on resource bounded instance complexity we prove the instance complexity conjecture of ko orponen schoning and watanabe for all recursive tally sets and for all recursive sets which are np hard under honest reductions in particular it holds for all natural np hard problems on the other hand the conjecture is shown to be oracle dependent additional results concern the nonrecursiveness of the instance complexity measure and a comparison of time bounded c and cd complexity

gamow vectors and borel summability in a class of quantum systems we analyze the detailed time dependence of the wave function x t for one dimensional hamiltonians h partial x 2 v x where v for example modeling barriers or wells and x 0 are compactly supported

using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

gamow states as continuous linear functionals over analytical test functions the space of analytical test functions rapidly decreasing on the real axis i e schwartz test functions of the type s on the real axis is used to construct the rigged hilbert space rhs h gamow states gs can be defined in rhs starting from dirac s formula it is shown that the expectation value of a self adjoint operator acting on a gs is real we have computed exactly the probability of finding a system in a gs and found that it is finite the validity of recently proposed approximations to calculate the expectation value of self adjoint operators in a gs is discussed

space of test functions for higher order field theories the fundamental space is defined as the set of entire analytic functions test functions z which are rapidly decreasing on the real axis the variable z corresponds to the complex energy plane the conjugate or dual space is the set of continuous linear functionals distributions on among those distributions are the propagators determined by the poles implied by the equations of motion and the contour of integration implied by the boundary conditions all propagators can be represented as linear combinations of elementary one pole functionals the algebra of convolution products is also determined the fourier transformed space contains test functions x these functions are extra rapidly decreasing so that the exponentially increasing solutions of higher order equations are distributions on

the closure properties on real numbers under limits and computable operators in effective analysis various classes of real numbers are discussed for example the classes of computable semi computable weakly computable recursively approximable real numbers etc all these classes correspond to some kind of weak computability of the real numbers in this paper we discuss mathematical closure properties of these classes under the limit effective limit and computable function among others we show that the class of weakly computable real numbers is not closed under effective limit and partial computable functions while the class of recursively approximable real numbers is closed under effective limit and partial computable functions

robust characterizations of polynomials withapplications to program testing the study of self testing and self correcting programs leads to the search for robust characterizations of functions here the authors make this notion precise and show such a characterization for polynomials from this characterization the authors get the following applications simple and efficient self testers for polynomial functions are constructed the characterizations provide results in the area of coding theory by giving extremely fast and efficient error detecting schemes for some well known codes this error detection scheme plays a crucial role in subsequent results on the hardness of approximating some np optimization problems

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

chaitin ω numbers solovay machines and godel incompleteness computably enumerable c e reals can be coded by chaitin machines through their halting probabilities tuning solovay s construction of a chaitin universal machine for which zfc if arithmetically sound cannot determine any single bit of the binary expansion of its halting probability we show that every c e random real is the halting probability of a universal chaitin machine for which zfc cannot determine more than its initial block of 1 bits as soon as you get a 0 it is all over finally a constructive version of chaitin information theoretic incompleteness theorem is proven

on the rhs description of resonances and virtual states we discuss the role of the m ller wave operators in a rigged hilbert space rhs model for resonances we also construct a new rhs of analytic functions valid for the description of resonances and virtual states these become generalized state vectors in a general physically realizable representation

fredholm s equations for subwavelength focusing subwavelength focusing sf is a very useful tool that can be carried out with the use of left hand materials for optics that involve the range of the microwaves many recent works have described a successful alternative procedure using time reversal methods the advantage is that we do not need devices which require the complicated manufacture of left hand materials nevertheless the theoretical mathematical bases are far from complete because before now we lacked an adequate easy to apply frame in this work we give for a broad class of discrete systems a solid support for the theory of electromagnetic sf that can be applied to communications and nanotechnology the very central procedure is the development of vector matrix formalism vmf based on exploiting both the inhomogeneous and homogeneous fredholm s integral equations in cases where the last two kinds of integral equations are applied to some selected discrete systems to this end we first establish a generalized newmann series for the fourier transform of the green s function in the inhomogeneous fredholm s equation of the problem then we go from an integral operator equation to a vector matrix algebraic one in this way we explore the inhomogeneous case and later on also the very interesting one about the homogeneous equation thus on the one hand we can relate in a simple manner the arriving electromagnetic signals with those at their sources and we can use them to perform a sf on the other hand we analyze the homogeneous version of the equations finding resonant solutions that have analogous properties to their counterparts in quantum mechanical scattering that can be used in a proposed very powerful way in communications also we recover quantum mechanical operator relations that are identical for classical electromagnetics finally we prove two theorems that formalize the relation between the theory of fredholm s integral equations and the vmf we present here

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

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

a general classification rule for probability measures abstract we consider the problem of classifying an unknown probability distribution based on a sequence of random samples drawn according to this distribution specifically if a is a subset of the space of all probability measures m1 sigma over some compact polish space e we want to decide whether or not the unknown distribution belongs to a or its complement we propose an algorithm which leads a s to a correct decision for any a satisfying certain structural assumptions a refined decision procedure is also presented which given a countable collection ai c m1 sigma i 1 2 each satisfying the structural assumption will eventually determine a s the membership of the distribution in any finite number of the ai applications to density estimation and the problem of order determination of markov processes are discussed

lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers the kucera gacs theorem is a landmark result in algorithmic randomness asserting that every real is computable from a martin lof random real if the computation of the first n bits of a sequence requires n h n bits of the random oracle then h is the redundancy of the computation kucera implicitly achieved redundancy n log n while gacs used a more elaborate coding procedure which achieves redundancy n log n a similar bound is implicit in the later proof by merkle and mihailovic in this paper we obtain optimal strict lower bounds on the redundancy in computations from martin lof random oracles we show that any nondecreasing computable function g such that n 2 g n is not a general upper bound on the redundancy in computations from martin lof random oracles in fact there exists a real x such that the redundancy g of any computation of x from a martin lof random oracle satisfies n 2 g n moreover the class of such reals is comeager and includes a 2 0 real as well as all weakly 2 generic reals on the other hand it has been recently shown that any real is computable from a martin lof random oracle with redundancy g provided that g is a computable nondecreasing function such that n 2 g n hence our lower bound is optimal and excludes many slow growing functions such as log n from bounding the redundancy in computations from random oracles for a large class of reals our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop

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

initial segment complexities of randomness notions schnorr famously proved that martin lof randomness of a sequence a can be characterised via the complexity of a s initial segments nies stephan and terwijn as well as independently miller showed that a set is 2 random that is martin lof random relative to the halting problem k iff there is no function f such that for all m and all n f m it holds that c a 0 a 1 a n f m it holds that c a 0 a 1 a n

satisfiability allows no nontrivial sparsification unless the polynomial time hierarchy collapses consider the following two player communication process to decide a language l the first player holds the entire input x but is polynomially bounded the second player is computationally unbounded but does not know any part of x their goal is to decide cooperatively whether x belongs to l at small cost where the cost measure is the number of bits of communication from the first player to the second player for any integer d 3 and positive real e we show that if satisfiability for n variable d cnf formulas has a protocol of cost o nd e then conp is in np poly which implies that the polynomial time hierarchy collapses to its third level the result even holds when the first player is conondeterministic and is tight as there exists a trivial protocol for e 0 under the hypothesis that conp is not in np poly our result implies tight lower bounds for parameters of interest in several areas namely sparsification kernelization in parameterized complexity lossy compression and probabilistically checkable proofs by reduction similar results hold for other np complete problems for the vertex cover problem on n vertex d uniform hypergraphs this statement holds for any integer d 2 the case d 2 implies that no np hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of o k2 e edges unless conp is in np poly where k denotes the size of the deletion set kernels consisting of o k2 edges are known for several problems in the class including vertex cover feedback vertex set and bounded degree deletion

kolmogorov complexity and instance complexity of recursively enumerable sets the way in which way kolmogorov complexity and instance complexity affect properties of recursively enumerable r e sets is studied the well known 2 log n upper bound on the kolmogorov complexity of initial segments of r e sets is shown to be optimal and the turing degrees of r e sets which attain this bound are characterized the main part of the paper is concerned with instance complexity introduced by ko orponen schoning and watanabe in 1986 as a measure of the complexity of individual instances of a decision problem they conjectured that for every r e nonrecursive set the instance complexity is infinitely often at least as high as the kolmogorov complexity the conjecture is refuted by constructing an r e nonrecursive set with instance complexity logarithmic in the kolmogorov complexity this bound is optimal up to an additive constant in the other extreme the conjecture is established for many classes of complete sets such as weak truth table complete wtt complete and q complete sets however there is a turing complete set for which it fails

the minimum consistent dfa problem cannot be approximated within any polynomial the minimum consistent dfa problem is that of finding a dfa with as few states as possible that is consistent with a given sample a finite collection of words each labeled as to whether the dfa found should accept or reject assuming that p np it is shown that for any constant k no polynomial time algorithm can be guaranteed to find a consistent dfa with fewer than opt k states where opt is the number of states in the minimum state dfa consistent with the sample this result holds even if the alphabet is of constant size two and if the algorithm is allowed to produce an nfa a regular expression or a regular grammar that is consistent with the sample a similar nonapproximability result is presented for the problem of finding small consistent linear grammars for the case of finding minimum consistent dfas when the alphabet is not of constant size but instead is allowed to vary with the problem specification the slightly stronger lower bound on approximability of opt 1 e log log opt is shown for any e 0

the rigged hilbert space approach to the lippmann schwinger equation ii the analytic continuation of the lippmann schwinger bras and kets the analytic continuation of the lippmann schwinger bras and kets is obtained and characterized it is shown that the natural mathematical setting for the analytic continuation of the solutions of the lippmann schwinger equation is the rigged hilbert space rather than just the hilbert space it is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution resulting in a semigroup time evolution physically the semigroup time evolution is simply a retarded or advanced propagator

natural halting probabilities partial randomness and zeta functions we introduce the zeta number natural halting probability and natural complexity of a turing machine and we relate them to chaitin s omega number halting probability and program size complexity a classification of turing machines according to their zeta numbers is proposed divergent convergent and tuatara we prove the existence of universal convergent and tuatara machines various results on algorithmic randomness and partial randomness are proved for example we show that the zeta number of a universal tuatara machine is c e and random a new type of partial randomness asymptotic randomness is introduced finally we show that in contrast to classical algorithmic randomness which cannot be naturally characterised in terms of plain complexity asymptotic randomness admits such a characterisation

recursive computational depth in the 1980 s bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object s structure in particular bennett identified the classes of weakly deep and strongly deep sequences and showed that the halting problem is strongly deep juedes lathrop and lutz subsequently extended this result by defining the class of weakly useful sequences and proving that every weakly useful sequence is strongly deep

dimension 1 sequences are close to randoms abstract we show that a sequence has effective hausdorff dimension 1 if and only if it is coarsely similar to a martin lof random sequence more generally a sequence has effective dimension s if and only if it is coarsely similar to a weakly s random sequence further for any s t every sequence of effective dimension s can be changed on density at most h 1 t h 1 s of its bits to produce a sequence of effective dimension t and this bound is optimal

characterizing the strongly jump traceable sets via randomness abstract we show that if a set a is computable from every superlow 1 random set then a is strongly jump traceable together with a result of greenberg and nies noam greenberg andre nies benign cost functions and lowness properties j symbolic logic 76 1 2011 289 312 this theorem shows that the computably enumerable c e strongly jump traceable sets are exactly the c e sets computable from every superlow 1 random set we also prove the analogous result for superhighness a c e set is strongly jump traceable if and only if it is computable from every superhigh 1 random set finally we show that for each cost function c with the limit condition there is a 1 random 2 0 set y such that every c e set a t y obeys c to do so we connect cost function strength and the strength of randomness notions together with a theorem of greenberg and nies ibd this result gives a full correspondence between obedience of cost functions and being computable from 2 0 1 random sets

efficient testing of bipartite graphs for forbidden induced subgraphs alon et al n alon e fischer m krivelevich and m szegedy combinatorica 20 2000 pp 451 476 showed that every property that is characterized by a finite collection of forbidden induced subgraphs is epsilon testable however the complexity of the test is double tower with respect to 1 epsilon as the only tool known to construct such tests uses a variant of szemeredi s regularity lemma here we show that any property of bipartite graphs that is characterized by a finite collection of forbidden induced subgraphs is epsilon testable with a number of queries that is polynomial in 1 epsilon our main tool is a new conditional version of the regularity lemma for binary matrices which may be interesting on its own

local monotonicity reconstruction we investigate the problem of monotonicity reconstruction as defined by ailon et al 2004 in a localized setting we have oracle access to a nonnegative real valued function f defined on the domain n d 1 dots n d where d is viewed as a constant we would like to closely approximate f by a monotone function g this should be done by a procedure a filter that given as input a point x in n d outputs the value of g x and runs in time that is polylogarithmic in n the procedure can indeed must be randomized but we require that all of the randomness be specified in advance by a single short random seed we construct such an implementation where the time and space per query is log n o 1 and the size of the seed is polynomial in log n and d furthermore with high probability the ratio of the hamming distance between g and f to the minimum possible hamming distance between a monotone function and f is bounded above by a function of d independent of n this allows for a local implementation one can initialize many copies of the filter with the same short random seed and they can autonomously handle queries while producing outputs that are consistent with the same approximating function g

testing eulerianity and connectivity in directed sparse graphs property testing problems are relaxations of decision problems a property testing algorithm referred to as a testing algorithm or tester has to decide if a given object has a prespecified property or is far from the property for a given distance parameter and for a prespecified distance measure the tester is given query access to the input and is required to run in sublinear time in this paper we focus on testing properties of directed graphs digraphs in particular we present the following results where n is the number of vertices in the graph d is the maximum degree and davg is the average degree we present a testing algorithm for the property of eulerianity in bounded degree digraphs which runs in time11the notation o g k stands for o g k polylog g k o 1 for unbounded degree digraphs we show a lower bound of n and give a testing algorithm that runs in time o n 3 2 we study the property of k vertex connectivity and give testing algorithms for both bounded degree and unbounded degree digraphs that run in time o ck d kd and o ck davg k 1 respectively where c 1 is a constant in addition we give a simpler analysis of the testing algorithm for k vertex connectivity in bounded degree undirected graphs that was shown by yoshida and ito y yoshida h ito property testing on k vertex connectivity of graphs in icalp 08 proceedings of the 35th international colloquium on automata languages and programming part i springer verlag berlin heidelberg 2008 pp 539 550 and extend the result to unbounded degree undirected graphs we consider the property of k edge connectivity in digraphs and simplify the analysis of the algorithm of yoshida and ito y yoshida h ito testing k edge connectivity of digraphs journal of system science and complexity 23 1 2010 91 101 for this property in addition we give a simpler analysis for the correctness of the testing algorithm for k edge connectivity in undirected graphs that was introduced by goldreich and ron o goldreich d ron property testing in bounded degree graphs algorithmica 2002 302 343

reply to comment on on the inconsistency of the bohm gadella theory with quantum mechanics in this reply we show that when we apply standard distribution theory to the lippmann schwinger equation the resulting spaces of test functions would comply with the hardy axiom only if classic results of paley and wiener of gelfand and shilov and of the theory of ultradistributions were wrong also we point out several differences between the standard method of constructing rigged hilbert spaces in quantum mechanics and the method used in time asymmetric quantum theory

direct demonstration of the completeness of the eigenstates of the schrodinger equation with local and nonlocal potentials bearing a coulomb tail demonstrating the completeness of wave function solutions of the radial schrodinger equation is a very difficult task existing proofs relying on operator theory are often very abstract and far from intuitive comprehension however it is possible to obtain rigorous proofs amenable to physical insight if one restricts the considered class of schrodinger potentials one can mention in particular unbounded potentials yielding a purely discrete spectrum and short range potentials however those possessing a coulomb tail very important for physical applications have remained problematic due to their long range character the method proposed in this paper allows to treat them correctly provided that the non coulomb part of potentials vanishes after a finite radius nonlocality of potentials can also be handled the main idea in the proposed demonstration is that regular solutions behave like sine cosine functions for large momenta so that their expansions verify fourier transform properties the high

the rigged hilbert space of the algebra of the one dimensional rectangular barrier potential the rigged hilbert space of the algebra of the one dimensional rectangular barrier potential is constructed the one dimensional rectangular potential provides another opportunity to show that the rigged hilbert space fully accounts for dirac s bra ket formalism the analogy between dirac s formalism and fourier methods is pointed out

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

resonance poles and gamow vectors in the rigged hilbert space formulation of quantum mechanics after a summary of the rigged hilbert space formulation of quantum mechanics and a brief statement of its advantages over von neumann s formulation a mathematically correct definition of gamow s exponentially decaying vectors as generalized energy eigenvectors is suggested it is shown that exponentially decaying vectors are obtained from the s matrix poles in the lower half of the second sheet and exponentially growing vectors from the s matrix poles in the upper half of the second sheet decaying state vectors are defined as functionals over half of the space of physical states and growing state vectors are defined as functionals over the other half on functionals over these subspaces the dynamical group of time development splits into two semigroups one for t 0 and the other for t 0 the generalized basis system connected with the spectrum of the hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated

the complexity of colouring problems on dense graphs abstract we consider the problem of deciding whether or not a graph has a vertex k colouring restricted to the class of graphs in which each vertex has degree at least n where is a fixed positive constant and n is the number of vertices of the graph we show that there is a deterministic polynomial time algorithm for this problem when k 3 k 2 and that this is best possible we also consider the corresponding enumeration problems

the fractionary schrodinger equation green functions and ultradistributions abstract in this work we generalize previous results about the fractionary schrodinger equation within the formalism of the theory of tempered ultradistributions several examples of the use of this theory are given in particular we evaluate the green function for a free particle in the general case for an arbitrary order of the derivative index

small bias probability spaces efficient constructions and applications it is shown how to efficiently construct a small probability space on n binary random variables such that for every subset its parity is either zero or one with almost equal probability they are called epsilon biased random variables the number of random bits needed to generate the random variables is o log n log frac 1 epsilon thus if epsilon is polynomially small then the size of the sample space is also polynomial random variables that are epsilon biased can be used to construct almost k wise independent random variables where epsilon is a function of k these probability spaces have various applications l derandomization of algorithms many randomized algorithms that require only k wise independence of their random bits where k is bounded by o log n can be derandomized by using epsilon biased random variables 2 reducing the number of random bits required by certain randomized algorithms e g verification of matrix multiplication 3 exhaustive tes

approximate hypergraph partitioning and applications szemeredi s regularity lemma is a cornerstone result in extremal combinatorics it roughly asserts that any dense graph is composed of a finite number of pseudorandom graphs the regularity lemma has found many applications in theoretical computer science and thus a lot of attention was given to designing algorithmic versions of this lemma our main results in this paper are the following i we introduce a new approach to the problem of constructing regular partitions of graphs which results in a surprisingly simple o n time algorithmic version of the regularity lemma thus improving over the previous o n 2 time algorithms furthermore unlike all the previous approaches for this problem see n alon and a naor siam j comput 35 2006 pp 787 803 r a duke h lefmann and v rodl siam j comput 24 1995 pp 598 620 a frieze and r kannan electron j combin 6 1999 article 17 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 and y kohayakawa v rodl and l thoma siam j comput 32 2003 pp 1210 1235 which only guaranteed to find tower size partitions our algorithm will find a small regular partition if one exists in the graph ii for any constant r geq3 we give an o n time randomized algorithm for constructing regular partitions of r uniform hypergraphs thus improving the previous o n 2r 1 time deterministic algorithms a czygrinow and v rodl siam j comput 30 2000 pp 1041 1066 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 these two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here given a directed hypergraph with bounded edge arities a set of constraints on the set sizes and densities of a possible partition of its vertex set and an approximation parameter we provide in o n time a partition approximating the constraints if a partition satisfying them exists we can also test in o 1 time for the existence of such a partition given the approximation parameter this algorithm extends the result of goldreich goldwasser and ron for graph partition problems o goldreich s goldwasser and d ron j acm 45 1998 pp 653 750 and encompasses more recent hypergraph related results such as the maximal constraint satisfaction approximation of g andersson and l engebretsen random structures algorithms 21 2002 pp 14 32

feasible reductions to kolmogorov loveland stochastic sequences for every binary sequence a there is an infinite binary sequence s such that a ttp s and s is stochastic in the sense of kolmogorov and loveland

on the inconsistency of the bohm gadella theory with quantum mechanics the bohm gadella theory sometimes referred to as the time asymmetric quantum theory of scattering and decay is based on the hardy axiom the hardy axiom asserts that the solutions of the lippmann schwinger equation are functionals over spaces of hardy functions the preparation registration arrow of time provides the physical justification for the hardy axiom in this paper it is shown that the hardy axiom is incorrect because the solutions of the lippmann schwinger equation do not act on spaces of hardy functions it is also shown that the derivation of the preparation registration arrow of time is flawed thus hardy functions neither appear when we solve the lippmann schwinger equation nor should they appear it is also shown that the bohm gadella theory does not rest on the same physical principles as quantum mechanics and that it does not solve any problem that quantum mechanics cannot solve the bohm gadella theory must therefore be abandoned

a description of virtual scattering states in the rigged hilbert space formulation of quantum mechanics we define a pair of gel fand triplets i hi i i 1 2 where the linear spaces i are formed by suitably chosen entire analytic functions on the riemann surface associated to the transformation z2 virtual scattering states are well defined as pairs of functionals respectively in 1 and 2

on the autoreducibility of random sequences a binary sequence a a 0 a 1 ldots is called infinitely often i o turing au to re duc ible if a is reducible to itself via an oracle turing machine that never queries its oracle at the current input outputs either a x or a don t know symbol on any given input x and outputs a x for infinitely many x if in addition the oracle turing machine terminates on all inputs and oracles a is called i o truth table autoreducible we obtain the somewhat counterintuitive result that every martin l of random sequence in fact even every rec random or p random sequence is i o truth table autoreducible furthermore we investigate the question of how dense the set of guessed bits can be when i o autoreducing a random sequence we show that rec random sequences are never i o truth table autoreducible such that the set of guessed bits has positive constant density in the limit and that a similar assertion holds for martin l of random sequences and i o turing autoreducibility on the other hand we show that for any rational valued computable function r that goes nonascendingly to zero any rec random sequence is i o truth table autoreducible such that on any prefix of length m at least a fraction of r m of the m bits in the prefix are guessed we include a self contained account of the hat problem a puzzle that has received some attention outside of theoretical computer science the hat problem asks for guessing bits of a finite sequence thus illustrating the notion of i o autoreducibility in a finite setting the solution to the hat problem is then used as a module in the proofs of the positive results on i o autoreducibility

constant round interactive proofs for delegating computation the celebrated sf ip sf pspace theorem lund fortnow karloff and nisan j acm 39 1992 pp 859 868 shamir j acm 39 1992 pp 869 877 allows an all powerful but untrusted pro

compression of data streams down to their information content according to the kolmogorov complexity every finite binary string is compressible to a shortest code its information content from which it is effectively recoverable we investigate the extent to which this holds for the infinite binary sequences streams we devise a new coding method that uniformly codes every stream x into an algorithmically random stream y in such a way that the first n bits of x are recoverable from the first i x upharpoonright n bits of y where i is any partial computable information content measure that is defined on all prefixes of x and where x upharpoonright n is the initial segment of x of length n as a consequence if g is any computable upper bound on the initial segment prefix free complexity of x then x is computable from an algorithmically random y with oracle use at most g alternatively making no use of such a computable bound g one can achieve an the oracle use bounded above by k x upharpoonright n log n this provides a strong analogue of shannon s source coding theorem for the algorithmic information theory

things that can be made into themselves one says that a property p of sets of natural numbers can be made into itself iff there is a numbering a 0 a 1 of all left r e sets such that the index set e a e satisfies p has the property p as well for example the property of being martin lof random can be made into itself herein we characterize those singleton properties which can be made into themselves a second direction of the present work is the investigation of the structure of left r e sets under inclusion modulo a finite set in contrast to the corresponding structure for r e sets which has only maximal but no minimal members both minimal and maximal left r e sets exist moreover our construction of minimal and maximal left r e sets greatly differs from friedberg s classical construction of maximal r e sets finally we investigate whether the properties of minimal and maximal left r e sets can be made into themselves

forbidden information godel incompleteness theorem leaves open a way around it vaguely perceived for a long time but not clearly identified thus godel believed informal arguments can answer any math question closing this loophole does not seem obvious and involves kolmogorov complexity this is unrelated to well studied before complexity quantifications of the usual godel effects i consider extensions u of the universal partial recursive predicate or say peano arithmetic i prove that any u either leaves an n bit input statement unresolved or contains nearly all information about the n bit prefix of any r e real which is n bits for some i argue that creating significant information about a specific math sequence is impossible regardless of the methods used similar problems and answers apply to other unsolvability results for tasks allowing multiple solutions for example nonrecursive tilings

recursively enumerable reals and chaitin ω numbers a real is called recursively enumerable if it is the limit of a recursive increasing converging sequence of rationals following solovay unpublished manuscript ibm thomas j watson research center yorktown heights new york may 1975 215 pp and chaitin ibm j res develop 21 1977 350 359 496 we say that an r e real dominates an r e real if from a good approximation of from below one can compute a good approximation of from below we shall study this relation and characterize it in terms of relations between r e sets solovay s unpublished manuscript ibm thomas j watson research center yorktown heights new york may 1975 215 pp like numbers are the maximal r e real numbers with respect to this order they are random r e real numbers the halting probability of a universal self delimiting turing machine chaitin s number j assoc comput mach 22 1975 329 340 is also a random r e real solovay showed that any chaitin number is like in this paper we show that the converse implication is true as well any like real in the unit interval is the halting probability of a universal self delimiting turing machine

computational depth and reducibility this paper reviews and investigates bennett s notions of strong and weak computational depth also called logical depth for infinite binary sequences roughly an infinite binary sequence x is defined to be weakly useful if every element of nonnegligible set of decidable sequences is reducible to x in recursively bounded time it is shown that every weakly useful sequence is strongly deep this result which generalizes bennett s observation that the halting problem is strongly deep implies that every high turing degree contains strongly deep sequences it is also shown that in the sense of baire category almost every infinite binary sequence is weakly deep but not strongly deep

lowness for bounded randomness in 3 brodhead downey and ng introduced some new variations of the notions of being martin lof random where the tests are all clopen sets we explore the lowness notions associated with these randomness notions while these bounded notions seem far from classical notions with infinite tests like martin lof and demuth randomness the lowness notions associated with bounded randomness turn out to be intertwined with the lowness notions for these two concepts in fact in one case we get a new and likely very useful characterization of k triviality

approximate graph coloring by semidefinite programming we consider the problem of coloring k colorable graphs with the fewest possible colors we present a randomized polynomial time algorithm that colors a 3 colorable graph on n vertices with min o 1 3 log 1 2 log n o n 1 4 log 1 2 n colors where is the maximum degree of any vertex besides giving the best known approximation ratio in terms of n this marks the first nontrivial approximation result as a function of the maximum degree this result can be generalized to k colorable graphs to obtain a coloring using min o 1 2 k log 1 2 log n o n 1 3 k 1 log 1 2 n colors our results are inspired by the recent work of goemans and williamson who used an algorithm for semidefinite optimization problems which generalize linear programs to obtain improved approximations for the max cut and max 2 sat problems an intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the lovasz t function we show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number by duality this also demonstrates interesting new facts about the t function

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

a rigged hilbert space of hardy class functions applications to resonances the explicit construction of a dense subspace of square integrable functions on the positive half of the real line is given this space has the properties that 1 it is endowed with a metrizable nuclear topology 2 it is stable under multiplication by x and 3 the functions in have suitable analytical continuation to a half plane the space of functions which are conjugate to elements of is also considered then the triplets l2 0 and l2 0 are used to give a description of resonances

one channel time reversal in chaotic cavities theoretical limits one channel time reversal experiments in closed chaotic cavities produce excellent but not perfect time reversed focusing this paper investigates such experiments by a simple eigenmode analysis of the system it shows that the process is even for long reversed signals subject to an information loss during recording and re emission which prevents perfect time reversal this fact can be expressed by a simple equation called the cavity equation which states that the signal of a one channel time reversal is equal to the signal of a perfect time reversal after convolution with the backscattering impulse response of the reversal point i e from this point to this point the latter convolution describes the introduction of the loss of information furthermore arguments are presented suggesting that one third of the total energy of the time reversed wave field is actually contained in the refocusing wavefront the predictions are verified by numerical finite difference time domain simulations

locally testable codes and pcps of almost linear length we initiate a systematic study of locally testable codes that is error correcting codes that admit very efficient membership tests specifically these are codes accompanied with tests that make a constant number of random queries into any given word and reject non codewords with probability proportional to their distance from the code locally testable codes are believed to be the combinatorial core of pcps however the relation is less immediate than commonly believed nevertheless we show that certain pcp systems can be modified to yield locally testable codes on the other hand we adapt techniques that we develop for the construction of the latter to yield new pcps our main results are locally testable codes and pcps of almost linear length specifically we prove the existence of the following constructs locally testable binary linear codes in which k information bits are encoded by a codeword of length k exp o log k this improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet pcp systems of almost linear length for sat the length of the proof is n exp o log n and verification in performed by a constant number i e 19 of queries as opposed to previous results that used proof length n 1 p o 1 q for verification by q queries the novel techniques in use include a random projection of certain codewords and pcp oracles that preserves local testability an adaptation of pcp constructions to obtain linear pcp oracles for proving conjunctions of linear conditions and design of pcps with some new soundness properties a direct construction of locally testable linear codes of subexponential length

a polynomial regularity lemma for semialgebraic hypergraphs and its applications in geometry and property testing in this paper we prove several extremal results for geometrically defined hypergraphs in particular we establish an improved lower bound single exponentially decreasing in k on the best constant delta 0 such that the vertex classes p 1 ldots p k of every k partite k uniform semialgebraic hypergraph h p 1 cup cdots cup p k e with e ge varepsilon pi j 1 k p i have for 1 leq i leq k delta p i element subsets p i subseteq p i satisfying p 1 times cdots times p k subseteq e the best previously known lower bound on delta due to bukh and hubard decreased triple exponentially fast in k we give three geometric applications of our results in particular we establish the following strengthening of the so called same type lemma of barany and valtr any disjoint finite sets p 1 ldots p k subset mathbb r d k d have for 1 leq i leq k subsets p i of size at least 2 o d 3k log k p i with the property that every k tuple formed by taking on

comment on on the inconsistency of the bohm gadella theory with quantum mechanics rigged hilbert spaces of hardy functions lead to a consistent theory of resonance scattering and decay contrary to the claims of a recent article 8 the theory holds for a wide range of potentials and rigorously describes the asymmetric time evolution of resonances

computing halting probabilities from other halting probabilities the halting probability of a turing machine is the probability that the machine will halt if it starts with a random stream written on its one way input tape when the machine is universal this probability is referred to as chaitin s omega number and is the most well known example of a real which is random in the sense of martin lof although omega numbers depend on the underlying universal turing machine they are robust in the sense that they all have the same turing degree namely the degree of the halting problem this means that given two universal prefix free machines u v the halting probability u of u computes the halting probability v of v if this computation uses at most the first n g n bits of u for the computation of the first n bits of v we say that u computes v with redundancy g in this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number we show that for each 1 any pair of omega numbers compute each other with redundancy log n on the other hand this is not true for 1 in fact we show that for each omega number u there exists another omega number which is not computable from u with redundancy log n this latter result improves an older result of frank stephan

completeness in quantum mechanics and the weyl titchmarsh kodaira theorem we discuss the completeness of generalized eigenfunctions in quantum mechanics using the classical theory developed by weyl titchmarsh and kodaira as applications we rigorously prove the completeness of generalized eigenfunctions for the step and well potentials

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

universal recursively enumerable sets of strings the main topics of the present work are universal machines for plain and prefix free description complexity and their domains it is characterised when an r e set w is the domain of a universal plain machine in terms of the description complexity of the spectrum function s w mapping each non negative integer n to the number of all strings of length n in w furthermore a characterisation of the same style is given for supersets of domains of universal plain machines similarly the prefix free sets which are domains or supersets of domains of universal prefix free machines are characterised furthermore it is shown that the halting probability w v of an r e prefix free set v containing the domain of a universal prefix free machine is martin lof random while v may not be the domain of any universal prefix free machine itself based on these investigations the question whether every domain of a universal plain machine is the superset of the domain of some universal prefix free machine is discussed a negative answer to this question had been presented at cie 2010 by mikhail andreev ilya razenshteyn and alexander shen while this paper was under review

working with strong reducibilities above totally omega c e and array computable degrees we investigate the connections between the complexity of a c e set as calibrated by its strength as an oracle for turing computations of functions in the ershov hierarchy and how strong reducibilities allow us to compute such sets for example we prove that a c e degree is totally w c e iff every set in it is weak truth table reducible to a hypersimple or ranked set we also show that a c e degree is array computable iff every left c e real of that degree is reducible in a computable lipschitz way to a random left c e real an number

the rigged hilbert space approach to the gamow states we use the resonances of the spherical shell potential to present a thorough description of the gamow quasinormal states within the rigged hilbert space it will be concluded that the natural setting for the gamow states is a rigged hilbert space whose test functions fall off at infinity faster than gaussians

on computing the trace of the kernel of the homogeneous fredholm s equation a method for computing the trace of the kernel of the homogeneous fredholm s equation for resonant states arising from nonlocal potentials is proposed we show that this integral formulation is convergent

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

every sequence is reducible to a random one every infinite sequence is turing reducible to an infinite sequence which is random in the sense of martin lof

mathematical metaphysics of randomness abstract we consider various mathematical refinements of the notion of randomness of an infinite sequence and relations between them on the base of the game approach a new possible refinement of the notion of randomness is introduced unpredictability the case of finite sequences is also considered

the lippmann schwinger equations in the rigged hilbert space we give a rigorous meaning to the lippmann schwinger equations relating generalized eigenvectors of the free and perturbed hamiltonians in a scattering process some related formulae are also presented a discussion is made on the context of rigged hilbert space formulation of quantum mechanics

the rigged hilbert space approach to the lippmann schwinger equation part i we exemplify the way the rigged hilbert space deals with the lippmann schwinger equation by way of the spherical shell potential we explicitly construct the lippmann schwinger bras and kets along with their energy representation their time evolution and the rigged hilbert spaces to which they belong it will be concluded that the natural setting for the solutions of the lippmann schwinger equation and therefore for scattering theory is the rigged hilbert space rather than just the hilbert space

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given