exact solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far field or intermediate regions exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain a stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field it is found that the degree of ill posedness of the inverse problem is significantly reduced when considering a hemi circular observation arc with respect to a linear array of sensors by a factor scaling on the small angular aperture subtended by the observation line estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace

jamming model for the extremal optimization heuristic extremal optimization a recently introduced meta heuristic for hard optimization problems is analysed on a simple model of jamming the model is motivated first by the problem of finding lowest energy configurations for a disordered spin system on a fixed valence graph the numerical results for the spin system exhibit the same phenomenology found in all earlier studies of extremal optimization and our analytical results for the model reproduce many of these features

on hp convergence of prolate spheroidal wave functions and a new well conditioned prolate collocation scheme the first purpose of this paper is to provide further illustrations from both theoretical and numerical perspectives for the nonconvergence of h refinement in hp approximation by the prolate spheroidal wave functions pswfs a surprising convergence property that was first discovered by boyd et al 2013 3 the second purpose is to offer a new basis that leads to prolate collocation systems with condition numbers independent of c n the intrinsic bandwidth parameter and the number of collocation points we highlight that the collocation scheme together with a very practical rule for pairing up c n significantly outperforms the legendre polynomial based method and likewise other jacobi polynomial based methods in approximating highly oscillatory bandlimited functions

social balance as a satisfiability problem of computer science reduction of frustration was the driving force in an approach to social balance as it was recently considered by antal t antal p l krapivsky and s redner phys rev e 72 036121 2005 we generalize their triad dynamics to k cycle dynamics for arbitrary integer k we derive the phase structure determine the stationary solutions and calculate the time it takes to reach a frozen state the main difference in the phase structure as a function of k is related to k being even or odd as a second generalization we dilute the all to all coupling as considered by antal to a random network with connection probability w 1 interestingly this model can be mapped to a satisfiability problem of computer science the phase of social balance in our original interpretation then becomes the phase of satisfaction of all logical clauses in the satisfiability problem in common to the cases we study the ideal solution without any frustration always exists but the question actually is as to whether this solution can be found by means of a local stochastic algorithm within a finite time the answer depends on the choice of parameters after establishing the mapping between the two classes of models we generalize the social balance problem to a diluted network topology for which the satisfiability problem is usually studied on the other hand in connection with the satisfiability problem we generalize the random local algorithm to a p random local algorithm including a parameter p that corresponds to the propensity parameter in the social balance problem the qualitative effect of the inclusion of this parameter is a bias towards the optimal solution and a reduction of the needed simulation time

controllability and maximum matchings of complex networks previously the controllability problem of a linear time invariant dynamical system was mapped to the maximum matching mm problem on the bipartite representation of the underlying directed graph and the sizes of mms on random bipartite graphs were calculated analytically with the cavity method at zero temperature limit here we present an alternative theory to estimate mm sizes based on the core percolation theory and the perfect matching of cores our theory is much more simplified and easily interpreted and can estimate mm sizes on random graphs with or without symmetry between out and in degree distributions our result helps to illuminate the fundamental connection between the controllability problem and the underlying structure of complex systems

an exactly solvable random satisfiability problem we introduce a new model for the generation of random satisfiability problems it is an extension of the hyper sat model of ricci tersenghi weigt and zecchina which is a variant of the famous k sat model it is extended to q state variables and relates to a different choice of the statistical ensemble the model has an exactly solvable statistic the critical exponents and scaling functions of the sat unsat transition are calculable at zero temperature with no need of replicas also with exact finite size corrections we also introduce an exact duality of the model and show an analogy of thermodynamic properties with the random energy model of disordered spin system theory relations with error correcting codes are also discussed

lipschitz continuity of solutions of linear inequalities programs and complementarity problems it is shown that solutions of linear inequalities linear programs and certain linear complementarity problems e g those with p matrices or z matrices but not semidefinite matrices are lipschitz continuous with respect to changes in the right hand side data of the problem solutions of linear programs are not lipschitz continuous with respect to the coefficients of the objective function the lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right hand side perturbation

statistical mechanics of the vertex cover problem we review recent progress in the study of the vertex cover problem vc the vc belongs to the class of np complete graph theoretical problems which plays a central role in theoretical computer science on ensembles of random graphs vc exhibits a coverable uncoverable phase transition very close to this transition depending on the solution algorithm easy hard transitions in the typical running time of the algorithms occur we explain a statistical mechanics approach which works by mapping the vc to a hard core lattice gas and then applying techniques such as the replica trick or the cavity approach using these methods the phase diagram of the vc could be obtained exactly for connectivities c e the solution of the vc exhibits full replica symmetry breaking the statistical mechanics approach can also be used to study analytically the typical running time of simple complete and incomplete algorithms for the vc finally we describe recent results for the vc when studied on other ensembles of finite and infinite dimensional graphs

analytic solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number

on reduced convex qp formulations of monotone lcps techniques for transforming convex quadratic programs qps into monotone linear complementarity problems lcps and vice versa are well known we describe a class of lcps for which a reduced qp formulation one that has fewer constraints than the standard qp formulation is available we mention several instances of this class including the known case in which the coefficient matrix in the lcp is symmetric

eigenfunctions of the airyʼs integral transform properties numerical computations and asymptotic behaviors abstract this paper is devoted to the study of the spectrum of the integral operator with airy s kernel we provide the reader with some efficient methods of computing the eigenfunctions and the eigenvalues of this operator the first method is based on the use of a differential operator which commutes with this integral operator the second method is based on a gaussian quadrature method applied to the finite airy s transform the asymptotic behavior of the eigenfunctions is studied by the use of a wkb method some numerical examples are given to illustrate results of this work

texture classification using the support vector machines in recent years support vector machines svms have demonstrated excellent performance in a variety of pattern recognition problems in this paper we apply svms for texture classification using translation invariant features generated from the discrete wavelet frame transform to alleviate the problem of selecting the right kernel parameter in the svm we use a fusion scheme based on multiple svms each with a different setting of the kernel parameter compared to the traditional bayes classifier and the learning vector quantization algorithm svms and in particular the fused output from multiple svms produce more accurate classification results on the brodatz texture album

typical approximation performance for maximum coverage problem this study investigated the typical performance of approximation algorithms known as belief propagation the greedy algorithm and linear programming relaxation for maximum coverage problems in sparse biregular random graphs after we used the cavity method for a corresponding hard core lattice gas model results showed that two distinct thresholds of replica symmetry and its breaking exist in the typical performance threshold of belief propagation in the low density region the superiority of three algorithms in terms of a typical performance threshold is obtained by some theoretical analyses although the greedy algorithm and linear programming relaxation have the same approximation ratio in worst case performance their typical performance thresholds are mutually different indicating the importance of typical performance results of numerical simulations validate the theoretical analyses and imply further mutual relations of approximation algorithms

solving satisfiability problems by fluctuations the dynamics of stochastic local search algorithms stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems we give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems we find two different dynamical regimes depending on the number of constraints per variable for low constraintness the problems are solved efficiently i e in linear time for higher constraintness the solution times become exponential we observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium if the algorithm runs long enough an exponentially rare fluctuation towards a solution appears

grassmann integral representation for spanning hyperforests given a hypergraph g we introduce a grassmann algebra over the vertex set and show that a class of grassmann integrals permits an expansion in terms of spanning hyperforests special cases provide the generating functions for rooted and unrooted spanning hyper forests and spanning hyper trees all these results are generalizations of kirchhoff s matrix tree theorem furthermore we show that the class of integrals describing unrooted spanning hyper forests is induced by a theory with an underlying osp 1 2 supersymmetry

instability of one step replica symmetry broken phase in satisfiability problems we reconsider the one step replica symmetry breaking 1rsb solutions of two random combinatorial problems k xorsat and k sat we present a general method for establishing the stability of these solutions with respect to further steps of replica symmetry breaking our approach extends the ideas of montanari and ricci tersenghi 2003 eur phys j b 33 339 to more general combinatorial problems it turns out that 1rsb is always unstable at sufficiently small clause density or high energy in particular the recent 1rsb solution to 3 sat is unstable at zero energy for m with m 4 153 on the other hand the sat unsat phase transition seems to be correctly described within 1rsb

replica bounds for optimization problems and diluted spin systems in this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by guerra cond mat 0205123 to prove that the replica method generates variational bounds for disordered systems we analyze a family of models that includes the viana bray model the diluted p spin model or random xor sat problem and the random k sat problem showing that the replica cavity method at the various levels of approximation provides systematic schemes to obtain lower bounds of the free energy at all temperatures and of the ground state energy in the case of k sat and xor sat it thus gives upper bounds of the satisfiability threshold our analysis underlines deep connections with the cavity method which are not evident in the long range case

the linear rational collocation method we introduce the collocation method based on linear rational interpolation for solving general hyperbolic problems prove its stability and its convergence in weighted norms and give numerical examples for its use

asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions abstract for a fixed bandwidth c the classical prolate spheroidal wave functions pswfs n c and the circular ones cpswfs n c form remarkable hilbertian bases for the spaces of fourier and hankel band limited functions with bandwidth c respectively the prolate spheroidal wave functions have already found many applications from various scientific area such as signal processing numerical analysis physics and random matrix theory these concrete applications of the pswfs and cpswfs require the accurate computation of these special functions to this end we develop in this work two approximate procedures the first procedure is based on pushing forward the wkb method and provide accurate approximations of the pswfs and the cpswfs in terms of some classical special functions the second procedure is based on efficient and accurate quadrature methods for the computation of the values and the eigenvalues associated with pswfs and the cpswfs both procedures are valid for small as well as for large values of the frequency band width c also we provide the reader with some numerical examples that illustrate the different results of this work

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

complexity transitions in global algorithms for sparse linear systems over finite fields we study the computational complexity of a very basic problem namely that of finding solutions to a very large set of random linear equations in a finite galois field modulo q using tools from statistical mechanics we are able to identify phase transitions in the structure of the solution space and to connect them to the changes in the performance of a global algorithm namely gaussian elimination crossing phase boundaries produces a dramatic increase in memory and cpu requirements necessary for the algorithms in turn this causes the saturation of the upper bounds for the running time we illustrate the results on the specific problem of integer factorization which is of central interest for deciphering messages encrypted with the rsa cryptosystem

bounds for diluted mean fields spin glass models in an important recent paper 2 s franz and m leone prove rigorous lower bounds for the free energy of the diluted p spin model and the k sat model at any temperature we show that the results for these two models are consequences of a single general principle our calculations are significantly simpler than those of 2 even in the replica symmetric case

relaxation and metastability in a local search procedure for the random satisfiability problem an analysis of the average properties of a local search procedure randomwalksat for the satisfaction of random boolean constraints is presented depending on the ratio alpha of constraints per variable reaching a solution takes a time t res growing linearly t res approximately tau res alpha n alpha alpha d with the size n of the instance the relaxation time tau res alpha in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator for alpha alpha d the system is trapped in some metastable state and resolution occurs from escape from this state through crossing of a large barrier an annealed calculation of the height zeta alpha of this barrier is proposed the polynomial to exponential cross over alpha d approximately 2 7 is not related to the onset of clustering among solutions occurring at alpha approximately 3 86

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

top down induction of model trees with regression and splitting nodes model trees are an extension of regression trees that associate leaves with multiple regression models in this paper a method for the data driven construction of model trees is presented namely the stepwise model tree induction smoti method its main characteristic is the induction of trees with two types of nodes regression nodes which perform only straight line regression and splitting nodes which partition the feature space the multiple linear model associated with each leaf is then built stepwise by combining straight line regressions reported along the path from the root to the leaf in this way internal regression nodes contribute to the definition of multiple models and have a global effect while straight line regressions at leaves have only local effects experimental results on artificially generated data sets show that smoti outperforms two model tree induction systems m5 and retis in accuracy results on benchmark data sets used for studies on both regression and model trees show that smoti performs better than retis in accuracy while it is not possible to draw statistically significant conclusions on the comparison with m5 model trees induced by smoti are generally simple and easily interpretable and their analysis often reveals interesting patterns

spectral methods based on prolate spheroidal wave functions for hyperbolic pdes we examine the merits of using prolate spheroidal wave functions pswfs as basis functions when solving hyperbolic pdes using pseudospectral methods the relevant approximation theory is reviewed and some new approximation results in sobolev spaces are established an optimal choice of the band limit parameter for pswfs is derived for single mode functions our conclusion is that one might gain from using the pswfs over the traditional chebyshev or legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions

combining reconstructive and discriminative subspace methods for robust classification and regression by subsampling linear subspace methods that provide sufficient reconstruction of the data such as pca offer an efficient way of dealing with missing pixels outliers and occlusions that often appear in the visual data discriminative methods such as lda which on the other hand are better suited for classification tasks are highly sensitive to corrupted data we present a theoretical framework for achieving the best of both types of methods an approach that combines the discrimination power of discriminative methods with the reconstruction property of reconstructive methods which enables one to work on subsets of pixels in images to efficiently detect and reject the outliers the proposed approach is therefore capable of robust classification with a high breakdown point we also show that subspace methods such as cca which are used for solving regression tasks can be treated in a similar manner the theoretical results are demonstrated on several computer vision tasks showing that the proposed approach significantly outperforms the standard discriminative methods in the case of missing pixels and images containing occlusions and outliers

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

random subcubes as a toy model for constraint satisfaction problems we present an exactly solvable random subcube model inspired by the structure of hard constraint satisfaction and optimization problems our model reproduces the structure of the solution space of the random k satisfiability and k coloring problems and undergoes the same phase transitions as these problems the comparison becomes quantitative in the large k limit distance properties as well the x satisfiability threshold are studied the model is also generalized to define a continuous energy landscape useful for studying several aspects of glassy dynamics

two solutions to diluted p spin models and xorsat problems we derive analytical solutions for p spin models with finite connectivity at zero temperature these models are the statistical mechanics equivalent of p xorsat problems in theoretical computer science we give a full characterization of the phase diagram location of the phase transitions static and dynamic together with a description of the clustering phenomenon taking place in configurational space we use two alternative methods the cavity approach and a rigorous derivation

polynomial iterative algorithms for coloring and analyzing random graphs we study the graph coloring problem over random graphs of finite average connectivity c given a number q of available colors we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable depending on q we find with a one step replica symmetry breaking approximation the precise value of the critical average connectivity c q moreover we show that below c q there exists a clustering phase c in c d c q in which ground states spontaneously divide into an exponential number of clusters furthermore we extended our considerations to the case of single instances showing consistent results this leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region i e when c in c d c q

interior point methods for massive support vector machines we investigate the use of interior point methods for solving quadratic programming problems with a small number of linear constraints where the quadratic term consists of a low rank update to a positive semidefinite matrix several formulations of the support vector machine fit into this category an interesting feature of these particular problems is the volume of data which can lead to quadratic programs with between 10 and 100 million variables and if written explicitly a dense q matrix our code is based on ooqp an object oriented interior point code with the linear algebra specialized for the support vector machine application for the targeted massive problems all of the data is stored out of core and we overlap computation and input output to reduce overhead results are reported for several linear support vector machine formulations demonstrating that the method is reliable and scalable

statistical mechanics of error exponents for error correcting codes error exponents characterize the exponential decay when increasing message length of the probability of error of many error correcting codes to tackle the long standing problem of computing them exactly we introduce a general thermodynamic formalism that we illustrate with maximum likelihood decoding of low density parity check codes on the binary erasure channel and the binary symmetric channel in this formalism we apply the cavity method for large deviations to derive expressions for both the average and typical error exponents which differ by the procedure used to select the codes from specified ensembles when decreasing the noise intensity we find that two phase transitions take place at two different levels a glass to ferromagnetic transition in the space of codewords and a paramagnetic to glass transition in the space of codes

general duality for abelian group valued statistical mechanics models we introduce a general class of statistical mechanics models taking values in an abelian group which includes examples of both spin and gauge models both ordered and disordered the model is described by a set of variables and a set of interactions each interaction is associated with a linear combination of variables these are summarized in a matrix j a gibbs factor is associated with each variable one body term and with each interaction then we introduce a duality transformation for systems in this class the duality exchanges the abelian group with its dual the gibbs factors with their fourier transforms and the interactions with the variables high low couplings in the interaction terms are mapped into low high couplings in the one body terms if the matrix j is interpreted as a vector representation of a matroid duality exchanges the matroid with its dual we discuss some physical examples the main idea is to generalize the known models up to eventually include randomness into the pattern of interaction we introduce and study a random gaussian model a random potts like model and a random variant of discrete scalar qed although the classical procedure as given by kramers and wannier does not extend in a natural way to such a wider class of systems our weaker procedure applies to these models too we shortly describe the consequence of duality for each example

time dependent hermite galerkin spectral method and its applications a time dependent hermite galerkin spectral method thgsm is investigated in this paper for the nonlinear convection diffusion equations in the unbounded domains the time dependent scaling factor and translating factor are introduced in the definition of the generalized hermite functions ghf as a consequence the thgsm based on these ghf has many advantages not only in theoretical proofs but also in numerical implementations the stability and spectral convergence of our proposed method have been established in this paper the korteweg de vries burgers kdvb equation and its special cases including the heat equation and the burgers equation as the examples have been numerically solved by our method the numerical results are presented and it surpasses the existing methods in accuracy our theoretical proof of the spectral convergence has been supported by the numerical results

robust linear and support vector regression the robust huber m estimator a differentiable cost function that is quadratic for small errors and linear otherwise is modeled exactly in the original primal space of the problem by an easily solvable simple convex quadratic program for both linear and nonlinear support vector estimators previous models were significantly more complex or formulated in the dual space and most involved specialized numerical algorithms for solving the robust huber linear estimator numerical test comparisons with these algorithms indicate the computational effectiveness of the new quadratic programming model for both linear and nonlinear support vector problems results are shown on problems with as many as 20000 data points with considerably faster running times on larger problems

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

reduced support vector machines a statistical theory in dealing with large data sets the reduced support vector machine rsvm was proposed for the practical objective to overcome some computational difficulties as well as to reduce the model complexity in this paper we study the rsvm from the viewpoint of sampling design its robustness and the spectral analysis of the reduced kernel we consider the nonlinear separating surface as a mixture of kernels instead of a full model the rsvm uses a reduced mixture with kernels sampled from certain candidate set our main results center on two major themes one is the robustness of the random subset mixture model the other is the spectral analysis of the reduced kernel the robustness is judged by a few criteria as follows 1 model variation measure 2 model bias deviation between the reduced model and the full model and 3 test power in distinguishing the reduced model from the full one for the spectral analysis we compare the eigenstructures of the full kernel matrix and the approximation kernel matrix the approximation kernels are generated by uniform random subsets the small discrepancies between them indicate that the approximation kernels can retain most of the relevant information for learning tasks in the full kernel we focus on some statistical theory of the reduced set method mainly in the context of the rsvm the use of a uniform random subset is not limited to the rsvm this approach can act as a supplemental algorithm on top of a basic optimization algorithm wherein the actual optimization takes place on the subset approximated data the statistical properties discussed in this paper are still valid

clustering of solutions in the random satisfiability problem using elementary rigorous methods we prove the existence of a clustered phase in the random k sat problem for k geq 8 in this phase the solutions are grouped into clusters which are far away from each other the results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks it can be generalized to other systems of both physical and computational interest

phase diagram of the 1 in 3 satisfiability problem we study typical case properties of the 1 in 3 satisfiability problem the boolean satisfaction problem where a clause is satisfied by exactly one literal in an enlarged random ensemble parametrized by average connectivity and probability of negation of a variable in a clause random 1 in 3 satisfiability and exact 3 cover are special cases of this ensemble we interpolate between these cases from a region where satisfiability can be typically decided for all connectivities in polynomial time to a region where deciding satisfiability is hard in some interval of connectivities we derive several rigorous results in the first region and develop a one step replica symmetry breaking cavity analysis in the second one we discuss the prediction for the transition between the almost surely satisfiable and the almost surely unsatisfiable phase and other structural properties of the phase diagram in light of cavity method results

bicolouring random hypergraphs we study the problem of bicolouring random hypergraphs both numerically and analytically we apply the zero temperature cavity method to find analytical results for the phase transitions dynamic and static in the one step replica symmetry breaking 1rsb approximation these points appear to be in agreement with the results of the numerical algorithm in the second part we implement and test the survey propagation algorithm for specific bicolouring instances in the so called hard sat phase

nonlinear optimization quadrature and interpolation we present a nonlinear optimization procedure for the design of generalized gaussian quadratures for a fairly broad class of functions while some of the components of the algorithm have been published previously we introduce an improved procedure for the determination of an acceptable initial point for the continuation scheme that stabilizes the newton type process used to find the quadratures the resulting procedure never failed when applied to chebyshev systems for which the existence and uniqueness of generalized gaussian quadratures are well known it also worked for many non chebyshev systems for which the generalized gaussian quadratures are not guaranteed to exist the performance of the algorithm is illustrated with several numerical examples some of the presented quadratures integrate efficiently large classes of singular functions

an improved estimate of pswf approximation and approximation by mathieu functions abstract in this paper an error estimate of spectral approximations by prolate spheroidal wave functions pswfs with explicit dependence on the bandwidth parameter and optimal order of convergence is derived which improves the existing result in chen et al spectral methods based on prolate spheroidal wave functions for hyperbolic pdes siam j numer anal 43 5 2005 1912 1933 the underlying argument is applied to analyze spectral approximations of periodic functions by mathieu functions which leads to new estimates featured with explicit dependence on the intrinsic parameter

the linear l 1 estimator and the huber m estimator relationships between a linear l1 estimation problem and the huber m estimator problem can be easily established by their dual formulations the least norm solution of a linear programming problem studied by mangasarian and meyer siam j control optim 17 1979 pp 745 752 provides a key link between the dual problems based on the dual formulations we establish a local linearity property of the huber m estimators with respect to the tuning parameter gamma and prove that the solution set of the huber m estimator problem is lipschitz continuous with respect to perturbations of the tuning parameter gamma as a consequence the set of the linear l1 estimators is the limit of the set of the huber m estimators as gamma to 0 thus the huber m estimator problem has many solutions for small tuning parameter gamma if the linear l1 estimation problem has multiple solutions a recursive version of madsen and nielsen s algorithm siam j optim 3 1993 pp 223 235 based on computation of the huber m estimator is proposed for finding a linear l1 estimator

dynamics of social balance on networks we study the evolution of social networks that contain both friendly and unfriendly pairwise links between individual nodes the network is endowed with dynamics in which the sense of a link in an imbalanced triad a triangular loop with one or three unfriendly links is reversed to make the triad balanced with this dynamics an infinite network undergoes a dynamic phase transition from a steady state to paradise all links are friendly as the propensity p for friendly links in an update event passes through 1 2 a finite network always falls into a socially balanced absorbing state where no imbalanced triads remain if the additional constraint that the number of imbalanced triads in the network not increase in an update is imposed then the network quickly reaches a balanced final state i introduction in this work we investigate the role of friends and enemies on the evolution of social networks we represent individuals as nodes in a graph and a relationship between individuals as a link that joins the corresponding nodes to quantify a relationship we assign the binary variable sij 1 to link ij with sij 1 if nodes i and j are friends and sij 1 ifi and j are enemies fig 1 a basic characterization of relationships between mutual acquaintances is the notion of social balance 1 2 the triad ijk is defined as balanced if the sign of the product of the links in the triad sijsjkski equals 1 while the triad is imbalanced otherwise we define a triad to be of type k if it contains k negative links thus 0 and 2 are balanced while 1 and 3 are imbalanced a balanced triad therefore fulfills the adage that i a friend of my friend is my friend ii an enemy of my friend

simultaneous classification and feature selection via convex quadratic programming with application to hiv associated neurocognitive disorder assessment abstract support vector machines svms that utilize a mixture of the l 1 norm and the l 2 norm penalties are capable of performing simultaneous classification and selection of highly correlated features these svms typically set up as convex programming problems are re formulated here as simple convex quadratic minimization problems over non negativity constraints giving rise to a new formulation the pq svm method solutions to our re formulation are obtained efficiently by an extremely simple algorithm computational results on a range of publicly available datasets indicate that these methods allow greater classification accuracy in addition to selecting groups of highly correlated features these methods were also compared on a new dataset assessing hiv associated neurocognitive disorder in a group of 97 hiv infected individuals

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

a quantum lovasz local lemma the lovasz local lemma lll is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of weakly dependent criteria we show that the lll extends to a much more general geometric setting where events are replaced with subspaces and probability is replaced with relative dimension which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions our result immediately applies to the k qsat problem quantum analog of k sat for instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most 2k e k of them has a joint satisfiable state we then apply our results to the recently studied model of random k qsat recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit where the density is the ratio of projectors to qubits using a hybrid approach building on work by laumann et al 2009 2010 we greatly extend the known satisfiable region for random k qsat to a density of 2k k2 since our tool allows us to show the existence of joint satisfying states without the need to construct them we are able to penetrate into regions where the satisfying states are conjectured to be entangled avoiding the need to construct them which has limited previous approaches to product states

a new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations a new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested this method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions stability and convergence results are proven for the chebyshev approximation of linear scalar hyperbolic equations the eigenvalues of this method applied to parabolic equations are shown to be real and negative

coloring random graphs we study the graph coloring problem over random graphs of finite average connectivity c given a number q of available colors we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable depending on q we find the precise value of the critical average connectivity c q moreover we show that below c q there exists a clustering phase c e c d c q in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms

replica bounds for diluted non poissonian spin systems in this paper we extend replica bounds and free energy subadditivity arguments to diluted spin glass models on graphs with arbitrary non poissonian degree distribution the new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method as a byproduct we obtain self averaging identities that generalize the ghirlanda guerra ones to the multi overlap case

analysis of spectral approximations using prolate spheroidal wave functions in this paper the approximation properties of the prolate spheroidal wave functions of order zero pswfs are studied and a set of optimal error estimates are derived for the pswf approximation of non periodic functions in sobolev spaces these results serve as an indispensable tool for the analysis of pswf spectral methods a pswf spectral galerkin method is proposed and analyzed for elliptic type equations illustrative numerical results consistent with the theoretical analysis are also presented

new quadrature formulas from conformal maps gauss and clenshaw curtis quadrature like legendre and chebyshev spectral methods make use of grids strongly clustered at boundaries from the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal nevertheless such methods may waste a factor of pi 2 with respect to each space dimension we propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight sided domain the new methods are compared with related ideas of bakhvalov kosloff and tal ezer rokhlin and alpert and others an advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands for example one of the formulas presented is proved to converge 50 faster than gauss quadrature for functions analytic in an varepsilon neighborhood of 1 1

pairs of sat assignments in random boolean formulae we investigate geometrical properties of the random k satisfiability problem using the notion of x satisfiability a formula is x satisfiable is there exist two sat assignments differing in nx variables we show the existence of a sharp threshold for this property as a function of the clause density for large enough k we prove that there exists a region of clause density below the satisfiability threshold where the landscape of hamming distances between sat assignments experiences a gap pairs of sat assignments exist at small x and around x 12 but they do not exist at intermediate values of x this result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis the cavity method and its algorithmic counterpart the survey propagation algorithm our method uses elementary probabilistic arguments first and second moment methods and might be useful in other problems of computational and physical interest where similar phenomena appear

phase coexistence and finite size scaling in random combinatorial problems we study an exactly solvable version of the well known random boolean satisfiability sat problem the so called random xor sat problem rare events are shown to affect the combinatorial phase diagram leading to a coexistence of solvable and unsolvable instances of the combinatorial problem in a certain region of the parameters characterizing the model such instances differ by a non extensive quantity in the ground state energy of the associated diluted spin glass model we also show that the critical exponent controlling the size of the critical window where the probability of having solutions vanishes depends on the model parameters shedding light on the link between random hyper graph topology and universality classes in the case of random sat a similar behaviour was conjectured to be connected to the onset of computational intractability

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

regular article spectral collocation time domain modeling of diffractive optical elements a spectral collocation multi domain scheme is developed for the accurate and efficient time domain solution of maxwell s equations within multi layered diffractive optical elements special attention is being paid to the modeling of out of plane waveguide couplers emphasis is given to the proper construction of high order schemes with the ability to handle very general problems of considerable geometric and material complexity central questions regarding efficient absorbing boundary conditions and time stepping issues are also addressed the efficacy of the overall scheme for the time domain modeling of electrically large and computationally challenging problems is illustrated by solving a number of plane as well as non plane waveguide problems

on the freezing of variables in random constraint satisfaction problems the set of solutions of random constraint satisfaction problems zero energy groundstates of mean field diluted spin glasses undergoes several structural phase transitions as the amount of constraints is increased this set first breaks down into a large number of well separated clusters at the freezing transition which is in general distinct from the clustering one some variables spins take the same value in all solutions of a given cluster in this paper we introduce and study a message passing procedure that allows to compute for generic constraint satisfaction problems the sizes of the rearrangements induced in response to the modification of a variable these sizes diverge at the freezing transition with a critical behavior which is also investigated in details we apply the generic formalism in particular to the random satisfiability of boolean formulas and to the coloring of random graphs the computation is first performed in random tree ensembles for which we underline a connection with percolation models and with the reconstruction problem of information theory the validity of these results for the original random ensembles is then discussed in the framework of the cavity method

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

linear dependency between spl epsi and the input noise in spl epsi support vector regression in using the spl epsi support vector regression spl epsi svr algorithm one has to decide a suitable value for the insensitivity parameter spl epsi smola et al considered its optimal choice by studying the statistical efficiency in a location parameter estimation problem while they successfully predicted a linear scaling between the optimal spl epsi and the noise in the data their theoretically optimal value does not have a close match with its experimentally observed counterpart in the case of gaussian noise in this paper we attempt to better explain their experimental results by studying the regression problem itself our resultant predicted choice of spl epsi is much closer to the experimentally observed optimal value while again demonstrating a linear trend with the input noise

witness of unsatisfiability for a random 3 satisfiability formula the random 3 satisfiability 3 sat problem is in the unsatisfiable unsat phase when the clause density alpha exceeds a critical value alpha s approximate to 4 267 rigorously proving the unsatisfiability of a given large 3 sat instance is however extremely difficult in this paper we apply the mean field theory of statistical physics to the unsatisfiability problem and show that a reduction to 3 xorsat which permits the construction of a specific type of unsat witnesses feige kim ofek witnesses is possible when the clause density alpha 19 we then construct feige kim ofek witnesses for single 3 sat instances through a simple random sampling algorithm and a focused local search algorithm the random sampling algorithm works only when a scales at least linearly with the variable number n but the focused local search algorithm works for clause density alpha cn b with b approximate to 0 59 and prefactor c approximate to 8 the exponent b can be further decreased by enlarging the single parameter s of the focused local search algorithm

spectral analysis of the finite hankel transform and circular prolate spheroidal wave functions in this paper we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite hankel transform operator these different eigenfunctions are called circular prolate spheroidal wave functions cpswfs this work is motivated by the potential applications of the cpswfs as well as the development of practical methods for computing their values also in this work we should prove that the cpswfs form an orthonormal basis of the space of hankel band limited functions an orthogonal basis of l 2 0 1 and an orthonormal system of l 2 0 our computation of the cpswfs and their associated eigenvalues is done by the use of two different methods the first method is based on a suitable matrix representation of the finite hankel transform operator the second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials also we give two maple programs that implement the previous two methods finally we present some numerical results that illustrate the results of this work

universality class of triad dynamics on a triangular lattice we consider triad dynamics as it was recently considered by antal et al phys rev e 72 036121 2005 as an approach to social balance here we generalize the topology from all to all to the regular one of a two dimensional triangular lattice the driving force in this dynamics is the reduction of frustrated triads in order to reach a balanced state the dynamics is parametrized by a so called propensity parameter p that determines the tendency of negative links to become positive as a function of p we find a phase transition between different kinds of absorbing states the phases differ by the existence of an infinitely connected percolated cluster of negative links that forms whenever p ensuremath leqslant p c moreover for p ensuremath leqslant p c the time to reach the absorbing state grows powerlike with the system size l while it increases logarithmically with l for pg p c from a finite size scaling analysis we numerically determine the static critical exponents ensuremath beta and ensuremath nu ensuremath perp together with ensuremath gamma ensuremath tau ensuremath sigma and the dynamical critical exponents ensuremath nu ensuremath vert and ensuremath delta the exponents satisfy the hyperscaling relations we also determine the fractal dimension d f that satisfies a hyperscaling relation as well the transition of triad dynamics between different absorbing states belongs to a universality class with different critical exponents we generalize the triad dynamics to four cycle dynamics on a square lattice in this case again there is a transition between different absorbing states going along with the formation of an infinite cluster of negative links but the usual scaling and hyperscaling relations are violated

simplest random k satisfiability problem we study a simple and exactly solvable model for the generation of random satisfiability problems these consist of gamman random boolean constraints which are to be satisfied simultaneously by n logical variables in statistical mechanics language the considered model can be seen as a diluted p spin model at zero temperature while such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space still at least one solution may be superimposed by construction the statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method the geometrical and topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in the local search method are thoroughly analyzed numerical analysis on very large samples allows for a precise characterization of the critical scaling behavior

image registration under illumination variations using region based confidence weighted m estimators we present an image registration model for image sets with arbitrarily shaped local illumination variations between images any nongeometric variations tend to degrade the geometric registration precision and impact subsequent processing traditional image registration approaches do not typically account for changes and movement of light sources which result in interimage illumination differences with arbitrary shape in addition these approaches typically use a least square estimator that is sensitive to outliers where interimage illumination variations are often large enough to act as outliers in this paper we propose an image registration approach that compensates for arbitrarily shaped interimage illumination variations which are processed using robust m estimators tuned to that region each m estimator for each illumination region has a distinct cost function by which small and large interimage residuals are unevenly penalized since the segmentation of the interimage illumination variations may not be perfect a segmentation confidence weighting is also imposed to reduce the negative effect of mis segmentation around illumination region boundaries the proposed approach is cast in an iterative coarse to fine framework which allows a convergence rate similar to competing intensity based image registration approaches the overall proposed approach is presented in a general framework but experimental results use the bisquare m estimator with region segmentation confidence weighting a nearly tenfold improvement in subpixel registration precision is seen with the proposed technique when convergence is attained as compared with competing techniques using both simulated and real data sets with interimage illumination variations

extremal optimization at the phase transition of the three coloring problem we investigate the phase transition in vertex coloring on random graphs using the extremal optimization heuristic three coloring is among the hardest combinatorial optimization problems and is equivalent to a 3 state anti ferromagnetic potts model like many other such optimization problems it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter the graph s mean vertex degree this phase transition is often associated with the instances of highest complexity we use extremal optimization to measure the ground state cost and the backbone an order parameter related to ground state overlap averaged over a large number of instances near the transition for random graphs of size n up to 512 for these graphs benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about osn 3 5 d update steps finite size scaling yields a critical mean degree value ac 4 703s28d furthermore the exploration of the degenerate ground states indicates that the backbone order parameter measuring the constrainedness of the problem exhibits a first order phase transition

robust convex quadratically constrained programs in this paper we study robust convex quadratically constrained programs a subset of the class of robust convex programs introduced by ben tal and nemirovski 4 in contrast to 4 where it is shown that such robust problems can be formulated as semidefinite programs our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second order cone programs socp we propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit socp and present examples where these classes of uncertainty sets are natural

generalized holomorphic szego kernel in 3d spheroids monogenic orthogonal polynomials over 3d prolate spheroids were previously introduced and shown to have some remarkable properties in particular the underlying functions take values in the quaternions identified with r 4 and are generally assumed to be nullsolutions of the well known moisil theodoresco system in this paper we show that these polynomial functions play an important role in defining the szego kernel function over the surface of 3d prolate spheroids as a concrete application we prove an explicit expression of the monogenic szego kernel function over 3d prolate spheroids and present two numerical examples

minimal kernel classifiers a finite concave minimization algorithm is proposed for constructing kernel classifiers that use a minimal number of data points both in generating and characterizing a classifier the algorithm is theoretically justified on the basis of linear programming perturbation theory and a leave one out error bound as well as effective computational results on seven real world datasets a nonlinear rectangular kernel is generated by systematically utilizing as few of the data as possible both in training and in characterizing a nonlinear separating surface this can result in substantial reduction in kernel data dependence over 94 in six of the seven public datasets tested on and with test set correctness equal to that obtained by using a conventional support vector machine classifier that depends on many more data points this reduction in data dependence results in a much faster classifier that requires less storage to eliminate data points the proposed approach makes use of a novel loss function the pound function which is a linear combination of the 1 norm and the step function that measures both the magnitude and the presence of any error