two combinatorial models with identical statics yet different dynamics motivated by the problem of sorting we introduce two simple combinatorial models with distinct hamiltonians yet identical spectra and hence partition function and show that the local dynamics of these models is very different after a deep quench one model slowly relaxes to the sorted state whereas the other model becomes blocked by the presence of stable local minima

traveling salesman problem with a center we study a traveling salesman problem where the path is optimized with a cost function that includes its length l as well as a certain measure c of its distance from the geometrical center of the graph using simulated annealing sa we show that such a problem has a transition point that separates two phases differing in the scaling behavior of l and c in efficiency of sa and in the shape of minimal paths

statistical mechanics of combinatorial optimization problems with site disorder we study the statistical mechanics of a class of problems whose phase space is the set of permutations of an ensemble of quenched random positions specific examples analyzed are the finite temperature traveling salesman problem on several different domains and various problems in one dimension such as the so called descent problem we first motivate our method by analyzing these problems using the annealed approximation then in the limit of a large number of points we develop a formalism to carry out the quenched calculation this formalism does not require the replica method and its predictions are found to agree with monte carlo simulations in addition our method reproduces an exact mathematical result for the maximum traveling salesman problem in two dimensions and suggests its generalization to higher dimensions the general approach may provide an alternative method to study certain systems with quenched disorder

a torelli theorem for simply connected elliptic surfaces with a section and p_g geqslant 2 0 the purpose of this note is to announce some recent results on the period mapping for simply connected elliptic surfaces with a section and p 2 among other things we are able to prove the period mapping for such surfaces has degree one that is to say it is generically injective in the terminology of c g this is called the weak global torelli theorem in order to make a precise statement of our results we must introduce some notation

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

experimental complexity analysis of continuous constraint satisfaction problems continuous constraint satisfaction problems are at the core of many real world applications including planning scheduling control and diagnosis of physical systems such as car planes and factories yet in order to be effective in such resource constrained environments constraint based techniques must take into account the complexity of continuous constrained problems in this paper we study complexity phase transition phenomena of continuous constraint satisfaction problems csps first we analyze three continuous constraint satisfaction formulations based on discrete 3 sat problems which have a strong relation between structure and search cost then we propose a generic benchmarking model for comparing continuous csps and algorithms and present two example problems based on sine functions in our experiments these problems are solved using both local and global search methods besides comparing the complexities of different continuous csps and search algorithms we also connect these back to results from similar studies on sat problems in solving continuous 3 sat and sine based csps we find that the median search cost is characterized by simple parameters such as the constraint to variable ratio and constraint tightness and that discrete search algorithms such as gsat have continuous counter parts with similar behavior regarding local versus global search techniques for constraint solving our results show that local search methods are more efficient for weakly constrained problems whereas global search methods work better on highly constrained problems

epsilon transformation exploiting phase transitions to solve combinatorial optimization problems abstract it has been shown that there exists a transition in the average case complexity of tree search problems from exponential to polynomial in the search depth we develop a new method called e transformation which makes use of this complexity transition to find a suboptimal solution with a random tree model we show that as the tree depth approaches infinity the expected number of nodes expanded by branch and bound bnb using e transformation is at most cubic in the search depth and that the relative error of the solution cost found with respect to the optimal solution cost is bounded above by a small constant we also present an iterative version of e transformation that can be used to find both suboptimal and optimal goal nodes depth first bnb dfbnb using iterative e transformation significantly improves upon dfbnb on random trees with large branching factors and deep goal nodes finding a better solution sooner we then present experimental results for e transformation and iterative e transformation on the asymmetric traveling salesman problem atsp and the maximum boolean satisfiability problem and identify the conditions under which these two methods are effective on the atsp dfbnb using e transformation outperforms a well known local search method and dfbnb using iterative e transformation improves upon dfbnb

gaussian mixture parameter estimation with known means and unknown class dependent variances this paper develops a recursive convergent estimator for some parameters of gaussian mixtures the m class conditional component densities of the mixture random variable are gaussian with known and distinct means and unknown and possibly different variances a joint estimator of m prior mixing probabilities and m class conditional variances is derived sufficient conditions on the data and control parameters are derived for the estimator to converge convergence of the estimator follows from the use of a stochastic approximation theorem techniques to extend the estimators for the case of successive class labels forming a markov chain are mentioned the estimator has applications in blind parameter estimation in digital communication with symbol dependent noise variance and in image compression

unsupervised context estimation in a mesh of pattern classes for image recognition abstract this paper defines a wide class of models for the statistical dependency of class labels context in a neighborhood within an image a computationally efficient closed form estimator for these context defining parameters is developed the estimator is unbiased convergent in the mean square sense and can be recursively implemented parameters of several practical context models within the class of models are estimated as functions of the estimates of the parameters in the general model the appropriate model can also be decided with the help of the context parameter estimates applications of the estimator in existing context classifiers are mentioned a limitation of the estimator is also pointed out iterative approaches for decision making in markov models are outlined to highlight the use of our estimator to ensure the convergence of iterative decisions to a local optimum

parameter estimation known vector signals in unknown gaussian noise this paper develops recursive convergent estimators for the parameters of finite gaussian mixtures with a common covariance matrix the mean vectors signals of the component densities are assumed to be known the motivation for the study stems from digital communication the basic approach is first illustrated for the case of an independent identically distributed sequence of samples from a univariate mixture of m classes symbols this is accomplished through the development of a convergent stochastic approximation form of estimator for the common variance value the asymptotic variance of the estimated variance is derived a batch processing alternative that possesses a sufficient statistic is developed for the case of a fixed size sample set three generalizations are studied the first extends from the case of the univariate data to multivariate data the second generalization allows for the statistical dependence of successive vector signals finally the case of dependent successive vector signals along with dependent successive additive noise vectors is treated in each case convergent estimators for all unknown parameters are developed many cases are illustrated with simulation experiments results presented are applicable to communication engineering pattern recognition and some special image processing problems

performance of linear space search algorithms abstract search algorithms that use space linear in the search depth are widely employed in practice to solve difficult problems optimally such as planning and scheduling in this paper we study the average case performance of linear space search algorithms including depth first branch and bound dfbnb iterative deepening id and recursive best first search rbfs to facilitate our analyses we use a random tree t b d that has mean branching factor b depth d and node costs that are the sum of the costs of the edges from the root to the nodes we prove that the expected number of nodes expanded by dfbnb on a random tree is no more than bd times the expected number of nodes expanded by best first search bfs on the same tree which usually requires space that is exponential in depth d we also show that dfbnb is asymptotically optimal when bfs runs in exponential time and id and rbfs are asymptotically optimal when the edge costs of t b d are integers if bp 0 is the expected number of children of a node whose costs are the same as that of their parent then the expected number of nodes expanded by these three linear space algorithms is exponential when bp o d 4 when bp 0 1 and at most quadratic when bp 0 1 in addition we study the heuristic branching factor of t b d and the effective branching factor of bfs dfbnb id and rbfs on t b d furthermore we use our analytic results to explain a surprising anomaly in the performance of these algorithms and to predict the existence of a complexity transition in the asymmetric traveling salesman problem

a complete anytime algorithm for number partitioning abstract given a set of numbers the two way number partitioning problem is to divide them into two subsets so that the sum of the numbers in each subset are as nearly equal as possible the problem is np complete based on a polynomial time heuristic due to karmarkar and karp we present a new algorithm called complete karmarkar karp ckk that optimally solves the general number partitioning problem and significantly outperforms the best previously known algorithms for large problem instances for numbers with twelve significant digits or less ckk can optimally solve two way partitioning problems of arbitrary size in practice for numbers with greater precision ckk first returns the karmarkar karp solution then continues to find better solutions as time allows over seven orders of magnitude improvement in solution quality is obtained in less than an hour of running time rather than building a single solution one element at a time or modifying a complete solution ckk constructs subsolutions and combines them together in all possible ways this approach may be effective for other np hard problems as well

phase transitions and the search problem we describe how techniques that were originally developed in statistical mechanics can be applied to search problems that arise commonly in artificial intelligence this approach is useful for understanding the typical behavior of classes of problems in particular these techniques predict that abrupt changes in computational cost analogous to physical phase transitions should occur universally as heuristic effectiveness or search space topology is varied we also present a number of open questions raised by these studies

searching for an optimal path in a tree with random costs we consider the problem of finding an optimal path leading from the root of a tree to any of its leaves the tree is known to be uniform binary and of height n and each branch independently may have a cost of 1 or 0 with probability p and 1 p respectively we show that for p 1 2 every algorithm which guarantees finding an exact cheapest path or even a path within a fixed cost ratio of the cheapest must run in exponential average time if however we are willing to accept a near optimal solution almost always then a pruning algorithm exists which finds such a solution in linear expected time the algorithm employs a depth first strategy which stops at regular intervals to appraise its progress and if the progress does not meet a criterion based on domain specific knowledge the current node is irrevocably pruned

higher noether lefschetz loci of elliptic surfaces we calculate the dimension of the locus of jacobian elliptic surfaces over p 1 with a given picard number in the corresponding moduli space

a geometric approach to non parametric density estimation a novel non parametric density estimator is developed based on geometric principles a penalised centroidal voronoi tessellation forms the basis of the estimator which allows the data to self organise in order to minimise estimate bias and variance this approach is a marked departure from usual methods based on local averaging and has the advantage of being naturally adaptive to local sample density scale invariance the estimator does not require the introduction of a plug in kernel thus avoiding assumptions of symmetricity and morphology a numerical experiment is conducted to illustrate the behaviour of the estimator and it s characteristics are discussed

variational torelli implies generic torelli david cox 1 ron donagi 2 and loring tu 3 1 department of mathematics amherst college amherst ma 01002 usa z department of mathematics northeastern university boston ma 02115 usa 3 department of mathematics tufts university medford ma 02155 usa introduction the methods of infinitesimal variation of hodge structure cf 3 provide a powerful tool for proving generic torelli theorems see 6 for the case of projective hypersurfaces however the success of this method depends on the implication variational torelli generic torelli in the literature various forms of this implication have appeared sometimes with proof cf 2 3 5 7 9 14 and 17 all previous precise formulations have required several auxiliary hypotheses including the existence of a coarse moduli space for the varieties in question the existence of a regular value for the period map i e a

the centred travelling salesman at high temperature a recently formulated statistical mechanics method is used to study the phase transition occurring in a generalization of the travelling salesman problem tsp known as the centred tsp the method shows that the problem has clear signs of a crossover but is only able to access unscaled finite temperatures above the transition point the solution of the problem using this method displays a curious duality

a study of complexity transitions on the asymmetric traveling salesman problem abstract the traveling salesman problem tsp is one of the best known combinatorial optimization problems branch and bound bnb is the best method for finding an optimal solution of the tsp previous research has shown that there exists a transition in the average computational complexity of bnb on random trees we show experimentally that when the intercity distances of the asymmetric tsp are drawn uniformly from 0 1 2 r the complexity of bnb experiences an easy hard transition as r increases we also observe easy hard easy complexity transitions when asymmetric intercity distances are chosen from a log normal distribution this transition pattern is similar to one previously observed on the symmetric tsp we then explain these different transition patterns by showing that the control parameter that determines the complexity is the number of distinct intercity distances