adsorption of pulled directed paths in a slanted boundary in this paper the generating function free energy and phase diagram of a class of models of directed paths related to the models of paths adsorbing in the boundary y nx n 2 for and pulled by a vertical force in its endpoint are determined these models are related to the special classes of models of random walks in the quarter plane and which interact with the x axis generating functions are determined by solving a functional recurrence using the kernel method and are algebraic the generating function is determined for general values of n and the solutions for n 1 and n 2 are examined explicitly together with the derivation of the phase diagram and locations of critical curves three phases including a free or bulk phase an adsorbed phase and a ballistic or stretched phase are identified and the nature of the transitions between these phases are examined order parameters are defined and asymptotic expressions are determined for the critical adsorption surface as a function of the pulling force some of these results are also briefly generalized to arbitrary values of n

adsorbing motzkin paths the adsorption of a motzkin path with lifted endpoints onto an adsorbing line is examined as a model of polymer adsorption the partition function of this model is determined exactly and the free energy is derived by examining the partition function the location of the critical point in the model is given exactly in one case and as the solution of a nonlinear equation in another case the phase diagram of the model is examined and i show that the transition is in general a first order adsorption transition except in cases where one or both endpoints of the path are restricted to be close to the adsorbing line

two non holonomic lattice walks in the quarter plane we present two classes of random walks restricted to the quarter plane with non holonomic generating functions the non holonomicity is established using the iterated kernel method a variant of the kernel method this adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions m mishna classifying lattice walks in the quarter plane j combin theory ser a 116 2009 460 477 the method also yields an asymptotic expression for the number of walks of length n

hilbert kunz density function and asymptotic hilbert kunz multiplicity for projective toric varieties abstract for a toric pair x d where x is a projective toric variety of dimension d 1 1 and d is a very ample t cartier divisor we show that the hilbert kunz density function hkd x d is the d 1 dimensional volume of p d z where p d r d is a compact d dimensional set which is a finite union of convex polytopes we also show that for k 1 the function hkd x k d can be replaced by another compactly supported continuous function k d which is linear in k this gives the formula for the associated coordinate ring r m lim k e h k r m k e 0 r m k d k d 1 e 0 r m d 1 0 d d where d see proposition 1 2 is solely determined by the shape of the polytope p d associated to the toric pair x d moreover d is a multiplicative function for segre products this yields explicit computation of d and hence the limit for smooth fano toric surfaces with respect to anticanonical divisor in general due to this formulation in terms of the polytope p d one can explicitly compute the limit for two dimensional toric pairs and their segre products we further show that theorem 6 3 the renormalized limit takes the minimum value if and only if the polytope p d tiles the space m r r d 1 with the lattice m z d 1 as a consequence one gets an algebraic formulation of the tiling property of any rational convex polytope

gain bounds of higher order nonlinear transfer functions bounds on the gain for high order nonlinear transfer functions of a wide class of nonlinear systems are derived it is shown that the gain bounds can be expressed explicitly in terms of the coefficients of discrete time nonlinear models the bounds can be used to check the convergence of the transfer function series and hence to determine the truncation order in practical applications

pulled motzkin paths in this paper the models of pulled dyck paths in janse van rensburg 2010 j phys a math theor 43 215001 are generalized to pulled motzkin path models the generating functions of pulled motzkin paths are determined in terms of series over trinomial coefficients and the elastic response of a motzkin path pulled at its endpoint see orlandini and whittington 2004 j phys a math gen 37 5305 14 is shown to be r f 0 for forces pushing the endpoint toward the adsorbing line and r f f 1 2cosh f 2sinh f f as f for forces pulling the path away from the x axis in addition the elastic response of a motzkin path pulled at its midpoint is shown to be r f 0 for forces pushing the midpoint toward the adsorbing line and r f f 1 2cosh f 2 sinh f 2 2f as f for forces pulling the path away from the x axis formal combinatorial identities arising from pulled motzkin path models are also presented these identities are the generalization of combinatorial identities obtained in directed paths models to their natural trinomial counterparts

nonlinear model predictive control of a simulated multivariable polymerization reactor using second order volterra models abstract two formulations of a nonlinear model predictive control scheme based on the second order volterra series model are presented the first formulation determines the control action using successive substitution and the second method directly solves a fourth order nonlinear programming problem on line one case study is presented for the siso control of an isothermal reactor which utilizes the first controller formulation a second case study is presented for the multivariable control of a large reactor and uses the nonlinear programming formulation for the controller the model coefficients for both examples are obtained by discretizing the bilinear taylor series approximation of the fundamental model and calculating markov parameters the relationships between discrete and continuous time bilinear model matrices using an explicit fourth order runge kutta method are also included the responses to setpoint changes of both reactors controlled with a linear model predictive control scheme and the second order volterra model predictive control scheme are compared to desired linear reference trajectories in the majority of the cases examined the responses obtained by the volterra controller followed the reference trajectories more closely practical issues including the reduction of the number of model parameters are addressed in both case studies

transformation of finite degree discrete volterra systems with cross products to finite input output forms in this paper the problem of finite input output representation is solved for a class of finite degree discrete volterra systems containing cross products among inputs and outputs via the so called spl delta spl epsiv operators a corresponding algorithm is supplied an application concerning the bilinear systems is also given

the random generation of directed animals abstract in this paper we propose an algorithm to randomly generate a directed animal directed animals are well known combinatorial objects and have been widely used for modelling the physical phenomenon of percolation the algorithm consists of three steps and we prove that each of them is performed in linear time finally we report the results of our experiments made by means of appropriate computer programs in order to give empirical evidence that our algorithm really works

an elementary solution of gessel s walks in the quadrant abstract around 2000 ira gessel conjectured that the number of lattice walks in the quadrant n 2 starting and ending at the origin 0 0 and taking their steps in had a simple hypergeometric form in the following decade this problem was recast in the systematic study of walks with small steps that is steps in 1 0 1 2 confined to the quadrant the generating functions of such walks are archetypal solutions of partial discrete differential equations a complete classification of quadrant walks according to the nature of their generating function algebraic d finite or not is now available but gessel s walks remained mysterious because they were the only model among the 23 d finite ones that had not been given an elementary solution instead gessel s conjecture was first proved using an inventive computer algebra approach in 2008 a year later the associated three variate generating function was proved to be algebraic by a computer algebra tour de force this was re proved recently using elaborate complex analysis machinery we give here an elementary and constructive proof our approach also solves other quadrant models with multiple steps recently proved to be algebraic via computer algebra

pulled directed lattice paths the generating functions of models of directed walks pulled by an external force f are determined using the kernel method these paths are models of linear polymers subject to an external force the generating function is related to the generating function of ballot paths and has an unexpected and non physical singularity for the model of dyck paths pulled at its central vertex in each model the force extension curve is determined exactly and has a sigmoid shape partially given by ctanh cf if f is the applied force and where c and c are the model dependent constants this result is consistent with previous results for models of pulled directed paths and also with data obtained from the numerical simulation of self avoiding walk models of linear polymers

parameterized convergence bounds for volterra series expansion of narx models evaluation of the convergence bound in the frequency domain for volterra series expansion of nonlinear systems described by narx models is studied this provides new convergence criteria under which the nonlinear system of interest has a convergent volterra series expansion and the new criteria are expressed explicitly in terms of the input magnitude model parameters and frequency variable the new convergence criteria are firstly developed for harmonic inputs and then extended to multi tone and general input cases based on the theoretical analysis a general procedure for calculating the convergence bound is provided the results provide a fundamental basis for nonlinear signal processing using the volterra series theory

computable convergence bounds of series expansions for infinite dimensional linear analytic systems and application this paper deals with the convergence of series expansions of trajectories for semi linear infinite dimensional systems which are analytic in state and affine in input a special case of such expansions corresponds to volterra series which are extensively used for the analysis the simulation and the control of weakly nonlinear finite dimensional systems the main results of this paper give computable bounds for both the convergence radius and the truncation error of the series these results can be used for model simplification and analytic approximation of trajectories with a guaranteed quality they are available for distributed and boundary control systems as an illustration these results are applied to an epidemic population dynamic model in this example it is shown that the truncation of the series at order 2 yields an accurate analytic approximation which can be used for time simulation and control issues the relevance of the method is illustrated by simulations

the observation space and realizations of finite volterra series in this paper we extend the results on the theory of realizations of finite volterra series by exploiting the structural properties of the observation space in general two distinct minimal realizations of a finite volterra series may be constructed one based on the properties of the observation space and the other based on the properties of the lie algebra both these realizations display the canonical structure found earlier for such systems the results given here also yield information on the observation algebra generated by functions in the observation space just as previous work gave information on the lie algebra as an application the structure found here is applied to the finite dimensional filtering problem for these systems

analytic models and ambiguity of context free languages we establish that several classical context free languages are inherently ambiguous by proving that their counting generating functions when considered as analytic functions exhibit some characteristic form of transcendental behaviour to that purpose we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages in particular the paper contains a general density theorem for unambiguous context free languages

symmetries of statistics on lattice paths between two boundaries abstract we prove that on the set of lattice paths with steps n 0 1 and e 1 0 that lie between two fixed boundaries t and b which are themselves lattice paths the statistics number of e steps shared with b and number of e steps shared with t have a symmetric joint distribution to do so we give an involution that switches these statistics preserves additional parameters and generalizes to paths that contain steps s 0 1 at prescribed x coordinates we also show that a similar equidistribution result for path statistics follows from the fact that the tutte polynomial of a matroid is independent of the order of its ground set we extend the two theorems to k tuples of paths between two boundaries and we give some applications to dyck paths generalizing a result of deutsch to watermelon configurations to pattern avoiding permutations and to the generalized tamari lattice finally we prove a conjecture of nicolas about the distribution of degrees of k consecutive vertices in k triangulations of a convex n gon to achieve this goal we provide a new statistic preserving bijection between certain k tuples of non crossing paths and k flagged semistandard young tableaux which is based on local moves reminiscent of jeu de taquin

real symmetric random matrices and path counting exact evaluation of mathrm tr s p is here performed for real symmetric matrices s of arbitrary order n up to some integer p where the matrix entries are independent identically distributed random variables with an arbitrary probability distribution these expectations are polynomials in the moments of the matrix entries they provide useful information on the spectral density of the ensemble in the large n limit they also are a straightforward tool to examine a variety of rescalings of the entries in the large n limit

reconstructing convex polyominoes from horizontal and vertical projections in 1 we studied the problem of reconstructing a discrete set 5 from its horizontal and vertical projections we defined an algorithm that establishes the existence of a convex polyomino whose horizontal and vertical projections are equal to a pair of assigned vectors h v with h m and v n its computational cost is o n4m4 in this paper we introduce some operations for recontructing convex polyominoes by means of vectors h s and v s partial sums these operations allows us to define a new algorithm whose complexity is less than o n2m2

technical communique a note on the optimal expansion of volterra models using laguerre functions this work tackles the problem of expanding volterra models using laguerre functions a strict global optimal solution is derived when each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases each of which parameterized by an individual laguerre pole intended for representing the dominant dynamic of the kernel along a particular dimension it is proved that the solution derived minimizes the upper bound of the squared norm of the error resulting from the practical truncation of the laguerre series expansion into a finite number of functions this is an extension of the results in campello favier and amaral 2004 optimal expansions of discrete time volterra models using laguerre functions automatica 40 815 822 where an optimal solution was obtained for the usual yet particular case in which a single laguerre pole is used for expanding a given kernel along all its dimensions it is also proved that the particular and extended solutions are equivalent to each other when the volterra kernels are symmetric

hilbert kunz functions of a hirzebruch surface abstract here we compute hilbert kunz functions of any nontrivial ruled surface over p k 1 with respect to all ample line bundles on it

consensusability of multi agent systems with time varying communication delay abstract this paper studies the consensusability problem of continuous time multi agent systems with time varying communication delay in undirected network we design a consensus protocol based on the low gain solution of a parametric algebraic riccati equation are and not use the precise information of the amount of time varying delay by studying the joint effect of agent dynamic and network topology sufficient conditions are given for consensus when all poles of the system matrix are on the closed left half plane in case of non zero poles on the imaginary axis maximal admissible upper bound of the time varying delay is given in terms of both the agent dynamic and network topology otherwise consensus can be achieved for the time varying delay with arbitrarily large upper bound finally simulation results are presented to demonstrate the effectiveness of the theoretical results

technical communique magnitude bounds of generalized frequency response functions for nonlinear volterra systems described by narx model in order to reveal the relationship between system time domain model parameters and system frequency response functions new magnitude bounds of frequency response functions for nonlinear volterra systems described by narx model are established the magnitude bound of the nth order generalized frequency response function gfrf can be expressed as a simple n degree polynomial function of the magnitude of the first order gfrf whose coefficients are functions of the model parameters and frequency variables thus the system output spectrum can also be bounded by a polynomial function of the magnitude of the first order gfrf these results demonstrate explicitly the analytical relationship between model parameters and system frequency response functions and provide a significant insight into the magnitude based analysis and synthesis of nonlinear systems in the frequency domain

computation of convergence bounds for volterra series of linear analytic single input systems in this paper the volterra series decomposition of a class of single input time invariant systems analytic in state and affine in input is analyzed input to state convergence results are obtained for several typical norms l 0 t l r as well as exponentially weighted norms from the standard recursive construction of volterra kernels new estimates of the kernel norms are derived the singular inversion theorem is then used to obtain the main result of the paper namely an easily computable bound of the convergence radius guaranteed error bounds for the truncated series are also provided the relevance of the method is illustrated in several examples

volterra series for solving weakly non linear partial differential equations application to a dissipative burgers equation a method to solve weakly non linear partial differential equations with volterra series is presented in the context of single input systems the solution x z t is represented as the output of a z parameterized volterra system where z denotes the space variable but z could also have a different meaning or be a vector in place of deriving the kernels from purely algebraic equations as for the standard case of ordinary differential systems the problem turns into solving linear differential equations this paper introduces the method on an example a dissipative burgers equation which models the acoustic propagation and accounts for the dominant effects involved in brass musical instruments the kernels are computed analytically in the laplace domain as a new result writing the volterra expansion for periodic inputs leads to the analytic resolution of the harmonic balance method which is frequently used in acoustics furthermore the ability of the volterra system to treat other signals constitutes an improvement for the sound synthesis it allows the simulation for any regime including attacks and transients numerical simulations are presented and their validity are discussed

on the convergence of the volterra series representation of the duffing s oscillators subjected to harmonic excitations abstract this paper is concerned with the investigation of the convergence issue of the volterra series representations of the duffing s oscillator subjected to harmonic inputs a simple criterion is proposed to determine the upper limit of the magnitude of the harmonic inputs a comparison between the new proposed criterion and a criterion suggested by tomlinson et al g r tomlinson g manson g m lee a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series journal of sound and vibration 190 1996 751 762 was carried out the results show that the new criterion can provide a more accurate prediction about the convergence of the volterra series representation of the duffing s oscillator

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

hilbert kunz theory for nodal cubics via sheaves abstract suppose b f x y z h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve c with a node let j be a homogeneous x y z primary ideal and n e n be the hilbert kunz function of b with respect to j let q p n when j x y z it is known that e n 7 3 q 2 1 3 q r where r 5 3 if q 2 3 and is 1 otherwise we generalize this showing that e n q 2 q r where r only depends on q mod 3 we describe and r in terms of classification data for a vector bundle on c

the parametric characteristic of frequency response functions for nonlinear systems the characteristic of the frequency response functions of nonlinear systems can be revealed and analyzed by analyzing of the parametric characteristics of these functions to achieve these objectives a new operator is defined and several fundamental and important results about the parametric characteristics of the frequency response functions of nonlinear systems are developed these theoretical results provide a significant and novel insight into the frequency domain characteristics of nonlinear systems and circumvent a large amount of complicated integral and symbolic calculations which have previously been required to perform nonlinear system frequency domain analysis several new results for the analysis and synthesis of nonlinear systems are also developed examples are included to illustrate potential applications of the new results

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

sound synthesis of a nonlinear string using volterra series this paper proposes to solve and simulate various kirchhoff models of nonlinear strings using volterra series two nonlinearities are studied the string tension is supposed to depend either on the global elongation of the string first model or on the local strain located at x second and more precise model the boundary conditions are simple dirichlet homogeneous ones or general dynamic conditions allowing the string to be connected to any system typically a bridge for each model a volterra series is used to represent the displacement as a functional of excitation forces the volterra kernels are solved using a modal decomposition the first kernel of the series yields the standard modes of the linearized problem while the next kernels introduce the nonlinear dynamics as a last step systematic identification of the kernels lead to a structure composed of linear filters sums and products which are well suited to the sound synthesis using standard signal processing techniques the nonlinear dynamic introduced through this simulation is significant and perceptible in sound results for sufficiently large excitations

analysis of nonlinear oscillators using volterra series in the frequency domain the volterra series representation is a direct generalisation of the linear convolution integral and has been widely applied in the analysis and design of nonlinear systems both in the time and the frequency domain the volterra series is associated with the so called weakly nonlinear systems but even within the framework of weak nonlinearity there is a convergence limit for the existence of a valid volterra series representation for a given nonlinear differential equation barrett 1965 1 proposed a time domain criterion to prove that the volterra series converges within a given region for a class of nonlinear systems with cubic stiffness nonlinearity in this paper this time domain criterion is extended to the frequency domain to accommodate the analysis of nonlinear oscillators subject to harmonic excitation a common and severe nonlinear phenomenon called jump a behavior associated with the duffing oscillator and the multi valued properties of the response solution is investigated using the new frequency domain criterion of establishing the upper limits of the nonlinear oscillators to predict the onset point of the jump and the volterra time and frequency domain analysis of this phenomenon are carried out based on graphical and numerical techniques

walks confined in a quadrant are not always d finite we consider planar lattice walks that start from a prescribed position take their steps in a given finite subset of z2 and always stay in the quadrant x 0 y 0 we first give a criterion which guarantees that the length generating function of these walks is d finite that is satisfies a linear differential equation with polynomial coefficients this criterion applies among others to the ordinary square lattice walks then we prove that walks that start from 1 1 take their steps in 2 1 1 2 and stay in the first quadrant have a non d finite generating function our proof relies on a functional equation satisfied by this generating function and on elementary complex analysis

brief series expansions for analytic systems linear in control this paper presents a series expansion for the evolution of a class of nonlinear systems characterized by constant input vector fields we present a series expansion that can be computed via explicit recursive expressions and we derive sufficient conditions for uniform convergence over the finite and infinite time horizon furthermore we present a simplified series and convergence analysis for the setting of second order polynomial vector fields the treatment only relies on elementary notions on analytic functions number theory and operator norms

directed paths in a wedge models of directed paths have been used extensively in the scientific literature to model linear polymers in this paper we examine a directed path model of a linear polymer in a confining geometry a wedge we are particularly interested in cn the number of directed lattice paths of length n steps which take steps in the north east and south east directions and are confined in the wedge y x p where p is an integer we examine the case p 2 in detail and show that the generating function satisfies a functional equation with quartic kernel we give a formal solution by using the kernel method show that this model is not d finite and determine asymptotics to leading order for cn in particular the number of paths of length n in the wedge y x 2 is given by where the constant 0 67874 can be determined to arbitrary accuracy with little computational effort

output frequency response function of nonlinear volterra systems an expression for the output frequency response function ofrf which defines the explicit analytical relationship between the output spectrum and the system parameters is derived for nonlinear systems which can be described by a polynomial form differential equation model an effective algorithm is developed to determine the ofrf directly from system simulation or experimental data simulation studies demonstrate the significance of the ofrf concept and verify the effectiveness of the algorithm which evaluates the ofrf numerically these new results provide an important basis for the analytical study and design of a wide class of nonlinear systems in the frequency domain

comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis by using the duffing oscillator as a case study this paper shows that the harmonic components in the nonlinear system response to a sinusoidal input calculated using the nonlinear output frequency response functions nofrfs are one of the solutions obtained using the harmonic balance method hbm a comparison of the performances of the two methods shows that the hbm can capture the well known jump phenomenon but is restricted by computational limits for some strongly nonlinear systems and can fail to provide accurate predictions for some harmonic components although the nofrfs cannot capture the jump phenomenon the method has few computational restrictions for the nonlinear damping systems the nofrfs can give better predictions for all the harmonic components in the system response than the hbm even when the damping system is strongly nonlinear

counting lattice paths taking steps in infinitely many directions under special access restrictions we count lattice paths that are confined to the first quadrant by the nature of their step vectors if no further restrictions apply a path can go from any point to infinitely many others but each point on the path has only finitely many predecessors by further restrictions we mean a boundary line above which the paths may have to stay access privilege to the boundary line itself is granted from certain lattice points in the form of a special access step set which itself may be infinite we also count the number of paths that contact the weak boundary a given number of times we approach explicit solutions of such enumeration problems via sheffer polynomials and functionals using results of the umbral calculus

a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series abstract high order frequency response functions based on the volterra series are employed to represent the input output characteristics of the duffing oscillator subject to sinusoidal excitation from these a series representation of a first order frequency response function of the non linear system is obtained and used to derive a simple criterion which provides an estimate of the magnitude of the input sine wave above which the predicted response begins to diverge the criterion results in a small overestimate of the magnitude of the input force excitation at the undamped natural frequency condition although the derived criterion strictly applies only to the duffing oscillator the procedure can be applied generally and is extended to a two degree of freedom system with a non linear cubic stiffness producing a similar criterion to that of the single degree of freedom system

a bound for the magnitude characteristics of nonlinear output frequency response functions part 1 analysis and computation a bound for the magnitude frequency domain characteristics associated with the outputs of a wide class of nonlinear systems is derived as a relatively simple function of the generalized frequency response functions and properties of the system inputs it is shown how the practical computation of the new bound can be easily performed for nonlinear systems with finite but arbitrary order nonlinearities and worked examples are included the paper is divided into two parts in part 1 an expression for the output magnitude bound is derived properties of the result are discussed and general procedures for the practical computation of the bound are developed in part 2 the practical computations associated with applying the bound to the polynomial nonlinear autoregressive model with exogenous input are discussed

asymptotic hilbert kunz multiplicity abstract for a pair m i where m is finitely generated graded module over a standard graded ring r of dimension d 2 and i is a graded ideal with l r i and generated by elements of the same degree we prove that lim q e 1 m i q q d exists where e 1 m i q denotes the first coefficient of the hilbert samuel polynomial of m i q we use this to get an expression for lim k e h k m i k e 0 m i k d k d 1 where e h k denotes the hilbert kunz multiplicity in particular if dim m d then we deduce that the difference e h k m i k e m i k d grows at least as a fixed positive multiple of k d 1 as k this is proved using renormalized hk density functions

truncation of nonlinear system expansions in the frequency domain the truncation of volterra series expansions is studied using the output frequency characteristics of nonlinear systems to develop a new algorithm for determining the terms to include in a volterra series expansion the results show the influence of both the generalized frequency response functions and properties of the input spectra on the significance of individual terms in the series the effectiveness of the proposed method is demonstrated using simulation studies including the analysis of a single degree of freedom mechanical oscillator nonlinear system analyses using volterra series theory must always be based on a truncated volterra series description the present study provides an effective strategy for determining which terms to include in the analysis of practical nonlinear systems based on volterra series models

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

the representation of the symmetric group on m tamari intervals an m ballot path of size n is a path on the square grid consisting of north and east unit steps starting at 0 0 ending at mn n and never going below the line fx myg the set of these paths can be equipped with a lattice structure called them tamari lattice and denoted by t m n which generalizes the usual tamari lattice tn obtained when m 1 this lattice was introduced by f bergeron in connection with the study of diagonally coinvariant spaces in three sets ofn variables the representation of sn on these spaces is conjectured to be closely related to the natural representation of sn on labelled intervals of them tamari lattice studied in this paper an interval p q of t m n is labelled if the north steps of q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps the symmetric group sn acts on labelled intervals of t m n by permutation of the labels we prove an explicit formula conjectured by f bergeron and the third author for the character of the associated representation of sn in particular the dimension of the representation that is the number of labelledm tamari intervals of sizen is found to be m 1 n mn 1 n 2 these results are new even whenm 1 the form of these numbers suggests a connection with parking functions but our proof is not bijective the starting point is a recursive description of m tamari intervals it yields an equation for an associated generating function which is a refined version of the frobenius series of the representation the form of this equation is highly non standard it involves two additional variables x and y a derivative with respect to y and iterated divided differences with respect tox the hardest part of the proof consists in solving it and we develop original techniques to do so r

convergence analysis of volterra series response of nonlinear systems subjected to harmonic excitation volterra series provides a strong platform for non linear analysis and higher order frequency response functions however limited convergence is an inherent difficulty associated with the series and needs to be addressed rigorously prior to its application to a physical system the power series representation of the response of non linear systems subjected to harmonic excitation is investigated in this study the problem of convergence is addressed in terms of the convergence of individual frequency harmonics of the non linear response though the procedure is applicable to general polynomial form non linearity it is illustrated for a duffing oscillator subjected to harmonic excitation a general and structured series expression is obtained for amplitudes of all the response harmonics and convergence is investigated in terms of a non dimensional non linear parameter critical values of this parameter representing the upper limit of excitation level for the convergence are defined for a wide range of excitation frequencies zones of convergence and divergence of the response series are presented graphically for a range of the non dimensional non linear parameter and the number of terms included in the approximation of a response harmonic an algorithm based on ratio test is presented to compute the critical value of the non dimensional non linear parameter results obtained from the suggested algorithm are found to be in close agreement with the exact values the method gives better results compared to previous methods and has wider application in terms of excitation frequency the procedure is also investigated for a two degree of freedom system

output frequency characteristics of nonlinear systems some interesting properties of output frequencies of volterra type nonlinear systems are particularly investigated these results provide a very novel and useful insight into the super harmonic and inter modulation phenomena in output frequency response of nonlinear systems with consideration of the effects incurred by different nonlinear components in the system the new properties theoretically demonstrate several fundamental output frequency characteristics and unveil clearly the mechanism of the interaction or coupling effects between different harmonic behaviors in system output frequency response incurred by different nonlinear components these results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities they can provide an important guidance to modeling identification control and signal processing by using the volterra series theory in practice

frequency domain analysis and identification of block oriented nonlinear systems abstract block oriented nonlinear models including wiener models hammerstein models and wiener hammerstein models etc have been extensively applied in practice for system identification signal processing and control in this study analytical frequency response functions including generalized frequency response functions gfrfs and nonlinear output spectrum of block oriented nonlinear systems are developed which can demonstrate clearly the relationship between frequency response functions and model parameters and also the dependence of frequency response functions on the linear part of the model the nonlinear part of these models can be a more general multivariate polynomial function these fundamental results provide a significant insight into the analysis and design of block oriented nonlinear systems effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems compared with some existing frequency domain identification methods the new estimation algorithms do not necessarily require model structure information not need the invertibility of the nonlinearity and not restrict to harmonic inputs simulation examples are given to illustrate these new results

approximate observability of infinite dimensional bilinear systems using a volterra series expansion in this paper the approximate observability of a class of infinite dimensional bilinear systems is investigated the observability conditions are discussed based on a volterra series representation of this class of systems a testable observability criterion is derived for the case where the infinitesimal generator is self adjoint or riesz spectral operator the theoretical results are illustrated by examples

on the convergence of volterra series of finite dimensional quadratic mimo systems in this paper the volterra series decomposition of a class of quadratic time invariant single input finite dimensional systems is analyzed the kernels are given by a recursive sequence of linear pdes in the time domain and an equivalent algebraic recursion in the laplace domain this is used to prove the convergence of the volterra series to a possibly weak trajectory of the system to provide a practicable value for the radius of convergence of the input in l and to compute a guaranteed error bound in l for the truncated series the result is then extended to mimo systems a numerical simulation is performed on an academic siso example to illustrate how easily the truncated volterra series can be implemented

basic analytic combinatorics of directed lattice paths this paper develops a unified enumerative and asymptotic theory of directed two dimensional lattice paths in half planes and quarter planes the lattice paths are specified by a finite set of rules that are both time and space homogeneous and have a privileged direction of increase they are then essentially one dimensional objects the theory relies on a specific kernel method that provides an important decomposition of the algebraic generating functions involved as well as on a generic study of singularities of an associated algebraic curve consequences are precise computable estimates for the number of lattice paths of a given length under various constraints bridges excursions meanders as well as a characterization of the limit laws associated to several basic parameters of paths

fliess operators on lp spaces convergence and continuity fliess operators as input output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory in the classical analysis of these operators certain growth conditions on the coe3cients in their series representations insure uniform and absolute convergence provided every input is uniformly bounded by some 4xed upperbound in some emerging applications however it is more natural to consider other classes of inputs in this paper lp function spaces are considered in particular it is shown that the classic growth conditions also provide su3cient conditions for convergence and continuity when the admissible inputs are from a ball in lp t0 t 0 t where t is bounded and p 1 in addition stronger global growth conditions are given that apply even for the case where t is unbounded when the coe3cients of a fliess operator have a state space representation it is shown that the state space model will always locally realize the correspondinginput output map on lp t0 t 0 t for su3ciently small t 0 if certain well posedness conditions are satis4ed then the state space model will globally realized the input output mapping for unbounded t when the coe3cients satisfy the global growth condition c 2002 elsevier science b v all rights reserved

asymptotic properties of some underdiagonal walks generation algorithms abstract using classical properties of random walks and brownian motion we derive the asymptotic distribution of some characteristics of an algorithm proposed by barcucci et al 1995 to generate underdiagonal walks of size n the main parameters of interest are the cost of the algorithm in terms of calls to a random generator the height of the walk at its last step and the maximum of the walk we obtain various forms of probability generating functions laplace transforms of densities continuous distribution functions asymptotic densities and discrete stationary distributions

modal analysis of a cracked beam abstract this paper addresses the problem of vibrations of a cracked beam in general the motion of such a beam can be very complex this phenomenon can be attributed to the presence of the non linearity due to the opening and closing of cracks the focus of this paper is the modal analysis of a cantilever beam with a transverse edge crack the non linearity mentioned above has been modelled as a piecewise linear system in an attempt to define effective natural frequencies for this piecewise linear system the idea of a bilinear frequency is utilized the bilinear frequency is obtained by computing the associated frequencies of each of the linear pieces of the piecewise linear system the finite element method is used to obtain the natural frequencies in each linear region in order better to understand the essential non linear dynamics of the cracked beam a piecewise linear two degree of freedom model is studied perturbation methods are used to obtain the non linear normal modes of vibration and the associated period of the motion results of this piecewise linear model problem are shown to justify the definition of the bilinear frequency as the effective natural frequency it is therefore expected that calculating piecewise mode shapes and bilinear frequencies is useful for understanding the dynamics of the infinite degree of freedom cracked beam

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples