eigenvalue separation in some random matrix models the eigenvalue density for members of the gaussian orthogonal and unitary ensembles follows the wigner semicircle law if the gaussian entries are all shifted by a constant amount s 2n 1 2 where n is the size of the matrix in the large n limit a single eigenvalue will separate from the support of the wigner semicircle provided s 1 in this study using an asymptotic analysis of the secular equation for the eigenvalue condition we compare this effect to analogous effects occurring in general variance wishart matrices and matrices from the shifted mean chiral ensemble we undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s and higher rank analogs of this setting this is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a green function in the dyson brownian motion model for the shifted mean gaussian unitary ensemble an

fuzzy logic controller design utilizing multiple contending software agents the concept of fuzzy logic control based on a distributed multi agent system is described in this paper it is shown that a system composed of a number of cooperating and contending intelligent agents can perform well in terms of sensing the measured input data and producing output control signals each agent s implementation consists of a number of fuzzy logic control rules although at each time step all the agents compete for the control of the system by proposing a possible action the main controller chooses among the selections based on a number of evaluation criteria it is through the simultaneous cooperation to overall control the system and competition to take control at a given time that the system yields convincing results the effectiveness of the novel methodology is illustrated through the control of a robotic system that keeps an inverted pendulum in balance while being perturbed by a human force in addition the proposed methodology is applied to the practical control of a service mobile manipulator interacting with a human it is shown that fuzzy logic control combined with intelligent multi agent architecture can yield effective and efficient intelligent controllers

off diagonal terms in symmetric operators in this paper we provide a quantitative comparison of two obstructions for a given symmetric operator s with dense domain in hilbert space h to be self adjoint the first one is the pair of deficiency spaces of von neumann and the second one is of more recent vintage let p be a projection in h we say that it is smooth relative to s if its range is contained in the domain of s we say that smooth projections pi i 1 diagonalize s if a i pi spi 0 for all i and b supi pi i if such projections exist then s has a self adjoint closure i e s has a spectral resolution and so our second obstruction to self adjointness is defined from smooth projections pi with i pi spi 0 we prove results both in the case of a single operator s and a system of operators

index distribution of cauchy random matrices using a coulomb gas technique we compute analytically the probability that a large n n cauchy random matrix has n positive eigenvalues where n is called the index of the ensemble we show that this probability scales for large n as where is the dyson index of the ensemble the rate function c is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis we find that the rate function around its minimum at 1 2 has a quadratic behavior modulated by a logarithmic singularity as a consequence the variance of the index scales for large n as var n cln n where c 2 2 is twice as large as the corresponding prefactor in the gaussian and wishart cases the analytical results are checked by numerical simulations and against an exact finite n formula which for 2 can be derived using orthogonal polynomials

long time behaviour of the solution to non linear kraichnan equations we consider the solution of a nonlinear kraichnan equation open image in new window with a covariance kernel k and boundary condition h t t 1 we study the long time behaviour of h as the time parameters t s go to infinity according to the asymptotic behaviour of k this question appears in various subjects since it is related with the analysis of the asymptotic behaviour of the trace of non commutative processes satisfying a linear differential equation but also naturally shows up in the study of the so called response function and aging properties of the dynamics of some disordered spin systems

analysis on function duplicating capabilities of fuzzy controllers abstract fuzzy controllers have been successfully applied to many cases to which conventional control algorithms are difficult to be applied even though the representations and the processings of data and information in the fuzzy controller are quite different from other control algorithms the information processing operation that it carries out is basically a function f a rn rm from a bounded subset a of n dimensional euclidean space to a bounded subset f a of m dimensional euclidean space where n and m are the number of measured states and the number of control inputs of controlled system respectively under the assumptions of mamdani s direct reasoning method and c o g center of gravity defuzzification method the fuzzy controllers are proven to perform the mappings of any given functions with appropriately defined fuzzy sets the mapping capabilities of the fuzzy controllers are analyzed in detail for two cases f r1 r1 and g r2 r1 also it will be shown that the results can be extended to multiple dimensional cases

second order asymptotics for matrix models we study several matrix models and show that when the potential is convex and a small perturbation of the gaussian potential the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one in order to do that we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges

the complexity of spherical p spin models a second moment approach recently auffinger ben arous and cern initiated the study of critical points of the hamiltonian in the spherical pure pp spin spin glass model and established connections between those and several notions from the physics literature denoting the number of critical values less than nunu by crtn u crtn u they computed the asymptotics of 1nlog ecrtn u 1nlog ecrtn u as nn the dimension of the sphere goes to we compute the asymptotics of the corresponding second moment and show that for p 3p 3 and sufficiently negative uu it matches the first moment e crtn u 2 e crtn u 2 1 as an immediate consequence we obtain that crtn u e crtn u 1crtn u e crtn u 1 in l2l2 and thus in probability for any uu for which ecrtn u ecrtn u does not tend to 00 we prove that the moments match on an exponential scale

transport maps for beta matrix models and universality we construct approximate transport maps for non critical beta matrix models that is maps so that the push forward of a non critical beta matrix model with a given potential is a non critical beta matrix model with another potential up to a small error in the total variation distance one of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension in addition we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models

on the overlap in the multiple spherical sk models in order to study certain questions concerning the distribution of the overlap in sherrington kirkpatrick type models such as the chaos and ultrametricity problems it seems natural to study the free energy of multiple systems with constrained overlaps one can write analogues of guerra s replica symmetry breaking bound for such systems but it is not at all obvious how to choose informative functional order parameters in these bounds we were able to make some progress for spherical pure p spin sk models where many computations can be made explicitly for pure 2 spin model we prove ultrametricity and chaos in an external field for the pure p spin model for even p 4 without an external field we describe two possible values of the overlap of two systems at different temperatures we also prove a somewhat unexpected result which shows that in the 2 spin model the support of the joint overlap distribution is not always witnessed at the level of the free energy and for example ultrametricity holds only in a weak sense

yang mills measure and the master field on the sphere we study the yang mills measure on the sphere with unitary structure group in the limit where the structure group has high dimension we show that the traces of loop holonomies converge in probability to a deterministic limit which is known as the master field on the sphere the values of the master field on simple loops are expressed in terms of the solution of a variational problem we show that given its values on simple loops the master field is characterized on all loops of finite length by a system of differential equations known as the makeenko migdal equations we obtain a number of further properties of the master field on specializing to families of simple loops our results identify the high dimensional limit in non commutative distribution of the brownian bridge in the group of unitary matrices starting and ending at the identity

sum rules via large deviations in this paper a sum rule means a relationship between a functional defined on a subset of all probability measures on r involving the reverse kullback leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction the first sum rule is the szeg o verblunsky theorem see simon 2011 theorem 1 8 6 p 29 and concerns the case of trigonometrical polynomials on the torus more recently killip and simon killip and simon 2003 have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution indeed this sum rule includes a contribution of the atomic part of the measure away from the support of the circular law in this paper we recover this sum rules by using only large deviations tools on random matrices furthermore we obtain new up to our knowledge magic sum rules for the reverse kullback leibler divergence with respect to the pastur marchenko or kesten mckay distributions as in the semicircular case these formulas generally include a contribution of the atomic part appearing away from the support of the reference measure from the point of view of large deviations we obtain these formulas by showing a large deviation principle for the point mass measure concentrated on random eigenvalues of classical ensembles and combine it with a large deviation principle due to gamboa and rouault 2011 for the recursion coefficients

statistics of the number of minima in a random energy landscape dated february 6 2008 we consider random energy landscapes constructed from d dimensional lattices or trees thedistribution of the number of local minima in such landscapes follows a large deviation principleand we derive the associated law exactly for dimension 1 also of interest is the probability of themaximum possible number of minima this probability scales exponentially with the number of sites we calculate analytically the corresponding exponent for the cayley tree and the two leg ladder for 2 to 5 dimensional hypercubic lattices we compute the exponent numerically and compare tothe cayley tree case

some properties of defuzzification neural networks abstract fuzzy systems have gained more and more attention from researchers and practitioners of various fields in such systems the output represented by a fuzzy set sometimes needs to be transformed into a scalar value and this task is known as the defuzzification process several analytic methods have been proposed for this problem but lately the neural network approach has been used for this purpose when employed as defuzzifiers a neural network is called a defuzzification neural network in this paper some preliminary results on properties of such defuzzification networks will be reported

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

random matrices and complexity of spin glasses cern y abstract we give an asymptotic evaluation of the complexity of spherical p spin spin glass models via random matrix theory this study enables us to obtain detailed infor mation about the bottom of the energy landscape including the absolute minimum the ground state the other local minima and describe an interesting layered structure of the low critical values for the hamiltonians of these models we also show that our ap proach allows us to compute the related tap complexity and extend the results known in the physics literature as an independent tool we prove a ldp for the k th largest eigenvalue of the goe extending the results of bdg01 how many critical values of given index and below a given level does a typical random morse function have on a high dimensional manifold our work addresses this question in a very special case we look at certain natural random gaussian functions on the n dimensional sphere known as p spin spherical spin glass models we cannot yet answer the question above about the typical number but we can study thoroughly the mean number which we show is exponentially large in n we introduce a new identity based on the classical kac rice formula relating random matrix theory and the problem of counting these critical values using this identity and tools from random matrix theory we give an asymptotic evaluation of the complexity of these spherical spin glass models the complexity mentioned here is defined as the mean number of critical points of given index whose value is below or above a given level this includes the important question of counting the mean number of local minima below a given level and in particular the question of finding the ground state energy the minimal value of the hamiltonian we show that this question is directly related to the study of the edge of the spectrum of the gaussian orthogonal ensemble goe the question of computing the complexity of mean field spin glass models has recently been thoroughly studied in the physics literature see for example clr03 and the refer ences therein mainly for a different measure of the complexity i e the mean number of solutions to the thouless anderson palmer equations or tap complexity our approach to the complexity enables us to recover known results in the physics literature about tap complexity to compute the ground state energy when p is even and to describe an interesting layered structure of the low energy levels of the hamiltonians of these models which might prove useful for the study of the metastability of langevin dynamics for these models in longer time scales than those studied in bdg01 the paper is organised as follows in section 2 we give our main results in section 3 we prove two main formulas theorem 2 1 and 2 2 relating random matrix theory specif ically the goe and spherical spin glasses these formulas are direct consequences of the kac rice formula we learned the version needed here in the book at07 for another modern account see aw09 the main new ingredient is the fact that for spherical spin glass models the hessian of the hamiltonian at a critical point conditioned on the value of the hamiltonian is a symmetric gaussian random matrix with independent entries up to symmetry plus a diagonal matrix this implies in particular that it is possible to

invariants for e 0 semigroups on ii 1 factors we introduce four new cocycle conjugacy invariants for e0 semigroups on ii1 factors a coupling index a dimension for the gauge group a super product system and a c semiflow using noncommutative ito integrals we show that the dimension of the gauge group can be computed from the structure of the additive cocycles we do this for the clifford flows and even clifford flows on the hyperfinite ii1 factor and for the free flows on the free group factor l f infty in all cases the index is 0 which implies they have trivial gauge groups we compute the super product systems for these families and using this we show they have trivial coupling index finally using the c semiflow and the boundary representation of powers and alevras we show that the families of clifford flows and even clifford flows contain infinitely many mutually non cocycle conjugate e0 semigroups

a neuro fuzzy system architecture for behavior based control of a mobile robot in unknown environments abstract a neuro fuzzy system architecture for behavior based control of a mobile robot in unknown environments is presented a neural network is used to understand environments its inputs are a heading angle between the robot and a specified target and range information acquired by an array of ultrasonic sensors the output from the neural network is a trained reference motion direction for robot navigation the methodology of the behavior based control approach proposed in this paper is 1 to analyze and to decompose a complex task based on stimulus response behavior and 2 to quantitatively formulate each type of behavior with a simple feature by fuzzy sets and fuzzy rules as well as to coordinate conflicts and competition among multiple types of behavior by fuzzy reasoning an advantage is that building fuzzy sets and rules for each simple featured type of behavior is much easier than for a complex task based upon a reference motion direction and distances between the robot and obstacles different types of behavior are fused by fuzzy logic to control the velocities of the two rear wheels of the robot simulation experiments show that the proposed neuro fuzzy system can improve navigation performance in complex and unknown environments in addition this architecture is suitable for multisensor fusion and integration

free energy and complexity of spherical bipartite models we investigate both free energy and complexity of the spherical bipartite spin glass model we first prove a variational formula in high temperature for the limiting free energy based on the well known crisanti sommers representation of the mixed p spin spherical model next we show that the mean number of local minima at low levels of energy is exponentially large in the size of the system and we derive a bound on the location of the ground state energy

replica trick to calculate means of absolute values applications to stochastic equations a version of the replica trick can be used to evaluate means of the absolute values of functions that are not positive definite and with some modifications the mean logarithm of the absolute value this trick can be applied in the context of stochastic equations where the absolute value of the jacobian of the stochastic function in each solution must be computed the calculation of the average number of solutions of the naive mean field equations for the sk spin glass is discussed an extension of the considerations to other mean field equations such as tap is straightforward the brst supersymmetry is found to provide a useful tool in uncovering some puzzling aspects in these calculations the trick presented provides a possible solution to these problems

about a solvable mean field model of a gaussian spin glass in a series of papers we have studied a modified hopfield model of a neural network with learned words characterized by a gaussian distribution the model can be represented as a bipartite spin glass with one party described by dichotomic ising spins and the other party by continuous spin variables with an a priori gaussian distribution by application of standard interpolation methods we have found it useful to compare the neural network model bipartite from one side with two spin glass models each monopartite from the other side of these the first is the usual sherrington kirkpatrick model the second is a spin glass model with continuous spins and inbuilt highly nonlinear smooth cut off interactions this model is an invaluable laboratory for testing all techniques which have been useful in the study of spin glasses the purpose of this paper is to give a synthetic description of the most peculiar aspects by stressing the necessary novelties in the treatment in particular it will be shown that the control of the infinite volume limit according to the well known guerra toninelli strategy requires in addition one to consider the involvement of the cut off interaction in the interpolation procedure moreover the control of the ergodic region the annealed case cannot be directly achieved through the standard application of the borel cantelli lemma but requires previous modification of the interaction this remark could find useful application in other cases the replica symmetric expression for the free energy can be easily reached through a suitable version of the doubly stochastic interpolation technique however this model shares the unique property that the fully broken replica symmetry ansatz can be explicitly calculated a very simple sum rule connects the general expression of the fully broken free energy trial function with the replica symmetric one the definite sign of the error term shows that the replica solution is optimal then we follow a deep strategy developed by talagrand to conclude that the replica symmetric ansatz not only gives a bound for the free energy which cannot be improved by symmetry breaking but gives also the true value for the free energy for the sake of completeness we study also the fluctuations of the centered and rescaled overlaps by following a method similar to that exploited for the sherrington kirkpatrick model in the replica symmetric region

universality of the rem for dynamics of mean field spin glasses we consider a version of glauber dynamics for a p spin sherrington kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the n dimensional hypercube we show that for all p 3 and all inverse temperatures 0 there exists a constant p 0 such that for all exponential time scales exp n with p the properly rescaled clock process time change process converges to an stable subordinator where 2 1 moreover the dynamics exhibits aging at these time scales with a time time correlation function converging to the arcsine law of this stable subordinator in other words up to rescaling on these time scales that are shorter than the equilibration time of the system the dynamics of p spin models ages in the same way as the rem and by extension bouchaud s rem like trap model confirming the latter as a universal aging mechanism for a wide range of systems the sk model the case p 2 seems to belong to a different universality class

basins of attraction of metastable states of the spherical p spin model we study the basins of attraction of metastable states in the spherical p spin spin glass model starting the relaxation dynamics at a given distance from a thermalized condition weighting the initial condition with the boltzmann distribution we find a finite size for the basins in contrast a white weighting of the initial condition implies vanishing basins of attraction we correspond our results to those of a recently constructed effective potential

critical points and supersymmetric vacua i supersymmetric vacua universes of string m theory may be identified with certain critical points of a holomorphic section the superpotential of a hermitian holomorphic line bundle over a complex manifold an important physical problem is to determine how many vacua there are and how they are distributed as the superpotential varies over physically relevant ensembles in several papers over the last few years m r douglas and co workers have studied such vacuum statistics problems for a variety of physical models at the physics level of rigor do ad dd the present paper is the first of a series by the present authors giving a rigorous mathematical foundation for the vacuum statistics problem it sets down basic results on the statistics of critical points s 0 of random holomorphic sections of hermitian holomorphic line bundles with respect to a metric connection when the sections are endowed with a gaussian measure the principal results give formulas for the expected density and number of critical points of fixed morse index of gaussian random sections relative to they are particularly concrete for riemann surfaces in our subsequent work the results will be applied to the vacuum statistics problem and to the purely geometric problem of studying metrics which minimize the expected number of critical points

tap complexity the cavity method and supersymmetry we compute the bray and moore bm tap complexity for the sherrington kirkpatrick model through the cavity method showing that some essential modifications are needed with respect to the standard formulation of the method this allows us to understand various features recently discovered and to unveil at last the physical meaning of the parameters of the bm theory we also reconsider the supersymmetric susy formulation of the problem finding that the bm solution satisfies some proper susy ward identities that are different from the standard ones the susy relationships encode the physical meaning of the parameters obtained through the cavity method the problem of the vanishing prefactor is addressed showing how it can be avoided

complex dynamics in learning complicated games game theory is the standard tool used to model strategic interactions in evolutionary biology and social science traditionally game theory studies the equilibria of simple games however is this useful if the game is complicated and if not what is we define a complicated game as one with many possible moves and therefore many possible payoffs conditional on those moves we investigate two person games in which the players learn based on a type of reinforcement learning called experience weighted attraction ewa by generating games at random we characterize the learning dynamics under ewa and show that there are three clearly separated regimes i convergence to a unique fixed point ii a huge multiplicity of stable fixed points and iii chaotic behavior in case iii the dimension of the chaotic attractors can be very high implying that the learning dynamics are effectively random in the chaotic regime the total payoffs fluctuate intermittently showing bursts of rapid change punctuated by periods of quiescence with heavy tails similar to what is observed in fluid turbulence and financial markets our results suggest that at least for some learning algorithms there is a large parameter regime for which complicated strategic interactions generate inherently unpredictable behavior that is best described in the language of dynamical systems theory

glauber dynamics of the random energy model i metastable motion on the extreme states we investigate the long time behavior of the glauber dynamics for the random energy model below the critical temperature we give very precise estimates on the motion of the process to and between the states of extremal energies we show that when disregarding time the consecutive steps of the process on these states are governed by a markov chain that jumps uniformly on all possible states the mean times of these jumps are also computed very precisely and are seen to be asymptotically independent of the terminal point a first indicator of aging is the observation that the mean time of arrival in the set of states that have waiting times of order t is itself of order t the estimates proven in this paper will furnish crucial input for a follow up paper where aging is analysed in full detail

advanced fuzzy logic control of a model car in extreme situations abstract state of the art in fuzzy logic control applications is the simple calculus of fuzzy if then rules predominantly using min max operators for the implementation mainly precompiler based tools are used though this concept is sufficient for simple control applications or consumer goods it is often insufficient for complex control applications this work investigates which advanced methods render the application of fuzzy control in complex problems possible and how fuzzy controllers may be built more efficiently the results of the theoretical work is demonstrated by the real model of a high performance race car

performance of statistical tests for single source detection using random matrix theory this paper introduces a unified framework for the detection of a single source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver the generalized maximum likelihood test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace using recent results from random matrix theory a practical way to evaluate the threshold and the p value of the test is provided in the asymptotic regime where the number k of sensors and the number n of observations per sensor are large but have the same order of magnitude the theoretical performance of the test is then analyzed in terms of receiver operating characteristic roc curve it is in particular proved that both type i and type ii error probabilities converge to zero exponentially as the dimensions increase at the same rate and closed form expressions are provided for the error exponents these theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix

phase space geometry and slow dynamics we describe a non arrhenius mechanism for the slowing down of dynamics that is inherent to the high dimensionality of the phase space we show that such a mechanism is at work both in a family of mean field spin glass models without any domain structure and in the case of ferromagnetic domain growth the marginality of spin glass dynamics as well as the existence of a quasi equilibrium regime can be understood within this scenario we discuss the question of ergodicity in an out of equilibrium situation

interactions in noncommutative dynamics a mathematical notion of interaction is introduced for noncommutative dynamical systems i e for one parameter groups of automorphisms of endowed with a certain causal structure with any interaction there is a well defined state of the past and a well defined state of the future we describe the construction of many interactions involving cocycle perturbations of the car ccr flows and show that they are nontrivial the proof of nontriviality is based on a new inequality relating the eigenvalue lists of the past and future states to the norm of a linear functional on a certain c algebra

the extremal process of critical points of the pure p spin spherical spin glass model recently sharp results concerning the critical points of the hamiltonian of the p spin spherical spin glass model have been obtained by means of moments computations in particular these moments computations allow for the evaluation of the leading term of the ground state i e of the global minimum in this paper we study the extremal point process of critical points that is the point process associated to all critical values in the vicinity of the ground state we show that the latter converges in distribution to a poisson point process of exponential intensity in particular we identify the correct centering of the ground state and prove the convergence in distribution of the centered minimum to a minus gumbel variable these results are identical to what one obtains for a sequence of i i d variables correctly normalized namely we show that the model is in the universality class of rem

markov dilations on w algebras abstract in this paper a systematic study of markov dilations is begun for completely positive operators on w algebras which leave a faithful normal state invariant it is shown that a minimal markov dilation preserves important properties of the underlying completely positive operator afterwards some results are proved concerning the construction of dilations which lead to markov dilations for large classes of operators finally some of the ideas developed here are applied to the study of a simple example over the 2 2 matrices

partition of networks into basins of attraction we study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree each node is associated with or attracted to its neighbor of maximal degree as long as the degree is increasing a node that has no neighbors of higher degree is a peak attracting all the nodes in its basin maximally random scale free networks exhibit different behavior based on their degree distribution exponent gamma for small gamma broad distribution networks are dominated by a giant basin whereas for large gamma narrow distribution there are numerous basins with peaks attracting mainly their nearest neighbors we derive expressions for the first two moments of the number of basins we also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale free networks that resemble random nets finally we generalize the problem to regular networks and lattices where all degrees are equal and thus the attractiveness of a node must be determined by an assigned weight rather than the degree we derive the complete distribution of basins of attraction resulting from randomly assigned weights in one dimensional chains

distributions of eigenvalues of large euclidean matrices generated from lp balls and spheres abstract let x 1 x n be points randomly chosen from a set g r n and f x be a function the euclidean random matrix is given by m n f x i x j 2 n n where is the euclidean distance when n is fixed and n we prove that m n the empirical distribution of the eigenvalues of m n converges to 0 for a big class of functions of f x assuming both n and n go to infinity proportionally we obtain the explicit limit of m n when g is the l p unit ball or sphere with p 1 as corollaries we obtain the limit of a n with a n d x i x j n n and d being the geodesic distance on the ordinary unit sphere in r n we also obtain the limit of a n for the euclidean distance matrix a n x i x j n n the limits are a b v where a and b are constants and v follows the marcenko pastur law the same are also obtained for other examples appeared in physics literature including exp x i x j n n and exp d x i x j n n our results partially confirm a conjecture by do and vu 14

a fourier view on the r transform and related asymptotics of spherical integrals abstract we estimate the asymptotics of spherical integrals of real symmetric or hermitian matrices when the rank of one matrix is much smaller than its dimension we show that it is given in terms of the r transform of the spectral measure of the full rank matrix and give a new proof of the fact that the r transform is additive under free convolution these asymptotics also extend to the case where one matrix has rank one but complex eigenvalue a result related with the analyticity of the corresponding spherical integrals

aging of spherical spin glasses sompolinski and zippelius 1981 propose the study of dynamical systems whose invariant measures are the gibbs measures for hard to analyze statistical physics models of interest in the course of doing so physicists often report of an aging phenomenon for example aging is expected to happen for the sherrington kirkpatrick model a disordered mean field model with a very complex phase transition in equilibrium at low temperature we shall study the langevin dynamics for a simplified spherical version of this model the induced rotational symmetry of the spherical model reduces the dynamics in question to an n dimensional coupled system of ornstein uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices we obtain the limiting dynamics for n approaching infinity and by analyzing its long time behavior explain what is aging mathematically speaking what causes this phenomenon and what is its relationship with the phase transition of the corresponding equilibrium invariant measures

retention capacity of random surfaces we introduce a water retention model for liquids captured on a random surface with open boundaries and investigate the model for both continuous and discrete surface heights 0 1 n 1 on a square lattice with a square boundary the model is found to have several intriguing features including a nonmonotonic dependence of the retention on the number of levels for many n the retention is counterintuitively greater than that of an n 1 level system the behavior is explained using percolation theory by mapping it to a 2 level system with variable probability results in one dimension are also found

obstacle avoidance for a mobile robot a neuro fuzzy approach in this paper a neuro fuzzy approach is presented in order to guide a mobile robot this task could be carried out specifying a set of fuzzy rules taking into account the different situations found by the mobile robot this set of fuzzy rules could be optimised according to different criteria however the approach shown in this paper is able to extract a set of fuzzy rules set from a set of trajectories provided by a human these trajectories guide the mobile robot towards the target in different cases thus it has been possible to obtain the rules and membership functions automatically whereas other approaches need a previous definition of the rules and membership functions in order to verify that the obtained behaviour is satisfactory the neuro fuzzy approach has been implemented in two mobile robots

fuzzy drive control of an autonomous mobile robot abstract this paper deals with the fuzzy drive control of an autonomous mobile robot first a fuzzy model fuzzy expert system for the human thought process is designed by using the idea of fuzziness and secondly it is applied to the autonomous mobile robot the autonomous mobile robot is being investigated very actively in many research laboratories of companies and universities around the world for example there are the unmanned automated carrier in a factory the working robot in extreme situations and the house service robot 1 3 a human makes a proper decision and puts the decision into a proper action instantly under fuzzy and insufficient information the fuzzy drive control system architecture is a hierarchy that is constructed by a visual sensor subsystem a motor drive unit and a fuzzy drive expert system which manages these subsystems the visual sensor subsystem which perceives the outer environment such as the road the obstacles etc is comprised of a tv camera and an image processing unit the motor drive unit motor controller has the function of controlling the speed and the steering of the mobile robot by changing the revolution rate of motors mounted on the robot the fuzzy drive expert system which is the core of the fuzzy drive control system recognizes the environment with the image data and sends control commands to the subsystems by these commands from the drive expert system each subsystem motivates the self subsisting control function to achieve that purpose this expert system has an inference unit and rules such as recognition rules and control rules which were obtained experimentally as an application of the fuzzy drive control experiments of driving an autonomous mobile robot on a straight road and one with corners are performed from the experimental results it was found that the robot control system has the faculty of flexible circumstantial decision about equal to what a human is able to realize

a self tuning fuzzy controller abstract the aim of a fuzzy controller is to compensate the dynamic characteristics of the controlled system it is the same purpose as for the fuzzy logic controller flc now the best design of the flc is required for achievement of this purpose but such a controller has not yet been completely designed because the fuzzy controller is constructed of control rules which are ambiguously described by conventional controller strategies and expert knowledge and because its structure theory is not well established yet that is in order to construct the best rules a highly skilled technique and a tuning technique using trial and error are required as a way to easily design a flc a method which improves a given standard flc is considered in this paper we propose a self tuning algorithm of the flc it has two functions in adjusting the scaling factors which are the parameters of the flc and in improving the control rules of flc by evaluating the control response at real time and the control results after operations we add these functions to the flc and design the self tuning fuzzy controller stfc this makes the flc a good controller

phase diagram of restricted boltzmann machines and generalized hopfield networks with arbitrary priors restricted boltzmann machines are described by the gibbs measure of a bipartite spin glass which in turn can be seen as a generalized hopfield network this equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities both at low and high load of pure states we study the paramagnetic spin glass and the spin glass retrieval phase transitions as the pattern i e weight distribution and spin i e unit priors vary smoothly from gaussian real variables to boolean discrete variables our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard hopfield model with boolean patterns the retrieval region becomes larger when the pattern entries and retrieval units get more peaked and conversely when the hidden units acquire a broader prior and therefore have a stronger response to high fields moreover at low load retrieval always exists below some critical temperature for every pattern distribution ranging from the boolean to the gaussian case

how many eigenvalues of a gaussian random matrix are positive we study the probability distribution of the index n sub i e the number of positive eigenvalues of an nxn gaussian random matrix we show analytically that for large n and large n sub with the fraction 0 c n sub n 1 of positive eigenvalues fixed the index distribution p n sub cn n approx exp beta n sup 2 phi c where beta is the dyson index characterizing the gaussian ensemble the associated large deviation rate function phi c is computed explicitly for all 0 c 1 it is independent of beta and displays a quadratic form modulated by a logarithmic singularity around c 1 2 as a consequence the distribution of the index has a gaussian form near the peak but with a variance delta n of index fluctuations growing as delta n approx lnn beta pi sup 2 for large n for beta 2 this result is independently confirmed against an exact finite n formula yielding delta n lnn 2 pi sup 2 c o n sup 1 for large n where the constant c for even n has the nontrivial value c gamma 1 3ln2 2 pi sup 2 approx equal 0 185 248 and gamma 0 5772 is the euler constant we also determine for large n the probability that the interval zeta sub 1 zeta sub 2 is free of eigenvalues some of these results have been announced in a recent letter phys rev lett 103 220603 2009

the ugc hardness threshold of the lp grothendieck problem for p 2 we consider the problem of given an n n matrix a aij whose diagonal entries vanish approximating in polynomial time the number rm opt p a max left sum i j 1 na ij x ix j x 1 ldots x n in mathbb r n wedge left sum i 1 n x i p right 1 p le 1 right when p 2 this is simply the problem of computing the maximum eigenvalue of a whereas for p actually it suffices to take p log n it is the grothendieck problem on the complete graph which was shown to have a o log n approximation algorithm in nemirovski et al nemirovski a c roos t terlaky 1999 on maximization of quadratic form over intersection of ellipsoids with common center math program ser a86 3 463 473 megretski megretski a 2001 relaxations of quadratic programs in operator theory and system analysis systems approximation singular integral operators and related topics bordeaux 2000 vol 129 operator theory advances and applications birkhauser basel 365 392 charikar and wirth charikar m a wirth 2004 maximizing quadratic programs extending grothendieck s inequality proc 45th annual sympos foundations comput sci ieee computer society 54 60 and was used in the work of charikar and wirth noted above to design the best known algorithm for the problem of computing the maximum correlation in correlation clustering thus the problem of approximating optp a interpolates between the spectral p 2 case and the correlation clustering p case from a physics point of view this problem corresponds to computing the ground states of spin glasses in a hard wall potential well we design a polynomial time algorithm which given p 2 and an n n matrix a aij with zeros on the diagonal computes optp a up to a factor p e 30 log p on the other hand assuming the unique games conjecture ugc we show that it is np hard to approximate optp a up to a factor smaller than p e hence as p the ugc hardness threshold for computing optp a is exactly p e 1 o 1

large deviation bounds for matrix brownian motion we prove large deviation bounds for the convergence of hermitian matrix valued brownian motion towards free brownian motion as a consequence we obtain upper and lower bounds on the microstates entropy introduced by voiculescu 24

on the energy landscape of spherical spin glasses abstract we investigate the energy landscape of the spherical mixed even p spin model near its maximum energy we relate the distance between pairs of near maxima to the support of the parisi measure at zero temperature we then provide an algebraic relation that characterizes one step replica symmetric breaking parisi measures for these measures we show that any two nonparallel spin configurations around the maximum energy are asymptotically orthogonal to each other in sharp contrast we study models with full replica symmetry breaking and show that all possible values of the asymptotic distance are attained near the maximum energy

topology trivialization and large deviations for the minimum in the simplest random optimization finding the global minimum of a cost function given by the sum of a quadratic and a linear form in n real variables over n 1 dimensional sphere is one of the simplest yet paradigmatic problems in optimization theory known as the trust region subproblem or constraint least square problem when both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field we first identify and study two distinct large n scaling regimes in which the linear term magnetic field leads to a gradual topology trivialization i e reduction in the total number mathcal n tot of critical stationary points in the cost function landscape in the first regime mathcal n tot remains of the order n and the cost function energy has generically two almost degenerate minima with the tracy widom tw statistics in the second regime the number of critical points is of the order of unity with a finite probability for a single minimum in that case the mean total number of extrema minima and maxima of the cost function is given by the laplace transform of the tw density and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the tw law though the full form of that distribution is not yet known to us one of its far tails can be inferred from the large deviation theory for the global minimum in the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large deviation approximation by finding both the rate function and the leading pre exponential factor

an investigation of the hidden structure of states in a mean field spin glass model we study the geometrical structure of the states in the low temperature phase of a mean field model for generalized spin glasses the p spin spherical model this structure cannot be revealed by the standard methods mainly due to the presence of an exponentially high number of states each one having a vanishing weight in the thermodynamic limit performing a purely entropic computation based on the tap equations for this model we define a constrained complexity which gives the overlap distribution of the states we find that this distribution is continuous non random and highly dependent on the energy range of the considered states furthermore we show which is the geometrical shape of the threshold landscape giving some insight into the role played by threshold states in the dynamical behaviour of the system

glauber dynamics of the random energy model ii aging below the critical temperature we investigate the long time behavior of the glauber dynamics for the ran dom energy model below the critical temperature we establish that for a suitably chosen timescale that diverges with the size of the system one can prove that a natural auto correlation function exhibits aging moreover we show that the long time asymptotics of this function coincide with those of the so called rem like trap model proposed by bouchaud and dean our results rely on very precise estimates on the distribution of transition times of the process between different states of extremely low energy

quasi equilibrium interpretation of ageing dynamics we develop an interpretation of the off equilibrium dynamical solution of mean field glassy models in terms of quasi equilibrium concepts we show that the relaxation of the thermo remanent magnetization follows a generalized version of the onsager regression postulate of induced fluctuations we then find the rationale for the equality between the fluctuation dissipation ratio and the rate of growth of the configurational entropy close to the asymptotic state found empirically in mean field solutions

the geometry of the gibbs measure of pure spherical spin glasses we analyze the statics for pure p spin spherical spin glass models with p ge 3 at low enough temperature with f n beta denoting the free energy we compute the logarithmic second order term of nf n beta and prove that for an appropriate centering c n beta nf n beta c n beta is a tight sequence we further establish the absence of temperature chaos those results follow from the following geometric picture we prove for the gibbs measure of interest by itself asymptotically the measure splits into infinitesimal spherical bands centered at deep minima playing the role of pure states for the pure models the latter makes precise the picture of many valleys separated by high mountains and significant parts of the tap analysis from the physics literature