iterative time limited signal restoration the purpose of this paper is to show that time limited restoration of shift invariant blurred signals can be done by means of fixed point solutions of contraction mappings under rather general conditions for the distortion operator all our results are valid for multidimensional signals an application of these results to the iterative extrapolation of band limited discrete images is shown

a unified approach to noniterative linear signal restoration the main goal of this paper is to describe a unified framework for several noniterative algorithms for signal extrapolation reported in the literature this unification is achieved through integral equation and hilbert space theories the importance of this unification is that we can bring to bear the vast body of techniques in these theories to the solution of the extrapolation problem we will show that the so called two step procedures for extrapolation with different underlying models can be unified by means of noniterative algorithms for solving optimization problems in hilbert spaces in particular we show that two step procedures under a discrete continuous model 1 2 belong to a general class of well known algorithms for solving linear integral equations of the first kind given g x x in a find f t t in omega such that g x int min omega k x t f t dt x in a 1 in addition we will show that the prolate spheroidal expansion technique is also a special case of the well known picard s eigenfunction procedure for the general integral problem 1 this theoretical unification together with that presented in 3 for iterative least squares algorithms demonstrates that most of the well known procedures for band limited extrapolation can be considered as special cases of standard techniques in integral equations and operator theory

improving newton raphson method for nonlinear equations by modified adomian decomposition method in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method the modified adomian decomposition method is applied to construct the numerical algorithms some numerical illustrations are given to show the efficiency of algorithms

superresolution reconstruction through object modeling and parameter estimation a method based on object modeling and parameter estimation is proposed to achieve superresolution reconstruction an efficient method for solving for the model parameters is given that uses linear prediction theory and linear least squares fitting reconstruction results from simulated and real magnetic resonance data are also presented to demonstrate its capability for gibbs ringing reduction and resolution enhancement

extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

computational complexity of holant problems we propose and explore a novel alternative framework to study the complexity of counting problems called holant problems compared to counting constraint satisfaction problems csp it is a refinement with a more explicit role for the constraint functions both graph homomorphism and csp can be viewed as special cases of holant problems we prove complexity dichotomy theorems in this framework our dichotomy theorems apply to local constraint functions which are symmetric functions on boolean input variables and evaluate to arbitrary real or complex values we discover surprising tractable subclasses of counting problems which could not easily be specified in the csp framework when all unary functions are assumed to be free mathrm holant problems the tractable ones consist of functions that are degenerate or of arity at most two or holographic transformations of fibonacci gates when only two special unary functions the constant zero and constant one functions are assumed to be free mathrm holant c problems we further identify three special families of tractable cases then we prove that all other cases are p hard the main technical tool we use and develop is holographic reductions another technical tool used in combination with holographic reductions is polynomial interpolations

energy concentration in band limited extrapolation the authors present a novel bandlimited extrapolation method that incorporates qualitative prior knowledge of the distribution of signal energy in both the time domain spatial domain and the frequency domain the extrapolation is easily computed by simple noniterative algorithm the prior information is incorporated by means of a weighted energy measure and the resulting solution is derived by straightforward hilbert space optimization the algorithm accommodates complex valued sequences nonuniform sampling and arbitrary spectral support and is applicable in one or several dimensions many existing extrapolation algorithms are noted as special cases examples of signal extrapolation and spectrum estimation using the new algorithm are given in one and two dimensions algorithm modification for noisy data is investigated and error bounds are presented

generalizing constraint satisfaction on trees hybrid tractability and variable elimination the constraint satisfaction problem csp is a central generic problem in artificial intelligence considerable progress has been made in identifying properties which ensure tractability in such problems such as the property of being tree structured in this paper we introduce the broken triangle property which allows us to define a novel tractable class for this problem which significantly generalizes the class of problems with tree structure we show that the broken triangle property is conservative i e it is preserved under domain reduction and hence under arc consistency operations and that there is a polynomial time algorithm to determine an ordering of the variables for which the broken triangle property holds or to determine that no such ordering exists we also present a non conservative extension of the broken triangle property which is also sufficient to ensure tractability and can also be detected in polynomial time we show that both the broken triangle property and its extension can be used to eliminate variables and that both of these properties provide the basis for preprocessing procedures that yield unique closures orthogonal to value elimination by enforcement of consistency finally we also discuss the possibility of using the broken triangle property in variable ordering heuristics

clifford gates in the holant framework abstract we show that the clifford gates and stabilizer circuits in the quantum computing literature which admit efficient classical simulation are equivalent to affine signatures under a unitary condition the latter is a known class of tractable functions under the holant framework

combinatorial problems raised from 2 semilattices abstract the constraint satisfaction problem csp provides a general framework for many combinatorial problems in a a bulatov a a krokhin p g jeavons classifying the complexity of constraints using finite algebras siam j comput 34 3 2005 720 742 p g jeavons on the algebraic structure of combinatorial problems theoret comput sci 200 1998 185 204 and then in a a bulatov p g jeavons algebraic structures in combinatorial problems technical report math al 4 2001 technische universitat dresden dresden germany 2001 a new approach to the study of the csp has been developed which uses properties of universal algebras assigned to certain subclasses of the csp such that the time complexity and other properties of subclasses can be derived from the properties of the assigned algebras in this paper we briefly survey this approach and then prove that problem classes corresponding to finite 2 semilattices that is groupoids satisfying the identities x x x x y y x x x y x x y can be solved in polynomial time making use of this result we classify finite conservative groupoids and 4 element algebras with minimal clone of term operations with respect to the complexity of the corresponding csp class

block interpolation a framework for tight exponential time counting complexity we devise a framework for proving tight lower bounds under the counting exponential time hypothesis mathsf eth introduced by dell et al our framework allows to convert many known mathsf p hardness results for counting problems into results of the following type if the given problem admits an algorithm with running time 2 o n on graphs with n vertices and mathcal o n edges then mathsf eth fails as exemplary applications of this framework we obtain such tight lower bounds for the evaluation of the zero one permanent the matching polynomial and the tutte polynomial on all non easy points except for two lines

some aspects of band limited signal extrapolation models discrete approximations and noise we present some theoretical results on the band limited signal extrapolation problem in section i we describe four basic models for the extrapolation problem these models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in 1 and 2 one of these models was shown to approximate the continuous band limited extrapolation problem 3 another model is obtained when the discrete fourier transform dft is used to implement the well known iterative algorithm given in 4 and 5 which was designed for solving the continuous extrapolation problem in section ii this model is related to the continuous model by means of an interesting approximation theorem also an important conjecture is presented section iii shows some approximation results specifically we prove that some discrete discrete and discrete continuous extrapolations of noisy signals converge to solutions of a certain continuous continuous noisy extrapolation problem when the noise is bounded by a known number max eta x leq epsilon this convergence is obtained by using normal families of entire functions in nand some other complex analysis tools we also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired this result indicates that in the presence of noise extrapolation techniques should be used judiciously in order to obtain reasonable results

a quadratically convergent algorithm for convex set constrained signal recovery this paper addresses the problem of recovering a signal that is constrained to lie in a convex set from linear measurements the current standard is the alternating projections paradigm pocs which has only first order convergence in general we present a quadratically convergent iterative algorithm newton algorithm for signal recovery from linear measurements and convex set constraints a new result on the existence and construction of the derivative of the projection operator onto a convex set is obtained which is used in the newton algorithm an interesting feature of the new algorithm is that each iteration requires the solution of a simpler subspace constrained reconstruction problem a computation and memory efficient version of the algorithm is also obtained by using the conjugate gradient algorithm within each newton iteration to avoid matrix inversion and storage from a computational point of view the computation per iteration of this algorithm is similar to the computation per iteration of the standard alternating projections algorithm the faster rate of convergence compared to alternating projections enables us to obtain a high resolution reconstruction with fewer computations the algorithm is thus well suited for large scale problems that typically arise in image recovery applications the algorithm is demonstrated in several applications

constraint satisfaction problems solvable by local consistency methods we prove that constraint satisfaction problems without the ability to count are solvable by the local consistency checking algorithm this settles three equivalent conjectures feder vardi sicomp 98 bulatov lics 04 and larose zadori au 07

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

an iterative method with quartic convergence for solving nonlinear equations in this study a new two step and three step iterative methods are derived for solving nonlinear equations it is shown that the three step iterative method has fourth order convergence several numerical examples are given to illustrate the efficiency and performance of the new method

classifying the complexity of constraints using finite algebras many natural combinatorial problems can be expressed as constraint satisfaction problems this class of problems is known to be np complete in general but certain restrictions on the form of the constraints can ensure tractability here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra hence we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra we introduce a notion of tractable algebra and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it including its subalgebras and homomorphic images this allows us to reduce significantly the types of algebras which need to be classified using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently then so can the corresponding search problem we then classify all finite strictly simple surjective algebras with respect to tractability obtaining a dichotomy theorem which generalizes schaefer s dichotomy for the generalized satisfiability problem finally we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem

a dichotomy theorem for constraint satisfaction problems on a 3 element set the constraint satisfaction problem csp provides a common framework for many combinatorial problems the general csp is known to be np complete however certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes there is therefore a fundamental research direction aiming to separate those subclasses of the csp that are tractable and those which remain np complete schaefer gave an exhaustive solution of this problem for the csp on a 2 element domain in this article we generalise this result to a classification of the complexity of the csp on a 3 element domain the main result states that every subproblem of the csp is either tractable or np complete and the criterion separating them is that conjectured in bulatov et al 2005 and bulatov and jeavons 2001b we also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure finally we exhibit a polynomial time algorithm which for a given set of allowed constraints outputs if this set gives rise to a tractable problem class to obtain the main result and the algorithm we extensively use the algebraic technique for the csp developed in jeavons 1998b bulatov et al 2005 and bulatov and jeavons 2001b

a generic arc consistency algorithm and its specializations consistency techniques have been studied extensively in the past as a way of tackling constraint satisfaction problems csp in particular various arc consistency algorithms have been proposed originating from waltz s filtering algorithm and culminating in the optimal algorithm ac 4 of mohr and henderson ac 4 runs in o ed 2 in the worst case where e is the number of arcs or constraints and d is the size of the largest domain being applicable to the whole class of binary csp these algorithms do not take into account the semantics of constraints in this paper we present a new generic arc consistency algorithm ac 5 the algorithm is parameterized on two specified procedures and can be instantiated to reduce to ac 3 and ac 4 more important ac 5 can be instantiated to produce an o ed 2 algorithm for a number of important classes of constraints functional anti functional monotonic and their generalization to functional anti functional and monotonic piecewise constraints we also show that ac 5 has an important application in constraint logic programming over finite domains the kernel of the constraint solver for such a programming language is an arc consistency algorithm for a set of basic constraints we prove that ac 5 in conjuction with node consistency provides a decision procedure for these constraints running in time o ed

the complexity of quantified constraint satisfaction collapsibility sink algebras and the three element case the constraint satisfaction probem csp is a well acknowledged framework in which many combinatorial search problems can be naturally formulated the csp may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints in front of which all variables are existentially quantified the quantified constraint satisfaction problem qcsp is the generalization of the csp where universal quantification is permitted in addition to existential quantification the general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language which is a set of relations that can be used to express constraints this paper introduces collapsibility a technique for deriving positive complexity results on the qcsp in particular this technique allows one to show that for a particular constraint language the qcsp reduces to the csp we show that collapsibility applies to three known tractable cases of the qcsp that were originally studied using disparate proof techniques in different decades quantified 2 sat aspvall plass and tarjan in 1979 quantified horn sat karpinski kleine buning and schmitt in 1987 and quantified affine sat creignou khanna and sudan in 2001 this reconciles and reveals common structure among these cases which are describable by constraint languages over a two element domain in addition to unifying these known tractable cases we study constraint languages over domains of larger size

on blockwise symmetric matchgate signatures and higher domain csp abstract for any n 3 and q 3 we prove that the equality function n on n variables over a domain of size q cannot be realized by matchgates under holographic transformations this is a consequence of our theorem on the structure of blockwise symmetric matchgate signatures this has the implication that the standard holographic algorithms based on matchgates a methodology known to be universal for csp over the boolean domain cannot produce p time algorithms for planar csp over any higher domain q 3

a comparison of structural csp decomposition methods abstract we compare tractable classes of constraint satisfaction problems csps we first give a uniform presentation of the major structural csp decomposition methods we then introduce a new class of tractable csps based on the concept of hypertree decomposition recently developed in database theory and analyze the cost of solving csps having bounded hypertree width we provide a framework for comparing parametric decomposition based methods according to tractability criteria and compare the most relevant methods we show that the method of hypertree decomposition dominates the others in the case of general csps i e csps of unbounded arity we also make comparisons for the restricted case of binary csps finally we consider the application of decomposition methods to the dual graph of a hypergraph in fact this technique is often used to exploit binary decomposition methods for nonbinary csps however even in this case the hypertree decomposition method turns out to be the most general method

on counting homomorphisms to directed acyclic graphs it is known that if p and np are different then there is an infinite hierarchy of different complexity classes that lie strictly between them thus if p np it is not possible to classify np using any finite collection of complexity classes this situation has led to attempts to identify smaller classes of problems within np where dichotomy results may hold every problem is either in p or is np complete a similar situation exists for counting problems if p p there is an infinite hierarchy in between and it is important to identify subclasses of p where dichotomy results hold graph homomorphism problems are a fertile setting in which to explore dichotomy theorems indeed feder and vardi have shown that a dichotomy theorem for the problem of deciding whether there is a homomorphism to a fixed directed acyclic graph would resolve their long standing dichotomy conjecture for all constraint satisfaction problems in this article we give a dichotomy theorem for the problem of counting homomorphisms to directed acyclic graphs let h be a fixed directed acyclic graph the problem is given an input digraph g determine how many homomorphisms there are from g to h we give a graph theoretic classification showing that for some digraphs h the problem is in p and for the rest of the digraphs h the problem is p complete an interesting feature of the dichotomy which is absent from previously known dichotomy results is that there is a rich supply of tractable graphs h with complex structure

a modified chebyshev s iterative method with at least sixth order of convergence this paper is devoted to the construction and analysis of a high order variant of the classical chebyshev method the method has order of convergence at least six for simple roots the extension to system of equations and its semilocal convergence for nonlinear equations are presented finally an application to well known algebraic riccati equation is considered

experimental evaluation of preprocessing algorithms for constraint satisfaction problems abstract this paper presents an experimental evaluation of two orthogonal schemes for pre processing constraint satisfaction problems csps the first of these schemes involves a class of local consistency techniques that includes directional arc consistency directional path consistency and adaptive consistency the other scheme concerns the prearrangement of variables in a linear order to facilitate an efficient search in the first series of experiments we evaluated the effect of each of the local consistency techniques on backtracking and backjumping surprisingly although adaptive consistency has the best worst case complexity bounds we have found that it exhibits the worst performance unless the constraint graph was very sparse directional arc consistency followed by either backjumping or backtracking and backjumping without any preprocessing outperformed all other techniques moreover the former dominated the latter in computationally intensive situations the second series of experiments suggests that maximum cardinality and minimum width are the best preordering i e static ordering strategies while dynamic search rearrangement is superior to all the preorderings studied

on newton type methods with cubic convergence recently there has been some progress on newton type methods with cubic convergence that do not require the computation of second derivatives weerakoon and fernando appl math lett 13 2000 87 derived the newton method and a cubically convergent variant by rectangular and trapezoidal approximations to newton s theorem while frontini and sormani j comput appl math 156 2003 345 140 2003 419 derived further cubically convergent variants by using different approximations to newton s theorem homeier j comput appl math 157 2003 227 169 2004 161 independently derived one of the latter variants and extended it to the multivariate case here we show that one can modify the werrakoon fernando approach by using newton s theorem for the inverse function and derive a new class of cubically convergent newton type methods

spin systems on k regular graphs with complex edge functions let k 1 be an integer and let h h 0 0 h 0 1 h 1 0 h 1 1 be a complex valued symmetric function on domain 0 1 i e where h 0 1 h 1 0 we introduce a new technique called a syzygy and prove a dichotomy theorem for the following class of problems specified by k and h given an arbitrary k regular graph g v e where the function h is attached to each edge compute z g email protected s v 0 1 u v eh s u s v z is known as the partition function of the spin system also known as counting graph homomorphisms on domain size two and is a special case of holant problems the dichotomy theorem gives a complete classification of the computational complexity of this problem depending on k and h the dependence on k and h is explicit

extrapolation and spectral estimation with iterative weighted norm modification an algorithm is developed to define from the data samples themselves a frequency weighted norm to use in minimum weighted norm extrapolation the iterative procedure developed consists of using a periodogram spectrum estimate obtained from some samples of the signal estimate extrapolation found at one iteration to define the weight that is used to estimate at the next iteration this algorithm usually converges in less than 10 iterations to an extrapolation which is characterized as a nonparametric frequency stationary extension of the data the frequency resolution and extrapolation length are controlled by the length of a time domain window used to obtain smooth spectral estimates between iterations examples are provided to illustrate the use of the algorithm for interpolation extrapolation the examples give comparable results to nonadaptive extrapolation methods without the need for a priori knowledge for a certain spectral estimation example the algorithm provides comparable resolution to the parametric methods with more accurate values of the relative strengths of the narrowband components

a fourth order iterative method for solving nonlinear equations the present paper illustrates an iterative numerical method to solve nonlinear equations of the form f x 0 especially those containing the partial and non partial involvement of transcendental terms comparative analysis shows that the present method is faster than newton raphson method hybrid iteration method new hybrid iteration method and others cost is also found to be minimum than these methods the beauty in our method can be seen because of the optimization in important effecting factors i e lesser number of iteration steps lesser number of functional evaluations and lesser value of absolute error in final as well as in individual step as compared to the other methods this work also demonstrates the higher order convergence of the present method as compared to others without going to the computation of second derivative

a linear time varying system framework for noniterative discrete time band limited signal extrapolation a technique for the analysis and design of noniterative algorithms for discrete time band limited signal extrapolation is described the approach involves modeling the extrapolation process as a linear time varying ltv system or filter together with a previously developed fourier theory for ltv systems this model provides a frequency domain transfer function representation for the extrapolation system this representation serves as a powerful tool for characterizing and comparing the reconstruction properties of several well known least squares optimal algorithms for band limited extrapolation moreover the frequency domain setting provides a conceptually attractive means for understanding the process of extrapolation itself additionally a least squares approximation methodology for designing ltv filters for band limited extrapolation is developed the design technique is shown to unify a broad class of algorithms for extrapolating discrete time data and further to provide a means for designing new and improved extrapolation algorithms

a complexity dichotomy for partition functions with mixed signs partition functions also known as homomorphism functions form a rich family of graph invariants that contain combinatorial invariants such as the number of k colorings or the number of independent sets of a graph and also the partition functions of certain spin glass models of statistical physics such as the ising model building on earlier work by dyer and greenhill random structures algorithms 17 2000 pp 260 289 and bulatov and grohe theoret comput sci 348 2005 pp 148 186 we completely classify the computational complexity of partition functions our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or p complete partition functions are described by symmetric matrices with real entries and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or p complete while in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases for partition functions described by hadamard matrices these turn out to be central in our proofs we obtain a simple algebraic tractability criterion which says that the tractable cases are those representable by a quadratic polynomial over the field mathbb f 2

continuation techniques for a certain class of analytic functions this paper deals with the numerical continuation problem of analytic functions g z given by g z int omega k z t f t dt where k defines a bounded operator on l 2 omega it is assumed that g is known over a finite segment a of the real line where g is to be sampled our continuation techniques emerged partly from a generalization of the landweber iteration we show that a certain discrete approximation of the proposed iterative technique yields two step noniterative algorithms for solving the continuation problem we also prove convergence of these approximations to the sought function g special emphasis on the continuation problem for the case k z t e 2 pi izt is given and some related numerical examples are presented the continuation problem when the known part of g z is contaminated with some noise is addressed and some techniques for solving this problem are also provided

an optimal k consistency algorithm abstract this paper generalizes the arc consistency algorithm of mohr and henderson 4 and the path consistency algorithm of han and lee 2 to a k consistency algorithm arc consistency and path consistency being 2 consistency and 3 consistency respectively the algorithm is a development of freuder s synthesis algorithm 1 it simultaneously establishes i consistency for each 1 i k it has worst case time and space complexity which is optimal when k is a constant and almost optimal for all other values of k in the case that all order i constraints exist for all 1 i n this algorithm is a solution to the consistent labeling problem with almost optimal worst case time and space complexity

simulating quantum computation by contracting tensor networks the treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree we prove that a quantum circuit with t gates whose underlying graph has a treewidth d can be simulated deterministically in t o 1 exp o d time which in particular is polynomial in t if d o log t among many implications we show efficient simulations for log depth circuits whose gates apply to nearby qubits only a natural constraint satisfied by most physical implementations we also show that one way quantum computation of raussendorf and briegel phys rev lett 86 2001 pp 5188 5191 a universal quantum computation scheme with promising physical implementations can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small treewidth graph with a constant maximum degree the requirement on the maximum degree was removed in i l markov and y shi preprint quant ph 0511069

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

lee yang theorems and the complexity of computing averages we study the complexity of computing average quantities related to spin systems such as the mean magnetization and susceptibility in the ferromagnetic ising model and the average dimer count or average size of a matching in the monomer dimer model by establishing connections between the complexity of computing these averages and the location of the complex zeros of the partition function we show that these averages are p hard to compute and hence under standard assumptions computationally intractable in the case of the ising model our approach requires us to prove an extension of the famous lee yang theorem from the 1950s

the analysis of the continuous time lms algorithm a continuous time analog adaptive filter is suggested using the digital prototype the continuous time lms least mean squares algorithm is then described by a set of simultaneous first order equations the adaptive gain is shown to be unbounded theoretically

fifth order iterative methods for solving nonlinear equations abstract in this paper we suggest and analyze a new two step iterative method for solving nonlinear equation f x 0 by rewriting the given nonlinear equation as a coupled system of equations and using the taylor series it is shown that this new iterative method is of fifth order several examples are given to illustrate its performance and efficiency comparison with other methods is also given this method can be viewed as an improvement of the halley s method

holographic algorithms by fibonacci gates abstract we introduce fibonacci gates as a polynomial time computable primitive and develop a theory of holographic algorithms based on these gates the fibonacci gates play the role of matchgates in valiant s theory valiant 2008 19 they give rise to polynomial time computable counting problems on general graphs while matchgates mainly work over planar graphs only we develop a signature theory and characterize all realizable signatures for fibonacci gates for bases of arbitrary dimensions we prove a basis collapse theorem we apply this theory to give new polynomial time algorithms for certain counting problems we also use this framework to prove that some slight variations of these counting problems are p hard holographic algorithms with fibonacci gates prove to be useful as a general tool for classification results of counting problems dichotomy theorems cai et al 2009 7

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

closure properties of constraints many combinatorial search problems can be expressed as constraint satisfaction problems and this class of problems is known to be np complete in general in this paper we investigate the subclasses that arise from restricting the possible constraint types we first show that any set of constraints that does not give rise to an np complete class of problems must satisfy a certain type of algebraic closure condition we then investigate all the different possible forms of this algebraic closure property and establish which of these are sufficient to ensure tractability as examples we show that all known classes of tractable constraints over finite domains can be characterized by such an algebraic closure property finally we describe a simple computational procedure that can be used to determine the closure properties of a given set of constraints this procedure involves solving a particular constraint satisfaction problem which we call an indicator problem

fourth order variants of cauchy s method for solving non linear equations abstract in this paper we present some new variants of cauchy s method in which the second derivative is replaced by a finite difference between the first derivatives analysis of convergence shows that the new methods have fourth order convergence per iteration the new methods require one evaluations of the function and two of its first derivative numerical results show that the new methods are efficient

interpolation and the discrete papoulis gerchberg algorithm analyze the performance of an iterative algorithm similar to the discrete papoulis gerchberg algorithm and which can be used to recover missing samples in finite length records of band limited data no assumptions are made regarding the distribution of the missing samples in contrast with the often studied extrapolation problem in which the known samples are grouped together indeed it is possible to regard the observed signal as a sampled version of the original one and to interpret the reconstruction result studied as a sampling result the authors show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data they give upper and lower bounds for the error as a function of the number of iterations together with the signals for which the bounds are attained also they analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix from this analysis they infer the optimum value of the relaxation constant they also point out among all sampling sets with the same density those for which the convergence rate of the recovery algorithm is maximum or minimum for low pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer the worst convergence rates generally occur when the missing samples are contiguous

third order iterative methods free from second derivative for nonlinear systems in this work we present a family of predictor corrector methods free from second derivative for solving nonlinear systems we prove that the methods of this family are of third order convergence we also perform numerical tests that allow us to compare these methods with newton s method in addition the numerical examples improve theoretical results showing super cubic convergence for some methods of this family

new iterative technique for solving a system of nonlinear equations various problems of pure and applied sciences can be studied in the unified frame work of the system of nonlinear equations in this paper a new family of iterative methods for solving a system of nonlinear equations is developed by using a new decomposition technique the convergence of the new methods is proved efficiency index of the proposed methods is discussed and compared with some other well known methods the upper bounds of the error and the radius of convergence of the methods are also found for the implementation and performance of the new methods the combustion problem streering problem and van der pol equation are solved and the results are compared with some existing methods several new iterative methods are derived from the general iterative scheme using the ideas and techniques of this paper one may be able to suggest and investigate a wide class of iterative methods for solving the system of nonlinear equations this is another direction of future research

discrete and continuous band limited signal extrapolation this paper has two main purposes first some new algorithms for the extrapolation of multidimensional band limited sequences are presented these algorithms extend those given in 1 in two ways 1 we do not impose any restrictions on either the shape of the region containing the set of samples of the two dimensional signal to be extrapolated or on the shape of its passband zone 2 we obtain a class of sequences which are band limited extrapolations of the given data including the minimum norm solution of 1 as a special case the second objective of out paper is to relate the discrete extrapolation problem to the continuous signal extrapolation problem 2 specifically we prove that the solution obtained by using our approach for the discrete extrapolation problem tends to the solution of the continuous problem when the sampling rate used in the known part of the continuous signal to be extrapolated approaches infinity

on the algebraic structure of combinatorial problems abstract we describe a general algebraic formulation for a wide range of combinatorial problems including satisfiability graph colorability and graph isomorphism in this formulation each problem instance is represented by a pair of relational structures and the solutions to a given instance are homomorphisms between these relational structures the corresponding decision problem consists of deciding whether or not any such homomorphisms exist we then demonstrate that the complexity of solving this decision problem is determined in many cases by simple algebraic properties of the relational structures involved this result is used to identify tractable subproblems of satisfiability and to provide a simple test to establish whether a given set of boolean relations gives rise to one of these tractable subproblems

a complete dichotomy rises from the capture of vanishing signatures we prove a complexity dichotomy theorem for holant problems over an arbitrary set of complex valued symmetric constraint functions mathcal f on boolean variables this extends and unifies all previous dichotomies for holant problems on symmetric constraint functions taking values without a finite modulus we define and characterize all symmetric vanishing signatures they turn out to be essential to the complete classification of holant problems the dichotomy theorem has an explicit tractability criterion expressible in terms of holographic transformations a holant problem defined by a set of constraint functions mathcal f is solvable in polynomial time if it satisfies this tractability criterion and is p hard otherwise the tractability criterion can be intuitively stated as follows a set mathcal f is tractable if 1 every function in mathcal f has arity at most two or 2 mathcal f is transformable to an affine type or 3 mathcal f is transformable to a product type o

an iterative method with cubic convergence for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations we show that this new iterative method has third order convergence several numerical examples are given to illustrate the efficiency and performance of this new method new method can be viewed as an improvement of the previously known iterative methods

constraints consistency and closure although the constraint satisfaction problem is np complete in general a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency in this paper we describe a simple algebraic property which characterises all possible constraint types for which strong k consistency is sufficient to ensure global consistency for each k 2 we give a number of examples to illustrate the application of this result

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

resolution limits in signal recovery resolution analysis for the problem of signal recovery from finitely many linear measurements is the subject of this paper the classical rayleigh limit serves only as a lower bound on resolution since it does not assume any recovery strategy and is based only on observed data we show that details finer than the rayleigh limit can be recovered by simple linear processing that incorporates prior information we first define a measure of resolution based on allowable levels of error that is more appropriate for current signal recovery strategies than the rayleigh definition in the practical situation in which only finitely many noisy observations are available we have to restrict the class of signals in order to make the resolution measure meaningful we consider the set of bandlimited and essentially timelimited signals since it describes most signals encountered in practice for this set we show how to precompute resolution limits from knowledge of measurement functionals signal to noise ratio passband energy concentration regions energy concentration factor and a prescribed level of error tolerance in the process we also derive an algorithm for high resolution signal recovery we illustrate the results with examples in one and two dimensions

beyond csp a dichotomy for counting weighted eulerian orientations with ars abstract we define and explore a notion of unique prime factorization for constraint functions and use this as a new tool to prove a complexity classification for counting weighted eulerian orientation problems with arrow reversal symmetry ars we prove that all such problems are either polynomial time computable or p hard we show that the class of weighted eulerian orientation problems subsumes all weighted counting constraint satisfaction problems csp on boolean variables more significantly we establish a novel connection between csp and counting weighted eulerian orientation problems that is global in nature this connection is based on a structural determination of all half weighted affine linear subspaces over z 2 which is proved using mobius inversion

an iterative method for solving nonlinear functional equations an iterative method for solving nonlinear functional equations viz nonlinear volterra integral equations algebraic equations and systems of ordinary differential equation nonlinear algebraic equations and fractional differential equations has been discussed

computational complexity of covering three vertex multigraphs a covering projection from a graph g onto a graph h is a mapping of the vertices of g onto the vertices of h such that for every vertex v of g the neighborhood of v is mapped bijectively onto the neighborhood of its image moreover if g and h are multigraphs then this local bijection has to preserve multiplicities of the neighbors as well the notion of covering projection stems from topology but has found applications in areas such as the theory of local computation and construction of highly symmetric graphs it provides a restrictive variant of the constraint satisfaction problem with additional symmetry constraints on the behavior of the homomorphisms of the structures involved we investigate the computational complexity of the problem of deciding the existence of a covering projection from an input graph g to a fixed target graph h among other partial results this problem has been shown np hard for simple regular graphs h of valency greater than 2 and a full characterization of computational complexity has been shown for target multigraphs with 2 vertices we extend the previously known results to the ternary case i e we give a full characterization of the computational complexity in the case of multigraphs with 3 vertices we show that even in this case a p np completeness dichotomy holds

third and fourth order iterative methods free from second derivative for nonlinear systems in this work we develop a family of predictor corrector methods free from second derivative for solving systems of nonlinear equations in general the obtained methods have order of convergence three but in some particular cases the order is four we also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with newton s classical method and with other recently published methods

the approximability of max csp with fixed value constraints in the maximum constraint satisfaction problem max csp one is given a finite collection of possibly weighted constraints on overlapping sets of variables and the goal is to assign values from a given finite domain to the variables so as to maximize the number or the total weight for the weighted case of satisfied constraints this problem is np hard in general and therefore it is natural to study how restricting the allowed types of constraints affects the approximability of the problem in this article we show that any max csp problem with a finite set of allowed constraint types which includes all fixed value constraints i e constraints of the form x e a is either solvable exactly in polynomial time or else is apx complete even if the number of occurrences of variables in instances is bounded moreover we present a simple description of all polynomial time solvable cases of our problem this description relies on the well known algebraic combinatorial property of supermodularity

maximum entropy image reconstruction in x ray and diffraction tomography the authors propose a bayesian approach with maximum entropy me priors to reconstruct an object from either the fourier domain data the fourier transform of diffracted field measurements in the case of diffraction tomography or directly from the original projection data in the case of x ray tomography the objective function obtained is composed of a quadratic term resulting from chi sup 2 statistics and an entropy term that is minimized using variational techniques and a conjugate gradient iterative method the computational cost and practical implementation of the algorithm are discussed some simulated results in x ray and diffraction tomography are given to compare this method to the classical ones

a computational proof of complexity of some restricted counting problems we explore a computational approach to proving the intractability of certain counting problems these problems can be described in various ways and they include concrete problems such as counting the number of vertex covers or independent sets for 3 regular graphs the high level principle of our approach is algebraic which provides sufficient conditions for interpolation to succeed another algebraic component is holographic reductions we then analyze in detail polynomial maps on r 2 induced by some combinatorial constructions these maps define sufficiently complicated dynamics of r 2 that we can only analyze them computationally in this paper we use both numerical computation as intuitive guidance and symbolic computation as proof theoretic verification to derive that a certain collection of combinatorial constructions in myriad combinations fulfills the algebraic requirements of proving p hardness the final result is a dichotomy theorem for a class of counting problems this includes a class of generic holant problems with an arbitrary real valued edge signature over 2 3 regular undirected graphs in particular it includes all partition functions with 0 1 vertex assignments and an arbitrary real valued edge function over all 3 regular undirected graphs

bandlimited extrapolation using time bandwidth dimension the problem of extrapolating discrete index bandlimited signals from a finite number of samples is addressed in this paper the algorithm presented in this paper exploits the fact that the set of bandlimited signals that are also essentially time limited is approximated well by a low dimensional linear subspace this fact which is well known for one dimensional 1 d signals with contiguous passbands and time concentration intervals is established for a more general class of multidimensional m d signals with discontiguous passbands and discontiguous time concentration regions a criterion is presented for determining the dimension of the approximating subspace and the minimax optimal subspace itself based on knowledge of the passband time concentration regions energy concentration factor and bounds on the tolerable extrapolation error the extrapolation is constrained to lie in this subspace and parameters characterizing the extrapolation are obtained from the data by solving a linear system of equations for certain sampling patterns the system is ill conditioned and a second rank reduction is needed to reduce the deleterious effects of observation noise and modeling error a novel criterion for rank selection based on known bounds on noise power and modeling error is presented the effectiveness of the new algorithm and the rank selection criterion are demonstrated by means of computer simulations

convergence of iterative nonexpansive signal reconstruction algorithms iterative algorithms for signal reconstruction from partial time and frequency domain knowledge have proven useful in a number of application areas in this paper a general convergence proof applicable to a general class of such iterative reconstruction algorithms is presented the proof relies on the concept of a nonexpansive mapping in both the time and frequency domains two examples studied in detail are time limited extrapolation equivalently band limited extrapolation and phase only signal reconstruction the proof of convergence for the phase only iteration is a new result obtained by this method of proof the generality of the approach allows the incorporation of nonlinear constraints such as time or space domain positivity or minimum and maximum value constraints finally the underrelaxed form of these iterations is also shown to converge even when the solution is not guaranteed to be unique

the complexity of weighted boolean csp this paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted boolean constraint satisfaction problem the problem is parameterized by a finite set mathcal f of nonnegative functions that may be used to assign weights to the configurations feasible solutions of a problem instance classical constraint satisfaction problems correspond to the special case of 0 1 valued functions we show that computing the partition function i e the sum of the weights of all configurations is text sf fp text sf p complete unless either 1 every function in mathcal f is of product type or 2 every function in mathcal f is pure affine in the remaining cases computing the partition function is in p

on the realization of band limited power spectra from partial covariance sequence the set m omega of all discrete time power spectra supported on a known interval omega omega with omega

analytical solutions to the pulsed klein gordon equation using modified variational iteration method mvim and boubaker polynomials expansion scheme bpes in this study we propose an analytical solution to the klein gordon equation in a pulsed stationary regime the performed protocols are based on the modified variational iteration method mvim and boubaker polynomials expansion scheme bpes the results are presented and compared with some solutions proposed later in order to confirm the good accuracy of the protocols used

from local to global consistency abstract in reasoning tasks involving the maintenance of consistent databases so called qqconstraint networks q q it is customary to enforce local consistency conditions in order to simplify the subsequent construction of a globally coherent model of the data in this paper we present a relationship between the sizes of the variables domains the constraints arity and the level of local consistency sufficient to ensure global consistency based on these parameters a new tractability classification of constraint networks is presented we also show based on this relationship that any relation on bi valued variables which is not representable by a network of binary constraints cannot be represented by networks with any number of hidden variables

holographic algorithm with matchgates is universal for planar csp over boolean domain we prove a complexity classification theorem that classifies all counting constraint satisfaction problems csp over boolean variables into exactly three classes 1 polynomial time solvable 2