic hypergeometric husimi distributions and corresponding phase distributions as well as new analytic basis representations of arbitrary quantum states in bargmann and hardy spaces the quantum statistical properties of the states such as photon counting statistics and quadrature squeezing are analytically and numerically discussed in the framework of conventional quantum optics

technical communique impulse observability and impulse controllability of linear time varying singular systems impulse controllability and impulse observability of linear time varying singular systems in a standard canonical form are examined we focus on the fast subsystems with constant rank n matrices sufficient and necessary conditions are derived and presented directly in terms of system parameters conditions obtained for these two properties are algebraically dual to each other

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

a deformed oscillator with coulomb energy spectrum a deformed oscillator with eigenvalues equal to the eigenvalues of the schrodinger equation with the coulomb potential is constructed the deformed oscillator algebra has a polynomial realization and the associate deformed operations of integration and differentiation are studied

the truncated stieltjes moment problem solved by using kernel density functions in this work the problem of the approximate numerical determination of a semi infinite supported continuous probability density function pdf from a finite number of its moments is addressed the target space is carefully defined and an approximation theorem is proved establishing that the set of all convex superpositions of appropriate kernel density functions kdfs is dense in this space a solution algorithm is provided based on the established approximate representation of the target pdf and the exploitation of some theoretical results concerning moment sequence asymptotics the solution algorithm also permits us to recover the tail behavior of the target pdf and incorporate this information in our solution a parsimonious formulation of the proposed solution procedure based on a novel sequentially adaptive scheme is developed enabling a very efficient moment data inversion the whole methodology is fully illustrated by numerical examples

disturbance impulse controllability in descriptor system the descriptor system ex ax gu hv with external disturbance v is considered to characterize the possibility to cancel the impulses produced by external disturbance v through using control u the concept of disturbance impulse controllability is introduced

generalized deformed oscillator and nonlinear algebras the harmonic oscillator is deformed arbitrarily and the properties of the deformed oscillator algebra are studied the eigenstates and eigenvalues of the deformed oscillator are calculated this generalized deformed oscillator algebra is used to create nonlinear deformation of the classical su 2 algebra the deformed boson realizations of an arbitrary nonlinear su 2 algebra are calculated the algebra of the q deformed oscillator is a special case of the generalized deformed oscillator algebra

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

dynamical symmetry approach to periodic hamiltonians we show that dynamical symmetry methods can be applied to hamiltonians with periodic potentials we construct dynamical symmetry hamiltonians for the scarf potential and its extensions using representations of su 1 1 and so 2 2 energy bands and gaps are readily understood in terms of representation theory we compute the transfer matrices and dispersion relations for these systems and find that the complementary series plays a central role as well as nonunitary representations

strong convergence of compensated split step theta methods for sdes with jumps under monotone condition abstract this paper is concerned with strong convergence of the compensated split step theta csst method for autonomous stochastic differential equations sdes with jumps under weaker conditions more precisely the diffusion coefficient and the drift coefficient are both locally lipschitz and the jump diffusion coefficient is globally lipschitz while they all satisfy the monotone condition it is proved that the csst method is strongly convergent of order 1 2 finally the obtained results are supported by numerical experiments

convergence and stability of the split step theta method for stochastic differential equations with piecewise continuous arguments abstract in this paper we consider the stochastic differential equations with piecewise continuous arguments sdepcas in which both the drift and the diffusion coefficients do not satisfy the global lipschitz and linear growth conditions especially the diffusion coefficients are highly non linear growing it is proved that the split step theta sst method with 1 2 1 is strongly convergent to sdepcas under the local lipschitz monotone and one sided lipschitz conditions it is also obtained that the sst method with 1 2 1 preserves the exponential mean square stability of sdepcas under the monotone condition and some condition on the step size without any restriction on the step size there exists 1 2 1 such that the sst method with 1 is exponentially stable in mean square moreover for sufficiently small step size the rate constant can be reproduced some numerical simulations are presented to illustrate the analytical theory

new nonlinear coherent states based on hypergeometric type operators the main goal of this paper is to present an alternative method to construct new kinds of nonlinear coherent states to do this we first establish a class of hypergeometric type of generalized displacement operators 1fr 0 0 1 r 1 za act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the fock space which admit a resolution of the identity through a positive definite measure on the complex plane furthermore realization of the compact form of these states as functions of the position coordinate x for r 2 leads to a generating function of the hermite polynomials in terms of the modified bessel function finally studying some statistical characters reveals that they have indeed non classical features such as squeezing an anti bunching effect and sub poissonian statistics too

strong convergence rates for backward euler on a class of nonlinear jump diffusion problems we generalise the current theory of optimal strong convergence rates for implicit euler based methods by allowing for poisson driven jumps in a stochastic differential equation sde more precisely we show that under one sided lipschitz and polynomial growth conditions on the drift coefficient and global lipschitz conditions on the diffusion and jump coefficients three variants of backward euler converge with strong order of one half the analysis exploits a relation between the backward and explicit euler methods

the quantum mechanical schrodinger picture of a q oscillator a q deformed version of standard quantum mechanics in the coordinate schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so called q discrete derivative as the representative of the momentum operator the chosen q discrete derivative is symmetric with respect to the exchange of q and q 1 under the usually adopted assumptions a q deformed schrodinger equation is derived for a harmonic oscillator the complete set of eigenfunctions can be explicitly constructed as special q functions and the corresponding energy eigenvalues are identical to those obtained by biedenharn in his pioneering work 1989 this q deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions

one dimensional schrodinger operators with interactions singular on a discrete set on etudie le caractere auto adjoint des operateurs de schrodinger sur y u j i x j y j ou on suppose y j x j 1

generalized deformed oscillator corresponding to the modified poschl teller energy spectrum a generalized deformed oscillator with eigenvalues equal to the eigenvalues of the schrodinger equation with the modified poschl teller potential is constructed for special values of the potential depth the deformed oscillator algebra has a finite dimensional irreducible representation the polynomial representation and the associate deformed operations of integration and differentiation are studied the results of this study are general and they can be applied directly to the case of the q deformed oscillator with q being a root of unity