on the uniqueness problems of entire functions and their derivatives in this paper we obtain some uniqueness theorems for entire functions and their derivatives sharing the same fixed points with the same multiplicities

entire functions that share one value with one or two of their derivatives we prove two uniqueness theorems for entire functions of finite order that share one finite value with one or two of their derivatives

an entire function and its derivatives sharing a polynomial in this paper we study the growth of solutions of a first order linear differential equation and that of a second order linear differential equation from this we obtain some uniqueness theorems of a nonconstant entire function and its first derivative having the same fixed points with the same multiplicities the results in this paper also improve some known results some examples show that the results in this paper are best possible

entire functions that have the same fixed points with their first derivatives let f be a nonconstant entire function and let k 2 be an integer we denote by ef z z c f z z counting multiplicities the set consisting of all the fixed points of f this paper proves that if f and f have the same fixed points namely ef z ef z and if f k z z whenever f z z then f f

further results on entire functions that share one value with their derivatives abstract this paper deals with problems of the uniqueness of entire functions that share one value with their derivatives the results in this paper generalize a result of jank mues and volkmann and answer a question posed by h zhong and a question of yi and yang

entire functions that share a small function with its derivative abstract in this paper by studying the properties of meromorphic functions which have few zeros and poles we find all the entire functions f z which share a small and finite order meromorphic function a z with its derivative and f n z a z 0 whenever f z a z 0 n 2 this result is a generalization of several previous results

thermodynamics of random number generation we analyze the thermodynamic costs of the three main approaches to generating random numbers via the recently introduced information processing second law given access to a specified source of randomness a random number generator rng produces samples from a desired target probability distribution this differs from pseudorandom number generators prng that use wholly deterministic algorithms and from true random number generators trng in which the randomness source is a physical system for each class we analyze the thermodynamics of generators based on algorithms implemented as finite state machines as these allow for direct bounds on the required physical resources this establishes bounds on heat dissipation and work consumption during the operation of three main classes of rng algorithms including those of von neumann knuth and yao and roche and hoshi and for prng methods we introduce a general trng and determine its thermodynamic costs exactly for arbitrary target distributions the results highlight the significant differences between the three main approaches to random number generation one is work producing one is work consuming and the other is potentially dissipation neutral notably trngs can both generate random numbers and convert thermal energy to stored work these thermodynamic costs on information creation complement landauer s limit on the irreducible costs of information destruction

meromorphic functions sharing three weighted values in this paper we deal with the relation between the characteristics of two meromorphic functions that share three weighted values which improve the results of q c zhang and through the results two conjectures which proposed by mues and osgood yang are solved in a weaker condition

interval algorithm for random number generation the problem of generating a random number with an arbitrary probability distribution by using a general biased m coin is studied an efficient and very simple algorithm based on the successive refinement of partitions of the unit interval 0 1 which we call the interval algorithm is proposed a fairly tight evaluation on the efficiency is given generalizations of the interval algorithm to the following cases are investigated 1 output sequence is independent and identically distributed i i d 2 output sequence is markov 3 input sequence is markov 4 input sequence and output sequence are both subject to arbitrary stochastic processes

uniqueness theorems on entire functions and their derivatives abstract this paper studies uniqueness problems on entire functions that share a finite nonzero value counting multiplicities with their derivatives and gives a proper answer to the problem proposed by l z yang proceedings of the 6th international colloquium on complex analysis 1998 pp 176 183

tree algorithms for unbiased coin tossing with a biased coin we give new algorithms for simulating a flip of an unbiased coin by flipping a coin of unknown bias we are interested in efficient algorithms where the expected number of flips is our measure of efficiency other authors have represented algorithms as lattices but by representing them instead as trees we are able to produce an algorithm more efficient than any previously appearing we also prove a conjecture of hoeffding and simons that there is no optimal algorithm further we consider generalizations where the input is a sequence of iid discrete random variables and the output is a uniform random variable with n possible outcomes in this setting we provide an algorithm significantly superior to those previously published

synthesis of stochastic flow networks a stochastic flow network is a directed graph with incoming edges inputs and outgoing edges outputs tokens enter through the input edges travel stochastically in the network and can exit the network through the output edges each node in the network is a splitter namely a token can enter a node through an incoming edge and exit on one of the output edges according to a predefined probability distribution stochastic flow networks can be easily implemented by beam splitters or by dna based chemical reactions with promising applications in optical computing molecular computing and stochastic computing in this paper we address a fundamental synthesis question given a finite set of possible splitters and an arbitrary rational probability distribution design a stochastic flow network such that every token that enters the input edge will exit the outputs with the prescribed probability distribution the problem of probability transformation dates back to von neumann s 1951 work and was followed among others by knuth and yao in 1976 most existing works have been focusing on the simulation of target distributions in this paper we design optimal sized stochastic flow networks for synthesizing target distributions it shows that when each splitter has two outgoing edges and is unbiased an arbitrary rational probability a over b with a leq b leq 2 n can be realized by a stochastic flow network of size n that is optimal compared to the other stochastic systems feedback cycles in networks strongly improves the expressibility of stochastic flow networks

a note on ω regular languages abstract this paper is a continuation of the work by landweber 5 to clarify the relationship between classes of languages accepted by finite automata we study their properties in the topological framework introduced by boasson and nivat 1 we characterize the classes in terms of borel hierarchy and show some interrelation of them

entire functions that share one value cm with their derivatives this paper studies the uniqueness problem on entire function that share a finite nonzero value cm with their derivatives and proves two main theorems which generalize some results given by jank mues and volkmann p li and c c yang h l zhong etc an example shows that the condition of one of our theorems is necessary

an entire function sharing one nonzero value with its linear differential polynomials abstract we study the uniqueness of an entire function when it shares a nonzero finite value with its first derivative and two linear differential polynomials

generating random bits from an arbitrary source fundamental limits suppose we are given a random source and want to use it as a random number generator at what rate can we generate fair bits from it we address this question in an information theoretic setting by allowing for some arbitrarily small but nonzero deviation from ideal random bits we prove our results with three different measures of approximation between the ideal and the obtained probability distributions the variational distance the d bar distance and the normalized divergence two different contexts are studied fixed length and variable length random number generation the fixed length results of this paper provide an operational characterization of the inf entropy rate of a source defined in han and verdu see ibid vol 39 no 3 p 752 772 1993 and the variable length results characterize the liminf of the entropy rate thereby establishing a pleasing duality with the fundamental limits of source coding a feature of our results is that we do not restrict ourselves to ergodic or to stationary sources

on results of h ueda and g brosch concerning the unicity of meromorphic functions by emplying the main ideas from g brosch two results of r nevanlinna s four value theorem type on the unicity of meromorphic functions in the open complex plane sharing three distinct values and a fourth value or a pair of values weakly are proved which extend and generalize previous results of r nevanlinna g g gundersen h ueda and g brosch

entire functions that share a polynomial with their derivatives abstract let f be a nonconstant entire function k and q be positive integers satisfying k q and let q be a polynomial of degree q this paper studies the uniqueness problem on entire functions that share a polynomial with their derivatives and proves that if the polynomial q is shared by f and f cm and if f k z q z 0 whenever f z q z 0 then f f we give two examples to show that the hypothesis k q is necessary

efficient generation of random bits from finite state markov chains the problem of random number generation from an uncorrelated random source of unknown probability distribution dates back to von neumann s 1951 work elias 1972 generalized von neumann s scheme and showed how to achieve optimal efficiency in unbiased random bits generation hence a natural question is what if the sources are correlated both elias and samuelson proposed methods for generating unbiased random bits in the case of correlated sources of unknown probability distribution specifically they considered finite markov chains however their proposed methods are not efficient or have implementation difficulties blum 1986 devised an algorithm for efficiently generating random bits from degree 2 finite markov chains in expected linear time however his beautiful method is still far from optimality on information efficiency in this paper we generalize blum s algorithm to arbitrary degree finite markov chains and combine it with elias s method for efficient generation of unbiased bits as a result we provide the first known algorithm that generates unbiased random bits from an arbitrary finite markov chain operates in expected linear time and achieves the information theoretic upper bound on efficiency

a generalization of peres s algorithm for generating random bits from loaded dice peres s algorithm produces unbiased random bits from biased coin tosses recursively using the famous von neumann s method as its base the algorithm is simple and elegant but at first glance appears to work almost like magic and its generalization is elusive we generalize the method to generate unbiased random bits from loaded dice i e many valued bernoulli source the generalization is asymptotically optimal in its output rate as is the original peres s algorithm three valued case is discussed in detail and then other many faced cases are considered

randomizing functions simulation of a discrete probability distribution using a source of unknown distribution in this paper we characterize functions that simulate independent unbiased coin flips from independent coin flips of unknown bias we call such functions randomizing our characterization of randomizing functions enables us to identify the functions that generate the largest average number of fair coin flips from a fixed number of biased coin flips we show that these optimal functions are efficiently computable then we generalize the characterization and we present a method to simulate an arbitrary rational probability distribution optimally in terms of the average number of output digits and efficiently in terms of computational complexity from outputs of many faced dice of unknown distribution we also study randomizing functions on exhaustive prefix free sets

language theoretical representations of ω languages abstract in this paper we give language theoretical representations for some classes of languages defined by finite automata the classes considered in this paper are proved to be the classes of languages represented as f f pf or p p where p and p are prefix free regular languages and f and f are prefix free finite languages

normal families and uniqueness theorems for entire functions abstract there exists a set s with 3 elements such that if f is a non constant entire function satisfying e s f e s f then f f the number 3 is best possible the proof uses the theory of normal families in an essential way

on the characteristics of meromorphic functions that share three values cm in this paper by studying the counting functions of the common 1 points of meromorphic functions a more precise relation between the characteristics of meromorphic functions that share three values cm has been obtained as applications of this many known results can be improved

accepting conditions for automata on ω languages abstract in this paper we study the languages accepted by finite automata under a new type of accepting conditions which are based on whether the automaton enters all of its accepting states infinitely many times or not we investigate the classes of languages defined thereby in relation to other well known subclasses of the class of regular languages

streaming universal distortion free entanglement concentration this paper presents a streaming sequential protocol for universal entanglement concentration at the shannon bound alice and bob begin with n identical but unknown two qubit pure states each containing e ebits of entanglement they each run a reversible algorithm on their qubits and end up with y perfect epr pairs where y ne o n our protocol is streaming so the n input systems are fed in one at a time and perfect epr pairs start popping out almost immediately it matches the optimal block protocol exactly at each stage so the average yield after n inputs is y ne o logn so somewhat surprisingly there is no tradeoff between yield and lag our protocol optimizes both in contrast the optimal n qubit block protocol achieves the same yield but since no epr pairs are produced until the entire input block is read its lag is o n finally our algorithm runs in o log n space so a lot of entanglement can be efficiently concentrated using a very small e g current or near future technology quantum processor along the way we find an optimal streaming protocol for extracting randomness from classical i i d sources and a more space efficient implementation of the schur transform

how to turn loaded dice into fair coins we present a new technique for simulating fair coin flips using a biased stationary source of randomness sequences of random numbers are of pervasive importance in cryptography and vital to many other computing applications many sources of randomness such as radioactive or quantum mechanical sources possess the property of stationarity in other words they produce independent outputs over fixed probability distributions the output of such sources may be viewed as the result of rolling a biased or loaded die while a biased die may be a good source of entropy many applications require input in the form of unbiased bits rather than biased ones for this reason von neumann 1951 presented a now well known and extensively investigated technique for using a biased coin to simulate a fair coin we describe a new generalization of von neumann s algorithm distinguished by its high level of practicality and amenability to analysis in contrast to previous efforts we are able to prove our algorithm optimally efficient in the sense that it simulates the maximum possible number of fair coin flips for a given number of die rolls in fact we are able to prove that in an asymptotic sense our algorithm extracts the full entropy of its input moreover we demonstrate experimentally that our algorithm achieves a high level of computational and output efficiency in a practical setting

source and channel simulation using arbitrary randomness necessary and sufficient conditions for approximation of a general channel by a general source are proved for the special case of a deterministic channel input which corresponds to source simulation we prove a stronger necessary condition as the approximation criteria vanishing variational distance between the original and the approximated quantity is used for both of the problems both necessary and sufficient conditions for the two problems are based on some individual properties of the sources and the channel and are relatively easy to evaluate in particular unlike prior results for this problem our results do not require solving an optimization problem to test simulatability the results are illustrated with several nonergodic examples

constructing perfect steganographic systems abstract we propose steganographic systems for the case when covertexts containers are generated by a finite memory source with possibly unknown statistics the probability distributions of covertexts with and without hidden information are the same this means that the proposed stegosystems are perfectly secure i e an observer cannot determine whether hidden information is being transmitted the speed of transmission of hidden information can be made arbitrary close to the theoretical limit the shannon entropy of the source of covertexts an interesting feature of the suggested stegosystems is that they do not require any secret or public key at the same time we outline some principled computational limitations on steganography we show that there are such sources of covertexts that any stegosystem that has linear in the length of the covertext speed of transmission of hidden text must have an exponential kolmogorov complexity this shows in particular that some assumptions on the sources of covertext are necessary

small functions and uniqueness of meromorphic functions we discuss the uniqueness of meromorphic functions sharing three weighted values and provide a complete answer to a question of t c alzahary

binarization trees and random number generation an m extracting procedure produces unbiased random bits from a loaded dice with m faces and its output rate the average number of output per input is bounded by the shannon entropy of the source this information theoretic upper bound can be achieved only asymptotically as the input size increases by certain extracting procedures that we call asymptotically optimal although a computationally efficient asymptotically optimal 2 extracting procedure has been known for a while its counterparts for m ary input m 2 was found only recently and they are still relatively complicated to describe a binarization takes inputs from an m faced dice and produce bit sequences to be fed into a binary extracting procedure to obtain random bits thus binary extracting procedures give rise to an m extracting procedure via a binarization a binarization is to be called complete if it preserves the asymptotic optimality and such a procedure has been proposed by zhou and bruck we show that a complete binarization naturally arises from a binary tree with m leaves therefore there exist complete binarizations in abundance and zhou bruck scheme is an instance of them we now have a relatively simple way to obtain an asymptotically optimal and computationally efficient m extracting procedure from a binary one because these binarizations are both conceptually and computationally simple the well known leaf entropy theorem and a closely related structure lemma play important roles in the arguments

universal finitary codes with exponential tails in 1977 keane and smorodinsky showed that there exists a nitary homomorphism from any nite alphab et bernoulli process to any other nite alphab et bernoulli process of strictly lower entropy in 1996 seran proved the existence of a nitary homomorphism with nite expected coding length in this paper we construct such a homomorphism in which the coding length has exponential tails our construction is source universal in the sense that it does not use any information on the source distribution other than the alphabet size and a bound on the entropy gap between the source and target distributions we also indicate how our methods can be extended to prove a source specic version of the result for markov chains

non contextual chocolate balls versus value indefinite quantum cryptography some quantum cryptographic protocols can be implemented with specially prepared metaphorical chocolate balls representing local hidden variables others protected by value indefiniteness cannot this latter feature which follows from bell and kochen specker type arguments is only present in systems with three or more mutually exclusive outcomes conversely there exist local hidden variable models based on chocolate ball configurations utilizable for cryptography which cannot be realized by quantum systems the possibility that quantum cryptography supported by value indefiniteness contextuality has practical advantages over more conventional quantum cryptographic protocols remains highly speculative

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

finite acceptance of infinite words abstract in this paper we consider the following two types of finite acceptance of infinite words by finite automata an infinite word is accepted if and only if there is a run on input for which 1 1 an accepting state is visited at least once or 2 2 an accepting state is visited at least once but only finitely often the resulting classes of regular languages are characterized by language theoretic means and they are positioned into the known hierarchies of regular languages

reasoning about infinite computations we investigate extensions of temporal logic by connectives defined by finite automata on infinite words we consider three different logics corresponding to three different types of acceptance conditions finite looping and repeating for the automata it turns out however that these logics all have the same expressive power and that their decision problems are all pspace complete we also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem