practical internal stability of n d discrete systems this paper deals with the problem of asymptotic stability for n d discrete systems in the practical sense that the system input and output signals are unbounded in at most one dimension a definition of practical internal stability is introduced and necessary and sufficient conditions are derived the obtained results show that practical internal stability is less restrictive and more relevant for practical applications than the conventional two dimensional 2 d internal stability

the klein gordon equation of generalized hulthen potential in complex quantum mechanics we have investigated the reality of exact bound states of complex and or pt symmetric non hermitian exponential type generalized hulthen potential the klein gordon equation has been solved by using the nikiforov uvarov method which is based on solving the second order linear differential equations by reduction to a generalized equation of hypergeometric type in many cases of interest negative and positive energy states have been discussed for different types of complex potentials

polynomial matrices with given determinant abstract let a be a q n polynomial matrix 1 q a if suitable polynomials 1 n exist such that one can write 1 a i 1 n a i n where ai 1 ai n is one row in a and 1 n is a unimodular row then a can be row bordered to an n n polynomial matrix m with det m some examples are discussed

random walk and the heat equation on superspace and anyspace random walks are used to study diffusion on anyspace anyspace is characterized by coordinate with n 0 and statistics e2 i n between independent copies anyonic integration and anyonic dirac functions are introduced and reduced to familiar results for supersymmetry when n 2 these ingredients are then used to formulate and solve the resulting anyonic diffusion equation

alternative descriptions and bipartite compound quantum systems we analyse some features of alternative hermitian and quasi hermitian quantum descriptions of simple and bipartite compound systems we show that alternative descriptions of two interacting subsystems are possible if and only if the metric operator of the compound system can be obtained as a tensor product of positive operators on component spaces some examples also show that such property could be strictly connected with symmetry properties of the non hermitian hamiltonian

asymptotic stability of two dimensional digital filters under quantization the stability of 2 d digital filters implemented with two s complement quantization is investigated the difference equation model is considered for the following cases i quantization after each product and ii quantization after a complete sum of products for both cases sufficient conditions are established for asymptotic stability the 2 d state space roesser s 1975 model is also considered under two s complement quantization and a sufficient condition is derived for its stability some examples are given to illustrate the results

exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator is studied in a region of the coupling constant plane in which there is a sequence of exponentially small subseries beyond the standard rayleigh schrodinger perturbation theory rspt power series we give a simple algorithm for the calculation of these subseries to any desired order in terms of the rspt coefficients expressed as polynomials in the quantum number and illustrate our results with numerical borel pade summations of the expansion up to third exponentially small order

stability analysis of 1 d and 2 d fixed point state space digital filters using any combination of overflow and quantization nonlinearities a new criterion together with its frequency domain interpretation for the global asymptotic stability of zero input one dimensional 1 d state space digital filters under various combinations of overflow and quantization nonlinearities and for the situation where quantization occurs after summation only is presented a condition in closed form involving solely the parameters of the state transition matrix for the nonexistence of limit cycles in second order digital filters is derived improved versions of some of the stability results due to leclerc and bauer 1994 are established finally the approach is extended to two dimensional 2 d digital filters described by the roesser and the fornasini marchesini second local state space models

rational and semi rational solutions of the y nonlocal davey stewartson i equation abstract the nonlocal davey stewartson ds i equation with a parity time symmetric potential with respect to the y direction which is called the y nonlocal ds i equation is a two dimensional analogue of the nonlocal nonlinear schrodinger nls equation the multi breather solutions for the y nonlocal ds i equation are derived by using the hirota bilinear method lump type solutions and hybrid solutions consisting of lumps sitting on periodic line waves are generated by long wave limits of the obtained soliton solutions also various types of analytical solutions for the nonlocal nls equation with negative nonlinearity including both the akhmediev breathers and the peregrine rogue waves sitting on periodic line waves can be generated with appropriate constraints on the parameters of the obtained exact solutions of the y nonlocal ds i equation particularly we show that a family of hybrid solitons describing the peregrine rogue wave that coexists with the akhmediev breather both of them sitting on a spatially periodic background can be thus obtained

an exhaustive search algorithm for checking limit cycle behavior of digital filters the presence of limit cycles that may arise in fixed point arithmetic implementation of a digital filter can significantly impair its performance this paper presents an algorithm to determine the presence absence of such limit cycles for generality the filter is taken to be in its state space formulation the algorithm is applicable independent of filter order type of quantization nonlinearity and whether the accumulator is single or double length it may be utilized to construct limit cycle free regions in filter coefficient space once a filter is determined to be limit cycle free a technique that provides a robustness region in coefficient space where all filters remain limit cycle free is also presented

two dimensional dissipative control and filtering for roesser model this paper investigates the problems of two dimensional 2 d dissipative control and filtering for a linear discrete time roesser model first a novel sufficient condition is proposed such that the discrete time roesser system is asymptotically stable and 2 d q s r hbox alpha dissipative special cases such as 2 d passivity performance and 2 d h infty performance and feedback interconnected systems are also discussed based on this condition new 2 d q s r hbox alpha dissipative state feedback and output feedback control problems are defined and solved for a discrete time roesser model the design problems of 2 d q s r hbox alpha dissipative filters of observer form and general form are also considered using a linear matrix inequality lmi approach two examples are given to illustrate the effectiveness and potential of the proposed design techniques

symmetrically regularized eckart poschl teller and hulthen potentials parity times time reversal symmetry of complex hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for hermiticity perhaps an equally important role is played by the related strengthened analyticity assumptions in a constructive illustration we complexify a few potentials solvable only in s wave then we continue their domain from the semi axis to the whole axis and obtain new exactly solvable models their energies turn out to be real as expected the new one dimensional spectra themselves differ quite significantly from their s wave predecessors

extremal non compactness of composition operators with linear fractional symbol abstract we realize the norms of certain composition operators c with linear fractional symbol acting on the hardy space in terms of the roots of associated hypergeometric functions this realization leads to simple necessary and sufficient conditions on for c to exhibit extremal non compactness establishes equivalence of cohyponormality and cosubnormality of composition operators with linear fractional symbol and yields a complete classification of those linear fractional that induce composition operators whose norms are determined by the action of the adjoint c on the normalized reproducing kernels in h 2

a computer aided test for the absence of limit cycles in fixed point digital filters the problem of determining the existence of limit cycles in direct form digital filters with quantization nonlinearities is considered a method based on an exhaustive search is introduced it is applicable to all direct form digital filters regardless of the type of quantization arithmetic the number of quantizers and the order of the filter the equivalence between the concept of global asymptotic stability and the exclusion of limit cycles is proved for linearly stable finite wordlength digital filters the robustness of this test with respect to coefficient perturbation in the digital filter is presented the computational aspects of the algorithm are also discussed the algorithm is shown to be capable of always determining the global asymptotic stability of the tested realization

2d systems feedback compensation an approach based on commutative linear transformations abstract algebraic properties of a pair of commutative matrices associated with an ideal in r z 1 z 2 are exploited for characterizing the closed loop polynomial variety of a 2d system also algorithms are given to find under what constraints the closed loop variety can be assigned and to compute the mfd of a compensator

algebraic and scattering aspects of a symmetric solvable potential we study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions we determine the transmission and reflection coefficients for the symmetric case and also formulate the problem in terms of an su 1 1 potential group which allows unified treatment of the discrete and the continuous spectra in a natural way we find that bound and scattering states of the symmetric problem supply a basis for the unitary irreducible representations of the su 1 1 potential group and this gives a straightforward group theoretical interpretation of the fact that the complex invariant potential has a real energy spectrum

bibo stability of multidimensional md shift invariant discrete systems bibo stability of 1 d and multidimensional md shift varying discrete systems is analyzed the test for one particular shift varying md system involves one md linear shift invariant stability test only the stability conditions derived also provides insight into robustness and margin of stability of the shift varying system two classes of systems are investigated 1 shift varying discrete direct realization systems and 2 shift varying discrete state space realization systems the results extend an implication of the perron frobenius theorem from 1 d shift invariant systems to shift invariant and or multidimensional systems

corrigendum to on norms of composition operators acting on bergman spaces j math anal appl 291 2004 189 202 abstract a theorem and a corollary in the paper cited in the title were stated incorrectly as was pointed out by christopher hammond we now state correctly and prove both of them these results still generalize and explain the geometric meaning of the cowen hurst norm formula we also include additional references and provide an example relevant for further study

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

nonlinear n pseudo fermions nonlinear pseudo fermions of degree n n pseudo fermions are introduced as pseudo particles with creation and annihilation operators b and a b a obeying the simple nonlinear anticommutation relation ab bnan 1 the n 1 order nilpotency of these operators follows from the existence of unique up to a bi normalization factor a vacuum supposing appropriate n 1 order nilpotent para grassmann variables and integration rules the sets of n pseudo fermion number states right and left ladder operator bi overcomplete sets of coherent states are constructed explicit examples of n pseudo fermion ladder operators are provided and the relation of pseudo fermions to finite level pseudo hermitian systems is briefly considered this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

a stability analysis of two dimensional nonlinear digital state space filters two different methods for a stability analysis of 2 d nonlinear discrete state space systems under zero input conditions are provided the first method reduces the task of testing a 2 d discrete nonlinear or shift varying system to a single 2 d linear stability test of a system matrix with nonnegative system matrix entries the second method is based on a number of norm tests for products of extreme matrices and can be considered the 2 d counterpart of the method for 1 d systems described by k t erickson and a n michel 1985 both of the introduced methods are based on the sector description of the nonlinearity and can be used to analyze digital filter stability under finite word length effects

the norm of a composition operator with linear symbol acting on the dirichlet space abstract we obtain a representation for the norm of a composition operator on the dirichlet space induced by a map of the form z a z b we compare this result to an upper bound for c that is valid whenever is univalent our work relies heavily on an adjoint formula recently discovered by gallardo gutierrez and montes rodriguez

graded gauge theory the mathematical background for a graded extension of gauge theories is investigated after discussing the general properties of graded lie algebras and what may serve as a model for a graded lie group the graded fiber bundle is constructed its basis manifold is supposed to be the so called superspace i e the product of the minkowskian space time with the grassmann algebra spanned by the anticommuting lorentz spinors the vertical subspaces tangent to the fibers are isomorphic with the graded extension of the su n lie algebra the connection and curvature are defined then on this bundle the two different gradings are either independent of each other or may be unified in one common grading which is equivalent to the choice of the spin statistics dependence the yang mills lagrangian is investigated in the simplified case the conformal symmetry breaking is discussed as well as some other physical consequences of the model

large order perturbation theory for a non hermitian pt symmetric hamiltonian a precise calculation of the ground state energy of the complex pt symmetric hamiltonian h p2 14x2 i x3 is performed using high order rayleigh schrodinger perturbation theory the energy spectrum of this hamiltonian has recently been shown to be real using numerical methods here we present convincing numerical evidence that the rayleigh schrodinger perturbation series is borel summable and show that pade summation provides excellent agreement with the real energy spectrum pade analysis provides strong numerical evidence that the once subtracted ground state energy considered as a function of 2 is a stieltjes function the analyticity properties of this stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable hamiltonian h p2 14 x2 ex3

limit cycles due to roundoff in state space digital filters a deterministic analysis is presented for limit cycles due to roundoff in state space digital filters two new sufficient conditions are established for the roundoff stability i e suppression of limit cycles in digital filters using these results restrictions on the location of poles are found for various realizations to be free from roundoff limit cycles

elimination of limit cycles due to two s complement quantization in normal form digital filters normal form digital filters are attractive due to their desirable properties when implemented in finite wordlength arithmetic these filters are free from all overflow limit cycles and quantization limit cycles when magnitude truncation is used however when two s complement truncation tct quantization is used limit cycles can still exist in this paper it is shown that when block structures are used normal form digital filters can be made free of limit cycles due to tct quantization it is shown that this can be done with a small block size an algorithm is also presented to find the minimum block size required for a given filter some examples are given to illustrate the results

spectral theory for algebraic combinations of toeplitz and composition operators abstract we determine the essential spectra of algebraic combinations of toeplitz operators with continuous symbol and composition operators induced by a class of linear fractional non automorphisms of the unit disk the operators in question act on the hardy space h 2 on the unit disk our method is to realize the c algebra that they generate as an extension of the compact operators by a concrete c algebra whose invertible elements are easily characterized

subnormality and composition operators on h2 if is a non constant analytic function defined on the unit disk d such that d d the composition operator c is the operator on the hardy space h2 of the unit disk defined by c f f in this paper we investigate the relationship between properties of the symbol and the subnormality of the operators c and c the main theorem shows under a regularity hypothesis that c is subnormal if and only if is in a restricted class of linear fractional transformations the two previously known special cases the affine transformations associated with the cesaro operator and the automorphisms are included as the extreme examples the proof of the main theorem consists of verifying a moment condition and yields in addition the construction of a measure that exhibits cgj as multiplication by an analytic function on p2 and the identification of the minimal normal extension as a weighted sum of shifts

comments on a robust mimo terminal sliding mode control scheme for rigid robotic manipulators the authors state that there are two errors in the above paper zhihong et al ibid vol 39 no 12 p 2464 9 1994 the purpose of this note is to give correct versions of remark 2 1 and remark 3 3

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

p mathcal t symmetric non self adjoint operators diagonalizable and non diagonalizable with a real discrete spectrum consider in the operator family h0 a 1a1 a dad d 2 is the quantum harmonic oscillator with rational frequencies w is a symmetric bounded potential and g is a real coupling constant we show that if g being an explicitly determined constant the spectrum of h g is real and discrete moreover we show that the operator has a real discrete spectrum but is not diagonalizable

generalized grassmannian coherent states for pseudo hermitian n level systems the purpose of this paper is to generalize fermionic coherent states for two level systems described by pseudo hermitian hamiltonian cherbal et al 2007 j phys a math theor 40 1835 to n level systems central to this task is the expression of the coherent states in terms of generalized grassmann variables these kinds of grassmann coherent states satisfy the bi overcompleteness condition instead of the overcompleteness one as it is reasonably expected because of the biorthonormality of the system choosing an appropriate grassmann weight function the resolution of identity is examined moreover grassmannian coherent and squeezed states of the deformed group suq 2 for the three level pseudo hermitian system are presented

pt symmetric cubic anharmonic oscillator as a physical model there is a factor of 2 error in equation 61 of this paper correcting this error leads to minor changes in equations 62 and 63 please see pdf for details

uniform asymptotic and jwkb expansions for anharmonic oscillators we show explicitly the relation between the uniform asymptotic and the jeffreys wentzel kramers brillouin jwkb wavefunctions and between the matching of uniform asymptotic expansions and the complete jwkb connection formulae written in terms of stokes multipliers and loop integrals as an application we give a unified derivation of the asymptotic behaviour of the imaginary part of the resonances in anharmonic oscillators and via dispersion relations the corresponding asymptotic behaviour of the rayleigh schrodinger perturbation theory coefficients

multidimensional discrete stability by serre categories and the construction and parametrization of observers via gabriel localizations immediate predecessors of this work were a paper on two dimensional deadbeat observers by bisiacco and valcher multidimens systems signal process 19 2008 pp 287 306 and one on one dimensional functional observers by blumthaler linear algebra appl 432 2010 pp 1560 1577 compare also fuhrmann s comprehensive paper linear algebra appl 428 2008 pp 44 136 the present paper extends blumthaler s results to continuous or discrete multidimensional behaviors i e constructs and parametrizes all controllable observers of a given multidimensional behavior and for this purpose also discusses the required multidimensional stability such an observer produces a signal that approximates or estimates a desired component of the behavior such that the signal difference is negligible in a suitable sense this definition thus presupposes that of negligible or stable autonomous systems in the standard one dimensional case these are the asymptotically stable behaviors we define and investigate

stability of distributed 3 d systems implemented on grid sensor networks using the givone roesser and the fornasini marchesini state space models for m d systems a method for distributed information processing in rectangular grid based wireless sensor networks is presented it can be used for distributed implementation of any general linear system on a grid sensor network nonlinearities caused by finite precision number representation schemes used for in node computations and inter node communication are modeled stability of the system is analyzed with special consideration given to the influence of inter node communication on system dynamics a necessary and sufficient condition for global asymptotic stability under finite precision arithmetic is established simulation results are presented to illustrate the theoretical results

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

norms of linear fractional composition operators we obtain a representation for the norm of the composition operator c on the hardy space h 2 whenever is a linear fractional mapping of the form z b cz d the representation shows that for such mappings the norm of c always exceeds the essential norm of c moreover it shows that a formula obtained by cowen for the norms of composition operators induced by mappings of the form z sz t has no natural generalization that would yield the norms of all linear fractional composition operators for rational numbers s and t cowen s formula yields an algebraic number as the norm we show e g that the norm of c 1 2 z is a transcendental number our principal results are based on a process that allows us to associate with each non compact linear fractional composition operator c for which c c e an equation whose maximum real solution is c 2 our work answers a number of questions in the literature for example we settle an issue raised by cowen and maccluer concerning co hyponormality of a certain family of composition operators

delay dependent stability for 2d systems with time varying delay subject to state saturation in the roesser model this paper addresses the problem of delay dependent stability of 2d systems with time varying delay subject to state saturation in the roesser model by introducing diagonally dominant matrices new delay dependent conditions are obtained in terms of linear matrix inequalities lmis where the lower and upper delay bounds along horizontal and vertical directions respectively are known numerical examples are provided to demonstrate the proposed results

two s complement quantization in periodic digital filters the stability of periodically shift variant psv filters are studied when implemented with two s complement truncation tct quantization block form and standard nonblock form implementations are considered and two sufficient conditions are established for stability as a special case second order coupled form psv filters are then investigated under tct quantization stability regions are established within the parameter space for block implementations examples are given to illustrate the results

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

z3 graded algebras and the cubic root of the supersymmetry translations a generalization of supersymmetry is proposed based on z3 graded algebras introducing the objects whose ternary commutation relations contain the cubic roots of unity e2 i 3 e4 i 3 and 1 the operators whose trilinear combinations yield the supersymmetric translation generators can be constructed cubic matrices forming a ternary algebra are the generalization of pauli s matrices the general properties of z3 grading some other representations of such algebras and their possible pertinence with regard to the quark model are briefly discussed

mathematical properties of a new levin type sequence transformation introduced by cižek zamastil and skala i algebraic theory ci ek zamastil and skala j math phys 44 962 968 2003 introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence sn n 0 of partial sums but also explicit estimates n n 0 for the truncation errors the explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than for instance pade approximants special cases of the new transformation are sequence transformations introduced by levin int j comput math b 3 371 388 1973 and weniger comput phys rep 10 189 371 1989 secs 7 9 numer algor 3 477 486 1992 and also a variant of richardson extrapolation philos trans r soc london ser a 226 299 349 1927 the algebraic theory of these transformations explicit expressions recurrence formulas explicit expres

noninteracting control of 2 d systems necessary and sufficient conditions for the existence of a decoupling bicausal precompensator for multivariable 2 d systems are derived in state space and frequency domains in general the decoupling problem for 2 d systems can be solved by feedback compensators if suitable injectivity assumptions are introduced on the input state matrices the structure of dynamic compensators is derived for this case and the 2 d decoupling problem with stability is solved

non hermitian matrix schrodinger equation backlund darboux transformation weyl functions and script pscript t symmetry the non hermitian matrix schrodinger equation obtained by the backlund darboux transformation bdt is treated the potentials fundamental solutions and weyl functions are constructed explicitly a symmetric reduction of the bdt is introduced and this case is studied in greater detail including potentials fundamental solutions bound states the reality of the discrete spectrum and spontaneous break of the symmetry the sign indefinite scalar product and examples

distributional borel summability of odd anharmonic oscillators it is proved that the divergent rayleigh schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are borel summable in the distributional sense to the resonances naturally associated with the system

which linear fractional composition operators are essentially normal abstract we characterize the essentially normal composition operators induced on the hardy space h2 by linear fractional maps they are either compact normal or the nontrivial case induced by parabolic nonautomorphisms these parabolic maps induce the first known examples of nontrivially essentially normal composition operators in addition we characterize those linear fractionally induced composition operators on h2 that are essentially self adjoint and present a number of results for composition operators induced by maps that are not linear fractional

numerical evidence that the perturbation expansion for a non hermitian pt symmetric hamiltonian is stieltjes recently several studies of non hermitian hamiltonians having pt symmetry have been conducted most striking about these complex hamiltonians is how closely their properties resemble those of conventional hermitian hamiltonians this paper presents further evidence of the similarity of these hamiltonians to hermitian hamiltonians by examining the summation of the divergent weak coupling perturbation series for the ground state energy of the pt symmetric hamiltonian h p2 14x2 i x3 recently studied by bender and dunne for this purpose the first 193 nonzero coefficients of the rayleigh schrodinger perturbation series in powers of 2 for the ground state energy were calculated pade summation and pade prediction techniques recently described by weniger are applied to this perturbation series the qualitative features of the results obtained in this way are indistinguishable from those obtained in the case of the perturbation series for the quartic anharmonic oscillator which is known to be a stieltjes se

one dimensional crystal with a complex periodic potential a one dimensional crystal model is constructed with a complex periodic potential a wave function solution for the crystal model is derived without relying on bloch functions the new wave function solution of this model is shown to correspond to the solution for the probability amplitude of a two level system the energy discriminant is evaluated using an analytic formula derived from the probability amplitude solution and based on an expansion parameter related to the energy and potential amplitude from the wave function energy discriminant the crystal band structure is derived and related to standard energy bands and gaps it is also shown that several of the properties of the two level system apply to the one dimensional crystal model the two level system solution which evolves in time is shown to manifest as a spatial configuration of the one dimensional crystal model the sensitivity of the wave function probability density is interpreted in the context of the new solution the spatial configurat

spectral analysis of the complex cubic oscillator using the exact semiclassical analysis we study the spectrum of a one parameter family of complex cubic oscillators the invariance property of the complex hamiltonians and the reality property of the spectrum are discussed analytic continuations of the spectrum in the complex parameter and their connections with the resonance problem for the real cubic oscillator are investigated the global analytic structure of the spectrum yields a branch point structure similar to the multivalued analytic structure discovered by bender and wu for the quartic oscillator

complex variable hvpt complex variable hypervirial perturbation theory is applied to the case of oscillator and coulomb potentials perturbed by a single term potential of the form vxn or vrn respectively the trial calculations reported show that this approach can produce accurate complex energies for resonant states via a simple and speedy calculation and can also be useful in studies of pt symmetry and tunnelling resonance effects

new criteria for asymptotic stability of one and multidimensional state space digital filters in fixed point arithmetic this paper addresses the problem of global asymptotic stability of one dimensional 1 d and multidimensional m d digital filters with any combination of overflow and quantization nonlinearities the stability analysis is carried out using 1 d and m d state space representations the approach introduced allows one to determine the stability behavior of single input single output systems with overflow and quantization nonlinearities the new criteria based on previous stability results of digital filters with quantization schemes are applicable to all arithmetic schemes for the first time results concerning general state variable representations of 1 d and m d digital filters with the naturally occurring combination of two s complement truncation quantization and overflow are reported furthermore significantly improved stability regions are obtained for digital filters with roundoff nonlinearities

criteria for the absence of limit cycles in two dimensional discrete systems sufficient conditions for the nonexistence of zero input limit cycles in two dimensional systems which contain a finite number of memoryless nonlinearities are presented two different conditions are given the first one is based on the frequency domain representation of the linear part of the discrete system and it is shown to be more general than the one already presented in 1 the second condition is formulated using the state space representation and is based on the properties of quasidominant matrices and on partial results on the 2 d lyapunov equation

algebraic aspects of two dimensional convolutional codes two dimensional 2d codes are introduced as linear shift invariant spaces of admissible signals on the discrete plane convolutional and in particular basic codes are characterized both in terms of their internal properties and by means of their input output representations the algebraic structure of the class of all encoders that correspond to a given convolutional code is investigated and the possibility of obtaining 2d decoders free from catastrophic errors as veil as efficient syndrome decoders is considered some aspects of the state space implementation of 2d encoders and decoders via finite memory 2d system are discussed

stability analysis of multidimensional m d direct realization digital filters under the influence of nonlinearities a stability test of m d direct realization digital filters with nonlinearities such as quantization overflow saturation etc is presented the method presented reduces a given nonlinear m d system to a nonlinear 1 d system such that the stability of the latter is sufficient for the stability of the former to check stability of the nonlinear 1 d system some new results on 1 d recursive systems with positive coefficients are applied the case of multiple nonlinearities in multidimensions is also addressed examples are given illustrating the application of the method

a note on output feedback stabilizability of multivariable 2d systems abstract the output feedback stabilizability conditions of 2d systems are expressed in term of structural properties of a pair of commuting linear transformations an algorithm is given for obtaining a stable closed loop 2d characteristic polynomial

norms of composition operators with rational symbol we compute the norms of composition operators with rational symbols that satisfy certain properties extending christopher hammond s methods on operators with linear fractional symbols this leads to a host of new examples of composition operators whose norms are calculable

dynamic regulation of 2d systems a state space approach abstract the possibilities of modifying the dynamical behavior of 2d state space models by output feedback compensation are investigated and a complete characterization of the closed loop polynomial varieties is given it turns out that plant hidden modes and rank singularities of the transfer function are the unique constraints we have to cope with in the compensator synthesis the proof of this result is based on algebraic manipulations of 2d mfds and on a coprime realization algorithm

adjoints of rationally induced composition operators abstract we give an elementary proof of a formula recently obtained by hammond moorhouse and robbins for the adjoint of a rationally induced composition operator on the hardy space h 2 christopher hammond jennifer moorhouse marian e robbins adjoints of composition operators with rational symbol j math anal appl 341 2008 626 639 we discuss some variants and implications of this formula and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator

norms and spectral radii of composition operators acting on the dirichlet space we obtain new upper bounds on the norms of univalently induced composition operators acting on the dirichlet space and compute explicitly the norms for univalent symbols whose range is the disk minus a set of measure zero as an application we show that the spectral radius of every univalently induced composition operator on the dirichlet space is equal to one

a comment on the gelfand yaglom theorem zeta functions and heat kernels for pt symmetric hamiltonians we propose a numerical method for computing zeta functions for general classes of hamiltonians based on the gelfand yaglom theorem which computes the determinant without needing to compute the eigenvalues we verify that this agrees with known analytic results for certain hermitean hamiltonians and we present numerical evidence that it may extend also to non hermitean pt symmetric hamiltonians

finite word length effects in m d digital filters with singularities on the stability boundary the author addresses stability robustness with respect to finite word length conditions for multidimensional m d digital filters with singularities on the distinguished boundary in particular it is shown that all purely first order digital filters with nonessential singularities of the second kind nssks are asymptotically stable under certain types of nonlinearities even bounded input bounded output bibo unstable m d digital filters with singularities on the distinguished boundary proved to be asymptotically stable in the first order case the results are then extended to certain classes of higher order m d digital filters it is demonstrated that although linear m d digital filters with nssks on the distinguished boundary have no margin of stability in the l sub 1 l sub 2 or asymptotic sense they might exhibit a rather robust asymptotic behavior with respect to nonideal effects such as finite word length conditions

on the eigenproblems of pt symmetric oscillators we consider the non hermitian hamiltonian h d2 dx2 p x2 ix 2n 1 on the real line where p x is a polynomial of degree at most n 1 with all non negative real coefficients possibly p 0 it is proved that the eigenvalues must be in the sector arg 2n 3 also for the cubic case h d2 dx2 ix 3 we establish a zero free region of the eigenfunction u and its derivative u and we find some other interesting properties of eigenfunctions

controllability of 2 d systems the concept of controllability of 2 d systems is introduced in a behavioral framework this property is defined as a systems concept and is characterized in terms of system representations state space realizations for controllable 2 d systems are presented the structure of state space systems is analyzed and the notions of trimness reachability and observability are introduced and characterized these realizations do not depend on any assumptions concerning causality

stability of digital filters implemented with two s complement truncation quantization the stability of digital filters implemented with two s complement truncation quantization is investigated both the direct form and normal form realizations are considered and regions in the parameter plane are obtained where the filters are globally asymptotically stable regions within the stability triangle of a direct form digital filter are also studied where limit cycles of period one and two are possible the results are illustrated with several examples

l gain performance analysis for two dimensional roesser systems with persistent bounded disturbance and saturation nonlinearity the l gain approach has been an essential tool in one dimensional system theory however limited results have been presented in the literature for the two dimensional 2 d l gain approach this paper investigates the l gain performance for 2 d systems in the roesser model with persistent bounded disturbance input and saturation nonlinearity a linear matrix inequality lmi based condition is established to reduce the effect of persistent bounded disturbance input on 2 d systems within a given disturbance attenuation level based on the discrete jensen inequality lower bounds lemma and diagonally dominant matrices we apply the obtained results to the l gain performance analysis for 2 d digital filters with saturation arithmetic

trajectory control and interconnection of 1d and nd systems in this paper we examine the relationship between control viewed as concatenation of trajectories and control viewed as interconnection of systems we show that for one dimensional linear time invariant systems the ability to obtain a given subsystem by regular interconnection a prerequisite for any feedback type structure is equivalent to the ability to drive any trajectory into that subsystem however in the case of multidimensional systems the former is a stronger property than the latter trajectory controllability can however be expressed as a regular interconnection of behaviors in an extended variable space by introducing latent or auxiliary variables this leads as a by product to the notion of controlling a system by means of latent variables

generalized grassmann variables for quantum kit k level systems and barut girardello coherent states for su r 1 algebras this paper concerns the construction of su r 1 barut girardello coherent states in terms of generalized grassmann variables we first introduce a generalized weyl heisenberg algebra a r r 1 generated by r pairs of creation and annihilation operators this algebra provides a useful framework to describe qubit and qukit k level systems it includes the usual weyl heisenberg and su 2 algebras we investigate the corresponding fock representation space the generalized grassmann variables are introduced as variables spanning the fock bargmann space associated with the algebra a r the barut girardello coherent states for su r 1 algebras are explicitly derived and their over completion properties are discussed