consistent labeling problems and their algorithms expected complexities and theory based heuristics a new parameter is introduced to characterize a type of search problem of broad relevance in artificial intelligence operations research and symbolic logic this parameter which we call inter variable compatibility is particularly important in that complexity analyses incorporating it are able to capture the dependence of problem complexity on search order used by an algorithm thus compatibility based theories can provide a theoretical basis for the extraction of heuristics for choosing good search orderings a long sought goal for such problems since it can lead to significant savings during search we carry out expected complexity analyses for the traditional backtrack algorithm as well as for two more recent algorithms that have been found empirically to be significant improvements forward checking and word wise forward checking we extract compatibility based ordering heuristics from the theory for forward checking preliminary experimental results are presented showing the large savings that result from their use similar savings can be expected for other algorithms when heuristics taking account of inter variable compatibilities are used our compatibility based theories also provide a more precise way of predicting which algorithm is best for a given problem

application of the karhunen loeve procedure for the characterization of human faces the use of natural symmetries mirror images in a well defined family of patterns human faces is discussed within the framework of the karhunen loeve expansion this results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix without increasing the complexity of the calculation the resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average

curve like structure extraction using minimal path propagation with backtracking minimal path techniques can efficiently extract geometrically curve like structures by finding the path with minimal accumulated cost between two given endpoints though having found wide practical applications e g line identification crack detection and vascular centerline extraction minimal path techniques suffer from some notable problems the first one is that they require setting two endpoints for each line to be extracted endpoint problem the second one is that the connection might fail when the geodesic distance between the two points is much shorter than the desirable minimal path shortcut problem in addition when connecting two distant points the minimal path connection might become inefficient as the accumulated cost increases over the propagation and results in leakage into some non feature regions near the starting point accumulation problem to address these problems this paper proposes an approach termed minimal path propagation with backtracking we found that the information in the process of backtracking from reached points can be well utilized to overcome the above problems and improve the extraction performance the whole algorithm is robust to parameter setting and allows a coarse setting of the starting point extensive experiments with both simulated and realistic data are performed to validate the performance of the proposed method

probabilistic analysis of a generalization of the unit clause literal selection heuristics for the k satisfiability problem abstract two algorithms for the k satisfiability problem are presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics the algorithms assign values to literals appearing in a given instance of k satisfiability one at a time until a solution is found or it is discovered that further assignments cannot lead to finding a solution one algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise the other algorithm uses a generalization of the unit clause rule as a heuristic for selecting the next literal at each step a literal is chosen randomly from a clause containing the least number of literals the algorithms run in polynomial time and it is shown that they find a solution to a random instance of k satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values it is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values

an average time analysis of backtracking formulas are given for the expected number of nodes in the backtrack tree that is generated while searching for all the solutions of a random predicate the most general formulas apply to selection from any set of predicates that obeys the following conditions each predicate is the conjunction of t terms selected from a set of terms t for any subset t leqq t the probability that the predicate contains only terms from t depends only on the size of t the set t must remain unchanged if each variable x i is replaced by p i x i where p i is a permutation function the time needed to evaluate the general formulas is proportional to v the number of variables in the predicate more detailed consideration is given to predicates whose terms are random disjunctive clauses with s literals t v alpha for some 1 alpha s and the random selections are done with repetition for this case the expected number of nodes is exp o v s alpha s 1 complete enumeration r

temporal constraint networks abstract this paper extends network based methods of constraint satisfaction to include continuous variables thus providing a framework for processing temporal constraints in this framework called temporal constraint satisfaction problem tcsp variables represent time points and temporal information is represented by a set of unary and binary constraints each specifying a set of permitted intervals the unique feature of this framework lies in permitting the processing of metric information namely assessments of time differences between events we present algorithms for performing the following reasoning tasks finding all feasible times that a given event can occur finding all possible relationships between two given events and generating one or more scenarios consistent with the information provided we distinguish between simple temporal problems stps and general temporal problems the former admitting at most one interval constraint on any pair of time points we show that the stp which subsumes the major part of vilain and kautz s point algebra can be solved in polynomial time for general tcsps we present a decomposition scheme that performs the three reasoning tasks considered and introduce a variety of techniques for improving its efficiency we also study the applicability of path consistency algorithms as preprocessing of temporal problems demonstrate their termination and bound their complexities

components of moduli spaces of spin curves with the expected codimension we prove a conjecture of gavril farkas claiming that for all integers r ge 2 and g ge left begin array c r 2 2 end array right there exists a component of the locus mathcal s r g of spin curves with a theta characteristic l such that h 0 l ge r 1 and h 0 l equiv r 1 text mod 2 which has codimension left begin array c r 1 2 end array right inside the moduli space mathcal s g of spin curves of genus g

network based heuristics for constraint satisfaction problems abstract many ai tasks can be formulated as constraint satisfaction problems csp i e the assignment of values to variables subject to a set of constraints while some csps are hard those that are easy can often be mapped into sparse networks of constraints which in the extreme case are trees this paper identifies classes of problems that lend themselves to easy solutions and develops algorithms that solve these problems optimally the paper then presents a method of generating heuristic advice to guide the order of value assignments based on both the sparseness found in the constraint network and the simplicity of tree structured csps the advice is generated by simplifying the pending subproblems into trees counting the number of consistent solutions in each simplified subproblem and comparing these counts to decide among the choices pending in the original problem

probe order backtracking the algorithm for constraint satisfaction problems probe order backtracking has an average running time much faster than any previously analyzed algorithm under conditions where solutions are common the algorithm uses a probing assignment a preselected test assignment to unset variables to help guide the search for a solution if the problem is not satisfied when the unset variables are temporarily set to the probing assignment the algorithm selects one of the relations which is not satisfied by the probing assignment and selects an unset variable which affects the value of that relation it then does a backtracking splitting step where it generates subproblems by setting the selected variable each possible way each subproblem is simplified and then solved recursively for random problems with v variables t clauses and probability p that a literal appears in a clause the average time for probe order backtracking is no more than v n when p geq ln t v plus lower order terms the best previous result was p geq sqrt ln t v when the algorithm is combined with an algorithm of franco that makes selective use of resolution the average time for solving random problems is no more than v n for all values of p when t leq o n 1 3 v ln v 2 3 the best previous result was t leq o n 1 3 v ln v 1 6 probe order backtracking also runs in polynomial average time when p leq 1 v compared with the best previous result of p leq 1 2v with probe order backtracking the range of p that leads to more than polynomial time is much smaller than that for previously analyzed algorithms

comments on a counterexample to a diameter algorithm for convex polygons in this paper we report the results of experimental studies of zero level one level and two level search rearrangement backtracking we establish upper and lower limits for the size problem for which one level backtracking is preferred over zero level and two level methods thereby showing that the zero level method is best for very small problems the one level method is best for moderate size problems and the two level method is best for extremely large problems together with our theoretical asymptotic formulas these measurements provide a useful guide for selecting the best search rearrangement method for a particular problem

solving satisfiability with less searching a new technique complement searching is given for reducing the amount of searching required to solve satisfiability constraint satisfaction problems search trees for these problems often contain subtrees that have approximately the same shape when this occurs knowledge that the first subtree does not have a solution can be used to reduce the searching in the second subtree only the part of the second subtree which is different from the first needs to be searched the pure literal rule of the davis putnam procedure is a special case of complement searching the new technique greatly reduces the amount of searching required to solve conjunctive normal form predicates that contain almost pure literals literals with a small number of occurrences

grothendieck s pairing on component groups of jacobians let r be a discrete valuation ring with field of fractions k let ak be an abelian variety over k with dual ak denote by a and a the corresponding neron models and by a and a their component groups in gr exp vii ix grothendieck used the notion of biextension invented by mumford to investigate how the duality between ak and ak is reflected on the level of neron models in fact the essence of the relationship between a and a is captured by a bilinear pairing a a q z

a t o 2 n 2 s o 2 n 4 algorithm for certain np complete problems in this paper we develop a general purpose algorithm that can solve a number of np complete problems in time t o 2 n 2 and space s o 2 n 4 the algorithm can be generalized to a fam

an analysis of backtracking with search rearrangement the search rearrangement backtracking algorithm of bitner and reingold comm acm 18 1975 pp 651 655 introduces at each level of the backtrack tree a variable with a minimal number of remaining values search order may differ on different branches for conjunctive normal form formulas with v variables s literals per term s geqq 3 and v alpha terms s 2 alpha s the average number of nodes in a search rearrangement backtrack tree is exp theta v s alpha 1 s 2 i e for some positive constants a 1 a 2 and v 0 when v geqq v 0 the number of nodes is between exp a 1 v s alpha 1 s 2 and exp a 2 v s alpha 1 s 2 for 1 alpha leqq s 2 the average number of nodes is between exp theta v s alpha 1 s 2 and exp theta ln v s 1 s 2 v s alpha 1 s 2 this compares with exp theta v s alpha s 1 for ordinary backtracking for 1 alpha s s

search rearrangement backtracking and polynomial average time the average time required for simple search rearrangement backtracking is compared with that for ordinary backtracking when each algorithm is used to find all solutions for random conjunctive normal form predicates the sets of random predicates are characterized by v the number of variables t v the number of clauses and p v the probability that a literal appears in clause for large v if vp v 1n 2 there is a difficult region where the average number of solutions per problem is exponentially small but backtracking requires an exponentially large time the difficult region for search rearrangement backtracking is only slightly smaller than the difficult region for ordinary backtracking it is conjectured that search rearrangement backtracking is exponentially faster than ordinary backtracking for nearly all of the difficult region it is proved that there is no major advantage in using search rearrangement backtracking outside of the difficult region

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

plucker forms and the theta map let rm su x r 0 be the moduli space of semistable vector bundles of rank r and trivial determinant over a smooth irreducible complex projective curve x the theta map theta r rm su x r 0 rightarrow bbb p n is the rational map defined by the ample generator of rm pic rm su x r 0 the main result of the paper is that theta r is generically injective if g gg r and x is general this partially answers the following conjecture proposed by beauville theta r is generically injective if x is not hyperelliptic the proof relies on the study of the injectivity of the determinant map d e wedge r h 0 e rightarrow h 0 det e for a vector bundle e on x and on the reconstruction of the grassmannian g r rm from a natural multilinear form associated to it defined in the paper as the pl ucker form the method applies to other moduli spaces of vector bundles on a projective variety x

linear discriminant analysis for two classes via removal of classification structure a new method for two class linear discriminant analysis called removal of classification structure is proposed its novelty lies in the transformation of the data along an identified discriminant direction into data without discriminant information and iteration to obtain the next discriminant direction it is free to search for discriminant directions oblique to each other and ensures that the informative directions already found will not be chosen again at a later stage the efficacy of the method is examined for two discriminant criteria studies with a wide spectrum of synthetic data sets and a real data set indicate that the discrimination quality of these criteria can be improved by the proposed method

boosted discriminant projections for nearest neighbor classification in this paper we introduce a new embedding technique to find the linear projection that best projects labeled data samples into a new space where the performance of a nearest neighbor classifier is maximized we consider a large set of one dimensional projections and combine them into a projection matrix which is not restricted to be orthogonal the embedding is defined as a classifier selection task that makes use of the adaboost algorithm to find an optimal set of discriminant projections the main advantage of the algorithm is that the final projection matrix does not make any global assumption on the data distribution and the projection matrix is created by minimizing the classification error in the training data set also the resulting features can be ranked according to a set of coefficients computed during the algorithm the performance of our embedding is tested in two different pattern recognition tasks a gender recognition problem and the classification of manuscript digits

minimal resolutions chow forms and ulrich bundles on k3 surfaces the minimal resolution conjecture mrc for points on a projective varietyx p predicts that the minimal graded free resolution of a general set x of points is as simple as the geometry of x allows originally the most studied case has been that when x p see epsw the general form of the mrc for subvarieties x p was formulated in mus and fmp the betti diagram of a large enough set x consisting of general points is obtained from the betti diagram of x by adding two rows indexed by u 1 and u where u is an integer depending on all differences bi 1 u 1 bi u are known and depend on the hilbert polynomial px and i u and see fmp the minimal resolution conjecture for general points on x predicts that

generating hard satisfiability problems abstract we report results from large scale experiments in satisfiability testing as has been observed by others testing the satisfiability of random formulas often appears surprisingly easy here we show that by using the right distribution of instances and appropriate parameter values it is possible to generate random formulas that are hard that is for which satisfiability testing is quite difficult our results provide a benchmark for the evaluation of satisfiability testing procedures

the pure literal rule and polynomial average time for a simple parameterized model of conjunctive normal form predicates we show that a simplified version of the davis putnam procedure can for many values of the parameters solve the satisfiability problem in polynomial average time let v be the number of variables t v the number of clauses in a predicate and p v the probability that a given literal appears in a clause p v is the same for all literals let varepsilon be any small positive constant and n any large positive integer then a version of the davis putnam procedure that uses only backtracking and the pure literal rule uses average time that is polynomial in the problem size when any of the following conditions are true for large v 1 t v leqq n ln v 2 t v geqq exp varepsilon v 3 p v geqq varepsilon or 4 p v leqq n ln v v 3 2 until recently the best previous bounds for cases 1 and 4 were t v leqq ln ln v ln3 and p v leqq exp v ln ln v these results show that t

polynomial average time satisfiability problems a class of random conjunctive normal form cnf predicates is characterized by v the number of variables from which literals may be formed giving 2v literals p v the probability that a given literal is in a random clause and t v the number of random independently selected clauses in a random predicate determining satisfiability on such a class using backtracking takes polynomial average time under the following conditions 1 limv vp v 0 and t v ln2 v ln v l p v and 2 lim vp v lim p v 0 and t v ln2 ep v exp 2vp v for any fixed 0 additional cases are also covered an analysis by goldberg showed that problems for which p v is constant can be solved in polynomial average time by a simplified davis putnam procedure we give an algorithm for solving problems with t v ln lnv ln3 in polynomial worst case time thus all problem sets for which p v or t v grows extremely rapidly or extremely slowly can be solved in polynomial average time we also show that backtracking requires exponential average time when limvp v 0 and t v ln2 v ln v 1 p v and when limvp v limv vp v 0 and t v ln2 ge ep v exp 2vp v

the gonality of general smooth curves with a prescribed plane nodal model let c be a smooth curve of genus g it is well known that c possesses a linear system g l if q l g 2n g 2 0 moreover if c is general and q g 2 k l and 6 kd k l 2 3 theorem 2 1 in loc cit and for each d this bound on 6 is sharp examples 4 1 and 4 2 in loc cit in this paper we consider the question for a general choice of f since the space parametrizing integral plane nodal curves with prescribed d and 6 is irreducible famous conjecture of severi proved by harris 9 the meaning of general becomes represented by a point of a particular nonempty zariski open subset of that parameter space note that qn x 3 g 0 if and only if 6 d 2 7d 18 2

on the existence of components of the hilbert scheme with the expected number of moduli in the study of the geometry of families of algebraic curves one common tool is the comparison between the extrinsic geometry represented by properties of their projective embeddings and of the hilbert scheme and the intrinsic geometry represented by abstract properties and the moduli space this comparison is often best represented by the natural map hn g r mg from the hilbert scheme hn g r of smooth curves in ip r of degree n and genus g or the hurwitz scheme for r 1 the severi variety for r 2 to the moduli space of smooth curves of genus g the behaviour of is very strongly influenced by the nature of its fibers the brillnoether varieties now whilemg is an irreducible variety the hilbert scheme hn g r is in general reducible and has in fact many components for r 3 this difference has inspired many authors in the search of what could be a good component of the hilbert scheme one possible answer to such a question is to apply the above principle of comparison ser

irreducibility and components rigid in moduli of the hilbert scheme of smooth curves denote by mathcal h d g r the hilbert scheme of smooth curves that is the union of components whose general point corresponds to a smooth irreducible and non degenerate curve of degree d and genus g in mathbb p r a component of mathcal h d g r is rigid in moduli if its image under the natural map pi mathcal h d g r dashrightarrow mathcal m g is a one point set in this note we provide a proof of the fact that mathcal h d g r has no components rigid in moduli for g 0 and r 3 from which it follows that the only smooth projective curves embedded in mathbb p 3 whose only deformations are given by projective transformations are the twisted cubic curves in case r ge 4 we also prove the non existence of a component of mathcal h d g r rigid in moduli in a certain restricted range of d g 0 and r in the course of the proofs we establish the irreducibility of mathcal h d g 3 beyond the range which has been known before

backjump based backtracking for constraint satisfaction problems the performance of backtracking algorithms for solving finite domain constraint satisfaction problems can be improved substantially by look back and look ahead methods look back techniques extract information by analyzing failing search paths that are terminated by dead ends look ahead techniques use constraint propagation algorithms to avoid such dead ends altogether this paper describes a number of look back variants including backjumping and constraint recording which recognize and avoid some unnecessary explorations of the search space the last portion of the paper gives an overview of look ahead methods such as forward checking and dynamic variable ordering and discusses their combination with backjumping

backtracking techniques for the job shop scheduling constraint satisfaction problem abstract this paper studies a version of the job shop scheduling problem in which some operations have to be scheduled within non relaxable time windows i e earliest latest possible start time windows this problem is a well known np complete constraint satisfaction problem csp a popular method for solving this type of problems involves using depth first backtrack search in our earlier work we focused on the development of consistency enforcing techniques and variable value ordering heuristics that improve the efficiency of this search procedure in this paper we combine these techniques with new look back schemes that help the search procedure recover from so called deadend search states i e partial solutions that cannot be completed without violating some constraints more specifically we successively describe three intelligent backtracking schemes 1 dynamic consistency enforcement dynamically identifies critical subproblems and determines how far to backtrack by selectively enforcing higher levels of consistency among variables participating in these critical subproblems 2 learning ordering from failure dynamically modifies the order in which variables are instantiated based on earlier conflicts and 3 incomplete backjumping heuristic abandons areas of the search space that appear to require excessive computational efforts these schemes are shown to 1 further reduce the average complexity of the backtrack search procedure 2 enable our system to efficiently solve problems that could not be solved otherwise due to excessive computation cost and 3 be more effective at solving job shop scheduling problems than other look back schemes advocated in the literature

the geometry of the moduli space of curves of genus 23 the problem of describing the birational geometry of the moduli space mg of complex curves of genus g has a long history severi already knew in 1915 thatmg is unirational for g 10 cf sev see also ac1 for a modern proof in the same paper severi conjectured thatmg is unirational for all genera g then for a long period this problem seemed intractable mumford writes in mu p 51 whether more mg s g 11 are unirational or not is a very interesting problem but one which looks very hard too especially if g is quite large the breakthrough came in the eighties when eisenbud harris and mumford proved that mg is of general type as soon as g 24 and that the kodaira dimension of m23 is 1 see hm eh3 we note that mg is rational for g 6 see dol for problems concerning the rationality of various moduli spaces severi s proof of the unirationality of mg for small g was based on representing a general curve of genus g as a plane curve of degree d with nodes this is possible when d 2g 3 2 when the number of nodes is small i e d 1 d 2 6 the dominant map from the variety of plane curves of degree d and genus g to mg yields a rational parametrization of the moduli space the two conditions involving d and can be satisfied only when g 10 so severi s argument cannot be extended for other genera however using much more subtle ideas chang ran and sernesi proved the unirationality of mg for g 11 12 13 see cr1 se1 while for g 15 16 they proved that the kodaira dimension is see cr2 4 the remaining cases g 14 and 17 g 23 are still quite mysterious harris and morrison conjectured in hmo that mg is uniruled precisely when g 23 all these facts indicate that m23 is a very interesting transition case our main result is the following

the mercat conjecture for stable rank 2 vector bundles on generic curves we prove the mercat conjecture for rank 2 vector bundles on generic curves of every genus for odd genus we identify the effective divisor in mg where the mercat conjecture fails and compute its slope the clifford index of a smooth curve c defined as the non negative quantity cliff c min deg l 2r l l pic c h c l 2 deg l g 1 stratifies the moduli space mg of curves of genus g with the smallest stratum being that consisting of hyperelliptic curves which have clifford index zero while the definition is obviously inspired by the classical clifford theorem the notion came to real prominence in the works of green lazarsfeld g gl voisin v2 and many others in the context of syzygies of canonical curves for a general curve c of genus g one has cliff c bg 1 2 c the locus of curves c mg with cliff c bg 1 2 c is a subvariety of codimension 1 respectively 2 when g is odd respectively even much studied by harris and mumford hm it has been a long standing problem to find an adequate definition of the clifford index for higher rank vector bundles on curves a higher rank clifford index should not only capture the behavior of the generic curve from the point of view of special higher rank vector bundles but also provide a geometrically meaningful stratification ofmg an interesting definition has been put forward by lange and newstead ln1 for a semistable vector bundle e of rank 2 and slope e on a curve c of genus g one defines its clifford index as cliff e e h c e 2 0 the rank 2 clifford index of c is then defined as the quantity cliff2 c min cliff e e uc 2 d d 2g 2 h c e 4 observe that if l is a line bundle on c with deg l 2h0 c l 2 cliff c that is l computes the clifford index of c then cliff l l cliff c in particular the inequality cliff2 c cliff c holds for every curve c of genus g the main achievement of this paper is the proof of the following result theorem 1 for a general curve c of genus g the following equality holds cliff2 c cliff c g 1 2 equivalently if e is a semistable rank 2 vector bundle on c contributing to cliff2 c then h c e e 2 cliff c mercat me conjectured the equality cliff2 c cliff c for every smooth curve c mg counterexamples to this expectation have been found using noether lefschetz special k3 surfaces in fo1 fo2 and ln2 however these examples are special in moduli and theorem 1 proves mercat s conjecture for generic curves of every genus

brill noether loci and the gonality stratification of m g for an irreducible smooth projective complex curve c of genus g the gonality defined as gon c min d z 1 there exists a gd on c is perhaps the second most natural invariant it gives an indication of how far c is from being rational in a way different from what the genus does for g 3 we consider the stratification of the moduli space mg of smooth curves of genus g given by gonality mg 2 m 1 g 3 m 1 g k mg

on the existence of certain families of curves introduction 25 w 1 generalities on families of curves in ip 28 w the map 0 d 33 w 3 the number of moduli of a family 40 w 4 plane curves existence of families with the expected number of moduli 43 w 5 smoothing certain reducible curves in ip r r 3 47 w 6 curves in ip r 3 existence of families with the expected number of moduli 52 w curves in ip3 a more general existence theorem 55

average time for the full pure literal rule the simplified pure literal algorithm solves satisfiability problems by choosing variables in a fixed order and then generating subproblems for various values of the chosen variable if some value satisfies every relation that depends on the chosen variable then only the subproblem for that preferred value is generated otherwise a subproblem is generated for every value of the variable the full pure literal algorithm chooses variables that have a preferred value before choosing those that do not a recurrence equation is found for the average time used by the full pure literal rule algorithm when solving random conjunctive normal form satisfiability problems the random problems are characterized by the number of variables v the number of clauses t and the probability that a literal is in a clause p an asymptotic lower bound analysis shows that running time is more than polynomial in v when t increases more rapidly than ln v 2 when p is set to maximize the running time a numerical study indicates that the results of the lower bound analysis are close to the true results thus the full pure literal rule is faster than the simple pure literal rule where the polynomial time boundary occurs at t lnv but slower than franco s infrequent variable algorithm where the time is polynomial for all p when t o v16

convergence properties of optimization algorithms for the sat problem the satisfiability sat problem is a basic problem in computing theory presently an active area of research on sat problem is to design efficient algorithms to find a solution for a satisfiable conjunctive normal form cnf formula a new formulation the universal sat problem model which transforms the sat problem on boolean space into an optimization problem on real space has been developed j gu 1988 1992 1994 many optimization techniques such as the steepest descent method newton s method and the coordinate descent method can be used to solve the universal sat problem we prove that when the initial solution is sufficiently close to the optimal solution the steepest descent method has a linear convergence ratio spl beta 1 newton s method has a convergence ratio of order two and the convergence ratio of the steepest descent method is approximately 1 spl beta m for the universal sat problem with m variables an algorithm based on the coordinate descent method for the universal sat problem is also presented experimental results show that this algorithm is more efficient than some previous ones in finding a solution for certain classes of the satisfiable cnf formulas

nonparametric discriminant analysis a nonparametric method of discriminant analysis is proposed it is based on nonparametric extensions of commonly used scatter matrices two advantages result from the use of the proposed nonparametric scatter matrices first they are generally of full rank this provides the ability to specify the number of extracted features desired this is in contrast to parametric discriminant analysis which for an l class problem typically can determine at most l 1 features second the nonparametric nature of the scatter matrices allows the procedure to work well even for non gaussian data sets using the same basic framework a procedure is proposed to test the structural similarity of two distributions the procedure works in high dimensional space it specifies a linear decomposition of the original data space in which a relative indication of dissimilarity along each new basis vector is provided the nonparametric scatter matrices are also used to derive a clustering procedure which is recognized as a k nearest neighbor version of the nonparametric valley seeking algorithm the form which results provides a unified view of the parametric nearest mean reclassification algorithm and the nonparametric valley seeking algorithm

divisors on general curves and cuspidal rational curves in t roduc t ion 371 0 pre l iminar ies 377 1 a pl i icker formula 378 2 d imens iona l t ransversa l i ty 380 3 d i m e n s i o n of families of r a t iona l curves with fixed degree and ass igned cusps in a project ive space 383 4 l inear series on curves with universa l s ingular i t ies 386 5 l inear series and project ive embedd ings of general curves 391 6 embedd ings of general cuspidal curves 400 7 degenera t ion of schuber t in tersect ions some special cases 403 8 degene ra t ion of schuber t in tersect ions ma in results 407 9 g c is reduced for general cuspidal r a t iona l c 414

families of nodal curves on projective threefolds and their regularity via postulation of nodes the main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds precisely given x a smooth projective threefold e a rank two vector bundle on x l a very ample line bundle on x and k 0 0 integers and denoted by v e l k the subscheme of p h 0 e l k parametrizing global sections of e l k whose zero loci are irreducible and nodal curves on x we present a new cohomological description of the tangent space t s v e l k at a point s 2 v e l k this description enable us to determine effective and uniform upper bounds for which are linear polynomials in k such that the family v e lk is smooth and of the expected dimension regular for short the almost sharpness of our bounds is shown by some interesting examples furthermore when x is assumed to be a fano or a calaby yau threefold we study in detail the regularity property of a point s 2 v e lk related to the postulation of the nodes of its zero locus c v s x roughly speaking when the nodes of c are assumed to be in general position either on x or on an irreducible divisor of x having at worst log terminal singularities or to lie on a l c i and subcanonical curve in x we find upper bounds onwhich are respectively cubic quadratic and linear polynomials in k ensuring the regularity of v e l k at s finally when x p 3 we also discuss some interesting geometric properties of the curves given by sections parametrized by v e ox k