ranking the best binary trees the problem of ranking the k best binary trees with respect to their weighted average leaves levels is considered both the alphabetic case where the order of the weights in the sequence w 1 cdots w n must be preserved in the leaves of the tree and the nonalphabetic case where no such restriction is imposed are studied for the alphabetic case a simple algorithm is provided for ranking the k best trees based on a recursive formula of complexity o kn 3 for nonalphabetic trees two different ranking problems are considered and for each of them it is shown that the next best tree can be solved by a dynamic programming formula of low complexity order

asymptotic analysis of dichotomous search with search and travel costs abstract an object to be searched lies on 0 n the search starts from the left end of the interval a query at a point finds out whether the objects lies to the left or to the right of the point the problem is to determine a sequence of queries that minimizes the expected sum of query c q per attempt and travel c t per unit distance costs required to locate the object within an interval of unit length this work develops simple approximate solutions to the problem and analyzes their performance

green s function for an unbounded biaxial medium in cylindrical coordinates the dyadic green s function for an unbounded biaxial medium is treated analytically in the fourier domain the green s function is initially expressed as a triple fourier integral which is next reduced to a double one by performing the integration over the longitudinal fourier variable a delta type source term is extracted which is dependent on the particular coordinate system

a new implementation of the mellin transform and its application to radar classification of ships a modified mellin transform for digital implementation is developed and applied to range radar profiles of naval vessels the scale invariance property of the mellin transform provides a means for extracting features from the profiles which are insensitive to the aspect angle of the radar past implementations of the mellin transform based on the fft have required exponential sampling interpolation and the computation of a correction term all of which introduce errors into the transform in addition exponential sampling results in a factor of ln n increase in the number of data points an alternate implementation developed in the paper utilizes a direct expansion of the mellin integral definition this direct mellin transform dmt eliminates the implementation problems associated with the fft approach and does not increase the number of samples a scale and translation invariant transform is developed from a modification of the dmt the mdmt applied to the fft of the radar profiles results in the desired insensitivity without having the low pass filtering characteristic that exists in other fourier mellin implementations

sub concave and sub convex capacities abstract in this study we propose the concepts of sub concave and sub convex capacities first we investigate some properties of sub concave and sub convex capacities second we discuss the choquet integrals with respect to sub concave and sub convex capacities we then show that the upper and lower probabilities comprising the supremum and infimum over the set of risk neutral martingale measures in the asymmetrical case are sub concave and sub convex capacities respectively

mellin transform methods applied to integral evaluation taylor series and asymptotic approximations mellin transform methods have been used to obtain taylor series or asymptotic approximations to definite integrals whose integrands consist of the product of two generalized hypergeometric functions this approach has proved extremely useful in many applied physics problems including the analysis of electromagnetic propagation in a turbulent medium the main result in this article is a generalization of the method to integrands containing the product of an arbitrary number of hypergeometric functions providing a very powerful and general technique for integral evaluation with many applications in physics

unary prefixed huffman coding for a group of quantized generalized gaussian sources we propose a coding algorithm called unary prefixed huffman uph coding for a family of infinite sources called quantized generalized gaussian gg sources compared with the existing codes for these gg sources the uph algorithm provides a more adaptive approach and its coding efficiency is upper bounded by entropy 2

a general framework for codes involving redundancy minimization a framework with two scalar parameters is introduced for various problems of finding a prefix code minimizing a coding penalty function the framework encompasses problems previously proposed by huffman campbell nath and drmota and szpankowski shedding light on the relationships among these problems in particular nath s range of problems can be seen as bridging the minimum average redundancy problem of huffman with the minimum maximum pointwise redundancy problem of drmota and szpankowski using this framework two linear time huffman like algorithms are devised for the minimum maximum pointwise redundancy problem the only one in the framework not previously solved with a huffman like algorithm both algorithms provide solutions common to this problem and a subrange of nath s problems the second algorithm being distinguished by its ability to find the minimum variance solution among all solutions common to the minimum maximum pointwise redundancy and nath problems simple redundancy bounds are also presented

table erratum table of integrals series and products academic press new york 1980 mr0582453 81g 33001 by i s gradshteĭn and i m ryzhik

conditions for optimality of the huffman algorithm a new general formulation of huffman tree construction is presented which has broad application recall that the huffman algorithm forms a tree in which every node has some associated weight by specifying at every step of the construction which nodes are to be combined to form a new node with a new combined weight we characterize a wide class of weight combination functions the quasilinear functions for which the huffman algorithm produces optimal trees under correspondingly wide classes of cost criteria in addition known results about huffman tree construction and related concepts from information theory and from the theory of convex functions are tied together suggestions for possible future applications are given

properties of the dyadic green s function for an unbounded anisotropic medium radiation in an unbounded anisotropic medium is treated analytically by studying the dyadic green s function of the problem initially expressed as a triple fourier integral which is next reduced to a double one under certain conditions the existence of incoming waves is verified it is also found that exponentially decaying waves are possible in such media finally the existence of branch points in the remaining integrand function is investigated and appropriate branch cuts are proposed

fast and accurate computation of bessel functions with large complex order and argument abstract a fast and highly accurate computational scheme to compute a product of bessel functions with large complex order and argument is of utmost importance in some problems in physics involving cylindrical configurations since these products can occur in the kernel of multiple integrals and are subject to mutual cancellations speed and accuracy of their numerical computation are key moreover the order and argument can vary over a vast range of several orders of magnitude to deliver a high relative accuracy 10 11 over the entire range we use the integral representations of the bessel functions and introduce a deformation into steepest descent paths the numerical computation is accelerated by three orders of magnitude if these paths are replaced by well chosen piecewise linear segments while maintaining the set relative accuracy

on convergent series representations of mellin barnes integrals multiple mellin barnes integrals are often used for perturbative calculations in particle physics in this context the evaluation of such objects may be performed through residues calculations which lead to their expression as multiple power series and logarithms of the parameters involved in the problem under consideration however in most of the cases several convergent series representations exist for a given integral these series converge in different regions of values of the parameters and it is not obvious to obtain them for twofold integrals we present a method which allows to derive straightforwardly and systematically first different sets of poles which correspond to different convergent double series representations of a given integral second the regions of convergence of all these series without an a priori full knowledge of their general term and third the general term of each series this may be performed if necessary once the relevant domain of convergence has been found thi

on an integral related to biaxially anisotropic media an integral arising in certain studies of theoretical electromagnetics is evaluated and its properties are discussed in some detail the integral has three integer two real and one complex parameter the integrand involves a product of bessel functions of different argument and order several generalizations are discussed

choquet expectation and peng s g expectation in this paper we consider two ways to generalize the mathematical expectation of a random variable the choquet expectation and peng s g expectation an open question has been after making suitable restrictions to the class of random variables acted on by the choquet expectation for what class of expectation do these two definitions coincide in this paper we provide a necessary and sufficient condition which proves that the only expectation which lies in both classes is the traditional linear expectation this settles another open question about whether choquet expectation may be used to obtain monte carlo like solution of nonlinear pde it cannot except for some very special cases

bayes theorem for choquet capacities we give an upper bound for the posterior probability of a measurable set a when the prior lies in a class of probability measures p the bound is a rational function of two choquet integrals if p is weakly compact and is closed with respect to majorization then the bound is sharp if and only if the upper prior probability is 2 alternating the result is used to compute bounds for several sets of priors used in robust bayesian inference the result may be regarded as a characterization of 2 alternating choquet capacities

operations research applications of dichotomous search abstract an object is searched for in 1 n queries for the object are sequentially conducted a query at x reveals whether the object s location is greater than x the objective is to find the object within a minimal expected number of queries this problem is called the dichotomous search problem and has many versions this paper surveys dichotomous search problems with the emphasis on operations research applications

testing and characterizing properties of nonadditive measures through violations of the sure thing principle in expected utility theory risk attitudes are modeled entirely in terms of utility in the rank dependent theories a new dimension is added chance attitude modeled in terms of nonadditive measures or nonlinear probability transformations that are independent of utility most empirical studies of chance attitude assume probabilities given and adopt parametric fitting for estimating the probability transformation only a few qualitative conditions have been proposed or tested as yet usually quasi concavity or quasi convexity in the case of given probabilities this paper presents a general method of studying qualitative properties of chance attitude such as optimism pessimism and the inverse s shape pattern both for risk and for uncertainty these qualitative properties can be characterized by permitting appropriate relatively simple violations of the sure thing principle in particular this paper solves a hitherto open problem the preference axiomatization of convex pessimistic or uncertainty averse nonadditive measures under uncertainty the axioms of this paper preserve the central feature of rank dependent theories i e the separation of chance attitude and utility

optimal prefix codes for sources with two sided geometric distributions a complete characterization of optimal prefix codes for off centered two sided geometric distributions of the integers is presented these distributions are often encountered in lossless image compression applications as probabilistic models for image prediction residuals the family of optimal codes described is an extension of the golomb codes which are optimal for one sided geometric distributions the new family of codes allows for encoding of prediction residuals at a complexity similar to that of golomb codes without recourse to the heuristic approximations frequently used when modifying a code designed for nonnegative integers so as to apply to the encoding of any integer optimal decision rules for choosing among a lower complexity subset of the optimal codes given the distribution parameters are also investigated and the relative redundancy of the subset with respect to the full family of optimal codes is bounded

existence of optimal prefix codes for infinite source alphabets it is proven that for every random variable with a countably infinite set of outcomes and finite entropy there exists an optimal prefix code which can be constructed from huffman codes for truncated versions of the random variable and that the average lengths of any sequence of huffman codes for the truncated versions converge to that of the optimal code also it is shown that every optimal infinite code achieves kraft s inequality with equality

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

precise minimax redundancy and regret recent years have seen a resurgence of interest in redundancy of lossless coding the redundancy regret of universal fixed to variable length coding for a class of sources determines by how much the actual code length exceeds the optimal ideal over the class code length in a minimax scenario one finds the best code for the worst source either in the worst case called also maximal minimax or on average we first study the worst case minimax redundancy over a class of stationary ergodic sources and replace shtarkov s bound by an exact formula among others we prove that a generalized shannon code minimizes the worst case redundancy derive asymptotically its redundancy and establish some general properties this allows us to obtain precise redundancy for memoryless markov and renewal sources for example we present the exact constant of the redundancy for memoryless and markov sources by showing that the integer nature of coding contributes log logm m 1 logm o 1 where m is the size of the alphabet then we deal with the average minimax redundancy and regret our approach here is orthogonal to most recent research in this area since we aspire to show that asymptotically the average minimax redundancy is equivalent to the worst case minimax redundancy for some classes of sources after formulating some general bounds relating these two redundancies we prove our assertion for memoryless and markov sources nevertheless we provide evidence that maximal redundancy of renewal processes does not have the same leading term as the average minimax redundancy however our general results show that maximal and average regrets are asymptotically equivalent

optimal parallel inspection for finding the first nonconforming unit in a batch an information theoretic approach we consider the case of a batch of discrete units produced by a process subject to failures under a known probability distribution function and apply information theory to the problem of finding the first nonconforming unit in the batch at minimum cost two distinct but related aspects of this problem were treated determining which units should be inspected and determining how many units should be sent for inspection at the same time the solution is based on the principles of inspecting the product units that maximize the reduction in the uncertainty regarding the location of the first nonconforming unit and of minimizing the cost per unit of uncertainty reduced these principles are formalized by means of a series of theorems leading to an easy to implement algorithm for managing parallel inspection this approach is successfully compared with the optimal solution obtained with dynamic programming and with other heuristics

optimal prefix codes for infinite alphabets with nonlinear costs let p p i be a measure of strictly positive probabilities on the set of nonnegative integers although the countable number of inputs prevents usage of the huffman algorithm there are nontrivial p for which known methods find a source code that is optimal in the sense of minimizing expected codeword length for some applications however a source code should instead minimize one of a family of nonlinear objective functions beta exponential means those of the form loga sigmaip i an i where n i is the length of the ith codeword and a is a positive constant applications of such minimizations include a novel problem of maximizing the chance of message receipt in single shot communications a 1 this paper introduces methods for finding codes optimal for such exponential means one method applies to geometric distributions while another applies to distributions with lighter tails the latter algorithm is applied to poisson distributions and both are extended to alphabetic codes as well as to minimizing maximum pointwise redundancy the aforementioned application of minimizing the chance of buffer overflow is also considered

em field induced in inhomogeneous dielectric spheres by external sources the electromagnetic field induced in the interior of inhomogeneous dielectric bodies by external sources can be evaluated by solving the well known electric field integrodifferential equation efide for spheres with constant magnetic permeability but variable dielectric constant a direct mainly analytical solution can be used even in case when the inhomogeneity in renders separation of variables inapplicable this approach constitutes a generalization of the hybrid analytical numerical scalar method developed by the authors in two recent papers for the corresponding acoustic scalar field induced in spheres with variable density and or compressibility this extension by no means trivial owing to the vector and integrodifferential nature of the equation is based on field vector expansions using the set of three harmonic surface vectors orthogonal and complete over the surface of the sphere for their angular dependence and dini s expansions of a general type for their radial functions the use of the latter has been shown to be superior to other possible sets of orthogonal expansions and as far as its convergence is concerned it may further be improved by properly choosing a crucial parameter in their eigenvalue equation the restriction to the spherical shape is imposed here to allow use of the well known expansion of green s dyadic in spherical eigenvectors of the vector wave equation

asymptotic average redundancy of huffman and other block codes we study asymptotically the redundancy of huffman and other codes it has been known from the inception of the huffman 1952 code that in the worst case its redundancy defined as the excess of the code length over the optimal ideal code length is not more than one however to the best of our knowledge no precise asymptotic results have been reported in literature thus far we consider here a memoryless binary source generating a sequence of length n distributed as binomial n p with p being the probability of emitting 0 based on the results of stubley 1994 we prove that for p 1 2 1 m 1 2 sup 1 m 2 sup m o spl rho sup n spl alpha n m rational where spl alpha log sub 2 1 p p and spl beta log sub 2 1 p spl rho x x is the fractional part of x the appearance of the fractal like function explains the erratic behavior of the huffman redundancy and its resistance to succumb to a precise analysis as a side result we prove that the average redundancy of the shannon block code is as n spl rarr spl infin r sub n sup s 1 2 o 1 spl alpha irrational 1 2 1 m 1 2 o spl rho sup n spl alpha n m rational where spl rho 1 finally we derive the redundancy of the golomb 1966 code for the geometric distribution which can be viewed as a special case of the huffman and shannon codes golomb s code redundancy has only oscillating behavior i e there is not convergent mode these findings are obtained by analytic methods such as theory of distribution of sequences modulo 1 and fourier series

a dichotomous search for a geometric random variable we are given a two state system that starts in state 0 ends in state 1 and makes a single transition from state 0 to state 1 during n periods if the system is in state 0 it moves to state 1 in the next period with a known positive time independent probability once it reaches state 1 it remains there by observing the state of the system at some intermediate period we can learn whether this transition occurred earlier or not an optimal search strategy minimizes the expected number of observations needed to locate the exact transition time in this paper we show how to compute efficiently and how to approximate the optimal strategy applications to the problem arise in the areas of quality control and maintenance of communication and supply lines

lagrangian empirical design of variable rate vector quantizers consistency and convergence rates the lagrangian formulation of variable rate vector quantization is known to yield useful necessary conditions for quantizer optimality and generalized lloyd algorithms for quantizer design the lagrangian formulation is demonstrated to provide a convenient framework for analyzing the empirical design of variable rate vector quantizers in particular the consistency of empirical design based on minimizing the lagrangian performance over a stationary and ergodic training sequence is shown for sources with finite second moment the finite sample performance is also studied for independent training data and sources with bounded support

optimal prefix codes for pairs of geometrically distributed random variables optimal prefix codes are studied for pairs of independent integer valued symbols emitted by a source with a geometric probability distribution of parameter q 0 1 covering the range q 1 2 the described codes produce the expected reduction in redundancy with respect to the 1 d case while maintaining low complexity coding operations

a combinatorial approach to golomb forests optimal binary prefix free codes for infinite sources with geometrically distributed frequencies e g p pi 1 p i 0 0p1 were first implicitly suggested by golomb over 30 years ago in the context of run length encodings ten years later gallager and van voorhis exhibited such optimal codes for all values of p these codes were derived by using the hu2man encoding algorithm to build optimal codes for finite sources and then showing that the finite codes converge in a very specific sense to the infinite one in this note we present a new combinatorial approach to solve the same problem one that does not use the hu2man algorithm but instead treats a coding tree as an infinite sequence of integers and derives properties of the sequence one consequence of this new approach is a complete characterization of all of the optimal codes in particular it shows that for all p 0p1 except for an easily describable countable set there is a unique optimal code but for each p in this countable set there are an uncountable number of optimal codes another consequence is a derivation of infinite codes for geometric sources when the encoding alphabet is no longer restricted to be the binary one a final consequence is the extension of the results to optimal forests instead of being restricted to optimal trees

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

formulas for the force dipole interaction of surface line defects in homoepitaxy using asymptotics we derive explicit simplified formulas for integrals representing the force dipole interaction energy per unit length between line defects steps of the same sign that form perturbations of circles in homoepitaxy our starting point is continuum linear elasticity in accordance with the classic model by marchenko and parshin 1981 sov phys jetp 52 129 in the case with concentric circular steps we define a small geometric parameter 2 which expresses the smallness of interstep distance relative to the circle radii we invoke the mellin transform with respect to 2 and derive systematically an approximation for the requisite integral this technique offers an alternative to an exact evaluation in terms of elliptic integrals we then demonstrate the use of the mellin transform when calculating the force dipole interaction energy between smoothly slowly varying steps that form perturbations of circles we discuss the implications of our results for small amplitude modulations of circular step profiles

electromagnetic fields in air of traveling wave currents above the earth the problem of the electromagnetic field created by a thin straight conductor of infinite length carrying a forward traveling wave current with a complex propagation constant above a homogeneous and isotropic planar earth of wave number k1 is formulated in terms of contour integrals in the limit where becomes equal to the free space wave number k0 the component of the magnetic field in air normal to both conductor and interface is evaluated in closed form in terms of known special functions while the remaining components of the field in air are expressed as series expansions in k02 k14 k04 1 2 via the application of a contour integration technique the new analytical formulas involve familiar transcendental functions and are valid at any distance from the source the analysis sheds light on the intricate nature of low frequency electromagnetic fields generated by transmission lines in the presence of a conducting or a dielectric half space

dichotomous search for random objects on an interval a set of objects to be searched is represented by a set of points lying in an interval of integers we wish to identify these points within unit intervals through a dichotomous search minimizing the expected cost of the search the optimal search strategy may depend on the information gained at each stage of the search it is shown here that for some cases there exists a common optimal strategy this strategy is generally different from the bisection procedure and is independent of the number of objects in the interval as a matter of fact at each stage of the search a partition of the interval is given and the only information needed for searching a subinterval of this partition is that it still contains an unidentified point

integral evaluation using the mellin transform and generalized hypergeometric functions tutorial and applications to antenna problems this is a tutorial presentation of the mellin transform mt method for the exact calculation of one dimensional definite integrals and an illustration of the application of this method to antenna electromagnetics problems once the basics have been mastered one quickly realizes that the mt method is extremely powerful often yielding closed form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals yet as opposed to other methods the mt method is very straightforward to apply it usually requires laborious calculations but little ingenuity in fact the mt method is used by mathematica to symbolically calculate definite integrals the first part of this paper is a step by step tutorial proceeding from first principles it includes basic information on mellin barnes integrals and generalized hypergeometric functions and summarizes the key ideas of the mt method in the remaining parts the mt method is applied to three examples from the antenna area the results here are believed to be new at least in the antenna electromagnetics literature in our first example we obtain a closed form expression as a generalized hypergeometric function for the power radiated by a constant current circular loop antenna this quantity has been extensively discussed recently our second example concerns the admittance of a 2 d slot antenna in both these examples the exact closed form expressions are applied to improve upon existing formulas in standard antenna textbooks in our third example finally we obtain a very simple expression for an integral arising in recent unpublished studies of unbounded biaxially anisotropic media

two dimensional highly directive currents on large circular loops properties of idealized two dimensional current distributions on circular loops are investigated analytically via the solution of a constrained optimization problem the directivity in the far field is maximized under a fixed c n t where n is the integral of the square of the current magnitude and t is the total radiated power c enters the ensuing fourier series for the current implicitly through a lagrange multiplier for non negative and large electrical radius ka the directivity and the current are evaluated approximately via combined use of the poisson summation formula and the mellin transform technique as a result a geometrical ray representation for the current is derived for the case of directivities that are slightly larger than that of the uniform distribution the analysis indicates certain advantages of large radiating structures for moderate values of the constraint c in the limit c of oseen s einstein needle radiation an asymptotic formula for the directivity is obtained pos

on polychotomous search problems abstract the paper gives a unified treatment of those stochastic sequential search problems where in each state the searcher may decompose according to a given scheme the current search region in a finite number of parts upon which one obtains the error free information in which of the parts the object is hidden many examples are given to show the generality of the model the main results are a a rigorous proof of the dynamic programming approach b the reduction of the state space for problems with translation invariant data c a dynamic programming proof for the computationally very attractive property of monotonicity and lipschitz continuity of optimal search rules for two important special models one of which has been investigated by hassin 1984 using methods from information theory

source coding for quasiarithmetic penalties whereas huffman coding finds a prefix code minimizing mean codeword length for a given finite item probability distribution quasiarithmetic or quasilinear coding problems have the goal of minimizing a generalized mean of the form rho 1 sigmaipirho li where li denotes the length of the ith codeword p i denotes the corresponding probability and rho is a monotonically increasing cost function such problems proposed by campbell have a number of diverse applications several cost functions are shown here to yield quasiarithmetic problems with simple redundancy bounds in terms of a generalized entropy a related property also shown here involves the existence of optimal codes for well behaved cost functions optimal codes always exist for possibly infinite alphabet sources having finite generalized entropy an algorithm is introduced for finding binary codes optimal for convex cost functions this algorithm which can be extended to other minimization utilities can be performed using quadratic time and linear space this reduces the computational complexity of a problem involving minimum delay in a queue allows combinations of previously considered problems to be optimized and greatly expands the set of problems solvable in quadratic time and linear space

some properties and convergence theorems of set valued choquet integrals the paper aims at discussing the set valued choquet integral which is the integral of set valued random variables with respect to capacities we mainly present representation theorems of the set valued random variable by using a sequence of choquet integrable selections and then we investigate some properties of set valued choquet integrals especially subadditive property and inequality of the metric of set valued choquet integrals we also prove fatou s lemmas lebesgue dominated convergence theorem and monotone convergence theorem of set valued choquet integrals under the weaker conditions than that in previous works

table errata table of integrals series and products by i s gradshteĭn and i m ryzhik academic press boston 1994 mr1243179 94g 00008

optimal algorithms for universal random number generation from finite memory sources we study random number generators rngs both in the fixed to variable length fvr and the variable to fixed length vfr regimes in a universal setting in which the input is a finite memory source of arbitrary order and unknown parameters with arbitrary input and output finite alphabet sizes applying the method of types we characterize essentially unique optimal universal rngs that maximize the expected output respectively minimize the expected input length in the fvr respectively vfr case for the fvr case the rng studied is a generalization of elias s scheme while in the vfr case the general scheme is new we precisely characterize up to an additive constant the corresponding expected lengths which include second order terms similar to those encountered in universal data compression and universal simulation furthermore in the fvr case we consider also a twice universal setting in which the markov order k of the input source is also unknown

huffman coding with an infinite alphabet a new type of sufficient condition is provided for a probability distribution on the nonnegative integers to be given an optimal d ary prefix code by a huffman type algorithm in the justification of our algorithm we introduce two new essentially one concepts as the definition of the optimality of a prefix d ary code which are shown to be equivalent to that defined in the traditional way these new concepts of the optimality are meaningful even for the case where the shannon entropy h p diverges