distribution estimation consistent in total variation and in two types of information divergence the problem of the nonparametric estimation of a probability distribution is considered from three viewpoints the consistency in total variation the consistency in information divergence and consistency in reversed order information divergence these types of consistencies are relatively strong criteria of convergence and a probability distribution cannot be consistently estimated in either type of convergence without any restrictions on the class of probability distributions allowed histogram based estimators of distribution are presented which under certain conditions converge in total variation in information divergence and in reversed order information divergence to the unknown probability distribution some a priori information about the true probability distribution is assumed in each case as the concept of consistency in information divergence is stronger than that of convergence in total variation additional assumptions are imposed in the cases of informational divergences

dynamic stiffness analysis of follower force abstract the dynamic stiffness method can predict an infinite number of natural modes of a conservative structure by means of a finite number of co ordinates the method is extended to non conservative systems characterized by follower forces in this paper skeletal frame structures are taken as examples the resulting matrix equations are solved by the parametric inverse iteration method with the intensity of the follower forces as parameter the influences of shear deformation and rotatory inertia on the flutter instability are considered the higher order flutter mode critical follower forces for a slender cantilever are found to be 2n 0 5 2 2ei l2 approximately the flutter frequency and flutter load are calculated directly by a newtonian method with a romberg algorithm for the determinant derivatives it is found that flutter may occur by coalescence of non adjacent linear modes

fisher information and stochastic complexity by taking into account the fisher information and removing an inherent redundancy in earlier two part codes a sharper code length as the stochastic complexity and the associated universal process are derived for a class of parametric processes the main condition required is that the maximum likelihood estimates satisfy the central limit theorem the same code length is also obtained from the so called maximum likelihood code

laplace s law of succession and universal encoding this correspondence shows that the known add half rule is not asymptotically optimal for predicting the n 1 st symbol after a sequence of n symbols whereas the add spl beta sub 0 rule spl beta sub 0 0 50922 spl middot spl middot spl middot is

adaptive coding and prediction of sources with large and infinite alphabets the problem of predicting a sequence x1 x2 generated by a discrete source with unknown statistics is considered each letter xt 1 is predicted using the information on the word x1x2 hellip xt only this problem is of great importance for data compression because of its use to estimate probability distributions for ppm algorithms and other adaptive codes on the other hand such prediction is a classical problem which has received much attention its history can be traced back to laplace we address the problem where the sequence is generated by an independent and identically distributed i i d source with some large or even infinite alphabet and suggest a class of new methods of prediction

binary forms hypergeometric functions and the schwarz christoffel mapping formula in a previous paper it was shown that if f is a binary form with complex coefficients having degree n 3 and discriminant df 0 and if af is the area of the region if x y l 1 in the real affine plane then idlil n n i af 3b l i where b 1 1 denotes the beta function with arguments of 1 3 this inequality was derived by demonstrating that the sequence mn defined by mn maxidfi n n af where the maximum is taken over all forms of degree n with df 0 is decreasing and then by showing that m3 3b q 1 the resulting estimate af 3b 1 for such forms with integer coefficients has had significant consequences for the enumeration of solutions of thue inequalities this paper examines the related problem of determining precise values for the sequence mn by appealing to the theory of hypergeometric functions it is shown that m4 27 6b 2 and that m4 is attained for the form xy x2 y2 it is also shown that there is a correspondence arising from the schwarz christowfel mapping formula between a particular collection of binary forms and the set of equiangular polygons with the property that af is the perimeter of the polygon corresponding to f based on this correspondence and a representation theorem for idfii n n1 af it is conjectured that mn d lln n 1 f n where fn x y x sin kn y cos k i and that the limiting value of the sequence m is 27r

a lower bound on compression of unknown alphabets many applications call for universal compression of strings over large possibly infinite alphabets however it has long been known that the resulting redundancy is infinite even for i i d distributions it was recently shown that the redundancy of the strings patterns which abstract the values of the symbols retaining only their relative precedence is subliner in the blocklength n hence the per symbol redundancy diminishers to zero in this paper we show that pattern redundancy is at least 1 5log2e n1 3 bits to do so we construct a generating function whose coefficients lower bound the redundancy and use hayman s saddle point approximation technique to determine the coefficients asymptotic behavior

the performance of universal encoding universal coding theory is surveyed from the viewpoint of the interplay between delay and redundancy the price for universality turns out to be acceptably small

universal codes for finite sequences of integers drawn from a monotone distribution we offer two noiseless codes for blocks of integers x sup n x sub 1 x sub n we provide explicit bounds on the relative redundancy that are valid for any distribution f in the class of memoryless sources with a possibly infinite alphabet whose marginal distribution is monotone specifically we show that the expected code length l x sup n of our first universal code is dominated by a linear function of the entropy of x sup n further we present a second universal code that is efficient in that its length is bounded by nh sub f o nh sub f where h sub f is the entropy of f which is allowed to vary with n since these bounds hold for any n and any monotone f we are able to show that our codes are strongly minimax with respect to relative redundancy as defined by elias 1975 our proofs make use of the elegant inequality due to aaron wyner 1972

on optimal sequential prediction for general processes this paper considers several aspects of the sequential prediction problem for unbounded nonstationary processes under pth power loss spl lscr sub p u v u v sup p 1 p spl infin in the first part it is shown that bayes prediction schemes are cesaro optimal under general conditions that cesaro optimal prediction schemes are unique in a natural sense and that cesaro optimality is equivalent to a form of weak calibration connections between calibration and stronger forms of optimality are considered extensions of the existence and uniqueness results to generalized prediction and prediction from observations with additive noise are established for binary processes it is shown that thresholding an optimal prediction scheme for the squared loss yields an optimal binary prediction scheme for the hamming loss in the second part of the paper it is shown how to construct from a countable family of prediction schemes a single composite scheme whose asymptotic performance on any suitable process dominates the performance of each member of the family the construction is based on aggregating methods for individual binary sequences using the construction some results of algoet 1994 1995 on the existence of cesaro optimal schemes for families of ergodic processes are rederived in a direct way and extended to unbounded processes

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

speaking of infinity i i d strings we study the redundancy of three approaches to compression of independent and identically distributed i i d strings over large possibly infinite alphabets standard compression of the string itself and compression of the string s shape and pattern which describe its symbols relative magnitude and precedence respectively we determine the rate at which per symbol standard redundancy increases to infinity as the alphabet size increases show that the maximum per symbol shape redundancy is between 0 027 and 1 and compare these to results showing that per symbol pattern redundancy diminishes to zero for all alphabet sizes we relate these concepts to ordered and unordered partitions of integers and sets and use this framework to explore relations between several combinatorial quantities including the bell fubini and second type stirling numbers

weakly convergent nonparametric forecasting of stationary time series the conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past several schemes are known for constructing pointwise consistent estimates but they all demand prohibitive amounts of input data we consider real valued time series and construct conditional distribution estimates that make much more efficient use of the input data the estimates are consistent in a weak sense and the question whether they are pointwise consistent is still open for finite alphabet processes one may rely on a universal data compression scheme like the lempel ziv 1978 algorithm to construct conditional probability mass function estimates that are consistent in expected information divergence consistency in this strong sense cannot be attained in a universal sense for all stationary processes with values in an infinite alphabet but weak consistency can some applications of the estimates to on line forecasting regression and classification are discussed

on the representation of integers by binary forms let f be a binary form with integer coefficients non zero discriminant and degree d with d at least 3 let r f z denote the number of integers of absolute value at most z which are represented by f we prove that there is a positive number c f such that r f z is asymptotic to c f z frac 2 d

universal schemes for learning the best nonlinear predictor given the infinite past and side information let x sub t be a real valued time series the best nonlinear predictor of x sub 0 given the infinite past x sub spl infin sup 1 in the least squares sense is equal to the conditional mean e x sub 0 x sub spl infin sup 1 previously it has been shown that certain predictors based on growing segments of past observations converge to the best predictor given the infinite past whenever x sub t is a stationary process with values in a bounded interval the present paper deals with universal prediction schemes for stationary processes with finite mean we also discuss universal schemes for learning the conditional mean e x sub 0 x sub spl infin sup 1 y sub spl infin sup 1 y sub 0 from past observations of a stationary pair process x sub t y sub t and schemes for learning the repression function m y e x y y from independent samples of x y

divergence and flutter of a cantilever rod with an intermediate spring support abstract the equation of motion in matrix form of a cantilever euler beam subject to a tipconcentrated follower force at the free end is formulated based on the lagrangian approach and the assumed mode method the non conservative nature of the system is identified by the non symmetric matrix in the equation of motion the beam is assumed to rest on an intermediate spring support jump phenomenon for the critical follower load is found to occur for both variations in the support location as well as the stiffness of the spring support detailed history of the changes in the load frequency diagram due to variation in the support location and the stiffness of the spring support is presented to explain the occurrence of the jump phenomenon an interesting finding is that the rod is found to be extremely unstable for a certain combination of spring stiffness and support location for a cantilever rod

exact bernoulli euler dynamic stiffness matrix for a range of tapered beams bernoulli euler theory and bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial torsional and flexural vibrations of any beam which is tapered such that a varies as yn and gj and i both vary as y n 2 where a gj and i have their usual meanings y cx l 1 c is a constant such that c 1 l is the length of the beam and x is the distance from one end of the beam numerical checks give better than seven figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements a procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped this enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members

a probabilistic approach to some asymptotics in noiseless communication renewal theory is a powerful tool in the analysis of source codes we use renewal theory to obtain some asymptotic properties of finite state noiseless channels we discuss the relationship between these results and earlier uses of renewal theory to analyze the lempel ziv 1977 1978 codes and the tunstall 1967 code as a new application of our results we provide the asymptotic performance of two of the perl garey and even 1975 prefix condition codes

about adaptive coding on countable alphabets this paper sheds light on adaptive coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non decreasing hazard rate log concave envelope distributions we prove that the auto censuring ac code is adaptive with respect to the collection of such classes the analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy and on a careful analysis of the performance of the ac coding algorithm the latter relies on nonasymptotic bounds for maxima of samples from discrete distributions with finite and nondecreasing hazard rate

about adaptive coding on countable alphabets max stable envelope classes in this paper we study the problem of lossless universal source coding for stationary memoryless sources on countably infinite alphabets this task is generally not achievable without restricting the class of sources over which universality is desired building on our prior work we propose natural families of sources characterized by a common dominating envelope we particularly emphasize the notion of adaptivity which is the ability to perform as well as an oracle knowing the envelope without actually knowing it this is closely related to the notion of hierarchical universal source coding but with the important difference that families of envelope classes are not discretely indexed and not necessarily nested our contribution is to extend the classes of envelopes over which adaptive universal source coding is possible namely by including max stable heavy tailed envelopes which are excellent models in many applications such as natural language modeling we derive a minimax lower bound on the redundancy of any code on such envelope classes including an oracle that knows the envelope we then propose a constructive code that does not use knowledge of the envelope the code is computationally efficient and is structured to use an expanding threshold for auto censoring etac and we therefore dub it the etac code we prove that the etac code achieves the lower bound on the minimax redundancy within a factor logarithmic in the sequence length and can be therefore qualified as a near adaptive code over families of heavy tailed envelopes for finite and light tailed envelopes the penalty is even less and the same code follows closely previous results that explicitly made the light tailed assumption our technical results are founded on methods from regular variation theory and concentration of measure

there is no universal source code for an infinite source alphabet shows that a discrete infinite distribution with finite entropy cannot be estimated consistently in information divergence as a corollary the authors show that there is no universal source code for an infinite source alphabet over the class of all discrete memoryless sources with finite entropy

dynamic stiffness for structures with distributed deterministic or random loads abstract the dynamic stiffness method applies mainly to excitations of harmonic nodal forces for distributed loads modal analysis is generally required in the case of a clamped clamped beam the modal decomposition of a uniformly distributed load by the eigenbeam functions inherits slow convergence because the finite loads at the beam ends cannot be represented efficiently by the zero deflection and zero slope of the clamped clamped beam functions the computed reactions at the supports do not converge at all the problem is eliminated in this paper by using the finite element interpolation functions for the distributed load if the distributed load is adequately represented explicit exact solutions are found otherwise the residual load is expanded in the modal space as the residual modal force is much smaller and agrees well with the clamped clamped conditions fast convergence is achieved by means of the principle of superposition a structure with members having distributed loads can be analyzed by two systems one is associated with the individual members having distributed loads and the other is associated with resulting equivalent nodal forces the required frequency functions are given for all possible cases the results presented are exact if the load is interpolated adequately by finite element shape functions both deterministic and random loads are considered closed form solutions are obtained for the first time

damped second order rayleigh timoshenko beam vibration in space an exact complex dynamic member stiffness matrix a uniform linear beam in a uniform linear ambient medium is studied the beam performs stationary harmonic damped nonsynchronous space vibration in simultaneous tension torsion bending and shear in the presence of a large static axial load hysteretic and viscous dampings of the beam material and ambient medium are considered generalized complex kolousek functions are derived a 12 12 complex symmetric stiffness matrix is established for a supported beam member excited at its ends by prescribed harmonic translations and rotations which have the same frequency but may be out of phase this matrix allows for an exact analysis of nonproportionally damped built up beam structures thus avoiding assumed mode shapes and lumped or consistent masses a general notation is suggested numerical examples are given including applications of the computer program sfvibat damp

the practical computation of areas associated with binary quartic forms we derive formulas for practically computing the area of the region f z y 1 defined by a binary quartic form f x y r x y these formulas which involve a particular hypergeometric function are useful when estimating the number of lattice points in certain regions of the type f x y h and will likely find application in many contexts we also show that for forms f of arbitrary degree the maximal size of the area of the region f x y 1 normalized with respect to the discriminant of f and taken with respect to the number of conjugate pairs of f x 1 increases as the number of conjugate pairs decreases and we give explicit numerical values for these normalized maxima when f is a quartic form

universal portfolios with side information we present a sequential investment algorithm the spl mu weighted universal portfolio with side information which achieves to first order in the exponent the same wealth as the best side information dependent investment strategy the best state constant rebalanced portfolio determined in hindsight from observed market and side information outcomes this is an individual sequence result which shows the difference between the exponential growth wealth of the best state constant rebalanced portfolio and the universal portfolio with side information is uniformly less than d 2n log n 1 k n log 2 for every stock market and side information sequence and for all time n here d k m 1 is the number of degrees of freedom in the state constant rebalanced portfolio with k states of side information and m stocks the proof of this result establishes a close connection between universal investment and universal data compression

non linear vibration of frames by the incremental dynamic stiffness method the dynamic stiffness method is extended to large amplitude free and forced vibrations of frames when the steady state vibration is concerned the time variable is replaced by the frequency parameter in the fourier series sense and the governing partial differential equations are replaced by a set of ordinary differential equations in the spatial variables alone the frequency dependent shape functons are generated approximately for the spatial discretization these shape functions are the exact solutions of a beam element subjected to mono frequency excitation and constant axial force to minimize the spatial discretization errors the system of ordinary differential equations is replaced by a system of non linear algebraic equations with the fourier coefficients of the nodal displacements as unknowns the fourier nodal coefficients are solved by the newtonian algorithm in an incremental manner when an approximate solution is available an improved solution is obtained by solving a system of linear equations with the fourier nodal increments as unknowns the method is very suitable for parametric studies when the excitation frequency is taken as a parameter the free vibration response of various resonances can be obtained without actually computing the linear natural modes for regular points along the response curves the accuracy of the gradient matrix jacobian or tangential stiffness matrix is secondary cf the modified newtonian method however at the critical positions such as the turning points at resonances and the branching points at bifurcations the gradient matrix becomes important the minimum number of harmonic terms required is governed by the conditions of completeness and balanceability for predicting physically realistic response curves the evaluations of the newly introduced mixed geometric matrices and their derivatives are given explicitly for the computation of the gradient matrix

universal compression of memoryless sources over unknown alphabets it has long been known that the compression redundancy of independent and identically distributed i i d strings increases to infinity as the alphabet size grows it is also apparent that any string can be described by separately conveying its symbols and its pattern the order in which the symbols appear concentrating on the latter we show that the patterns of i i d strings over all including infinite and even unknown alphabets can be compressed with diminishing redundancy both in block and sequentially and that the compression can be performed in linear time to establish these results we show that the number of patterns is the bell number that the number of patterns with a given number of symbols is the stirling number of the second kind and that the redundancy of patterns can be bounded using results of hardy and ramanujan on the number of integer partitions the results also imply an asymptotically optimal solution for the good turing probability estimation problem