digital pid controller design via general discrete orthogonal polynomials based on the discrete orthogonal polynomials a new computer oriented algebraic method is proposed for designing a digital pid controller the pid controller can approximately achieve an arbitrarily specified closed loop response and is determined by solving a linear equation using the least squares method an example is given to demonstrate the availability of the proposed technique

fractal wavelet dimensions and localization in this paper we want to give a new definition of fractal dimensions as small scale behavior of theq energy of wavelet transforms this is a generalization of previous multi fractal approaches with this particular definition we will show that the 2 dimension correlation dimension of the spectral measure determines the long time behavior of the time evolution generated by a bounded self adjoint operator acting in some hilbert space it will be proved that for we have mathop lim inf limits t to infty frac log int 0 t d omega left left langle psi left e ia omega right phi right rangle right 2 log t kappa 2 and that mathop lim sup limits t to infty frac log int 0 t d omega left left langle psi left e ia omega right phi right rangle right 2 log t kappa 2 wherek 2 are the upper and lower correlation dimensions of the spectral measure associated with and a quantitative version of the rage theorem shall also be given

fourier asymptotics of fractal measures abstract a measure on r n will be called locally uniformly dimensional if br x cr for all r 1 and all x where br x denotes the ball of radius r about x for l 2 d the measure d is in i so d is well defined we show it is locally in l2 and sup r 1 r n b r y f d 2 d c f 2 under additional hypotheses we show that lim r r n b r y f d 2 2 d is comparable in size to 2 2 a number of other related results are established the special case when is an integer and is the surface measure on a c1 manifold was treated by s agmon and l hormander j analyse math 30 1976 1 38

application of chebyshev polynomials to the optimal control of time varying linear systems the operational matrix of backward integration for the shifted chebyshev polynomials is introduced in this study the general expression of the shifted chebyshev polynomial approximation for any two arbitrary functions is also presented a linear time varying optimal control system with a quadratic performance measure is solved by using the shifted chebyshev polynomials only a small number of chebyshev polynomials is needed to produce an excellent result and the outcome is much better than the solution obtained by using the block pulse function so computer memory capacity and computing time can be saved considerably

remarks on the relation between quantum dynamics and fractal spectra abstract we investigate the large time behaviour of various solutions of the schrodinger equation in terms of some local and global dimensions of the spectral measure we emphasize in particular the role of the hausdorff and correlation dimensions for the growth exponents of position moments we also discuss the stability of such exponents under local perturbations

singular continuous spectrum on a cantor set of zero lebesgue measure for the fibonacci hamiltonian it is rigorously proven that the spectrum of the tight binding fibonacci hamiltonian h mn m n 1 m n 1 m n n 1 n where 5 1 2 and means integer part is a cantor set of zero lebesgue measure for all real nonzero and the spectral measures are purely singular continuous this follows from a recent result by kotani coupled with the vanishing of the lyapunov exponent in the spectrum

meblin transforms associated with julia sets and physical applications we introduce the mellin transform of the balanced invariant measure associated to the julia set generated by a rational transformation we show that its analytic continuation is a meromorphic function the poles of which are on a semi infinite periodic lattice this allows one to have an understanding of the behavior of the measure near a repulsive fixed point trace identities corresponding to the fact that the analytically continued mellin transform vanishes at negative integers are derived for the polynomial case the quadratic map is first analyzed in detail and the analytic properties of the inverse of the green s function are exhibited of interest is the appearance of a dense set of spikes at dyadic points when the julia set is disconnected these results are used to study the residues of the mellin transform a certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems the excitation spectrum of which displays unusual oscillations the appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion

model reduction of digital systems using discrete walsh series this study discusses the application of discrete walsh series expansion to reduce the order of a linear time invariant digital system described by z transfer function the approach is based on matching the discrete walsh spectra to determine both the coefficients of the denominator and numerator of the reduced model the proposed method is simple for computation can preserve the dynamic characteristic of the original model satisfactorily and guarantees to have the same zero initial response as the original system

rigorous spectral analysis of the metal insulator transition in a limit periodic potential we consider the limit periodic jacobi matrices associated with the real julia sets of f a z z2 a for which ae 2 oo can be seen as the strength of the limit periodic coefficients the typical local spectral exponent of their spectral measures is shown to be a harmonic function in a decreasing logarithmically from 1 to 0

combined methods for model reduction via discrete laguerre polynomials abstract model reduction of linear discrete systems by matching the discrete laguerre spectra of the unit step responses of the original and reduced models is first discussed in order to preserve the stable property of the original model combined methods which use conventional stable methods to determine the parameters of denominator of the reduced models and use the discrete laguerre spectrum matching to determine the parameters of the numerator are discussed

optimal control using discrete laguerre polynomials this paper presents discrete laguerre polynomials which approximate the optimal control problems of digital systems the discrete variational principle in combination with the idea of a penalty function is used to construct the modified discrete euler lagrange equations the discrete laguerre series is then applied to simplify the modified equations into a set of linear algebraic ones for the approximate solutions of state and control variables an example is given and the results obtained are very accurate and attractive

application of discrete chebyshev polynomials to the optimal control of digital systems the shift transformation matrix for discrete chebyshev polynomials is introduced in this study the discrete variational principle combined with the idea of penalty function is taken to construct the modified discrete euler lagrange equations then the discrete chebyshev series are applied to simplify the modified equations into a set of linear algebraic ones for the approximations of state and control variables of digital systems it is seen that this technique is quite straightforward and simple and computing time can be saved considerably

the operational matrices of integration and differentiation for the fourier sine cosine and exponential series for the fourier sine cosine series basis vector varphi t and the fourier exponential series basis vector psi t a linear nonsingular transfor marion t is determined such that psi t t varphi t this result is then used to show that the operational matrices of integration p and q for varphi t and psi t respectively are related by the expression tp qt analogous results are derived for the corresponding operational matrices of differentiation d and r general expressions are derived for t p q d and r

discrete pulse orthogonal functions for the analysis parameter estimation and optimal control of linear digital systems discrete pulse orthogonal functions dpofs are introduced and their operational matrix is proposed to solve various control problems the applications of dpofs in digital control systems are analogous to those of block pulse functions bpfs in continuous systems applying the dpofs the approximate solutions of digital time invariant systems can be easily obtained by a convenient algorithm three examples are presented to demonstrate the applications of dpofs

reduced order iir approximation to fir digital filters an algorithm for designing an infinite impulse response iir stable filter using a finite impulse response fir given filter with the objective of reducing the delay and order is described the design is in the time domain using the least squares inverse algorithm which is briefly described in this method the numerator of the approximated filter is part of the fir filter itself and no calculations and minimization are needed to find the numerator coefficients except finding the fir roots an error analysis between the given fir and approximated iir filters is provided this error analysis enables the designer to fix a design parameter often unnoted keeping the energies of the approximated and original filters equal results and two illustrative examples are presented

on rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions we consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as hausdorff dimension box dimension etc we also introduce different ways to define and study the generalized spectrum for dimensions a one parameter family of characteristics of dimension type

solving related two and three dimensional linear programming problems in logarithmic time abstract given n linear inequalities in three variables we show how to construct a corresponding spherical subdivision using great circle arcs in time o n log n and space o n this subdivision in turn allows us to compute the point in space satisfying all inequalities and maximizing any desired linear objective function in time o log n

optimal search in planar subdivisions a planar subdivision is any partition of the plane into possibly unbounded polygonal regions the subdivision search problem is the following given a subdivision s with n line segments and a query point p determine which region of s contains p we present a practical algorithm for subdivision search that achieves the same optimal worst case complexity bounds as the significantly more complex algorithm of lipton and tarjan namely o log n search time with o n storage our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications

collision detection algorithm of a continuous type using spherical extreme vertex diagrams abstract most collision detection algorithms are ones of a discrete type which utilize convexity and local coherence to verify the closest points however the incremental algorithms have such critical drawbacks like failure to detect high speed collisions and dependence on the time complexity on the time step size of animation in this paper in order to find a new method which can resolve the drawbacks we are concerned with a more restricted but nontrivial version of the problem given a fixed infinite plane h and a moving convex polyhedron p compute their collision time our algorithm utilizes the spherical extreme vertex diagram that is the embedding of the dual graph p onto the gauss sphere exploiting the diagram we are able to efficiently compute the sequence of the extreme vertices v i s i 1 m and find the time interval t i t i 1 during which v i is the extreme vertex of p with the sequences of the consecutive time intervals and extreme vertices we construct a distance function between p and h hence we can compute their collision time within a given tolerance by applying the interval newton method to the distance function moreover the total time complexity is independent of the time step size of animation

fractal measures and mean p variations abstract recently strichartz proved that if is locally uniformly dimensional on r d then where 0 d and bt denotes the ball of radius t center at 0 if is self similar and satisfies a certain open set condition he also obtained a formula for the so that 0 lim sup t 1 t d b t f s 2 the can serve in some sense as the dimensional index of the measure by using the mean p variation and the tauberian theorems we extend the first inequality and its variants to p q forms and give necessary and sufficient conditions on for such inequalities to hold we then use the mean quadratic variation to study some self similar measures on r which do not satisfy the open set condition the s that are constructed from s 1 x x s 2 x x 1 1 2 with weights 1 2 each the index for corresponding to 5 1 2 is calculated the expression for such is significantly different from the one obtained by strichartz

spectral properties of one dimensional quasi crystals in this paper we prove that the one dimensional schrodinger operator onl 2 with potential given by upsilon n lambda chi 1 alpha 1 x n alpha alpha notin mathbb q has a cantor spectrum of zero lebesgue measure for any irrational and any 0 we can thus extend the kotani result on the absence of absolutely continuous spectrum for this model to all mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepge9fr xfr x frpewzqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiaadihaiiaacq wfiiiztuudjxwak1uy0hmmaehbfv3yslgzg0uy0hgiud3bagqbbiab nj8ubaa 4628 x in mathbb t

the dimension of the maximal measure for a polynomial map let f c c be a polynomial of degree d 2 the most interesting dynamics of f takes place on its julia set j f an invariant compact perfect set which can be defined for example as the closure of the repelling periodic points brolin constructed the equilibrium distribution m on j f an ergodic measure which is also the weak limit as n x of a probability measure distributed equally over the dn nth inverse images of almost any point we shall prove the following

discrete walsh polynomials in the optimal control of linear digital time varying systems the discrete walsh polynomials are applied to solve the digital optimal control of time varying systems in this study due to the elegant properties of operational matrices of discrete walsh polynomials the class of digital time varying problems can be analysed successfully the design algorithms of digital optimal control are based on the discrete variational principle combined with the idea of a penalty function to obtain directly computational formulations for evaluating the optimal control and state trajectory an example is given to illustrate the usefulness of the method and the results are very accurate

walsh function analysis of linear and bilinear discrete time systems a technique is presented for analysing linear and bilinear discrete systems in terms of walsh functions the method replaces the solution of a dynamical difference equation with the solution of an algebraic equation in the special case of a linear system a numerically stable recursive technique is derived to solve this algebraic equation the technique is illustrated with an example

spectral properties of one dimensional schrodinger operators with potentials generated by substitutions we investigate one dimensional discrete schrodinger operators whose potentials are invariant under a substitution rule the spectral properties of these operators can be obtained from the analysis of a dynamical system called the trace map we give a careful derivation of these maps in the general case and exhibit some specific properties under an additional easily verifiable ypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying schrodinger operator is singular and supported on a set of zero lebesgue measure a condition allowing to exclude point spectrum is also given the application of our theorems is explained on a series of examples

an efficient algorithm for determining the extreme vertices of a moving 3d convex polyhedron with respect to a plane abstract we present an efficient algorithm for finding the sequence of extreme vertices of a moving convex polyhedron p with respect to a fixed plane h using the spherical extreme vertex diagram due to the point plane duality we are able to find such a sequence in o log n sj 1 mj time where s is the number of extreme vertices in the sequence and mj 1 j s is the number of edges of the spherical region svj corresponding to an extreme vertex vj in the sequence

solution of discrete scaled systems via general discrete orthogonal polynomials abstract general discrete orthogonal polynomials are introduced to analyse and approximate the solution of a class of discrete scaled systems using the general discrete shift transformation matrix together with the general discrete scale matrix the discrete scaled system can be reduced to a set of simultaneous linear algebraic equations the coefficient vectors of general discrete orthogonal polynomials can be determined simply by the derived algorithm examples are included to show the applicability of the general discrete orthogonal polynomial approximations

simple methods for the shift transformation matrix direct product matrix and summation matrix of discrete walsh series this paper presents new easy methods for directly obtaining the shift transformation matrix direct product matrix and summation matrix of discrete walsh series the proposed methods can be conveniently implemented with a digital computer they will be very useful in the study of control systems via discrete walsh series

discrete walsh operational matrices for analysis and optimal control of linear digital systems abstract by the introduction of the shift transformation matrix direct product matrix and summation matrix of the discrete walsh series the analysis of time varying digital control systems is facilitated and the approximate solution of time invariant digital optimal control problems is achieved of this study the design algorithms of digital optimal control are based on the discrete variational principle combined with the idea of penalty functions to obtain the conveniently computational formulations for evaluating the optimal control and trajectory three examples are illustrated by using the discrete walsh approach

quantum intermittency in almost periodic lattice systems derived from their spectral properties abstract hamiltonian tridiagonal matrices characterized by multi fractal spectral measures in the family of iterated function systems can be constructed by a recursive technique described here we prove that these hamiltonians are almost periodic they are suited to describe quantum lattice systems with nearest neighbours coupling as well as chains of linear classical oscillators and electrical transmission lines we investigate numerically and theoretically the time dynamics of the systems so constructed we derive a relation linking the long time power law behaviour of the moments of the position operator expressed by a scaling function of the moment order and spectral multi fractal dimensions dq via d1 we show cases in which this relation is exact and cases where it is only approximate unveiling the reasons for the discrepancies

pointwise dimensions on real julia sets from a formula for the pointwise dimensions of bernoulli measures on real quadratic julia sets we deduce several aspects of their multifractal analysis in particular we prove a lower bound for the scaling function and give explicit formulae for its borders

optimal point location in a monotone subdivision point location often known in graphics as hit detection is one of the fundamental problems of computational geometry in a point location query we want to identify which of a given collection of geometric objects contains a particular point let mathcal s denote a subdivision of the euclidean plane into monotone regions by a straight line graph of m edges in this paper we exhibit a substantial refinement of the technique of lee and preparata siam j comput 6 1977 pp 594 606 for locating a point in mathcal s based on separating chains the new data structure called a layered dag can be built in o m time uses o m storage and makes possible point location in o log m time unlike previous structures that attain these optimal bounds the layered dag can be implemented in a simple and practical way and is extensible to subdivisions with edges more general than straight line segments

analysis and optimal control of discrete linear time varying systems via discrete general orthogonal polynomials general discrete orthogonal polynomials are introduced to analyse and to approximate the solution of a class of linear time varying discrete systems it is shown that the proposed method is very general in that it works for any discrete orthogonal polynomial that possess a recurrence relation hence the present method includes the results obtained by use of discrete chebyshev polynomials and discrete laguerre polynomials as special cases the analysis of linear time varying systems via general discrete orthogonal polynomials is presented general discrete orthogonal polynomials are then applied to approximate the solution of time varying optimal control problems illustrative examples are included and a comparison of the results obtained via several different orthogonal polynomials is presented