a unified method for the design of nonovershooting linear multivariable state feedback tracking controllers we consider the use of linear multivariable feedback control to achieve a nonovershooting step response a method is given for designing a linear time invariant state feedback controller to asymptotically track a constant step reference with zero overshoot and arbitrarily small rise time under some mild assumptions we present a unified design method that can be applied to continuous and discrete time systems square and non square systems minimum and nonminimum phase systems and also strictly proper and nonstrictly proper systems

orthogonal polynomial interpretation of δ toda equations in this paper a discretization of toda equations is analyzed the correspondence between these toda equations for the coefficients of the jacobi operator and its resolvent function is established it is shown that the spectral measure of these operators evolve in t like where is a given positive borel measure the lax pair for the toda equations is derived and characterized in terms of linear functionals where orthogonal polynomials which satisfy an appell condition with respect to the forward difference operator appear in a natural way in order to illustrate the results of the paper we work out two examples of toda equations related with jacobi and laguerre orthogonal polynomials

fundamental cycle of a periodic box ball system we investigate a box ball system with periodic boundary conditions since the box ball system is a deterministic dynamical system that takes only a finite number of states it will exhibit periodic motion we determine its fundamental cycle for a given initial state

combinatorial representation of invariants of a soliton cellular automaton abstract the structure of the soliton cellular automaton is studied by means of combinatorial techniques it is shown that the shape of the young tableaux gives an infinite number of time invariants of the automaton the employed combinatorial materials are the dyck language stack representable sequences and the robinson schensted algorithm

controller synthesis for sign invariant impulse response in this paper we consider the problem of designing controllers for discrete time linear time invariant lti plants that render the closed loop impulse response nonnegative such systems have a non undershooting and non overshooting step response we first show that the impulse response of any discrete time lti system changes sign at least r times if it has r real positive zeros outside a circular disk centered at the origin and containing all its poles we then show that a necessary and sufficient condition on the plant for the existence of a compensator that makes the closed loop impulse response sign invariant is that there be no real positive nonminimum phase plant zeros finally we show by construction how such a compensator may be synthesized when the plant does satisfy the existence condition

linearizable cellular automata the initial value problem for a class of reversible elementary cellular automata with periodic boundaries is reduced to an initial boundary value problem for a class of linear systems on a finite commutative ring moreover a family of such linearizable cellular automata is given

the box ball system and the n soliton solution of the ultradiscrete kdv equation any state of the box ball system bbs together with its time evolution is described by the n soliton solution with appropriate choice of n of the ultradiscrete kdv equation it is shown that simultaneous elimination of all 10 walls in a state of the bbs corresponds exactly to reducing the parameters that determine the size of a soliton by one this observation leads to an expression for the solution to the initial value problem ivp for the bbs expressions for the solution to the ivp for the ultradiscrete toda molecule equation and the periodic bbs are also presented

on a periodic soliton cellular automaton we propose a box and ball system with a periodic boundary condition periodic box and ball system pbbs the time evolution rule of the pbbs is represented as a boolean recurrence formula an inverse ultradiscretization of which is shown to be equivalent to the algorithm of the calculus for the 2nth root the relations to the pbbs of the combinatorial r matrix of u q an 1 are also discussed

correlation functions for a periodic box ball system we investigate correlation functions in a periodic box ball system for the two point functions of short distance we give explicit formulae obtained by combinatorial methods we give expressions for general n point functions in terms of ultradiscrete theta functions

clouds fibres and echoes a new approach to studying random walks on fractals up to now the general approach of constructing evolution differential equations to describe random walks on fractals has not succeeded is this because the true probability density function is inherently fractal when plotted in the appropriate similarity variable we find a cloud which is not too smooth further investigation shows that this cloud has a structure that might be overlooked if one is looking for the usual single valued probability density function the cloud is composed of an infinite family of smooth fibres each of which describes the behaviour of the walk on an infinite echo point class the fibres are individually smooth and so are naturally amenable to analysis with differential equations

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

shaping of the output response in a class of linear multivariable systems a state feedback controller is designed to regulate as desired the output response of minimum phase linear multivariable square plants with cb having full rank b input matrix and c output matrix properties of the controller are given and an example is worked out for illustration

optimal alternating direction implicit preconditioners for conjugate gradient methods in this work the extrapolated alternating direction implicit e adi preconditioners for the conjugate gradient cg method are applied for the solution of the second order elliptic partial differential equation pde in a rectangle under dirichlet boundary conditions it is shown that the spectral condition number for the discretized model problem when the adi cg method is used is an order of magnitude smaller than that of the cg method the pde is approximated by uniform 5 and 9 point stencils and it is shown that the determination of the optimal condition number is equivalent to that of the determination of the optimal eadi scheme analytic expressions for the optimal acceleration and extrapolation parameters are obtained in both cases of the 5 and 9 point schemes those for the 5 point complete known ones while those for the 9 point are novel finally numerical examples show that the eadi preconditioners proposed make the cg methods compete well against most of the fastest well known methods

asymptotic behaviour of fundamental cycle of periodic box ball systems we investigate asymptotic behaviour of fundamental cycle of periodic box ball systems pbbss when the system size n goes to infinity according to integrable nature of the pbbs the trajectory is confined to qualitatively smaller number of states than that of the total states we prove that although the maximum fundamental cycle is of order of exp n almost all fundamental cycle is less than exp logn 2

on the initial value problem of a periodic box ball system we show that the initial value problem of a periodic box ball system can be solved in an elementary way using simple combinatorial methods

simple strong glass forming models mean field solution with activation we introduce simple models inspired by previous models for froths and covalent glasses with trivial equilibrium properties but dynamical behaviour characteristic of strong glass forming systems these models are also a generalization of backgammon or urn models to a non constant number of particles where entropic barriers are replaced by energy barriers allowing for the existence of activated processes we formulate a mean field version of the models which keeps most of the features of the finite dimensional ones and solve analytically the out of equilibrium dynamics in the low temperature regime where activation plays an essential role

combinatorial aspects of the conserved quantities of the tropical periodic toda lattice the tropical periodic toda lattice trop p toda is a dynamical system attracting attention in the area of the interplay of integrable systems and tropical geometry we show that the young diagrams associated with trop p toda given by two very different definitions are identical the first definition is given via a lax representation of the discrete periodic toda lattice and the second one is associated with a generalization of the kerov kirillov reshetikhin bijection in the combinatorics of the bethe ansatz by means of this identification it is shown for the first time that the young diagrams given by the latter definition are preserved under time evolution this result is regarded as an important first step in clarifying the iso level set structure of this dynamical system in general cases i e not restricted to generic cases

matrix decomposition algorithms for modified spline collocation for helmholtz problems we consider the solution of various boundary value problems for the helmholtz equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimal fourth order approximations for the solution of the collocation equations we formulate matrix decomposition algorithms fast direct methods which employ fast fourier transforms and require o n2 log n operations on an n n uniform partition of the unit square a computational study confirms the published analysis for the dirichlet problem and indicates that similar results hold for neumann mixed and periodic boundary conditions the numerical results also exhibit superconvergence phenomena not reported in earlier studies

periodic multiwave solutions to the toda type cellular automaton an ultradiscretization of the riemann theta function is proposed the ultradiscretization satisfies an addition formula which is an ultradiscrete analogue of an addition formula for the riemann theta function using the addition formula periodic multiwave solutions to the toda type cellular automaton are obtained

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

edwards measures for powders and glasses can one construct a thermodynamics for compact slowly moving powders and grains a few years ago edwards proposed a possible step in this direction raising the fascinating perspective that such systems have a statistical mechanics of their own different from that of maxwell boltzmann and gibbs allowing us to have some information while still ignoring dynamic details recent developments in the theory of glasses have come to confirm these ideas within mean field in order to go beyond we explicitly generate edwards measure in a 3d model comparison of the results with the irreversible compaction data shows very good agreement the present framework immediately suggests new experimental checks

a tropical analogue of fay s trisecant identity and the ultra discrete periodic toda lattice we introduce a tropical analogue of fay s trisecant identity for a special family of hyperelliptic tropical curves we apply it to obtain the general solution of the ultra discrete toda lattice with periodic boundary conditions in terms of the tropical riemann s theta function

toric networks geometric r matrices and generalized discrete toda lattices we use the combinatorics of toric networks and the double affine geometric r matrix to define a three parameter family of generalizations of the discrete toda lattice we construct the integrals of motion and a spectral map for this system the family of commuting time evolutions arising from the action of the r matrix is explicitly linearized on the jacobian of the spectral curve the solution to the initial value problem is constructed using riemann theta functions

a geometric realization of the periodic discrete toda lattice and its tropicalization an explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete toda lattice pdtl is presented first the time evolution is realized as a point addition on a hyperelliptic curve which is the spectral curve of the pdtl then the point addition is translated into curve intersections next it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the pdtl finally the formulation is lifted to the framework of tropical geometry and a tropical geometric realization of the periodic box ball system is constructed via tropical curve intersections

linear matrix inequalities for globally monotonic tracking control this paper addresses the problem of achieving monotonic tracking control for any initial condition also referred to as global monotonic tracking control this property is shown to be equivalent to global non overshooting as well as to global non undershooting i e non overshooting and non undershooting for any initial condition respectively the main objective of this paper is to prove that a stable system is globally monotonic if and only if all the rows of the output matrix are left eigenvectors of the space transition matrix this property allows one to formulate the design of a controller which ensures global monotonic tracking as a convex optimization problem described by a set of linear matrix inequalities lmis

conserved quantities and generalized solutions of the ultradiscrete kdv equation we construct generalized solutions to the ultradiscrete kdv equation including the so called negative solition solutions the method is based on the ultradiscretization of soliton solutions to the discrete kdv equation with gauge transformation the conserved quantities of the ultradiscrete kdv equation are shown to be constructed in a similar way to those for the box ball system

conserved quantities of generalized periodic box ball systems constructed from the ndkp equation we investigate periodic box ball systems pbbss with several kinds of balls and box capacity greater than or equal to one conserved quantities of the pbbss are constructed from those of the nonautonomous discrete kp ndkp equation using the lax representation of the ndkp equation

ultradiscretization of solvable one dimensional chaotic maps we consider the ultradiscretization of a solvable one dimensional chaotic map which arises from the duplication formula of the elliptic functions it is shown that the ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously a geometric interpretation of the dynamics of the tent map is given in terms of the tropical jacobian of a certain tropical curve generalization to the maps corresponding to the mth multiplication formula of the elliptic functions is also discussed

on reversibility of cellular automata with periodic boundary conditions reversibility of one dimensional cellular automata with periodic boundary conditions is discussed it is shown that there exist exactly 16 reversible elementary cellular automaton rules for infinitely many cell sizes by means of a correspondence between elementary cellular automaton and the de bruijn graph in addition a sufficient condition for reversibility of three valued and two neighbour cellular automaton is given

nonovershooting control of strict feedback nonlinear systems a means of obtaining a nonovershooting output tracking response for single input single output siso strict feedback nonlinear systems is introduced with the proposed method arbitrary reference trajectories can be tracked from below for arbitrary initial conditions as long as the initial value of the plant output is strictly below the initial value of the reference trajectory in addition a design is presented for approximately nonovershooting control in the presence of disturbances where the amount of overshoot can be made arbitrarily small by appropriately choosing the control gains finally an output feedback example shows the ability of our approach to ensure arbitrarily small overshoot where the overshoot is caused by the initial condition of the unmeasured part of the state

path description of conserved quantities of generalized periodic box ball systems we investigate conserved quantities of periodic box ball systems pbbs with arbitrary kinds of balls and box capacity greater than or equal to 1 we introduce the notion of nonintersecting paths on the two dimensional array of boxes and give a combinatorial formula for the conserved quantities of the generalized pbbs using these paths

invariant measures for the box ball system based on stationary markov chains and periodic gibbs measures the box ball system bbs is a simple model of soliton interaction introduced by takahashi and satsuma in the 1990s recent work of the authors together with tsuyoshi kato and satoshi tsujimoto derived various families of invariant measures for the bbs based on two sided stationary markov chains d a croydon et al dynamics of the box ball system with random initial conditions via pitman s transformation arxiv 1806 02147 in this article we survey the invariant measures that were presented in d a croydon et al and also introduce a family of new ones for periodic configurations that are expressed in terms of gibbs measures moreover we show that the former examples can be obtained as infinite volume limits of the latter another aspect of d a croydon et al was to describe scaling limits for the bbs here we review the results of d a croydon et al and also present scaling limits other than those that were covered there one the zigzag process has previously been observed in the context of queuing another a periodic version of the zigzag process is apparently novel furthermore we demonstrate that certain palm measures associated with the stationary and periodic versions of the zigzag process yield natural invariant measures for the dynamics of corresponding versions of the ultradiscrete toda lattice the box ball system bbs is a simple model of soliton interaction introduced by takahashi and satsuma in the 1990s recent work of the authors together with tsuyoshi kato and satoshi tsujimoto derived various families of invariant measures for the bbs based on two sided stationary markov chains d a croydon et al dynamics of the box ball system with random initial conditions via pitman s transformation arxiv 1806 02147 in this article we survey the invariant measures that were presented in d a croydon et al and also introduce a family of new ones for periodic configurations that are expressed in terms of gibbs measures moreover we show that the former examples can be obtained as infinite volume limits of the latter another aspect of d a croydon et al was to describe scaling limits for the bbs here we review the results of d a croydon et al and also present scaling limits other than those that were covered there one the zigzag process has previously been observed in the context

the am 1 automata related to crystals of symmetric tensors a soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra uq am 1 is introduced it is a crystal theoretic formulation of the generalized box ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous scattering matrices of two solitons coincide with the combinatorial r matrices of uq am 1 1 a piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete kadomtsev petviashivili equation a class of n soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter

iterative line cubic spline collocation methods for elliptic partial differential equations in several dimensions this paper presents a new class of second and fourth order line cubic spline collocation methods the lcsc methods for multidimensional linear elliptic partial differential equations with no cross derivative terms the lcsc methods approximate the differential operator along lines in each dimension independently and then combine the results into one large linear system expressed in terms of discretization stencils for the operator these methods have nonzero entries only in the coordinate directions the advantage of this approach is that the discretization is much simpler to derive and analyze further iterative methods are easily applied to the resulting linear systems especially on parallel computers the disadvantage is that the resulting linear system is k times larger in k dimensions using the simplicity of the methods iterative schemes are analyzed and formulated in order to solve the resulting lcsc linear systems in the case of helmholtz problems block jacobi extrapolated jacobi ej and

glassy behaviour in a simple topological model in this paper we study a simple purely topological cellular model which is allowed to evolve through a glauber kawasaki process we find a non thermodynamic transition to a glassy phase in which the energy defined as the square of the local cell topological charge fails to reach the equilibrium value below a characteristic temperature which is dependent on the cooling rate we investigate a correlation function which exhibits ageing behaviour and follows a master curve in the stationary regime when time is rescaled by a factor of the relaxation time tr this master curve can be fitted by a von schweidler law in the late relaxation regime the relaxation times can be well fitted at all temperatures by an offset arrhenius law a power law can be fitted to an intermediate temperature regime the exponent of the power law and the von schweidler law roughly agree with the relationship predicted by mode coupling theory by defining a suitable response function we find that the fluctuation dissipation ratio is held until sometime later than the appearance of the plateaux non monotonicity of the response is observed after this ratio is broken a feature which has been observed in other models with dynamics involving activated processes