scalar equations for synchronous boolean networks with biological applications one way of coping with the complexity of biological systems is to use the simplest possible models which are able to reproduce at least some nontrivial features of reality although two value boolean models have a long history in technology it is perhaps a little bit surprising that they can also represent important features of living organizms in this paper the scalar equation approach to boolean network models is further developed and then applied to two interesting biological models in particular a linear reduced scalar equation is derived from a more rudimentary nonlinear scalar equation this simpler but higher order two term equation gives immediate information about both cycle and transient structure of the network

constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

selberg supertrace formula for super riemann surfaces iii bordered super riemann surfaces this paper is the third in a sequel to develop a super analogue of the classical selberg trace formula the selberg supertrace formula it deals with bordered super riemann surfaces the theory of bordered super riemann surfaces is outlined and the corresponding selberg supertrace formula is developed the analytic properties of the selberg super zeta functions on bordered super riemann surfaces are discussed and super determinants of dirac laplace operators on bordered super riemann surfaces are calculated in terms of selberg super zeta functions

dynamical properties of s gap shifts and other shift spaces abstract we study the dynamical properties of certain shift spaces to help study these properties we introduce two new classes of shifts namely boundedly supermultiplicative bsm shifts and balanced shifts it turns out that any almost specified shift is both bsm and balanced and any balanced shift is bsm however as we will demonstrate there are examples of shifts which are bsm but not balanced we also study the measure theoretic properties of balanced shifts we show that a shift space admits a gibbs state if and only if it is balanced restricting ourselves to s gap shifts we relate certain dynamical properties of an s gap shift to combinatorial properties from expansions in non integer bases this identification allows us to use the machinery from expansions in non integer bases to give straightforward constructions of s gap shifts with certain desirable properties we show that for any q 0 1 there is an s gap shift which has the specification property and entropy q we also use this identification to address the question for a given q 0 1 how many s gap shifts exist with entropy q for certain exceptional values of q there is a unique s gap shift with this entropy

on the computational complexity of ising spin glass models in a spin glass with ising spins the problems of computing the magnetic partition function and finding a ground state are studied in a finite two dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice in contrast to this fact the same problems are shown to belong to the class of np hard problems both in the two dimensional case within a magnetic field and in the three dimensional case np hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it

greedy expansions and sets with deleted digits we generalize a result of daroczy and katai on the characterization of univoque numbers with respect to a non integer base publ math debrecen 46 3 4 1995 385 by relaxing the digit alphabet to a generic set of real numbers we apply the result to derive the construction of a buchi automaton accepting all and only the greedy sequences for a given base and digit set in the appendix we prove a more general version of the fact that the expansion of an element x q q is ultimately periodic if q is a pisot number

superconformally covariant operators and super w algebras superdifferential operators of order 2n 1 which are covariant with respect to superconformal changes of coordinates on a compact super riemann surface are studied herein it is shown that all such operators arise from super mobius covariant ones a canonical matrix representation is presented and applications to classical super w algebras are discussed

hausdorff dimension of univoque sets and devil s staircase abstract we fix a positive integer m and we consider expansions in arbitrary real bases q 1 over the alphabet 0 1 m we denote by u q the set of real numbers having a unique expansion completing many former investigations we give a formula for the hausdorff dimension d q of u q for each q 1 furthermore we prove that the dimension function d 1 0 1 is continuous and has bounded variation moreover it has a devil s staircase behavior in q where q denotes the komornik loreti constant although d q d q for all q q we have d 0 a e in q during the proofs we improve and generalize a theorem of erd s et al on the existence of large blocks of zeros in expansions and we determine for all m the lebesgue measure and the hausdorff dimension of the set u of bases in which x 1 has a unique expansion

on computation with pulses we explore the computational power of formal models for computation with pulses such models are motivated by realistic models for biological neurons and by related new types of vlsi pulse stream vlsi in preceding work it was shown that the computational power of formal models for computation with pulses is quite high if the pulses arriving at a computational unit have an approximately linearly rising or linearly decreasing initial segment this property is satisfied by common models for biological neurons on the other hand several implementations of pulse stream vlsi employ pulses that are approximately piecewise constant i e step functions in this article we investigate the relevance of the shape of pulses in formal models for computation with pulses the results show that the computational power drops significantly if one replaces pulses with linearly rising or decreasing initial segments by piecewise constant pulses we provide an exact characterization of the latter model in terms of a weak version of a random access machine ram we also compare the language recognition capability of a recurrent version of this model with that of deterministic finite automata and turing machines

characterizations of fuzzy finite automata in this paper we establish some important concepts in fuzzy finite automata ffas with bifuzzy property and clarify their essential relationships first we present a number of basic definitions and properties in ffas we then define a bifuzzy family of subautomata bifuzzy source and successor operators as well as demonstrate the bifuzzy topological characterization of ffas also we clarify the relationships between our results and previous ones and thus derive a relation figure visualizing their connections afterwards we investigate further the bifuzzy separability retrievability and homomorphism fuzzy continuous mapping and open mapping are also incorporated in particular we prove a number of equivalent characterizations for these concepts and expound the relationships amongst them this also concludes that both the bifuzzy source and successor operators are fuzzy closure operators we discover that our investigation generalizes to some extent the algebraic fuzzy automata elaborated by malik mordeson and others finally the main results obtained are summarized the potential applications are indicated and a number of related problems for further study are addressed

selberg super trace formula for super riemann surfaces ii elliptic and parabolic conjugacy classes and selberg super zeta functions further contributions developing a super analogue of the classical selberg trace formula the selberg super trace formula are presented this paper deals with the calculation of contributions arising from elliptic and parabolic conjugacy classes to the selberg super trace formula for super riemann surfaces analytic properties and the functional equation for the corresponding selberg super zeta functionr0 r1 andzs respectively are derived and discussed in particular the elliptic contributions to a super fuchsian group only alter the multiplicities of the trivial zeros and poles of the selberg super zeta functionr0 r1 andzs respectively already due to the hyperbolic conjugacy classes the parabolic conjugacy classes introduce new features in the analytical structure

theory of neuromata a finite automaton the so called neuromaton realized by a finite discrete recurrent neural network working in parallel computation mode is considered both the size of neuromata i e the number of neurons and their descriptional complexity i e the number of bits in the neuromaton representation are studied it is proved that a constraint time delay of the neuromaton output does not play a role within a polynomial descriptional complexity it is shown that any regular language given by a regular expression of length n is recognized by a neuromaton with n neurons further it is proved that this network size is in the worst case optimal on the other hand generally there is not an equivalent polynomial length regular expression for a given neuromaton then two specialized constructions of neural acceptors of the optimal descriptional complexity n for a single n bit string recognition are described they both require o n1 2 neurons and either o n connections with constant weights or o n1 2 edges with weights of the o 2 n size furthermore the concept of hopfield languages is introduced by means of so called hopfield neuromata i e of neural networks with symmetric weights it is proved that the class of hopfield languages is strictly contained in the class of regular languages the necessary and sufficient so called hopfield condition stating when a regular language is a hopfield language is formulated a construction of a hopfield neuromaton is presented for a regular language satisfying the hopfield condition the class of hopfield languages is shown to be closed under union intersection concatenation and complement and it is not closed under iteration finally the problem whether a regular language given by a neuromaton or by a hopfield acceptor is nonempty is proved to be pspace complete as a consequence the same result for a neuromaton equivalence problem is achieved

a family of universal recurrent networks abstract hava siegelmann and eduardo sontag have shown that recurrent neural networks using the linear bounded sigmoid are computationally universal we show that this remains true if the linear bounded sigmoid is replaced by any function in a fairly large class

super riemann surfaces uniformization and teichmuller theory teichmuller theory for super riemann surfaces is rigorously developed using the supermanifold theory of rogers in the case of trivial topology in the soul directions relevant for superstring applications the following results are proven the super teichmuller space is a complex super orbifold whose body is the ordinary teichmuller space of the associated riemann surfaces with spin structure for genusg 1 it has 3g 3 complex even and 2g 2 complex odd dimensions the super modular group which reduces super teichmuller space to super moduli space is the ordinary modular group there are no new discrete modular transformations in the odd directions the boundary of super teichmuller space contains not only super riemann surfaces with pinched bodies but rogers supermanifolds having nontrivial topology in the odd dimensions as well we also prove the uniformization theorem for super riemann surfaces and discuss their representation by discrete supergroups of fuchsian and schottky type and by beltrami differentials finally we present partial results for the more difficult problem of classifying super riemann surfaces of arbitrary topology

moyennes de certaines fonctions multiplicatives sur les entiers friables 2 nous evaluons les moyennes indiquees dans le titre sous des hypotheses analytiques concernant la serie de dirichlet associee

graded riemann surfaces there has been a lot of activity directed at describing super riemann surfaces and the super teichmuller spaces that classify them most descriptions use a subcategory ofg supermanifolds in which the coordinate charts have a particularly simple form de witt supermanifolds this paper considers the more restrictive case of riemann surfaces in the category of graded manifolds the gain in doing this is the evident role of the double coversl 2 c of the lorentz group in the classification of graded riemann surfaces

general purpose computation with neural networks a survey of complexity theoretic results we survey and summarize the literature on the computational aspects of neural network models by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics the criteria of classification include the architecture of the network feedforward versus recurrent time model discrete versus continuous state type binary versus analog weight constraints symmetric versus asymmetric network size finite nets versus infinite families and computation type deterministic versus probabilistic among others the underlying results concerning the computational power and complexity issues of perceptron radial basis function winner take all and spiking neural networks are briefly surveyed with pointers to the relevant literature in our survey we focus mainly on the digital computation whose inputs and outputs are binary in nature although their values are quite often encoded as analog neuron states we omit the important learning issues

simple local search problems that are hard to solve many algorithms for np hard optimization problems find solutions that are locally optimal in the sense that the solutions cannot be improved by a polynomially computable perturbation very little is known about the complexity of finding locally optimal solutions either by local search algorithms or using other indirect methods johnson papadimitriou and yannakakis j comput system sci 37 1988 pp 79 100 studied this question by defining a complexity class pls that captures local search problems it was proved that finding a partition of a graph that is locally optimal into equal parts with respect to the acclaimed kernighan lin algorithm is pls complete it is shown here that several natural simple local search problems are pls complete and thus just as hard two examples are finding a partition that cannot be improved by a single swap of two vertices and finding a stable configuration for an undirected connectionist network when edges or other objects are unweighted then a local optimum

simulating threshold circuits by majority circuits we prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial size depth 2 majority circuit in general we show that a polynomial size depth d threshold circuit can be simulated uniformly by a polynomial size majority circuit of depth d 1 goldmann hastad and razborov showed in comput complexity 2 1992 pp 277 300 that a nonuniform simulation exists our construction answers two open questions posed by them we give an explicit construction whereas they use a randomized existence argument and we show that such a simulation is possible even if the depth d grows with the number of variables n their simulation gives polynomial size circuits only when d is constant

an algebraic framework to represent finite state machines in single layer recurrent neural networks in this paper we present an algebraic framework to represent finite state machines fsms in single layer recurrent neural networks slrnns which unifies and generalizes some of the previous proposals this framework is based on the formulation of both the state transition function and the output function of an fsm as a linear system of equations and it permits an analytical explanation of the representational capabilities of first order and higher order slrnns the framework can be used to insert symbolic knowledge in rnns prior to learning from examples and to keep this knowledge while training the network this approach is valid for a wide range of activation functions whenever some stability conditions are met the framework has already been used in practice in a hybrid method for grammatical inference reported elsewhere sanfeliu and alquezar 1994

moyennes de certaines fonctions multiplicatives sur les entiers friables 4 we provide two generic applications of previously established estimates for friable mean values of non negative multiplicative functions

bounds for the computational power and learning complexity of analog neural nets it is shown that high order feedforward neural nets of constant depth with piecewise polynomial activation functions and arbitrary real weights can be simulated for boolean inputs and outputs by neural nets of a somewhat larger size and depth with heaviside gates and weights from 1 0 1 this provides the first known upper bound for the computational power of the former type of neural nets it is also shown that in the case of first order nets with piecewise linear activation functions one can replace arbitrary real weights by rational numbers with polynomially many bits without changing the boolean function that is computed by the neural net in order to prove these results we introduce two new methods for reducing nonlinear problems about weights in multilayer neural nets to linear problems for a transformed set of parameters these transformed parameters can be interpreted as weights in a somewhat larger neural net as another application of our new proof technique we show that neural nets with piecewise polynomial activation functions and a constant number of analog inputs are probably approximately correct pac learnable in valiant s model for pac learning comm assoc comput mach 27 1984 pp 1134 1142

finite orbits in multivalued maps and bernoulli convolutions abstract bernoulli convolutions are certain measures on the unit interval depending on a parameter between 1 and 2 in spite of their simple definition they are not yet well understood we study their two dimensional density which exists by a theorem of solomyak to each bernoulli convolution there is an interval d called the overlap region and a map which assigns two values to each point of d and one value to all other points of 0 1 there are two types of finite orbits of these multivalued maps which correspond to zeros and potential singularities of the density respectively orbits which do not meet d belong to an ordinary map called transformation and exist for all 1 6182 they were studied by erdos joo komornik sidorov de vries and others as points with unique addresses and by jordan shmerkin and solomyak as points with maximal local dimension in the two dimensional view these orbits form address curves related to the milnor thurston itineraries in one dimensional dynamics the curves depend smoothly on the parameter and represent quantiles of all corresponding bernoulli convolutions finite orbits which intersect d have a network like structure and can exist only at perron parameters their points are intersections of extended address curves and can have finite or countable number of addresses as found by sidorov for an uncountable number of parameters the central point 1 2 has only two addresses the intersection of periodic address curves can lead to singularities of the measures we give examples which are not pisot or salem parameters

on the continuity of the hausdorff dimension of the univoque set abstract in a recent paper komornik et al 2017 19 proved a long conjectured formula for the hausdorff dimension of the set u q of numbers having a unique expansion in the non integer base q and showed that this hausdorff dimension is continuous in q unfortunately their proof contained a gap which appears difficult to fix this article gives a completely different proof of these results using a more direct combinatorial approach

depth size tradeoffs for neural computation the tradeoffs between the depth i e the time for parallel computation and the size i e the number of threshold gates in neural networks are studied the authors focus the study on the neural computations of symmetric boolean functions and some arithmetic functions it is shown that a significant reduction in the size is possible for symmetric functions and some arithmetic functions at the expense of a small constant increase in depth in the process several neural networks which have the minimum size among all the known constructions have been developed results on implementing symmetric functions can be used to improve results about arbitrary boolean functions in particular it is shown that any boolean function can be computed in a depth 3 neural network with o 2 sup n sup 2 threshold gates it is also proven that at least omega 2 sup n sup 3 threshold gates are required

multiple expansions of real numbers with digits set 0 1 q for q 1 we consider expansions in base q with digits set left 0 1 q right let mathcal u q be the set of points which have a unique q expansion for k 2 3 ldots aleph 0 let mathcal b k be the set of bases q 1 for which there exists x having precisely k different q expansions and for q in mathcal b k let mathcal u q k be the set of all such x s which have exactly k different q expansions in this paper we show that begin aligned mathcal b aleph 0 2 infty quad text and quad mathcal b k q c infty quad text for any quad k ge 2 end aligned where q c approx 2 32472 is the appropriate root of x 3 3x 2 2x 1 0 moreover we show that for any integer k ge 2 and any q in mathcal b k the hausdorff dimensions of mathcal u q k and mathcal u q are the same i e begin aligned dim h mathcal u q k dim h mathcal u q quad text for any quad k ge 2 end aligned finally we conclude that the set of points having a continuum of q expansions has full hausdorff dimension

attraction radii in binary hopfield nets are hard to compute we prove that it is an np hard problem to determine the attraction radius of a stable vector in a binary hopfield memory network and even that the attraction radius is hard to approximate under synchronous updating the problems are already np hard for two step attraction radii direct one step attraction radii can be computed in polynomial time

positive neural networks in discrete time implement monotone regular behaviors we study the expressive power of positive neural networks the model uses positive connection weights and multiple input neurons different behaviors can be expressed by varying the connection weights we show that in discrete time and in the absence of noise the class of positive neural networks captures the so called monotone regular behaviors which are based on regular languages a finer picture emerges if one takes into account the delay by which a monotone regular behavior is implemented each monotone regular behavior can be implemented by a positive neural network with a delay of one time unit some monotone regular behaviors can be implemented with zero delay and interestingly some simple monotone regular behaviors cannot be implemented with zero delay

holomorphic coordinates for supermoduli space two dimensional supergravity provides complex coordinates for supermoduli space compatible with its natural complex structure such coordinates are useful for investigating questions of holomorphic factorization of superdeterminants they also demonstrate explicitly that the complex structure on supermoduli space is indeed integrable

fuzzy finite state automata can be deterministically encoded into recurrent neural networks there has been an increased interest in combining fuzzy systems with neural networks because fuzzy neural systems merge the advantages of both paradigms on the one hand parameters in fuzzy systems have clear physical meanings and rule based and linguistic information can be incorporated into adaptive fuzzy systems in a systematic way on the other hand there exist powerful algorithms for training various neural network models however most of the proposed combined architectures are only able to process static input output relationships they are not able to process temporal input sequences of arbitrary length fuzzy finite state automats ffas can model dynamical processes whose current state depends on the current input and previous states unlike in the case of deterministic finite state automats dfas ffas are not in one particular state rather each state is occupied to some degree defined by a membership function based on previous work on encoding dfas in discrete time second order recurrent neural networks we propose an algorithm that constructs an augmented recurrent neural network that encodes a ffa and recognizes a given fuzzy regular language with arbitrary accuracy we then empirically verify the encoding methodology by correct string recognition of randomly generated ffas in particular we examine how the networks performance varies as a function of synaptic weight strengths

entropy topological transitivity and dimensional properties of unique q expansions let m be a positive integer and q in 1 m 1 we consider expansions of real numbers in base q over the alphabet 0 ldots m in particular we study the set mathcal u q of real numbers with a unique q expansion and the set mathbf u q of corresponding sequences it was shown by komornik kong and li that the function h which associates to each q in 1 m 1 the topological entropy of mathcal u q is a devil s staircase in this paper we explicitly determine the plateaus of h and characterize the bifurcation set mathscr e of q s where the function h is not locally constant moreover we show that mathscr e is a cantor set of full hausdorff dimension we also investigate the topological transitivity of a naturally occurring subshift mathbf v q sigma which has a close connection with open dynamical systems finally we prove that the hausdorff dimension and box dimension of mathcal u q coincide for all q in 1 m 1

number representation using generalized β transformation we study non standard number systems with negative base b instead of the ito sadahiro definition based on the transformation t b of the interval b b 1 1 b 1 into itself we suggest a generalization using an interval l l 1 with l 1 0 such numeration systems share many properties of positive base numeration introduced by renyi although the proofs are not always straightforward in this paper we focus on the description of admissible digit strings and their periodicity we address the question of the description of reference strings used in the admissibility condition we give examples which contradict a result of gora and show that in this aspect the negative base numeration significantly differs from the renyi numeration

the geometry of super riemann surfaces we define super riemann surfaces as smooth 2 2 dimensional supermanifolds equipped with a reduction of their structure group to the group of invertible upper triangular 2 2 complex matrices the integrability conditions for such a reduction turn out to be most of the torsion constraints of 2d supergravity we show that they are both necessary and sufficient for a frame to admit local superconformal coordinates the other torsion constraints are merely conditions to fix some of the gauge freedom in this description or to specify a particular connection on such a manifold analogous to the levi civita connection in riemannian geometry unlike ordinary riemann surfaces a super riemann surface cannot be regarded as having only one complex dimension nevertheless in certain important aspects super riemann surfaces behave as nicely as if they had only one dimension in particular they posses an analog widehat partial of the cauchy riemann operator on ordinary riemann surfaces a differential operator taking values in the bundle of half volume forms this operator furnishes a short resolution of the structure sheaf making possible a quillen theory of determinant line bundles finally we show that the moduli space of super riemann surfaces is embedded in the larger space of complex curves of dimension 1 1

analog neural nets with gaussian or other common noise distributions cannot recognize arbitary regular languages we consider recurrent analog neural nets where the output of each gate is subject to gaussian noise or any other common noise distribution that is nonzero on a sufficiently large part of the state space we show that many regular languages cannot be recognized by networks of this type and we give a precise characterization of languages that can be recognized this result implies severe constraints on possibilities for constructing recurrent analog neural nets that are robust against realistic types of analog noise on the other hand we present a method for constructing feedforward analog neural nets that are robust with regard to analog noise of this type

efficient simulation of finite automata by neural nets let k m denote the smallest number with the property that every m state finite automaton can be built as a neural net using k m or fewer neurons a counting argument shows that k m is at least o m log m 1 3 and a construction shows that k m is at most o m 3 4 the counting argument and the construction allow neural nets with arbitrarily complex local structure and thus may require neurons that themselves amount to complicated networks mild and in practical situations almost necessary constraints on the local structure of the network give again by a counting argument and a construction lower and upper bounds for k m that are both linear in m

arithmetics in number systems with a negative base we study the numeration system with a negative base introduced by ito and sadahiro we focus on arithmetic operations in the sets fin b and z b of numbers having finite resp integer b expansions we show that fin b is trivial if b is smaller than the golden ratio 12 1 5 for b 12 1 5 we prove that fin b is a ring only if b is a pisot or salem number with no negative conjugates we prove the conjecture of ito and sadahiro that fin b is a ring if b is a quadratic pisot number with positive conjugate for quadratic pisot units we determine the number of fractional digits that may appear when adding or multiplying two b integers

stable encoding of large finite state automata in recurrent neural networks with sigmoid discriminants we propose an algorithm for encoding deterministic finite state automata dfas in second order recurrent neural networks with sigmoidal discriminant function and we prove that the languages accepted by the constructed network and the dfa are identical the desired finite state network dynamics is achieved by programming a small subset of all weights a worst case analysis reveals a relationship between the weight strength and the maximum allowed network size which guarantees finite state behavior of the constructed network we illustrate the method by encoding random dfas with 10 100 and 1000 states while the theory predicts that the weight strength scales with the dfa size we find empirically the weight strength to be almost constant for all the random dfas these results can be explained by noting that the generated dfas represent average cases we empirically demonstrate the existence of extreme dfas for which the weight strength scales with dfa size

beta expansions natural extensions and multiple tilings associated with pisot units from the works of rauzy and thurston we know how to construct multiple tilings of some euclidean space using the conjugates of a pisot unit beta and the greedy beta transformation in this paper we consider different transformations generating expansions in base beta including cases where the associated subshift is not sofic under certain mild conditions we show that they give multiple tilings we also give a necessary and sufficient condition for the tiling property generalizing the weak finiteness property w for greedy beta expansions remarkably the symmetric beta transformation does not satisfy this condition when beta is the smallest pisot number or the tribonacci number this means that the pisot conjecture on tilings cannot be extended to the symmetric beta transformation closely related to these multiple tilings are natural extensions of the transformations which have many nice properties they are invariant under the lebesgue measure under certain conditions they provide markov partitions of the torus they characterize the numbers with purely periodic expansion and they allow determining any digit in an expansion without knowing the other digits

the structure of logarithmic advice complexity classes a nonuniform class called here full p log due to ko is studied it corresponds to polynomial time with logarithmically long advice its importance lies in the structural properties it enjoys more interesting than those of the alternative class p log specifically its introduction was motivated by the need of a logarithmic advice class closed under polynomial time deterministic reductions several characterizations of full p log are shown formulated in terms of various sorts of tally sets with very small information content a study of its inner structure is presented by considering the most usual reducibilities and looking for the relationships among the corresponding reduction and equivalence classes defined from these special tally sets

the computational power of discrete hopfield nets with hidden units we prove that polynomial size discrete hopfield networks with hidden units compute exactly the class of boolean functions pspace poly i e the same functions as are computed by polynomial space bounded nonuniform turing machines as a corollary to the construction we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions p poly i e the class computed by polynomial time bounded nonuniform turing machines

on univoque and strongly univoque sets abstract much has been written about expansions of real numbers in noninteger bases particularly for a finite alphabet 0 1 and a real number base 1 1 the so called univoque set of numbers which have a unique expansion in base has garnered a great deal of attention in recent years motivated by recent applications of expansions to bernoulli convolutions and a certain class of self affine functions we introduce the notion of a strongly univoque set we study in detail the set d of numbers which are univoque but not strongly univoque our main result is that d is nonempty if and only if the number 1 has a unique nonterminating expansion in base and in that case d is uncountable we give a sufficient condition for d to have positive hausdorff dimension and show that on the other hand there are infinitely many values of for which d is uncountable but of hausdorff dimension zero

the convergence of symmetric threshold automata we study the length of the transient in a threshold automata with values on the set 0 1 the original conjecture was that in the symmetric case this length was a linear function of the number of cells in the automata goles e 1981 comportement oscillatoire d une famille d automates cellulaires non uniformes these imag grenoble france and olivos j 1979 contribution a l algorithmique these orsay france we shall show that this length can be exponential

some mathematical remarks on the polynomial selection in nfs in this work we consider the proportion of smooth free of large prime factors values of a binary form f x 1 x 2 in mathbf z x 1 x 2 in a particular case we give an asymptotic equivalent for this proportion which depends on f this is related to murphy s alpha function which is known in the cryptographic community but which has not been studied before from a mathematical point of view our result proves that when alpha f is small f has a high proportion of smooth values this has consequences on the first step called polynomial selection of the number field sieve the fastest algorithm of integer factorization

on the complexity of learning for spiking neurons with temporal coding abstract spiking neurons are models for the computational units in biological neural systems where information is considered to be encoded mainly in the temporal patterns of their activity in a network of spiking neurons a new set of parameters becomes relevant which has no counterpart in traditional neural network models the time that a pulse needs to travel through a connection between two neurons also known as delay of a connection it is known that these delays are tuned in biological neural systems through a variety of mechanisms in this article we consider the arguably most simple model for a spiking neuron which can also easily be implemented in pulsed vlsi we investigate the vapnik chervonenkis vc dimension of networks of spiking neurons where the delays are viewed as programmable parameters and we prove tight bounds for this vc dimension thus we get quantitative estimates for the diversity of functions that a network with fixed architecture can compute with different settings of its delays in particular it turns out that a network of spiking neurons with k adjustable delays is able to compute a much richer class of functions than a threshold circuit with k adjustable weights the results also yield bounds for the number of training examples that an algorithm needs for tuning the delays of a network of spiking neurons results about the computational complexity of such algorithms are also given

on the computational complexity of binary and analog symmetric hopfield nets we investigate the computational properties of finite binary and analog state discrete time symmetric hopfield nets for binary networks we obtain a simulation of convergent asymmetric networks by symmetric networks with only a linear increase in network size and computation time then we analyze the convergence time of hopfield nets in terms of the length of their bit representations here we construct an analog symmetric network whose convergence time exceeds the convergence time of any binary hopfield net with the same representation length further we prove that the min energy problem for analog hopfield nets is np hard and provide a polynomial time approximation algorithm for this problem in the case of binary nets finally we show that symmetric analog nets with an external clock are computationally turing universal

a generalized convergence theorem for neural networks a neural network model is presented in which each neuron performs a threshold logic function the model always converges to a stable state when operating in a serial mode and to a cycle of length at most 2 when operating in a fully parallel mode this property is the basis for the potential applications of the model such as associative memory devices and combinatorial optimization the two convergence theorems for serial and fully parallel modes of operation are reviewed and a general convergence theorem is presented that unifies the two known cases new relations between the neural network model and the problem of finding a minimum cut in a graph are obtained

constructing deterministic finite state automata in recurrent neural networks recurrent neural networks that are trained to behave like deterministic finite state automata dfas can show deteriorating performance when tested on long strings this deteriorating performance can be attributed to the instability of the internal representation of the learned dfa states the use of a sigmoidel discriminant function together with the recurrent structure contribute to this instability we prove that a simple algorithm can construct second order recurrent neural networks with a sparse interconnection topology and sigmoidal discriminant function such that the internal dfa state representations are stable that is the constructed network correctly classifies strings of arbitrary length the algorithm is based on encoding strengths of weights directly into the neural network we derive a relationship between the weight strength and the number of dfa states for robust string classification for a dfa with n state and m input alphabet symbols the constructive algorithm generates a programmed neural network with o n neurons and o mn weights we compare our algorithm to other methods proposed in the literature

unique expansions of real numbers it was discovered some years ago that there exist non integer real numbers q 1 for which only one sequence ci of integers ci 0 q satisfies the equality i 1 ciq i 1 the set of such univoque numbers has a rich topological structure and its study revealed a number of unexpected connections with measure theory fractals ergodic theory and diophantine approximation in this paper we consider for each fixed q 1 the set uq of real numbers x having a unique representation of the form i 1 ciq i x with integers ci belonging to 0 q we carry out a detailed topological study of these sets for instance we characterize their closures and we determine those bases q for which uq is closed or even a cantor set we also study the set uq consisting of all sequences ci of integers ci 0 q such that i 1 ciq i uq we determine the numbers r 1 for which the map q uq defined on 1 is constant in a neighborhood of r and the numbers q 1 for which uq is a subshift or a subshift of finite type

some computations on the spectra of pisot and salem numbers properties of pisot numbers have long been of interest one line of questioning initiated by erdos joo and komornik in 1990 is the determination of l q for pisot numbers q where l q inf y y e0 e1q1 enqn ei 1 0 y 0 although the quantity l q is known for some pisot numbers q there has been no general method for computing l q this paper gives such an algorithm with this algorithm some properties of l q and its generalizations are investigated a related question concerns the analogy of l q denoted a q where the coefficients are restricted to 1 in particular for which non pisot numbers is a q nonzero this paper finds an infinite class of salem numbers where a q 0

a d conversion with imperfect quantizers this paper analyzes mathematically the effect of quantizer threshold imperfection commonly encountered in the circuit implementation of analog to digital a d converters such as pulse code modulation pcm and sigma delta sigmadelta modulation sigmadelta modulation which is based on coarse quantization of oversampled redundant samples of a signal enjoys a type of self correction property for quantizer threshold errors bias that is not shared by pcm although classical sigmadelta modulation is inferior to pcm in the rate distortion sense this robustness feature is believed to be one of the reasons why sigmadelta modulation is preferred over pcm in a d converters with imperfect quantizers motivated by these facts other encoders are constructed in this paper that use redundancy to obtain a similar self correction property but that achieve higher order accuracy relative to bit rate compared to classical sigmadelta more precisely two different types of encoders are introduced that exhibit exponential accuracy in the bit rate in contrast to the polynomial type accuracy of classical sigmadelta while possessing the self correction property

continuous time symmetric hopfield nets are computationally universal we establish a fundamental result in the theory of computation by continuous time dynamical systems by showing that systems corresponding to so called continuous time symmetric hopfield nets are capable of general computation as is well known such networks have very constrained lyapunov function controlled dynamics nevertheless we show that they are universal and efficient computational devices in the sense that any convergent synchronous fully parallel computation by a recurrent network of n discrete time binary neurons with in general asymmetric coupling weights can be simulated by a symmetric continuous time hopfield net containing only 18n 7 units employing the saturated linear activation function moreover if the asymmetric network has maximum integer weight size wmax and converges in discrete time t then the corresponding hopfield net can be designed to operate in continuous time t e for any e 0 such that wmax212n e21 e in terms of standard discrete computation models our result implies that any polynomially space bounded turing machine can be simulated by a family of polynomial size continuous time symmetric hopfield nets

comments on constructive learning of recurrent neural networks limitations of recurrent cascade correlation and a simple solution giles et al 1995 have proven that fahlman s recurrent cascade correlation rcc architecture is not capable of realizing finite state automata that have state cycles of length more than two under a constant input signal this paper extends the conclusions of giles et al by showing that there exists a corollary to their original proof which identifies a large second class of automata that is also unrepresentable by rcc

constructive learning of recurrent neural networks limitations of recurrent cascade correlation and a simple solution it is often difficult to predict the optimal neural network size for a particular application constructive or destructive methods that add or subtract neurons layers connections etc might offer a solution to this problem we prove that one method recurrent cascade correlation due to its topology has fundamental limitations in representation and thus in its learning capabilities it cannot represent with monotone i e sigmoid and hard threshold activation functions certain finite state automata we give a preliminary approach on how to get around these limitations by devising a simple constructive training method that adds neurons during training while still preserving the powerful fully recurrent structure we illustrate this approach by simulations which learn many examples of regular grammars that the recurrent cascade correlation method is unable to learn

quasi periodic β expansions and cut languages abstract motivated by the analysis of neural net models between integer and rational weights we introduce a so called cut language over a real digit alphabet which contains finite expansions i e base representations of the numbers less than a given threshold we say that an infinite expansion is eventually quasi periodic if its tail sequence formed by the numbers whose representations are obtained by removing leading digits contains an infinite constant subsequence we prove that a cut language is regular iff its threshold is a quasi periodic number whose all expansions are eventually quasi periodic by showing that altogether they have a finite number of tail values for algebraic bases we prove that there is an eventually quasi periodic expansion with an infinite number of tail values iff there is a conjugate of on the unit circle for transcendental combined with algebraic digits a expansion is eventually quasi periodic iff it has a finite number of tail values for a pisot base and digits from the smallest field extension q over rational numbers including we show that any number from q is quasi periodic in addition we achieve a dichotomy that a cut language is either regular or non context free and we show that any cut language with rational parameters is context sensitive

derandomizing approximation algorithms based on semidefinite programming remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental np hard problems namely max cut max k cut max sat max dicut max bisection k vertex coloring maximum independent set etc all these breakthroughs involve polynomial time randomized algorithms based upon semidefinite programming a technique pioneered by goemans and williamson in this paper we give techniques to derandomize the above class of randomized algorithms thus obtaining polynomial time deterministic algorithms with the same approximation ratios for the above problems at the heart of our technique is the use of spherical symmetry to convert a nested sequence of n integrations which cannot be approximated sufficiently well in polynomial time to a nested sequence of just a constant number of integrations which can be approximated sufficiently well in polynomial time

sur l inegalite de turan kubilius friable the exact value of the friable turan kubilius constant is determined when u log x log y tends to a finite limit for the set of y friable integers not exceeding x

pumping lemma in automata theory based on complete residuated lattice valued logic a note abstract automata theory based on complete residuated lattice valued logic called l valued automata was established in qiu automata theory based on completed residuated lattice valued logic i science in china f 44 6 2001 419 429 qiu automata theory based on completed residuated lattice valued logic ii science in china f 45 6 2002 442 452 in this note we deal with the pumping lemma in l valued automata theory after recalling some preliminaries related to complete residuated lattices and l valued automata we define a number of l valued accepting predicates in particular the pumping lemma for l valued automata theory is set up we show that if those related l valued predicates are defined by using connective instead of then the pumping lemma may not hold again furthermore we investigate the l valued automata with e transitions and present the equivalence between the l valued automata without e transitions and those with e transitions finally a number of related questions is addressed

on threshold circuits and polynomial computation a threshold circuit consists of an acyclic digraph of unbounded fanin where each node computes a threshold function or its negation this paper investigates the computational power of threshold circuits a surprising relationship is uncovered between threshold circuits and another class of unbounded fanin circuits which are denoted finite field z p n circuits where each node computes either multiple sums or products of integers modulo a prime p n in particular it is proved that all functions computed by threshold circuits of size s n x2265 n and depth d n can also be computed by z p n circuits of size o s n log s n np n log p n and depth o d n furthermore it is shown that all functions computed by z p n circuits of size s n and depth d n can be computed by threshold circuits of size o 1 x2208 2 s n log p n 1 x2208 and depth o 1 x2208 5 d n these are the main results of this paper there are many useful and quite surprising consequences of this result for example an integer reciprocal can be computed in size n o 1 m and depth o 1 more generally any analytic function with a convergent rational polynomial power series such as sine cosine exponentiation square root and logarithm can be computed within accuracy 2 nc for any constant c by threshold circuits of polynomial size and constant depth in addition integer and polynomial division fff polynomial interpolation chinese remaindering all the elementary symmetric functions banded matrix inverse and triangular toeplitz matrix inverse can be exactly computed by threshold circuits of polynomial size and constant depth all these results and simulations hold for polytime uniform circuits this paper also gives a corresponding simulation oflogspace uniform z p n circuits by logspace uniform threshold circuits requiring an additional multiplying factor of o log log log p n depth finally purely algebraic methods forlowerbounds for zp n circuits are developed using degree arguments a depth hierarchy theorem for z p n circuits is proved for any s n x2265 n d n o s n c for some constant c p n where 6 s n d n d n p n c 2 n there exist explicitly constructible functions computable by z p n circuits of size s n and depth d n but provably not computable by z p n circuits of size s n c and depth o d n for any constant c x2265 1

dynamics of discrete time continuous state hopfield networks the dynamics of discrete time continuous state hopfield networks is driven by an energy function in this paper we use this tool to prove under mild hypotheses that any trajectory converges to a fixed point for the sequential iteration and to a cycle of length 2 or a fixed point for the parallel iteration perhaps surprisingly it seems that no rigorous proof of these results was published before