direct sums of self small mixed groups abstract the class g of self small mixed abelian groups of finite rank has recently been the focus of a number of investigations it is known that up to quasi isomorphism g is dual to the category of locally free torsion free abelian groups of finite rank we utilize the reduced mixed groups in g as building blocks of a more extensive class g the smallest class containing the reduced groups in g that is closed under taking infinite direct sums and direct summands our central result is the determination of a complete set of isomorphism invariants for the groups in g we supplement this broad classification theorem with an investigation of the fine structure of completely decomposable groups in g

application of general orthogonal polynomials to the optimal control of time varying linear systems the operational matrices for forward or backward integration of general orthogonal polynomials are derived the tensor expression for the generalized orthogonal polynomial approximation of any two arbitrary functions is also introduced it is shown that the approximation solution obtained using the chebyshev polynomial is readily obtainable as a special case of the results derived thus the present results include results presented by shih in 1983 and chou and horng in 1984 a linear time varying optimal control system with a quadratic performance measure is solved by using the generalized orthogonal polynomials comparison of the results with those obtained using several different classical orthogonal polynomial approximations is also included

balanced projectives and axiom 3 abstract a global definition of k nice subgroup is formulated which generalizes both the notion of a nice subgroup in the context of torsion groups and the notion of a k nice submodule used in a recent study of p local balanced projective groups it is shown analogous to hill s treatment of totally projective p groups that the global balanced projective groups of warfield are precisely those mixed abelian groups that satisfy the third axiom of countability for k nice subgroups a variety of equivalent characterizations of balanced projectives are derived by methods independent of the uniqueness and existence theorems previously established for these groups

on intermittency in a cascade model for turbulence in this note we study the possibility of performing analytic computations of the exponents characterizing the multifractal behaviour of turbulence a simple analytic computation is presented in the framework of the cascade model or shell model

analysis of bilinear systems via single term walsh series this paper presents a new method of analysis of bilinear systems via the single term walsh series stws the stws method has definite advantages over the walsh function wf method and block pulse function bpf method both the discrete values and block pulse values of the results are obtained for any length of time and selecting any positive integer as the number of intervals the recursive relations are simple and save both computer time and storage illustrative examples are presented

gibbsian hypothesis in turbulence we show that kolmogorov multipliers in turbulence cannot be statistically independent of others at adjacent scales or even a finite range apart by numerical simulation of a shell model and by theory as the simplest generalization of independent distributions we suppose that the steady state statistics of multipliers in the shell model are given by a translation invariant gibbs measure with a short range potential when expressed in terms of suitable spin variables real valued spins that are logarithms of multipliers and xy spins defined by local dynamical phases numerical evidence is presented in favor of the hypothesis for the shell model in particular novel scaling laws and derivative relations predicted by the existence of a thermodynamic limit the gibbs measure appears to be in a high temperature unique phase regime with paramagnetic spin order

analysis and optimal control of time varying linear systems via block pulse functions succinct and clear sighted operational properties of block pulse functions are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index piecewise constant solutions equally distributed which are simple in form and convenient for use or implementation are consequently obtained another advantage of this method is that any positive integer can be chosen as the number of sub intervals whereas in the case of walsh function approximation the choice can only be made from 2 4 8 16 32 and so on the algorithms developed in the paper are illustrated by appropriate examples

application of taylor series in obtaining the orthogonal operational matrix in this research first we explicitly obtain the relation between the coefficients of the taylor series and jacobi polynomial expansions then we present a new method for computing classical operational matrices derivative integral and product for general jacobi orthogonal functions polynomial and rational this method can be used for many classes of orthogonal functions

application of shifted chebyshev series to the optimal control of linear distributed parameter systems by use of a particular property of the shifted chebyshev series the optimal control of a distributed parameter system is simplified into the optimal control of a linear time invariant lumped parameter system next a directly computational formulation for evaluating the optimal control and trajectory of a linear distributed parameter system is developed the formulation is straightforward and convenient for digital computation an illustrative example is given to demonstrate the applicability of the proposed method

analysis and parameter estimation of bilinear systems via block pulse functions problems of analysis and parameter estimation of bilinear systems via block pulse functions are discussed and recursive algorithms convenient for computers are presented compared with walsh function approaches the results are much simpler in form in this method any positive integer can be chosen as the number of sub intervals consequently much computing time and storage can be saved and the recursive algorithms developed also make it possible to identify on line the parameters of bilinear systems

non gibbsian states and their gibbs description the driving principle behind this paper is the following thesis every physically reasonable random field has to be a gibbs random field in this paper the so called non gibbsian random fields are considered the usual definition of the gibbs field is generalized in such a way so as to include some of the discovered non gibbsian fields the new definition is then used to show that the projection of the two dimensional ising model onto the one dimensional sublattice 1 falls into the class of the generalized gibbs fields

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

improved shell model of turbulence we introduce a shell model of turbulence that exhibits improved properties in comparison to the standard and very popular gledzer ohkitani and yamada goy model the nonlinear coupling is chosen to minimize correlations between different shells in particular the second order correlation function is diagonal in the shell index and the third order correlation exists only between three consecutive shells spurious oscillations in the scaling regime which are an annoying feature of the goy model are eliminated by our choice of nonlinear coupling we demonstrate that the model exhibits multiscaling similar to the goy model the scaling exponents are shown to be independent of the viscous mechanism as is expected for navier stokes turbulence and other shell models these properties of the model make it optimal for further attempts to achieve understanding of multiscaling in nonlinear dynamics

hierarchical optimal control for distributed parameter systems via block pulse operator this paper presents a development of the hierarchical optimal control for distributed parameter systems via block pulse operator the proposed algorithm enables one to solve optimal control of distributed parameter systems with hierarchical structure and as a result the resulting solutions are piecewise constant with minimal mean square error the algorithm is simple in form and computationally advantageous an illustrative example shows the applicability of the proposed method in practice

very fast linear and non linear observers part 1 theory the problem of the design of very fast observers is considered for linear and nonlinear systems four optimal criteria are introduced allowing a wide choice of observer parameters and an allocation of desired eigenvalues newly derived approaches are compared with known methods of optimal observer design numerical examples illustrate these methods

numerical solution of the controlled duffing oscillator by hybrid functions a numerical method for solving the controlled duffing oscillator is presented the method is based upon hybrid functions approximation the properties of hybrid functions which consists of block pulse functions plus legendre polynomials are given the method is based upon expanding various time functions in the system as their truncated hybrid functions the operational matrix of product is introduced this matrix together with the operational matrix of integration are utilized to reduce the solution of controlled duffing oscillator to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

shifted chebyshev series analysis and identification of time varying bilinear systems shifted chebyshev series are developed in this study to analyse and identify time varying bilinear systems effective computational formulations which are very straightforward and convenient for digital computations are achieved consequently the computational difficulty can be remarkably reduced two illustrative examples are given to demonstrate the superiority of the presently proposed method

fourier series direct method for variational problems a direct method for solving variational problems using fourier series is presented an operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations illustrative examples are also given

optimal subgrid scheme for shell models of turbulence we discuss a theoretical framework to define an optimal subgrid closure for shell models of turbulence the closure is based on the ansatz that consecutive shell multipliers are short range correlated following the third hypothesis of kolmogorov formulated for similar quantities for the original three dimensional navier stokes turbulence we also propose a series of systematic approximations to the optimal model by assuming different degrees of correlations across scales among amplitudes and phases of consecutive multipliers we show numerically that such low order closures work well reproducing all known properties of the large scale dynamics including anomalous scaling we found small but systematic discrepancies only for a range of scales close to the subgrid threshold which do not tend to disappear by increasing the order of the approximation we speculate that the lack of convergence might be due to a structural instability at least for the evolution of very fast degrees of freedom at small scales connections with similar problems for large eddy simulations of the three dimensional navier stokes equations are also discussed

analysis and optimal control of time varying systems via generalized orthogonal polynomials the method or generalized orthogonal polynomials gop is applied to the analysis and optimal control of time varying systems the proposed gop can represent all kinds of individual orthogonal polynomial and non orthogonal taylor series the operational matrix for the forward and backward integration of the generalized orthogonal polynomials and the operational matrix of the product of r and the generalized orthogonal polynomial vector are derived and applied to time varying systems by using these three kinds of operational matrices the computational algorithm for calculating the expansion coefficients is very simple and effective three satisfactory examples illustrate the usefulness of the method

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

comments on the properties of the operational matrices of integration and differentiation for fourier trigonometric functions the properties of the operational matrices of integration p and differentiation q are discussed for the case of fourier trigonometric functions in particular a theorem that establishes the relationship between the matrices p sup 1 and q is given and the meaning of using p sup 1 instead of q recently proposed in literature is clarified

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

the design of optimal observers via shifted chebyshev polynomials the shifted chebyshev polynomials to approximate the quadratic performance measure of estimation errors for the design of optimal observers with specified distinct and multiple eigenvalues is developed in this study this method is very simple as compared with other design techniques of optimal observers one example is illustrated and only a small number m 4 of shifted chebyshev series are needed to produce a much better result than that obtained by the convenient block pulse approach

analysis parameter estimation and optimal control of time delay systems via chebyshev series the chebyshev delay operational matrix is introduced first and then applied to approximate the solutions of linear time invariant and time varying delay systems with arbitrary time delay the parameter identification problem of the delay control system is also studied furthermore an approximate solution of the optimal control problem with quadratic performance measure is then discussed four examples are given and the results are shown to be very accurate and satisfactory

pseudo free energies and large deviations for non gibbsian fkg measures a large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on 0 1 zd is proven in this way a pseudo free energy and pressure is defined for ergodic systems the large deviations property holds with the usual scaling the case of non ergodic systems is also discussed a similar result holds for occupation times the perturbation by an external field is treated

optimal control via fourier series of operational matrix of integration the state equations of an optimal regulator problem are given in terms of the truncated fourier series and the associated operational matrix of integration an effective computational algorithm is developed to calculate the expansion coefficients of the derivatives of state variables for saving computer storage and time and minimizing the computational error an illustrative example is given and satisfactory computational results are obtained

the operational matrix of differentiation for orthogonal polynomial series in the present paper a new method is introduced for the approximate solution of ordinary linear differential equations using orthogonal polynomial series on a computer for this purpose an operational matrix of differentiation is introduced that essentially transforms the differential operation to a matrix multiplication additionally an operational matrix of polynomial series transformation is used that effectively transforms any orthogonal polynomial series to taylor series the use of these operational matrices simplifies considerably the structure of the computer codes used for the application of spectral methods for the solution of linear differential equations and allows the generation of a general program for all types of orthogonal polynomial series

solutions of convolution integral and fredholm integral equations via double fourier series double fourier series are developed to approximate the solutions of the convolution integral and fredholm integral equations properties of fourier series are first briefly presented and the operational matrix of integration together with the product operational matrix is utilized to reduce the computation of integral equations to a set of simultaneous linear algebraic equations the method is computationally very attractive and applications are demonstrated through illustrative examples

use of block pulse functions in the optimal control of deterministic systems abstract the synthesis of an optimal control function for deterministic systems described by intergrodifferential equations is investigated by using the elegant operational properties of block pulse functions a directly computable algorithm for evaluating the optimal control and trajectory of deterministic systems is developed an example is given to illustrate the utility of this method

analysis and optimal control of time varying systems via chebyshev polynomials abstract the operational properties of the forward and backward integration and the product of chebyshev polynomials are derived these properties are applied to the analysis and optimal control of time varying linear systems by the approximation of time functions by truncated chebyshev series due to the unique property of the chebyshev polynomials excellent results are obtained as demonstrated by examples

analysis and optimal control of time varying linear systems via taylor series the taylor series is used for the analysis of time varying linear systems with the introduction of the product matrix of the taylor polynomial vector and its transpose for the solution of the optimal control problem the operational matrix of backward integration for the taylor series is introduced in this way the state transition matrix of optimal control of linear systems with quadratic performance index may be integrated approximately via taylor series the results obtained indicate an excellent agreement with the exact solution for a small number of terms

double shifted chebyshev series for convolution integral and integral equations double shifted chebyshev polynomials are developed in this study to approximate the solutions of the convolution integral volterra integral equation and fredholm integral equation this method simplifies the computations of integral equations to the successive solutions of a linear algebraic equation in matrix form in addition the computational complexity can be reduced remarkably three examples are illustrated it is seen that the proposed approach is straightforward and convenient and converges faster in finding approximations than other existing orthogonal function methods

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

characterizations of warfield groups abstract warfield groups are defined to be direct summands of simply presented abelian groups by introducing a global definition of quasi sequentially nice subgroups we prove some characterizations of warfield groups involving these groups knice subgroups and valuated coproducts furthermore we exhibit a new proof of a characterization of warfield groups in terms of nice subgroups and valuated coproducts

legendre series approach to identification and analysis of linear systems in this paper the legendre operational matrix of integration is introduced and it is subsequently used for the identification and analysis of linear systems the algorithms proposed are easily implementable in a digital computer and they are analogous to those already derived for other types of orthogonal functions

analysis and optimal control of time varying linear systems via shifted legendre polynomials this paper extends the application of shifted legendre polynomial expansion to time varying systems the extension is achieved through representing the product of two shifted legendre series in a new shifted legendre series with this treatment of the product of two time functions the operational properties of the shifted legendre polynomials are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index

analysis of linear time varying systems and bilinear systems via fourier series a method for finding an approximate solution of linear time varying systems and bilinear systems via fourier series is proposed properties of fourier series are first briefly presented and the operational matrix for integration together with the product operational matrix are utilized to reduce the dynamical equation of a linear time varying system or bilinear system to a set of simultaneous linear algebraic equations illustrative examples are also given

shifted chebyshev series analysis of linear optimal control systems incorporating observers the analysis of optimal control systems incorporating observers has been approached by the shifted chebyshev series in this study the method simplifies the system of equations into the successive solution of a set of linear algebraic equations one example is illustrated and only a small number of the chebyshev polynomials is needed to obtain a very attractive and excellent result