some notes on varn coding a duality between varn 1971 coding and tunstall 1968 coding is demonstrated and new bounds on the expected transmission cost per source symbol for varn codes are established for binary channels the superiority of the new upper bound over krause s 1962 upper bound is proven

tunstall adaptive coding and miscoding in the first part of this paper we tackle the case where a variable length to block tunstall code is used to encode the wrong source miscoding it turns out that exactly as happens in the case with huffman coding the asymptotic excess rate is given by the informational divergence between the probability distribution ruling the source and the probability distribution for which the tunstall code had been devised we also prove asymptotic equality between the individual rate of a codeword and the corresponding self information this allows us to bound the maximal and the minimal length in a tunstall tree in the second part of the paper we study the case where the probability distribution of the source changes in time if this happens it is necessary to frequently update the current code to ensure that the optimality conditions concerning its rate are met this coding procedure is known as adaptive coding we propose some schemes for adaptive tunstall coding based on the structure of the coding tree on the ordering property analogous to gallager s 1978 sibling property of the huffman code and on the tunstall regions analogous to the attraction regions due to longo and galasso 1982

irreducible unitary representations of su 2 2 we give an explicit classification of the irreducible unitary representations of the simple lie group su 2 2

the space complexity of recognizing well parenthesized expressions in the streaming model the index function revisited we show an p n t lower bound for the space required by any unidirectional constanterror randomized t pass streaming algorithm that recognizes whether an expression over two types of parenthesis is well parenthesized this proves a conjecture due to magniez mathieu and nayak 2009 and rigorously establishes the peculiar power of bi directional streams over unidirectional ones observed in the algorithms they present the lower bound is obtained by analysing the information that is necessarily revealed by the players about their respective inputs in a two party communication protocol for a variant of the index function

variable to fixed length codes for markov sources petry s efficient and optimal variable to fixed length source code for discrete memoryless sources was described by schalkwijk by extending this coding technique we are able to give an algorithm for markov sources that is easy to implement we can bound the loss of efficiency as a function of the code complexity and the mismatch between the source and the code rates arbitrarily close to the source entropy are shown to be achievable in this sense the codes introduced are optimal

generalized tunstall codes for sources with memory tunstall codes are variable to fixed length codes that maximize the expected number of source letters per dictionary string for discrete memoryless sources we analyze a generalization of tunstall coding to sources with memory and demonstrate that as the dictionary size increases the number of code letters per source symbol comes arbitrarily close to the minimum among all variable to fixed length codes of the same size we also find the asymptotic relationship between the dictionary size and the average length of a dictionary entry

reversible arithmetic coding for quantum data compression we study the problem of compressing a block of symbols a block quantum state emitted by a memoryless quantum bernoulli source we present a simple to implement quantum algorithm for projecting with high probability the block quantum state onto the typical subspace spanned by the lending eigenstates of its density matrix we propose a fixed rate quantum shannon fano code to compress the projected block quantum state using a per symbol code rate that is slightly higher than the von neumann 1955 entropy limit finally we propose quantum arithmetic codes to efficiently implement quantum shannon fano 1948 codes our arithmetic encoder and decoder have a cubic circuit and a cubic computational complexity in the block size both the encoder and decoder are quantum mechanical inverses of each other and constitute an elegant example of reversible quantum computation

supermodularity and subadditivity properties of the entropy on the majorization lattice we prove that the entropy is a supermodular and subadditive function on the lattice of all n dimensional probability distributions ordered according to the partial order relation defined by majorization among vectors

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

the unitary irreducible representations of overline rm so _ 0 4 2 an exhaustive classification of all irreducible harish chandra mathfrak s mathfrak o 4 2 modules integrable to unitarizable projective representations of the conformal group is established by infinitesimal methods the classification is based 1 on the reduction upon the maximal compact subalgebra associated with a lattice of points in 3 and 2 on a set of additional parameters upon which the eigenvalues of central elements of the enveloping algebra depend polynomially

separation of random number generation and resolvability we consider the problem of determining when a given source can be used to approximate the output due to any input to a given channel we provide achievability and converse results for a general source and channel for the special case of a full rank discrete memoryless channel we give a stronger converse result than we can give for a general channel

on the security of pseudorandomized information theoretically secure schemes in this paper we discuss a naive method of randomness reduction for cryptographic schemes which replaces the required perfect randomness with output distribution of a computationally secure pseudorandom generator prg we propose novel ideas and techniques for evaluating the indistinguishability between the random and pseudorandom cases even against an adversary with computationally unbounded attack algorithm hence the prg based randomness reduction can be effective even for information theoretically secure cryptographic schemes especially when the amount of information received by the adversary is small in comparison to a preceding result of dubrov and ishai stoc 2006 our result removes the requirement of generalized notion of nb prgs and is effective for more general kinds of protocols we give some numerical examples to show the effectiveness of our result in practical situations and we also propose a further idea for improving the effect of the prg based randomness reduction

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

stock trading an optimal selling rule trading in stock markets consists of three major steps select a stock purchase a number of shares and eventually sell them to make a profit the timing to buy and sell is extremely crucial a selling rule can be specified by two preselected levels a target price and a stop loss limit this paper is concerned with an optimal selling rule based on the model characterized by a number of geometric brownian motions coupled by a finite state markov chain such a policy can be obtained by solving a set of two point boundary value differential equations moreover the corresponding expected target period and probability of making money and that of losing money are derived analytic solutions are obtained in one and two dimensional cases finally a numerical example is considered to demonstrate the effectiveness of our method

generation of discrete distributions from biased coins the procedure of knuth and yao 1976 to simulate random numbers with specified distribution by parsing sequences of fair coin tosses is generalized to employ discrete distributions of particular form instead of fair coins each probability in these distributions is an integral power of some fixed value t the parse tree for the simulation procedure is closely related to the code trees arising in karp s optimal variable length coding algorithm for code symbols of unequal cost

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

bounds on the entropy of a function of a random variable and their applications it is well known that the entropy h x of a discrete random variable x is always greater than or equal to the entropy h f x of a function f of x with equality if and only if f is one to one in this paper we give tight bounds on h f x when the function f is not one to one and we illustrate a few scenarios where this matters as an intermediate step toward our main result we derive a lower bound on the entropy of a probability distribution when only a bound on the ratio between the maximal and minimal probabilities is known the lower bound improves on previous results in the literature and it could find applications outside the present scenario

zero redundancy coding for unequal code symbol costs source distributions that can be encoded with zero redundancy for the case of unequal code symbol costs are examined these distributions provide a natural generalization of the binary equal costs case for which these distributions are the dyadic distributions these zero redundancy codes have the property that the expected proportion of codeword symbols given by a particular letter is equal to an experimental function of the code letter cost the converse is not true in general however partial converse results hold maximum entropy zero redundancy distributions are easily identified through their connection with unequal cost coding for uniform sources

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

optimal stock liquidation in a regime switching model with finite time horizon this paper is concerned with a finite horizon optimal selling rule a set of geometric brownian motions coupled by a finite state markov chain is used to characterize stock price movements given a fixed transaction fee the optimal selling rule can be obtained by solving an optimal stopping problem the corresponding value function is shown to be the unique viscosity solution to the associated hjb equations numerical solutions to these equations and their convergence are obtained a numerical example is presented to illustrate the results

a simple unpredictable pseudo random number generator two closely related pseudo random sequence generators are presented the 1 p generator with input p a prime outputs the quotient digits obtained on dividing 1 by p the x 2 bmod n generato

trading a mean reverting asset buy low and sell high this paper is concerned with an optimal trading buy and sell rule the underlying asset price is governed by a mean reverting model the objective is to buy and sell the asset so as to maximize the overall return slippage cost is imposed on each transaction the associated hjb equations quasi variational inequalities are used to characterize the value functions it is shown that the solution to the original optimal stopping problem can be obtained by solving two quasi algebraic equations sufficient conditions are given in the form of a verification theorem a numerical example is reported to demonstrate the results

recursive algorithms for stock liquidation a stochastic optimization approach this work develops stochastic optimization algorithms for a class of stock liquidation problems the stock liquidation rules are based on hybrid geometric brownian motion models allowing regime changes that are modulated by a continuous time finite state markov chain the optimal selling policy is of threshold type and can be obtained by solving a set of two point boundary value problems the total number of equations to be solved is the same as the number of states of the underlying markov chain to reduce the computational burden using a stochastic optimization approach recursive algorithms are constructed to approximate the optimal threshold values convergence and rates of convergence of the algorithm are studied simulation examples are presented and the computation results are compared with the analytic solutions finally the algorithms are tested using real market data

optimal switching for the pairs trading rule a viscosity solutions approach abstract this paper studies the problem of determining the optimal cut off for pairs trading rules we consider two cointegrated assets whose spread is modelled by a general mean reverting process and the optimal pair trading rule is formulated as an optimal switching problem between three regimes flat position no holding stocks one long and the other short and one short and the other long a fixed commission cost is charged with each transaction we use a viscosity solutions approach to prove the existence and the explicit characterisation of cut off points via the resolution of quasi algebraic equations we illustrate our results by numerical simulations

computing with very weak random sources we give an efficient algorithm to extract randomness from a very weak random source using a small additional number t of truly random bits our work extends that of nisan and zuckerman j comput system sci 52 1996 pp 43 52 in that t remains small even if the entropy rate is well below constant a key application of this is in running randomized algorithms using such a very weak source of randomness for any fixed gamma 0 we show how to simulate rp algorithms in time n o log n using the output of a ds with min entropy r gamma such a weak random source is asked once for r bits it outputs an r bit string according to any probability distribution that places probability at most 2 r gamma on each string if gamma 1 2 our simulation also works for bpp for gamma 1 1 k 1 our simulation takes time n o logk n log k is the logarithm iterated k times we also give a polynomial time bpp simulation using chor goldreich sources of min entropy r omega 1 which is optimal we present applications to time space tradeoffs expander constructions and to the hardness of approximation of independent interest is our randomness efficient leftover hash lemma a key tool for extracting randomness from weak random sources

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 approximation of uniform distributions from variable to fixed length codes in this correspondence we prove that the probability distribution induced on the leaves of a tunstall parse tree for a given source is a unique lower bound in the partially ordered set of the probability distributions induced by all possible parse trees with a same number of leaves and ordered according to the majorization partial order we apply this result to the problem of optimally approximating a uniform distribution with flips of a biased coin

on the quality of random sequences produced with a combined random bit generator in this paper we prove that a combined binary generator which combines modulo 2 bit streams produced by independent generators or by generators that can be grouped into independent pairs of dependent generators can provide unbiased and uncorrelated bit streams this result was obtained under the realistic assumption that each source generator produces a sequence of correlated and biased bits the proposed theorems and formulas are useful in designing random bit generators with satisfactory properties that are robust to external manipulations or to environmental changes up to a certain degree

universal delay limited simulation universal delay limited simulation of an unknown information source of a certain parametric family e g the family of memoryless sources or markov sources of a given order given a training sequence from that source and a stream of independent and uniformly distributed bits is considered the goal of universal simulation is that the probability law of the generated sequence be identical to that of the training sequence with minimum mutual information between the random processes generating both sequences in the delay limited setting the simulation algorithm generates a random sequence sequentially by delivering one symbol for each training symbol that is made available after a given initial delay whereas the random bits are assumed to be available on demand in this paper the optimal universal delay limited simulation scheme is characterized for broad parametric families and the mutual information achieved by the proposed scheme is analyzed the results are extended to a setting of variable delay

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 the selection of measures of distance between probability distributions abstract a notion of relative strength and weakness is defined for the comparison of measures of distance between probability distributions and is shown to be appropriate for use in the selection of distances as quasioptimal design criteria in problems such as signal selection and detector design when the preferred criteria probability of error and asymptotic relative efficiency are intractable csiszar s work on the topological properties of divergences a class of distances also investigated by ali and silvey enables one to identify strong distances and provides a justification for the experimental findings of previous investigators

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

the construction of huffman codes is a submodular convex optimization problem over a lattice of binary trees we show that the space of all binary huffman codes for a finite alphabet defines a lattice ordered by the imbalance of the code trees representing code trees as path length sequences we show that the imbalance ordering is closely related to a majorization ordering on real valued sequences that correspond to discrete probability density functions furthermore this tree imbalance is a partial ordering that is consistent with the total orderings given by either the external path length sum of tree path lengths or the entropy determined by the tree structure on the imbalance lattice we show the weighted path length of a tree the usual objective function for huffman coding is a submodular function as is the corresponding function on the majorization lattice submodular functions are discrete analogues of convex functions these results give perspective on huffman coding and suggest new approaches to coding as optimization over a lattice

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

entropy and an optimal random number transformation corresp a simple proof is given of the optimality of the procedure of d e knuth and a c yao for transforming a random number with distribution 1 2 1 2 into one with distribution p 1 cdots p r a direct representation of the cost average number of input symbols needed to output one symbol of the optimal transformation is given in terms of entropy a procedure that leads to a refined entropic upper bound to the cost

efficient generation of fair dice with few biased coins given a random variable x which takes n equiprobable values we consider several algorithmic questions related to the classical problem of simulating the outcomes of x by using a limited number of biased coins

simulating fair dice with biased coins this paper is concerned with simulating a fair die with a bounded number of coin flips and a bounded number of possibly biased coins as the main results this paper shows that for anyn 2 a set ofh n coins is sufficient to simulate a fairn sided die withind lgn coin flips whereh n is the number of 1 s of the binary representation of an integern and that for anyn 2d 1 d 3 a set ofd h n coins is necessary and sufficient to simulate a fairn sided die withind lgn coin flips

unbiased bits from sources of weak randomness and probabilistic communication complexity a new model for weak random physical sources is presented the new model strictly generalizes previous models e g the santha and vazirani model 27 the sources considered output strings according to probability distributions in which no single string is too probable the new model provides a fruitful viewpoint on problems studied previously such as extracting almost perfect bits from sources of weak randomness the question of possibility as well as the question of efficiency of such extraction schemes are addressed probabilistic communication complexity it is shown that most functions have linear communication complexity in a very strong probabilistic sense robustness of bpp with respect to sources of weak randomness generalizing a result of vazirani and vazirani 32 33

algorithms for generating probabilities with multivalued stochastic relay circuits the problem of random number generation dates back to von neumann s work in 1951 since then many algorithms have been developed for generating unbiased bits from complex correlated sources as well as for generating arbitrary distributions from unbiased bits an equally interesting but less studied aspect is the structural component of random number generation that is given a set of primitive sources of randomness and given composition rules induced by a device or nature how can we build networks that generate arbitrary probability distributions in this paper we study the generation of arbitrary probability distributions in multivalued relay circuits a generalization in which relays can take on any of n states and the logical and and or are replaced with min and max respectively these circuits can be thought of as modeling the timing of events which depend on other event occurences we describe a duality property and give algorithms that synthesize arbitrary rational probability distributions we prove that these networks are robust to errors and design a universal probability generator which takes input bits and outputs any desired binary probability distribution

a model for reversible investment capacity expansion we consider the problem of determining the optimal investment level that a firm should maintain in the presence of random price and or demand fluctuations we model market uncertainty by means of a geometric brownian motion and we consider general running payoff functions our model allows for capacity expansion as well as for capacity reduction with each of these actions being associated with proportional costs the resulting optimization problem takes the form of a singular stochastic control problem that we solve explicitly we illustrate our results by means of the so called cobb douglas production function the problem that we study presents a model in which the associated hamilton jacobi bellman equation admits a classical solution that conforms with the underlying economic intuition but does not necessarily identify with the corresponding value function which may be identically equal to infty thus our model provides a situation that highlights the need for rigorous mathematical analysis when addressing stochastic optimization applications in finance and economics as well as in other fields

elementary representations and intertwining operators for su 2 2 i the structure of the group su 2 2 and of its lie algebra is studied in detail the results will be applied in subsequent parts devoted to the explicit construction of elementary representations of su 2 2 induced from different parabolic subgroups and of the intertwining operators between these representations a summary of some results of parts ii and iii is given

folklore in source coding information spectrum approach information theory has several traditional folklore problems about data compression or channel coding with reference to random number generation problems here we focus on and reasonably formulate one of them from the viewpoint of information spectra specifically we verify the validity of the folklore that the output from any source encoder working at the optimal coding rate with asymptotically vanishing probability of error looks like almost completely random

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

optimal trend following trading rules we develop an optimal trend following trading rule in a bull bear switching market where the drift of the stock price switches between two parameters corresponding to an uptrend bull market and a downtrend bear market according to an unobservable markov chain we consider a flnite horizon investment problem and aim to maximize the expected return of the terminal wealth we start by restricting to allowing at and long positions only and describe the trading decisions using a sequence of stopping times indicating the time of entering and exiting long positions assuming trading all available funds we show that the optimal trading strategy is a trend following system characterized by the conditional probability in the uptrend crossing two threshold curves the thresholds can be obtained by solving the associated hjb equations in addition we examine trading strategies with short selling in terms of an approximation simulations and empirical experiments are conducted and reported

twice universal simulation of markov sources and individual sequences the problem of universal simulation given a training sequence is studied both in a stochastic setting and for individual sequences in the stochastic setting the training sequence is assumed to be emitted by a markov source of unknown order extending previous work where the order is assumed known and leading to the notion of twice universal simulation a simulation scheme which partitions the set of sequences of a given length into classes is proposed for this setting and shown to be asymptotically optimal this partition extends the notion of type classes to the twice universal setting in the individual sequence scenario the same simulation scheme is shown to generate sequences which are statistically similar in a strong sense to the training sequence for statistics of any order while essentially maximizing the uncertainty on the output

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

new methods for digital generation and postprocessing of random data a new method for digital true random number generation based on asynchronous logic circuits with feedback is introduced in particular a concrete technique using the so called galois and fibonacci ring oscillators is developed and analyzed both theoretically and experimentally the generated random binary sequences may have a very high speed and a higher and more robust entropy rate in comparison with previous proposals for digital random number generators a new method for digital postprocessing of random data based on irregularly clocked nonautonomous synchronous logic circuits with feedback is also introduced and a concrete technique using a self clock controlled linear feedback shift register is proposed the postprocessing can provide both randomness extraction and computationally secure speed increase of input random data

optimal selling rules in a regime switching exponential gaussian diffusion model this paper develops optimal selling rules in asset trading using a regime switching exponential gaussian diffusion model the optimization problem is solved by a combined approach of boundary value problems and probabilistic analysis a system of linear differential equations with variable coefficients and two point boundary conditions satisfied by the objective function of the problem is derived the existence and uniqueness of the solution are proved a closed form solution in terms of weber functions is obtained for one dimensional cases for m dimensional cases a stochastic recursive algorithm for numerically searching the optimal value is developed numerical results are reported

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

bounding the average length of optimal source codes via majorization theory we consider the problem of bounding the average length of an optimal huffman source code when only limited knowledge of the source symbol probability distribution is available for instance we provide tight upper and lower bounds on the average length of optimal source codes when only the largest or the smallest source symbol probability is known our results rely on basic results of majorization theory and on the schur concavity of the minimum average length of variable length source codes for discrete memoryless sources in the way to prove our main result we also give closed formula expressions for the average length of huffman codes for several classes of probability distributions

a trend following strategy conditions for optimality based on trend following trading strategies that are widely used in the investment world this work provides a set of sufficient conditions that determine the optimality of the traditional trend following strategies when the trends are completely observable a dynamic programming approach is used to verify the optimality under these conditions the value functions are characterized by the associated hjb equations and are shown to be either linear functions or infinity depending on the parameter values the results reveal two counterintuitive facts a trend following may not lead to optimal reward in some cases even when if the investor knows exactly when a trend change occurs b stock volatility is not relevant in trend following when trends are observable

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

tunstall code khodak variations and random walks a variable to fixed length encoder partitions the source string into variable length phrases that belong to a given and fixed dictionary tunstall and independently khodak designed variable to fixed length codes for memoryless sources that are optimal under certain constraints in this paper we study the tunstall and khodak codes using variety of techniques ranging from stopping times for sums of independent random variables to tauberian theorems and mellin transform after proposing an algebraic characterization of the tunstall and khodak codes we present new results on the variance and a central limit theorem for dictionary phrase lengths this analysis also provides a new argument for obtaining asymptotic results about the mean dictionary phrase length and average redundancy rates

on the combination of self organized systems to generate pseudo random numbers cellular automata ca is a self organizing structure with complex behavior which can be used to generating pseudo random numbers pure ca has a simple structure and has no ability to produce long sequences of random numbers but in order to resolve this problem a programmable controllable ca pcca using actuating factor or combination of different self organizing criticality phenomena can be used the purpose of the aforementioned methods is the use of the ca speed while preventing the automata self organizing factor for specific use in generating random numbers in this paper a pcca by using sandpile model is proposed the sandpile model is a complex system that operates at a critical state between chaos and order this state is known as self organized criticality soc and is characterized by displaying invariant scale behavior in the precise case of the sandpile model by randomly and continuously dropping sand grains on top of a two dimensional grid lattice a power law relationship between the frequency and size of sand avalanches is observed the avalanche behavior and the pure ca behavior are combined in a novel method which can be used as the pseudo random number generator prng experimental results show that this generator is able to reach a random behavior from a pseudo chaotic one by combining two self organized systems in addition to independency of the initial state it has been also indicated that the generated numbers are independent and its probability distribution is likely to be uniform accordingly some tests of prngs such as entropy chi square diehard and several basic statistical tests have been performed and all of them have been successfully passed

on the existence of pseudorandom generators pseudorandom generators suggested and developed by blum and micali and yao are efficient deterministic programs that expand a randomly selected k bit seed into a much longer pseudorandom bit sequ

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

relative uniformity of sources and the comparison of optimal code costs conditions on the probability distributions of two sources that describe their relative uniformity are shown to be sufficient for inequalities on their minimum average codeword lengths and entropies the huffman coding problem and variants involving renyi entropy are addressed

second order asymptotics in fixed length source coding and intrinsic randomness there is a difference between the optimal rates of fixed length source coding and intrinsic randomness when we care about the second order asymptotics we prove this difference for general information sources and then investigate independent and identically distributed i i d random variables and markovian variables as examples the difference is demonstrated through an investigation of the second order asymptotic behavior of the rates a universal fixed length source code attaining the second order optimal rate is also proposed the difference between the rates of fixed length source coding and intrinsic randomness proves that the outputs of fixed length source codes are not uniformly distributed

optimal selling rules in a regime switching model in this note we derive optimal selling rules under a regime switching model the optimal stopping rule is of a threshold type for each state derived via the modified smooth fit the proof is via the martingale theory numerical examples are reported to demonstrate the dependence of threshold levels with various parameters and to compare our result with some suboptimal selling rules

an optimal pairs trading rule abstract this paper is concerned with a pairs trading rule the idea is to monitor two historically correlated securities when divergence is underway i e one stock moves up while the other moves down a pairs trade is entered which consists of a pair to short the outperforming stock and to long the underperforming one such a strategy bets the spread between the two would eventually converge in this paper a difference of the pair is governed by a mean reverting model the objective is to trade the pair so as to maximize an overall return a fixed commission cost is charged with each transaction in addition a stop loss limit is imposed as a state constraint the associated hjb equations quasi variational inequalities are used to characterize the value functions it is shown that the solution to the optimal stopping problem can be obtained by solving a number of quasi algebraic equations we provide a set of sufficient conditions in terms of a verification theorem numerical examples are reported to demonstrate the results

optimal switching strategy of a mean reverting asset over multiple regimes we solve optimal iterative three regime switching problems with transaction costs with investment in a mean reverting asset that follows an ornstein uhlenbeck process and find the explicit solutions the investor can take either a long short or square position and can switch positions during the period modeling the short sales position is necessary to study optimal trading strategies such as the pair trading few studies provide explicit solutions to problems with multiple more than two regimes states the value function is proved to be a unique viscosity solution of a hamilton jacobi bellman variational inequality hjb vi multiple regime switching problems are more difficult to solve than conventional two regime switching problems because they need to identify not only when to switch but also where to switch therefore multiple regime switching problems need to identify the structure of the continuation switching regions in the free boundary problem for each regime if the number of the states n is two only two regions have to be identified but if n 3 n p 2 6 regions have to be detected we identify the structure of the switching regions for each regime using the theories related to the viscosity solution approach

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

extraction of optimally unbiased bits from a biased source we explore the problem of transforming n independent and identically biased 1 1 valued random variables x sub 1 x sub n into a single 1 1 random variable f x sub 1 x sub n so that this result is as unbiased as possible in general no function f produces a completely unbiased result we perform the first study of the relationship between the bias b of these x sub i and the rate at which f x sub 1 x sub n can converge to an unbiased 1 1 random variable as n spl rarr spl infin a 1 1 random variable has bias b if e x sub i b fixing a bias b we explore the rate at which the output bias e f x sub 1 x sub n can tend to zero for a function f 1 1 spl rarr 1 1 this is accomplished by classifying the behavior of the natural normalized quantity spl xi b spl delta inf sub f lim sub n spl rarr spl infin n spl radic e f x sub 1 x sub n this infimum taken over all such f we show that for rational b spl xi b 1 s where 1 b 2 r s r and s relatively prime developing the theory of uniform distribution of sequences to suit our problem we then explore the case where b is irrational we prove a new metrical theorem concerning multidimensional diophantine approximation type from which we show that for lebesgue almost all biases b spl xi b 0 finally we show that algebraic biases exhibit curious boundary behavior falling into two classes class 1 those algebraics b for which spl xi b 0 and furthermore c sub 1 spl les spl xi b spl les c sub 2 where c sub 1 and c sub 2 are positive constants depending only on b s algebraic characteristics class 2 those algebraics b for which there exist n 0 and f 1 1 sup n spl rarr 1 1 so that e f x sub 1 x sub n 0 notice that this classification excludes the possibility that n spl radic e f x sub 1 x sub n limits to zero for algebraics for rational and algebraic biases we also study the computational problem by restricting f to be a polynomial time computable function finally we discuss natural extensions where output distributions other than the uniform distribution on 1 1 are sought

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

explicit solutions for an optimal stock selling problem under a markov chain model abstract this paper is concerned with explicit solutions for a classical optimal stock selling problem in contrast to almost all market models treated in the literature the underlying market is solely determined by a two state markov chain such markov chain model is strikingly simple and yet appears capable capturing various market movements ranging from close to brownian motion to no so brownian ones the purpose of this paper is to study the optimal selling rule under such a model and develop a set of analysis techniques useful for related optimal stopping problems in this paper the goal of the problem under consideration is to find an optimal stopping time to sell the stock so as to maximize an expected return explicit solutions to the associated variational inequalities are obtained these solutions are given in terms of a set of threshold levels verification theorems are provided to justify their optimality finally numerical examples are provided to illustrate the results

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

the zero guards algorithm for general minimum distance decoding problems we present some properties of an improved version of the zero neighbors algorithm the zero guards algorithm these properties can be used to find a zero guards a new decoding procedure using a zero guards is also given