free energy and correlations of the number theoretical spin chain we analyze the number theoretical spin chain with partition function z 1 using the polymer model technique the finite grand canonical chains give bounds for the limit free energy and internal energy the correlation functions for inverse temperature 1 are products of two point functions a combinatorial result for general interval graphs is derived

randomness computability and density we study effectively given positive reals more specifically computably enumerable reals under a measure of relative randomness introduced by solovay manuscript ibm thomas j watson research center yorktown heights ny 1975 and studied by calude hertling khoussainov and wang theoret comput sci 255 2001 pp 125 149 calude theoret comput sci 271 2002 pp 3 14 kucera and slaman siam j comput 31 2002 pp 199 211 and downey hirschfeldt and laforte mathematical foundations of computer science 2001 springer verlag berlin 2001 pp 316 327 among others this measure is called domination or solovay reducibility and is defined by saying that alpha dominates beta if there are a constant c and a partial computable function varphi such that for all positive rationals q alpha we have varphi q downarrow beta and beta varphi q leqslant c alpha q the intuition is that an approximating sequence for alpha generates one for beta whose rate of convergence is not much slower than that of the original sequence it is not hard to show that if alpha dominates beta then the initial segment complexity of alpha is at least that of beta in this paper we are concerned with structural properties of the degree structure generated by solovay reducibility we answer a natural question in this area of investigation by proving the density of the solovay degrees we also provide a new characterization of the random computably enumerable reals in terms of splittings in the solovay degrees specifically we show that the solovay degrees of computably enumerable reals are dense that any incomplete solovay degree splits over any lesser degree and that the join of any two incomplete solovay degrees is incomplete so that the complete solovay degree does not split at all the methodology is of some technical interest since it includes a priority argument in which the injuries are themselves controlled by randomness considerations

phases of the number theoretic spin chain we present numerical and analytical evidence for a first order phase transition of the ferromagnetic spin chain with partition functionz 1 at the inverse temperature cr 2

universal computably enumerable sets and initial segment prefix free complexity we show that there are turing complete computably enumerable sets of arbitrarily low nontrivial initial segment prefix free complexity in particular given any computably enumerable set a with nontrivial prefix free initial segment complexity there exists a turing complete computably enumerable set b with complexity strictly less than the complexity of a on the other hand it is known that sets with trivial initial segment prefix free complexity are not turing complete moreover we give a generalization of this result for any finite collection of computably enumerable sets a i i

on a ferromagnetic spin chain the quotient s 1 s of riemann zeta functions is shown to be the partition function of a ferromagnetic spin chain for inverse temperatures

randomness and recursive enumerability one recursively enumerable real alpha dominates another one beta if there are nondecreasing recursive sequences of rational numbers a n n in omega approximating alpha and b n n in omega approximating beta and a positive constant c such that for all n c alpha a n geq beta b n see r m solovay draft of a paper or series of papers on chaitin s work manuscript ibm thomas j watson research center yorktown heights ny 1974 p 215 and g j chaitin ibm j res develop 21 1977 pp 350 359 we show that every recursively enumerable random real dominates all other recursively enumerable reals we conclude that the recursively enumerable random reals are exactly the omega numbers g j chaitin ibm j res develop 21 1977 pp 350 359 second we show that the sets in a universal martin lof test for randomness have random measure and every recursively enumerable random number is the sum of the measures represented in a universal martin lof test

process and truth table characterisations of randomness this paper uses quick process machines to provide characterisations of computable randomness schnorr randomness and weak randomness the quick process machine is a type of process machine first considered in work of levin and zvonkin a new technique for building process machines and quick process machines is presented this technique is similar to the kc theorem for prefix free machines using this technique a method of translating computable martingales to quick process machines is given this translation forms the basis for these new randomness characterisations quick process machines are also used to provide characterisations of computable randomness schnorr randomness and weak randomness in terms of truth table reducibility

on initial segment complexity and degrees of randomness one approach to understanding the fine structure of initial segment complexity was introduced by downey hirschfeldt and laforte they define x k y to mean that vn k x n k y n 0 1 the equivalence classes under this relation are the k degrees we prove that if x y is 1 random then x and y have no upper bound in the k degrees hence no join we also prove that n randomness is closed upward in the k degrees our main tool is another structure intended to measure the degree of randomness of real numbers the vl degrees unlike the k degrees many basic properties of the vl degrees are easy to prove we show that x k y implies x vl y so some results can be transferred the reverse implication is proved to fail the same analysis is also done for c the analogue of k for plain kolmogorov complexity two other interesting results are included first we prove that for any z 2 a 1 random real computable from a 1 z random real is automatically 1 z random second we give a plain kolmogorov complexity characterization of 1 randomness this characterization is related to our proof that x c y implies x vl y

a lattice gas of prime numbers and the riemann hypothesis abstract in recent years there has been some interest in applying ideas and methods taken from physics in order to approach several challenging mathematical problems particularly the riemann hypothesis most of these kinds of contributions are suggested by some quantum statistical physics problems or by questions originated in chaos theory in this article we show that the real part of the non trivial zeros of the riemann zeta function extremizes the grand potential corresponding to a simple model of one dimensional classical lattice gas the critical point being located at 1 2 as the riemann hypothesis claims

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

on kurtz randomness kurtz randomness is a notion of algorithmic randomness for real numbers in particular a real x is called kurtz random or weakly random iff it is contained in every computably enumerable set u of lebesgue measure 1 we prove a number of characterizations of this notion relating it to other notions of randomness such as the well known notions of computable randomness martin lof randomness and schnorr randomness for the first time we give machine characterizations of kurtz randomness whereas the turing degree of every martin lof random c e real is the complete degree and the degrees of schnorr random c e reals are all high we show that kurtz random c e reals occur in every non zero c e degree additionally we show that the sets that are low for kurtz randomness are all hyperimmune and include those that are low for schnorr randomness characterized previously by terwijn and email protected 2004 elsevier b v all rights reserved

capacity and power allocation for fading mimo channels with channel estimation error in this correspondence we investigate the effect of channel estimation error on the capacity of multiple input multiple output mimo fading channels we study lower and upper bounds of mutual information under channel estimation error and show that the two bounds are tight for gaussian inputs assuming gaussian inputs we also derive tight lower bounds of ergodic and outage capacities and optimal transmitter power allocation strategies that achieve the bounds under perfect feedback for the ergodic capacity the optimal strategy is a modified waterfilling over the spatial antenna and temporal fading domains this strategy is close to optimum under small feedback delays but when the delay is large equal powers should be allocated across spatial dimensions for the outage capacity the optimal scheme is a spatial waterfilling and temporal truncated channel inversion numerical results show that some capacity gain is obtained by spatial power allocation temporal power adaptation on the other hand gives negligible gain in terms of ergodic capacity but greatly enhances outage performance

on asymptotic properties of a number theoretic function arising out of a spin chain model in statistical mechanics let let n be a positive integer and define n as the number of matrices c which are products of a and b where both a and b must occur such that the trace tr c n it has been conjectured that see 10 in this note we consider the summatory function and show that

kolmogorov complexity of initial segments of sequences and arithmetical definability the structure of the k degrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments in thek degrees of infinite binary sequences x is below y if the prefix free kolmogorov complexity of the first n bits of x is less than the complexity of the first n bits of y for each n identifying infinite binary sequences with subsets of n we study the k degrees of arithmetical sets and explore the interactions between arithmetical definability and prefix free kolmogorov complexity we show that in the k degrees for each n 1 there exists a s n 0 non zero degree which does not bound any d n 0 non zero degree an application of this result is that in the k degrees there exists a s 2 0 degree which forms a minimal pair with all s 1 0 degrees this extends the work of csima and montalban 2006 8 and merkle and stephan 2007 17 our main result is that given any d 2 0 family c of sequences there is a d 2 0 sequence of non trivial initial segment complexity which is not larger than the initial segment complexity of any non trivial member of c this general theorem has the following surprising consequence there is a 0 computable sequence of non trivial initial segment complexity which is not larger than the initial segment complexity of any non trivial computably enumerable set our analysis and results demonstrate that examining the extend to which arithmetical definability interacts with the k reducibility and in general any weak reducibility is a fruitful way of studying the induced structure

on the asymptotic capacity of stationary gaussian fading channels we consider a peak power limited single antenna flat complex gaussian fading channel where the receiver and transmitter while fully cognizant of the distribution of the fading process have no knowledge of its realization upper and lower bounds on channel capacity are derived with special emphasis on tightness in the high signal to noise ratio snr regime necessary and sufficient conditions in terms of the autocorrelation of the fading process are derived for capacity to grow double logarithmically in the snr for cases in which capacity increases logarithmically in the snr we provide an expression for the pre log i e for the asymptotic ratio between channel capacity and the logarithm of the snr this ratio is given by the lebesgue measure of the set of harmonics where the spectral density of the fading process is zero we finally demonstrate that the asymptotic dependence of channel capacity on the snr need not be limited to logarithmic or double logarithmic behaviors we exhibit power spectra for which capacity grows as a fractional power of the logarithm of the snr

a rate splitting approach to fading channels with imperfect channel state information as shown by medard the capacity of fading channels with imperfect channel state information can be lower bounded by assuming a gaussian channel input x with power p and by upper bounding the conditional entropy h x y hat h by the entropy of a gaussian random variable with variance equal to the linear minimum mean square error in estimating x from y hat h we demonstrate that using a rate splitting approach this lower bound can be sharpened by expressing the gaussian input x as the sum of two independent gaussian variables x 1 and x 2 and by applying medard s lower bound first to bound the mutual information between x 1 and y while treating x 2 as noise and by applying it a second time to the mutual information between x 2 and y while assuming x 1 to be known we obtain a capacity lower bound that is strictly larger than medard s lower bound we then generalize this approach to an arbitrary number l of layers where x is expressed as the sum of l independent gaussian random variables of respective variances p ell ell 1 dotsc l summing up to p among all such rate splitting bounds we determine the supremum over power allocations p ell and total number of layers l this supremum is achieved for l rightarrow infty and gives rise to an analytically expressible capacity lower bound for gaussian fading this novel bound is shown to converge to the gaussian input mutual information as the signal to noise ratio snr grows provided that the variance of the channel estimation error h hat h tends to zero as the snr tends to infinity

fading channels how perfect need perfect side information be the analysis of flat fading channels is often performed under the assumption that the additive noise is white and gaussian and that the receiver has precise knowledge of the realization of the fading process these assumptions imply the optimality of gaussian codebooks and of scaled nearest neighbor decoding here we study the robustness of this communication scheme with respect to errors in the estimation of the fading process we quantify the degradation in performance that results from such estimation errors and demonstrate the lack of robustness of this scheme for some situations we suggest the rule of thumb that in order to avoid degradation the estimation error should be negligible compared to the reciprocal of the signal to noise ratio snr

a farey fraction spin chain we introduce a new number theoretic spin chain and explore its thermodynamics and connections with number theory the energy of each spin configuration is defined in a translation invariant manner in terms of the farey fractions and is also expressed using pauli matrices we prove that the free energy exists and a phase transition occurs for positive inverse temperature 2 the free energy is the same as that of related non translation invariant number theoretic spin chain using a number theoretic argument the low temperature 3 state is shown to be completely magnetized for long chains the number of states of energy e log n summed over chain length is expressed in terms of a restricted divisor problem we conjecture that its asymptotic form is n log n consistent with the phase transition at 2 and suggesting a possible connection with the riemann function the spin interaction coefficients include all even many body terms and are translation invariant computer results indicate that all the interaction coefficients except the constant term are ferromagnetic

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

generalized farey trees transfer operators and phase transitions we consider a family of markov maps on the unit interval interpolating between the tent map and the farey map the latter is not uniformly expanding each map being composed of two fractional linear transformations the family generalizes many particular properties which for the case of the farey map have been successfully exploited in number theory we analyze the dynamics through the spectral properties of the generalized transfer operator application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function

lowness properties and randomness abstract the set a is low for martin lof randomness if each random set is already random relative to a a is k trivial if the prefix complexity k of each initial segment of a is minimal namely n k a n k n o 1 we show that these classes coincide this answers a question of ambos spies and kucera in p cholak s lempp m lerman r shore eds computability theory and its applications current trends and open problems american mathematical society providence ri 2000 each low for martin lof random set is 2 0 our class induces a natural intermediate 3 0 ideal in the r e turing degrees which generates the whole class under downward closure answering a further question in p cholak s lempp m lerman r shore eds computability theory and its applications current trends and open problems american mathematical society providence ri 2000 we prove that each low for computably random set is computable

the low activity phase of some dirichlet series we show that a rigorous statistical mechanics description of some dirichlet series is possible using the abstract polymer model language of statistical mechanics and the polymer expansion theory we characterize the low activity phase by the suitable exponential decay of the truncated correlation functions

oscillation in the initial segment complexity of random reals abstract we study oscillation in the prefix free complexity of initial segments of 1 random reals for upward oscillations we prove that n 2 g n diverges iff n k x n n g n for every 1 random x 2 for downward oscillations we characterize the functions g such that n k x n n g n for almost every x 2 the proof of this result uses an improvement of chaitin s counting theorem we give a tight upper bound on the number of strings 2 n such that k n k n m the work on upward oscillations has applications to the k degrees write x k y to mean that k x n k y n o 1 the induced structure is called the k degrees we prove that there are comparable 2 0 1 random k degrees we also prove that every lower cone and some upper cones in the 1 random k degrees have size continuum finally we show that it is independent of zfc even assuming that the continuum hypothesis fails whether all chains of 1 random k degrees of size less than 2 0 have a lower bound in the 1 random k degrees

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

applications of statistical mechanics in number theory abstract the links between statistical physics and number theory are discussed first the attempts to prove the riemann hypothesis by means of the suitable spin model and the lee yang theorem about zeros of the partition function are shortly reviewed next the analogies between random walks and prime numbers are mentioned in the last section the partition function of the system whose energies are defined by the distances between consecutive primes is calculated the arguments are given that such a prime numbers gas behaves like a set of noninteracting harmonic oscillators

mismatched decoding revisited general alphabets channels with memory and the wide band limit the mismatch capacity of a channel is the highest rate at which reliable communication is possible over the channel with a given possibly suboptimal decoding rule this quantity has been studied extensively for single letter decoding rules over discrete memoryless channels dmcs here we extend the study to memoryless channels with general alphabets and to channels with memory with possibly non single letter decoding rules we also study the wide band limit and in particular the mismatch capacity per unit cost and the achievable rates on an additive noise spread spectrum system with single letter decoding and binary signaling

on a ferromagnetic spin chain ii thermodynamic limit the existence of the thermodynamic limit of the free energy for the ferromagnetic spin chain connected with the riemann zeta function is proven

chaitin s halting probability and the compression of strings using oracles if a computer is given access to an oracle the characteristic function of a set whose membership relation may or may not be algorithmically calculable this may dramatically affect its ability to compress information and to determine structure in strings which might otherwise appear random this leads to the basic question given an oracle a how many oracles can compress information at most as well as a this question can be formalized using kolmogorov complexity we say that b lka if there exists a constant c such that ka kb c for all strings where kx denotes the prefix free kolmogorov complexity relative to oracle x the formal counterpart to the previous question now is what is the cardinality of the set of lk predecessors of a we completely determine the number of oracles that compress at most as well as any given oracle a by answering a question of miller notre dame j formal logic 2010 50 381 391 which also appears in nies computability and randomness oxford uk oxford university press 2009 problem 8 1 13 the class of lk predecessors of a set a is countable if and only if chaitin s halting probability is martin lof random relative to a