determined theories and limit laws we provide a new model theoretic technique for proving 0 1 and convergence laws as an application we obtain a new slightly less computational proof of convergence laws due to spencer and thoma for the probability functions pnl in n n l in in n n c n

a faffa mlfma algorithm for electromagnetic scattering based on the multilevel fast multipole algorithm mlfma an efficient method is proposed to accelerate the solution of the combined field integral equation in electromagnetic scattering and radiation where the fast far field approximation faffa is combined with mlfma the translation between groups in mlfma is expensive because spherical hankel functions and legendre polynomials are involved and the translator is defined on an eward sphere with many k spl circ directions when two groups are in the far field region however the translation can be greatly simplified by faffa where only a single k spl circ direction is involved in the translator the condition for using faffa and the way to efficiently incorporate faffa with mlfma are discussed complexity analysis illustrates that the computational cost in faffa mlfma can be asymptotically cut by half compared to the conventional mlfma numerical results are given to verify the efficiency of the algorithm

efficient generation of method of moments matrices using the characteristic function method we present a new fast and efficient technique for computing the mom matrix elements from rao wilton glisson rwg bases one that does not require evaluating the usual surface integrals involving the expansion and testing functions the matrix elements are represented in terms of characteristic functions cfs which can be expressed in a closed canonical form these universal cfs make matrix generation more efficient and faster than the conventional triangular facet interaction scheme the proposed technique can be applied for any subsectional basis functions other the rwg if the size of the basis function is small enough compared to the wavelength and the distance between the source and the testing basis functions is beyond a nominal distance the proposed approach is validated for several canonical geometries and the accuracy of the generated matrix elements as well as the radar cross section rcs has been verified

spectral density of random graphs with topological constraints the spectral density of random graphs with topological constraints is analysed using the replica method we consider graph ensembles featuring generalised degree degree correlations as well as those with a community structure in each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question we highlight the effect of these topological constraints on the resulting spectral density

conditional probability logic lifted bayesian networks and almost sure quantifier elimination abstract we introduce a formal logical language called conditional probability logic cpl which extends first order logic and which can express probabilities conditional probabilities and which can compare conditional probabilities intuitively speaking although formal details are different cpl can express the same kind of statements as some languages which have been considered in the artificial intelligence community we also consider a way of making precise the notion of lifted bayesian network where this notion is a type of lifted probabilistic graphical model used in machine learning data mining and artificial intelligence a lifted bayesian network in the sense defined here determines in a natural way a probability distribution on the set of all structures in the sense of first order logic with a common finite domain d our main result theorem 3 14 is that for every noncritical cpl formula x there is a quantifier free formula x which is almost surely equivalent to x as the cardinality of d tends towards infinity this is relevant for the problem of making probabilistic inferences on large domains d because a the problem of evaluating by brute force the probability of x being true for some sequence d of elements from d has in general highly exponential time complexity in the cardinality of d and b the corresponding probability for the quantifier free x depends only on the lifted bayesian network and not on d some conclusions regarding the computational complexity of finding are given in remark 3 17 the main result has two corollaries one of which is a convergence law and zero one law for noncritial cpl formulas

analysis of transient electromagnetic scattering from composite conducting dielectric targets with the time domain fast dipole method an efficient time domain fast dipole method is proposed to accelerate the time domain volume surface integral equation for analyzing the transient electromagnetic scattering from composite conducting dielectric targets with arbitrary shape and inhomogeneity this scheme is based on the equivalent dipole moment and the taylor series expansion of the transient far field the computational complexity and memory requirement of the proposed algorithm can be reduced considerably by embedding the fast algorithm in a hierarchical framework several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method

on sufficient conditions for unsatisfiability of random formulas a descriptive complexity approach to random 3 sat is initiated we show that unsatisfiability of any significant fraction of random 3 cnf formulas cannot be certified by any property that is expressible in datalog combined with the known relationship between the complexity of constraint satisfaction problems and expressibility in datalog our result implies that any constraint propagation algorithm working with small constraints will fail to certify unsatisfiability almost always our result is a consequence of designing a winning strategy for one of the players in the existential pebble game the winning strategy makes use of certain extension axioms that we introduce and hold almost surely on a random 3 cnf formula the second contribution of our work is the connection between finite model theory and propositional proof complexity to make this connection explicit we establish a tight relationship between the number of pebbles needed to win the game and the width of the resolution refutations as a consequence to our result and the known size width relationship in resolution we obtain new proofs of the exponential lower bounds for resolution refutations of random 3 cnf formulas and the pigeonhole principle

dynamics of on line learning in radial basis function networks on line learning is examined for the radial basis function network an important and practical type of neural network the evolution of generalization error is calculated within a framework which allows the phenomena of the learning process such as the specialization of the hidden units to be analyzed the distinct stages of training are elucidated and the role of the learning rate described the three most important stages of training the symmetric phase the symmetry breaking phase and the convergence phase are analyzed in detail the convergence phase analysis allows derivation of maximal and optimal learning rates as well as finding the evolution of the mean system parameters the variances of these parameters are derived and shown to be typically small finally the analytic results are strongly confirmed by simulations

fast analysis of transient electromagnetic scattering using the taylor series expansion enhanced time domain integral equation solver this paper concentrates on an efficient fast algorithm for accelerating the marching on in time based time domain integral equation method for the analysis of the transient electromagnetic scattering from perfect electrically conducting bodies taking advantage of the approximate evaluation of the vector and scalar potential terms and the taylor series expansion of far field the computational complexity and memory requirement of the proposed algorithm scale as mathrm o n t n s log n s and mathrm o n s 1 5 respectively where n t and n s denote the number of simulation time steps and spatial basis functions respectively several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed algorithm

generating simple random graphs with prescribed degree distribution let f be a probability distribution with support on the non negative integers four methods for generating a simple undirected graph with approximate degree distribution f are described and compared two methods are based on the so called configuration model with modifications ensuring a simple graph one method is an extension of the classical erdos renyi graph where the edge probabilities are random variables and the last method starts with a directed random graph which is then modified to a simple undirected graph all methods are shown to give the correct distribution in the limit of large graph size but under different assumptions on the degree distribution f and also using different order of operations

four types of learning curves if machines are learning to make decisions given a number of examples the generalization error e t is defined as the average probability that an incorrect decision is made for a new example by a machine when trained with t examples the generalization error decreases as t increases and the curve e t is called a learning curve the present paper uses the bayesian approach to show that given the annealed approximation learning curves can be classified into four asymptotic types if the machine is deterministic with noiseless teacher signals then 1 e at 1 when the correct machine parameter is unique and 2 e at 2 when the set of the correct parameters has a finite measure if the teacher signals are noisy then 3 e at 1 2 for a deterministic machine and 4 e c at 1 for a stochastic machine

fast dipole method for electromagnetic scattering from perfect electric conducting targets a new fast dipole method fdm is proposed for the electromagnetic scattering from arbitrarily shaped three dimensional 3d electrically large perfect electric conducting pec targets in free space based on the concept of equivalent dipole moment method edm and the fast multipole method fmm the electric field magnetic field and combined field integral equations cfie for this algorithm have been developed and implemented although the basic acceleration idea in the fdm has been borrowed from the fmm the specific implementation of these two algorithms is completely different in the fdm a simple taylor s series expansion of the distance between two interacting equivalent dipoles is used which transforms the impedance element into another aggregation translation disaggregation form naturally furthermore this algorithm is very simple for numerical implementation for it does not involve the calculation of a number of bessel functions legendre functions for the addition theorem and complex integral operators the fdm can achieve o n1 5 computational complexity and memory requirement where n is the number of unknowns numerical results are presented to validate the efficiency and accuracy of this method through comparison with other rigorous solutions

finitely coordinated models for low temperature phases of amorphous systems we introduce models of heterogeneous systems with finite connectivity defined on random graphs to capture finite coordination effects on the low temperature behaviour of finite dimensional systems our models use a description in terms of small deviations of particle coordinates from a set of reference positions particularly appropriate for the description of low temperature phenomena a born von karman type expansion with random coefficients is used to model effects of frozen heterogeneities the key quantity appearing in the theoretical description is a full distribution of effective single site potentials which needs to be determined self consistently if microscopic interactions are harmonic the effective single site potentials turn out to be harmonic as well and the distribution of these single site potentials is equivalent to a distribution of localization lengths used earlier in the description of chemical gels for structural glasses characterized by frustration and anharmonicities in the microscopic interactions the distribution of single site potentials involves anharmonicities of all orders and both single well and double well potentials are observed the latter with a broad spectrum of barrier heights the appearance of glassy phases at low temperatures is marked by the appearance of asymmetries in the distribution of single site potentials as previously observed for fully connected systems double well potentials with a broad spectrum of barrier heights and asymmetries would give rise to the well known universal glassy low temperature anomalies when quantum effects are taken into account

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

phase transitions in number theory from the birthday problem to sidon sets in this work we show how number theoretical problems can be fruitfully approached with the tools of statistical physics we focus on g sidon sets which describe sequences of integers whose pairwise sums are different and propose a random decision problem which addresses the probability of a random set of k integers to be g sidon first we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit initially assuming independence we then develop a mean field theory for the g sidon decision problem we further improve the mean field theory which is only qualitatively correct by incorporating deviations from independence yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit connections between the generalized birthday problem in probability theory the number theory of sidon sets and the properties of q potts models in condensed matter physics are briefly discussed

replica theory for learning curves for gaussian processes on random graphs we use a statistical physics approach to derive accurate predictions for the challenging problem of predicting the performance of gaussian process regression performance is quantified by the learning curve defined as the average error versus number of training examples we assume the gaussian process prior is defined by a random walk kernel inputs are vertices on a random graph and the outputs are noisy function values we show that replica techniques can be used to obtain exact performance predictions in the limit of large graphs after first rewriting the average error in terms of a graphical model conventionally the gaussian process kernel is only globally normalized so that the prior variance can differ between vertices as a more principled alternative we also consider local normalization where the prior variance is uniform the normalization constants for the prior then have to be defined as thermal averages in an unnormalized model and this requires the introduction of a second auxiliary set of replicas our results for both types of kernel normalization apply generically to all random graph ensembles constrained by a fixed but arbitrary degree distribution we compare with numerically simulated learning curves and find excellent agreement a significant improvement over existing approximations

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

spectra of modular and small world matrices we compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter and intra module degree distributions subject only to the constraint of finite mean connectivities we also evaluate spectra of a certain class of small world matrices generated from random graphs by introducing shortcuts via additional random connectivity components both adjacency matrices and the associated graph laplacians are investigated for the laplacians we find lifshitz type singular behaviour of the spectral density in a localized region of small values in the case of modular networks we can identify contributions of local densities of state from individual modules for small world networks we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states exhibiting only localized states in the band gaps results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances and with direct diagonalization results

learning curves for gaussian process regression approximations and bounds we consider the problem of calculating learning curves i e average generalization performance of gaussian processes used for regression on the basis of a simple expression for the generalization error in terms of the eigenvalue decomposition of the covariance function we derive a number of approximation schemes we identify where these become exact and compare with existing bounds on learning curves the new approximations which can be used for any input space dimension generally get substantially closer to the truth we also study possible improvements to our approximations finally we use a simple exactly solvable learning scenario to show that there are limits of principle on the quality of approximations and bounds expressible solely in terms of the eigenvalue spectrum of the covariance function

reply to comments on fast dipole method for electromagnetic scattering from perfect electric conducting targets the effect of the phase approximation used in the above communication on the solution is discussed it can be found that this phase approximation can not achieve the second order accuracy however it can reduce the possible maximum phase error especially for far separated groups compared with only using linear term furthermore we give a new formulation of fast dipole method fdm in which the amplitude approximation error is reduced

the average number of distinct sites visited by a random walker on random graphs we study the linear large n behaviour of the average number of distinct sites s n visited by a random walker after n steps on a large random graph an expression for the graph topology dependent prefactor b in s n bn is proposed we use generating function techniques to relate this prefactor to the graph adjacency matrix and then devise message passing equations to calculate its value numerical simulations are performed to evaluate the agreement between the message passing predictions and random walk simulations on random graphs scaling with system size and average graph connectivity are also analysed

when does the zero one law hold in 1960 paul erdbs and alfred renyi er began the subject of random graphs the 1985 book of bela bollobas b provides the standard reference for this field in modem terminology the random graph g n p is a graph on vertex set n 1 n where each pair i j of vertices are adjacent with independent probability p more accurately g n p is a probability space over the space of graphs on vertex set n for any property a of graphs there is a probability denoted pr g n p l a that g n p satisfies a in their very title on the evolution of random graphs erd6s and renyi envisioned a dynamic process g n p changing character as p moved from zero to one they discovered as did their many successors that for many natural properties a pr g n p t a was usually near zero or near one and made the jump from near zero to near one or back again in a very narrow range the placement of this critical range of p depended on n for example let a be the property of containing a triangle there are n n3 6 3 potential triangles each is a triangle in g n p with probability p 3 and so the expected number of triangles in g n p is asymptotically n3p3 6 this suggests the critical range p 8 1 n indeed erdos and renyi proved that if p p n l n then limn gc pr g n p l a 1 notation f n g n means limn oo f n g n ox they called p n l n a threshold function for this property a as other examples connectivity has threshold function logn n containing a clique on four points has threshold function n 2 3 containing an edge has easily threshold function n 2 and every vertex lying in a triangle has threshold function log n 13n 23 it was the observation that threshold functions seemed to be of the form log n n fl with a f rational that motivated our current line of research what can we say about the possible threshold functions of properties a not much if we place no restrictions on a for example the property that the number of edges is even shows no threshold function behavior if we restrict

random walk kernels and learning curves for gaussian process regression on random graphs we consider learning on graphs guided by kernels that encode similarity between vertices our focus is on random walk kernels the analogues of squared exponential kernels in euclidean spaces we show that on large locally treelike graphs these have some counter intuitive properties specifically in the limit of large kernel lengthscales we consider using these kernels as covariance functions of gaussian processes in this situation one typically scales the prior globally to normalise the average of the prior variance across vertices we demonstrate that in contrast to the euclidean case this generically leads to significant variation in the prior variance across vertices which is undesirable from a probabilistic modelling point of view we suggest the random walk kernel should be normalised locally so that each vertex has the same prior variance and analyse the consequences of this by studying learning curves for gaussian process regression numerical calculations as well as novel theoretical predictions for the learning curves using belief propagation show that one obtains distinctly different probabilistic models depending on the choice of normalisation our method for predicting the learning curves using belief propagation is significantly more accurate than previous approximations and should become exact in the limit of large random graphs

an efficient fast algorithm for accelerating the time domain integral equation discontinuous galerkin method an efficient fast algorithm for accelerating the time domain integral equation discontinuous galerkin tdiedg method for analyzing the transient scattering from electrically large complex objects is proposed the tdiedg is discretized using the monopolar rao wilton glisson rwg function and shifted lagrange polynomial function in spatial and time domain respectively the final system of equations can be solved iteratively using the classical marching on in time scheme taking advantage of the approximate evaluation of the vector and scalar potential terms combined with the taylor series expansion of the transient far field the computational burden of the tdiedg solver has been significantly reduced in addition it is flexible and convenient to analyze object with nonconformal discretization due to the employment of the monopolar rwg functions

zero one laws for sparse random graphs for random graph theorists see e g bollobas 1 for general reference p any constant is not the only not even the most interesting case rather they consider p p n a function approaching zero in their seminal paper erd6s and renyi 5 showed that for many interesting a there is a function p n which they called a threshold function so that if r n p n then f n r n a 0 while if p n r n then f n r n a 1 notation p r means lim p r 0 all limits are as n approaches infinity let us say p p n satisfies the zero one law if for all a in gra limf n p a 0 or 1 we shall partially characterize those p p n which satisfy the zero one law