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12
Accidental Algorithms
 In Proc. 47th Annual IEEE Symposium on Foundations of Computer Science 2006
, 2004
"... Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in onetoone, or possibly onetomany, fashion. In this paper we propose a new ..."
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Cited by 31 (2 self)
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Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in onetoone, or possibly onetomany, fashion. In this paper we propose a new kind of reduction that allows for gadgets with manytomany correspondences, in which the individual correspondences among the solution fragments can no longer be identified. Their objective may be viewed as that of generating interference patterns among these solution fragments so as to conserve their sum. We show that such holographic reductions provide a method of translating a combinatorial problem to finite systems of polynomial equations with integer coefficients such that the number of solutions of the combinatorial problem can be counted in polynomial time if one of the systems has a solution over the complex numbers. We derive polynomial time algorithms in this way for a number of problems for which only exponential time algorithms were known before. General questions about complexity classes can also be formulated. If the method is applied to a #Pcomplete problem then polynomial systems can be obtained the solvability of which would imply P #P = NC2. 1
The Small Scale Structure of SpaceTime: A Bibliographical Review
, 1995
"... This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1 ..."
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Cited by 19 (0 self)
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This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1
On Valiant’s holographic algorithms
"... Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primit ..."
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Cited by 2 (2 self)
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Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.
Fisher Zeroes and Singular Behaviour of the Two Dimensional Potts Model in the Thermodynamic Limit
, 1997
"... The duality transformation is applied to the Fisher zeroes near the ferromagnetic critical point in the q ? 4 state two dimensional Potts model. A requirement that the locus of the duals of the zeroes be identical to the dual of the locus of zeroes in the thermodynamic limit (i) recovers the ratio o ..."
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Cited by 1 (0 self)
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The duality transformation is applied to the Fisher zeroes near the ferromagnetic critical point in the q ? 4 state two dimensional Potts model. A requirement that the locus of the duals of the zeroes be identical to the dual of the locus of zeroes in the thermodynamic limit (i) recovers the ratio of specific heat to internal energy discontinuity at criticality and the relationships between the discontinuities of higher cumulants and (ii) identifies duality with complex conjugation. Conjecturing that all zeroes governing ferromagnetic singular behaviour satisfy the latter requirement gives the full locus of such Fisher zeroes to be a circle. This locus, together with the density of zeroes is then shown to be sufficient to recover the singular part of the thermodynamic functions in the thermodynamic limit. 1 Supported by EU TMR Project No. ERBFMBICT961757 2 Current address Introduction The qstate Potts model [1], introduced in 1952 as a generalization of the Ising model [2], ha...
Universal VertexIRF Transformation for Quantum Affine Algebras
, 2008
"... We construct a universal VertexIRF transformation between Vertex type universal solution and Face type universal solution of the quantum dynamical YangBaxter equation. This universal VertexIRF transformation satisfies the generalized coBoundary equation) case. This solution has a simple Gauss dec ..."
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We construct a universal VertexIRF transformation between Vertex type universal solution and Face type universal solution of the quantum dynamical YangBaxter equation. This universal VertexIRF transformation satisfies the generalized coBoundary equation) case. This solution has a simple Gauss decomposition which is constructed using Sevostyanov’s characters of twisted quantum Borel algebras. We show that the evaluation of this universal solution in the evaluation representation of Uq(A (1) 1) gives the standard Baxter’s transformation between the 8Vertex model and the IRF height model. and is an extension of our previous work to the quantum affine Uq(A (1) r 1
Dublin Preprint: TCDMATH 9706 Liverpool Preprint: LTH 399 Fisher Zeroes and Singular Behaviour of the Two
"... The duality transformation is applied to the Fisher zeroes near the ferromagnetic critical point in the q ? 4 state two dimensional Potts model. A requirement that the locus of the duals of the zeroes be identical to the dual of the locus of zeroes in the thermodynamic limit (i) recovers the rati ..."
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The duality transformation is applied to the Fisher zeroes near the ferromagnetic critical point in the q ? 4 state two dimensional Potts model. A requirement that the locus of the duals of the zeroes be identical to the dual of the locus of zeroes in the thermodynamic limit (i) recovers the ratio of specific heat to internal energy discontinuity at criticality and the relationships between the discontinuities of higher cumulants and (ii) identifies duality with complex conjugation. Conjecturing that all zeroes governing ferromagnetic singular behaviour satisfy the latter requirement gives the full locus of such Fisher zeroes to be a circle. This locus, together with the density of zeroes is then shown to be sufficient to recover the singular part of the thermodynamic functions in the thermodynamic limit.
Symplectic Geometry on Quantum Plane
, 1999
"... A study of symplectic forms associated with two dimensional quantum planes and the quantum sphere in a three dimensional orthogonal quantum plane is provided. The associated Hamiltonian vector fields and Poissonian algebraic relations are made explicit. ..."
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A study of symplectic forms associated with two dimensional quantum planes and the quantum sphere in a three dimensional orthogonal quantum plane is provided. The associated Hamiltonian vector fields and Poissonian algebraic relations are made explicit.
unknown title
, 2008
"... The present paper focuses on the orderdisorder transition of an Ising model on a selfsimilar lattice. We present a numerical study, based on the Monte Carlo method in conjunction with the finite size scaling method, of the critical properties of the Ising model on a two dimensional deterministic f ..."
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The present paper focuses on the orderdisorder transition of an Ising model on a selfsimilar lattice. We present a numerical study, based on the Monte Carlo method in conjunction with the finite size scaling method, of the critical properties of the Ising model on a two dimensional deterministic fractal lattice of Hausdorff dimension dH = ln 8/ln 3 = 1.89278926.... We give evidence of the existence of an orderdisorder transition at finite temperature at a value βc ≃ 0.675. By comparing lattices of increasing size we obtain numerical estimates of the critical exponents. Finally we check the hyperscaling relation and find indications that the dimension that plays a role in this relation is the Haussdorff dimension. Key words: Lattice. Monte Carlo. Phase transitions. Fractals. Ising model. The present understanding of phase transitions has greatly benefited from the study of the spinlattice models, perhaps the simplest examples of extended systems showing non trivial cooperative behavior, such as spontaneously symmetry breaking. In most cases
Integration Approach to Ising Models
, 1996
"... An integral representation of the partition function for general ndimensional Ising models with nearest or nonnearest neighbours interactions is given. The representation is used to derive some properties of the partition function. An exact solution is given in a particular case. 1 On leave from I ..."
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An integral representation of the partition function for general ndimensional Ising models with nearest or nonnearest neighbours interactions is given. The representation is used to derive some properties of the partition function. An exact solution is given in a particular case. 1 On leave from Institute of Physics, Chinese Academy of Sciences, Beijing 1 As a simple prototype of a statistical mechanical system that undergoes a phase transition for spatial dimensionality n> 1, the Ising(Lenz) model [1] has been extensively studied in various ways. Since the exact solution of the free energy F and the spontaneous magnetization for the two dimensional zerofield Ising model (on a square lattice) were obtained more than fifty years ago [2, 3], many efforts have been made towards a detailed study of properties and the possible finding of exact solutions for the higher dimensional Ising model or for a twodimensional Ising model with nonzero magnetic field, for reviews see e.g., [47]. In this paper we study the Ising models by an “integration
Symmetry, Integrable Chain Models and Stochastic Processes
, 1996
"... A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with An symmetry and the related TemperleyLieb algebraic structures and representatio ..."
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A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with An symmetry and the related TemperleyLieb algebraic structures and representations are discussed. It is shown that corresponding to these An symmetric integrable chain models there are exactly solvable stationary discretetime (resp. continuoustime) Markov chains whose spectra of the transition matrices (resp. intensity matrices) are the same as the ones of the corresponding integrable models. 1