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

the theory of non linear elastic ship water interaction dynamics non linear mathematical models are developed to provide formulations of the equations of motion describing the dynamical interaction behaviour between an incompressible or compressible ideal fluid and a moving or fixed elastic or rigid structure the general theoretical approach is based on the fundamental equations of continuum mechanics the concept of hamilton s principle and suitably formulated variational principles the resultant mathematical model expressed in a fixed or a moving frame of reference allows the theoretical establishment of non linear problems associated with ship dynamics and offshore engineering through applications of the variational principles this is demonstrated by rigorously deriving the governing equations of motion for general non linear ship water interaction problems in particular the theory is applied to a rigid ship travelling in calm water or in waves a bottom fixed rigid rod or tower excited by an incident wave and a two dimensional elastic beam travelling in waves

a collapse theorem for holographic algorithms with matchgates on domain size at most 4 holographic algorithms with matchgates are a novel approach to design polynomial time computation they use kasteleyn s algorithm for perfect matchings and more importantly a holographic reduction the two fundamental parameters of a holographic reduction are the domain size k of the underlying problem and the basis size a holographic reduction transforms the computation to matchgates by a linear transformation that maps to a tensor product space of a linear space of dimension 2 we prove a sharp basis collapse theorem which shows that for domain size 3 and 4 all non trivial holographic reductions can be expressed with a basis of size 1 or 2 respectively the main proof techniques are matchgate identities and a group property of matchgate signatures

an effective dichotomy for the counting constraint satisfaction problem bulatov proceedings of the 35 th international colloquium on automata languages and programming part 1 lecture notes in comput sci 5125 springer new york 2008 pp 646 661 gave a dichotomy for the counting constraint satisfaction problem csp a problem from csp is characterized by a constraint language gamma a fixed finite set of relations over a finite domain d an instance of the problem uses these relations to constrain an arbitrarily large finite set of variables bulatov showed that the problem of counting the satisfying assignments of instances of any problem from csp is either in polynomial time fp or is p complete his proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field we give an elementary proof of bulatov s dichotomy based on succinct representations which we call frames of a class of highly structured relations which we call strongly rectangular we show that these are precisely the relation

the complexity of counting cuts and of computing the probability that a graph is connected several enumeration and reliability problems are shown to be p complete and hence at least as hard as np complete problems included are important problems in network reliability analysis namely computing the probability that a graph is connected and counting the number of minimum cardinality s t cuts or directed network cuts also shown to be p complete are counting vertex covers in a bipartite graph counting antichains in a partial order and approximating the probability that a graph is connected and the probability that a pair of vertices is connected

a mixed finite element method for the dynamic analysis of coupled fluid solid interaction problems on the basis of a variational principle a mixed finite element approach is developed to describe the linear dynamics of coupled fluid structure interactions the variables of acceleration in the elastic solid and pressure in the fluid are adopted as the arguments of the variational principle these are chosen since they directly relate to many practical fluid structure interaction dynamic problems involving free surface disturbances e g a dam water system a fuel cell in an aircraft etc matrix equations describing the motions are presented and four methods of solution discussed each simplifying and approximating the matrix equations for easier application to solve various types of engineering problems this is demonstrated by analysing a selection of fluid structure interaction problems of practical interest the examples illustrate the general principle and application of the described functional approach without need to resort to more complex dynamic problems which can be analysed in a similar manner

the complexity of symmetric boolean parity holant problems for certain subclasses of np oplus p or p characterized by local constraints it is known that if there exist any problems within that subclass that are not polynomial time computable then all the problems in the subclass are np complete oplus p complete or p complete such dichotomy results have been proved for characterizations such as constraint satisfaction problems and directed and undirected graph homomorphism problems often with additional restrictions here we give a dichotomy result for the more expressive framework of holant problems for example these additionally allow for the expression of matching problems which have had pivotal roles in the development of complexity theory as our main result we prove the dichotomy theorem that for the class oplus p every set of symmetric holant signatures of any arities that is not polynomial time computable is oplus p complete the result exploits some special properties of the class oplus p and characterizes four distinct tractable

fast multigrid method for solving incompressible hydrodynamic problems with free surfaces we develop of a finite volume multigrid euler scheme for solving three dimensional fully nonlinear ship wave problems the flowfield and the a priori unknown free surface location are calculated by coupling the free surface kinematic and dynamic equations with the equations of motion for the bulk flow the evolution of the free surface boundary condition is linked to the evolution of the bulk flow via a novel iteration strategy that allows temporary leakage through the surface before the solution is converged the method of artificial compressibility is used to enforce the incompressibility constraint for the bulk flow a multigrid algorithm is used to accelerate convergence to a steady state

hydrodynamic forces on submerged oscillating cylinders at forward speed the hydrodynamic problem of submerged oscillating cylinders at forward speed is analysed by linearized potential theory the numerical method used combines a finite element approximation of the velocity potential in a region surrounding the cylinder with a boundary integral equation representation of the outer region this method avoids the calculation of the second order derivatives of the steady potential due to forward speed which appear in the body surface condition for the unsteady potential due to the oscillation of the cylinder numerical results from the present method for test cases of a circular cylinder in an unbounded fluid and below a free surface are in excellent agreement with the analytical solutions further results for elliptic cylinders are provided and the influence of forward speed on the hydrodynamic force on a submerged cylinder is investigated

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

fundamental viscous solutions or transient oseenlets associated with a body manoeuvring in a viscous fluid a viscous boundary element method involving a convolution integral formulation is developed to determine directly the fluid actions and velocity flow field associated with a body manoeuvring in a viscous fluid the proposed general approach is analogous to a potential flow singularity distribution panel method with the singularity replaced by a fundamental viscous solution from the developed nonlinear mathematical model based on an integral identity relation it is seen that the oseen equation its variant or modified form is central to the theoretical development of the fundamental viscous solution analytical expressions are presented for the fundamental solutions or transient oseenlets appropriate to two dimensional and three dimensional steady and unsteady transient manoeuvring problems involving prescribed translational rotational and their combined motions the examples considered and hence the fundamental viscous solutions relate to model towing tank experiments i e steady state rotating arm planar motion mechanism oscillatory tests etc used extensively to derive fluid action data on manoeuvring ships and submersibles examples of combined translational and rotational motions are investigated and these relate to an elementary idealization of a propeller rotating in an axial flow and an idealization of a body moving ahead with parasitic oscillatory roll motion the predicted fluid flow patterns associated with the various fundamental solutions are presented and these clearly illustrate the generation shedding and decay of vortices in the wakes

the complexity of weighted boolean csp with mixed signs we give a complexity dichotomy for the problem of computing the partition function of a weighted boolean constraint satisfaction problem such a problem is parameterized by a set c of rational valued functions which generalize constraints each function assigns a weight to every assignment to a set of boolean variables our dichotomy extends previous work in which the weight functions were restricted to being non negative we represent a weight function as a product of the form 1 sg where the polynomial s determines the sign of the weight and the non negative function g determines its magnitude we show that the problem of computing the partition function the sum of the weights of all possible variable assignments is in polynomial time if either every function in c can be defined by a pure affine magnitude with a quadratic sign polynomial or every function can be defined by a magnitude of product type with a linear sign polynomial in all other cases computing the partition function is fp p complete

quantum circuits that can be simulated classically in polynomial time a model of quantum computation based on unitary matrix operations was introduced by feynman and deutsch it has been asked whether the power of this model exceeds that of classical turing machines we show here that a significant class of these quantum computations can be simulated classically in polynomial time in particular we show that two bit operations characterized by 4 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time this contrasts with the known universality of two bit operations and demonstrates that efficient quantum computation of restricted classes is reconcilable with the polynomial time turing hypothesis therefore it is possible that the techniques introduced bring the quantum computational model within the realm of algebraic complexity theory in a manner consistent with one view of quantum physics the wave function is simulated deterministically and randomization arises only in the course of making measurements the results generalize the quantum model in that they do not require the matrices to be unitary in a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read twice boolean formulae all our results are based on the use of gates that are defined in terms of their graph matching properties

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

reduced order modeling of fluid structure interaction and vortex induced vibration systems using an extension of jourdain s principle abstract a first principles variational approach is proposed for reduced order modeling of fluid structure interaction fsi systems specifically vortex induced vibration viv fsi has to be taken into account in the design and analysis of many engineering applications yet a comprehensive theoretical development where analytical equations are derived from first principles is nonexistent an approach where jourdain s principle is modified and extended for fsi is used to derive reduced order models from an extended variational formulation where assumptions are explicitly stated two viv models are considered an elastically supported inverted pendulum and a translating cylinder both immersed in a flow and allowed to move transversely to the flow direction their reduced order models are obtained in the form of i a single governing equation and ii two general coupled equations as well as the coupled lift oscillator model comparisons are made with three existing models based on our theoretical results and especially the reduced order model we conclude that the first principles development herein is a viable framework for the modeling of complex fluid structure interaction problems such as vortex induced oscillations

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

holographic algorithms the power of dimensionality resolved valiant s theory of holographic algorithms is a novel methodology to achieve exponential speed ups in computation a fundamental parameter in holographic algorithms is the dimension of the linear basis vectors we completely resolve the problem of the power of higher dimensional bases we prove that 2 dimensional bases are universal for holographic algorithms

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

holographic algorithms with matchgates capture precisely tractable planar csp valiant introduced matchgate computation and holographic algorithms a number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time we show that in a very strong sense matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in the statistical physics community for decades they capture precisely those problems which are p hard on general graphs but computable in polynomial time on planar graphs more precisely we prove complexity dichotomy theorems in the framework of counting csp problems the local constraint functions take boolean inputs and can be arbitrary real valued symmetric functions we prove that every problem in this class belongs to precisely three categories 1 those which are tractable i e polynomial time computable on general graphs or 2 those which are p hard on general graphs but tractable on planar graphs or 3 those which are p har

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

reflection positivity rank connectivity and homomorphism of graphs it is shown that a graph parameter can be realized as the number of homomorphisms into a fixed weighted graph if and only if it satisfies two linear algebraic conditions reflection positivity and exponential rank connectivity in terms of statistical physics this can be viewed as a characterization of partition functions of vertex models

expressiveness of matchgates matchgates have been used to simulate classically in polynomial time assemblies of quantum gates in this paper it is shown that every matchgate for 2 input 2 output functions has to obey a certain set of five polynomial identities it is also shown that no matchgate can realize a nontrivial control gate for any number of inputs on the other hand it is proved that classical boolean formulae can be expressed as matchcircuits of polynomial size

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

well posedness for equilibrium problems and for optimization problems with equilibrium constraints in this paper we generalize the concepts of well posedness to equilibrium problems and to optimization problems with equilibrium constraints we establish some metric characterizations of well posedness for equilibrium problems and for optimization problems with equilibrium constraints we prove that under suitable conditions the well posedness is equivalent to the existence and uniqueness of solutions the corresponding concepts of well posedness in the generalized sense are also introduced and investigated for equilibrium problems and for optimization problems with equilibrium constraints

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