interfacial adsorption phenomena of the three dimensional three state potts model the authors study the interfacial adsorption phenomena of the three state ferromagnetic potts model on the simple cubic lattice by the monte carlo method finite size scaling analyses of the net adsorption yield evidence of the phase transition being of first order and kbtc j 1 8166 2

finite size scaling of the three state potts model on a simple cubic lattice finite size scaling is studied for the three state potts model on a simple cubic lattice we show that the specific heat and the magnetic susceptibility scale accurately as the volume the correlation length exhibits behaviors expected for a genuine first order transition the one extracted from the unsubtracted correlation function shows a characteristic finite size behavior whereas the physical correlation length that characterizes the first excited state stays at a finite value and is discontinuous at the transition point

bootstrapping variables in algebraic circuits we show that for the blackbox polynomial identity testing pit problem it suffices to study circuits that depend only on the first extremely few variables one needs only to consider size s degree s circuits that depend on the first log c s variables where c is a constant and composes a logarithm with itself c times thus the hitting set generator hsg manifests a bootstrapping behavior a partial hsg against very few variables can be efficiently grown to a complete hsg a boolean analog or a pseudorandom generator property of this type is unheard of our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially w r t variables this is repeated c times in an efficient way pushing the envelope further we show that i a quadratic time blackbox pit for 6 913 variate degree s size s polynomials will lead to a near complete derandomization of pit and ii a blackbox pit for n variate degree s size s circuits in s n time for 1 2 will lead to a near complete derandomization of pit in contrast s n time is trivial our second idea is to study depth 4 circuits that depend on constantly many variables we show that a polynomial time computable o s 1 49 degree hsg for trivariate depth 4 circuits bootstraps to a quasipolynomial time hsg for general polydegree circuits and implies a lower bound that is a bit stronger than that of kabanets and impagliazzo kabanets v impagliazzo r 2003 proceedings of the thirty fifth annual acm symposium on theory of computing stoc 03

cech de rham theory for leaf spaces of foliations we present a new cech de rham model for the cohomology of the classifying space of a foliated manifold this model enables us to lift the construction of known characteristic classes in the cohomology of the manifold to the cohomology of the classifying space by standard geometric methods and without making any reference to necessarily non hausdorff groupoids and their classifying spaces we also show how the cech de rham model can be used to prove some other known formulas as well as a version of poincare duality for foliations

local index theory over foliation groupoids we give a local proof of an index theorem for a dirac type operator that is invariant with respect to the action of a foliation groupoid g i fm denotes the space of units of g then the input is a g equivariant fiber bundle p m along with a g invariant fiberwise dirac type operator d on p the index theorem is a formula for the pairing of the index of d a s an element of a certain k theory group with a closed graded trace on a certain noncommutative de rham algebra b associated to g the proof is by means of superconnections in the framework of noncommutative geometry

cyclic cohomology and the family lefschetz theorem any closed current on the base of a compact fibration gives rise to a cyclic cocycle on the smooth convolution algebra we prove that such cocycle furnishes additive maps from the vertically equivariant k theory to the scalars this enables to associate to any closed current on the base of the fibration a lefschetz formula for fiber preserving isometries using geometric operators on the base we deduce the integrality of some characteristic numbers

finite size effects for the potts model with weak boundary conditions using the pirogov sinai theory we study finite size effects for the ferromagnetic q state potts model in a cube with boundary conditions that interpolate between free and constant boundary conditions if the surface coupling is about half of the bulk coupling and q is sufficiently large we show that only small perturbations of the ordered and disordered ground states are dominant contributions to the partition function in a finite but large volume this allows a rigorous control of the finite size effects for these weak boundary conditions in particular we give explicit formulae for the rounding of the infinite volume jumps of the internal energy and magnetization as well as the position of the maximum of the finite volume specific heat while the width of the rounding window is of order l d the same as for periodic boundary conditions the shift is much larger of order l 1 for strong boundary conditions the surface coupling is either close to zero or close to the bulk coupling the finite size effects at the transition point are shown to be dominated by either the disordered or the ordered phase respectively in particular it means that sufficiently small boundary fields lead to the disordered and not to the ordered gibbs state this gives an explicit proof of a van enter s result that the phase transition in the potts model is not robust

the potts model and flows ii many spin correlation function for pt i see ibid vol 19 p 411 1985 a graph theoretic analysis is made of the m spin correlation functions of the lambda state potts model in paper i the correlation functions for m 2 were expressed in terms of rooted mod lambda flow polynomials the authors introduce a more general type of polynomial the partitioned m rooted flow polynomial which plays a fundamental role in the calculation of the multispin correlation functions the m rooted equivalent transmissivities of tsallis and levy 1981 are interpreted in terms of percolation theory and are expressed as linear combinations of the above correlation functions

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

the index of operators on foliated bundles abstract we compute the equivariant cohomology connes karoubi character of the index of elliptic operators along the leaves of the foliation of a flat bundle the proof uses techniques similar to those developed in algebraic topology in the study of non commutative algebras

transfer matrices and partition function zeros for antiferromagnetic potts models vi square lattice with extra vertex boundary conditions we study using transfer matrix methods the partition function zeros of the square lattice q state potts antiferromagnet at zero temperature square lattice chromatic polynomial for the boundary conditions that are obtained from an m n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column we provide numerical evidence that the partition function zeros are becoming dense everywhere in the complex q plane outside the limiting curve mathcal b infty mathrm sq for this model with ordinary e g free or cylindrical boundary conditions despite this the infinite volume free energy is perfectly analytic in this region

index theory for quasi crystals i computation of the gap label group abstract in this paper we give a complete solution to the gap labelling conjecture for quasi crystals the method adopted relies on the index theory for laminations and the main tools are the connes skandalis longitudinal k theory index morphism together with the measured index formula

a homology theory for etale groupoids etale groupoids arise naturally as models for leaf spaces of foliations for orbifolds and for orbit spaces of discrete group actions in this paper we introduce a sheaf homology theory for etale groupoids we prove its invariance under morita equivalence as well as verdier duality between haeiger cohomology and this homology we also discuss the relation to the cyclic and hochschild homologies of connes convolution algebra of the groupoid and derive some spectral sequences which serve as a tool for the computation of these homologies

systems with logarithmic specific heat finite size scaling a detailed study is presented of the finite size scaling in systems with vanishing critical exponent alpha which usually have logarithmic specific heat singularity the appropriate form of the finite size hyperuniversality is established recent results on complex temperature plane zeros of the partition function are extended to the alpha 0 case

the phase diagram of the o n model we investigate the properties of the zero field o n model on the honeycomb lattice outside the range of presently available exact solutions to this purpose we make use of finite size scaling and transfer matrix techniques our results confirm the expected phase diagram of the o n model in the range 0 n 2 it consists of a disordered phase separated by a second order transition from a low temperature phase with algebraically decaying correlations we confirm that in this low temperature phase the exponents are independent of the temperature more significantly we find evidence for this same universal behaviour even in the case n 2 this contrasts with the temperature dependent exponents of the generic xy model our numerical evidence in this case is supported by arguments based on a mapping from the o n 2 model to the ashkin teller model on the triangular lattice furthermore our results agree with the absence of a phase transition for n 2 for negative values of n we observe unphysical singularities in the free energies of finite systems

series analysis of the 3 state potts model in 2 1 dimensions both high temperature and low temperature series are used to locate and characterize the first order transition in the 3 state potts model in 2 1 dimensions on both square and triangular lattices estimates are presented for the vacuum energy latent heat magnetization susceptibility and mass gap at the transition the spontaneous magnetization and latent heat appear to display an approximate universality at this weak first order transition

the 3 state potts model in 2 1 dimensions the 3 state potts model in 2 1 dimensions is studied using series and monte carlo methods high temperature series for the vacuum energy mass gap and susceptibility are computed on the square and triangular lattices using linked cluster expansions monte carlo estimates are obtained by the stochastic truncation method the model is found to undergo a weak first order transition as in the euclidean version of the model

a higher lefschetz formula for flat bundles in this paper we prove a fixed point formula for flat bundles to this end we use cyclic cocycles which are constructed out of closed invariant currents we show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact lie groups this enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation

area time efficient architecture of fft based montgomery multiplication the modular multiplication operation is the most time consuming operation for number theoretic cryptographic algorithms involving large integers such as rsa and diffie hellman implementations reveal that more than 75 percent of the time is spent in the modular multiplication function within the rsa for more than 1 024 bit moduli there are fast multiplier architectures to minimize the delay and increase the throughput using parallelism and pipelining however such designs are large in terms of area and low in efficiency in this paper we integrate the fast fourier transform fft method into the mclaughlin s framework and present an improved fft based montgomery modular multiplication mmm algorithm achieving high area time efficiency compared to the previous fft based designs we inhibit the zero padding operation by computing the modular multiplication steps directly using cyclic and nega cyclic convolutions thus we reduce the convolution length by half furthermore supported by the number theoretic weighted transform the fft algorithm is used to provide fast convolution computation we also introduce a general method for efficient parameter selection for the proposed algorithm architectures with single and double butterfly structures are designed obtaining low area latency solutions which we implemented on xilinx virtex 6 fpgas the results show that our work offers a better area latency efficiency compared to the state of the art fft based mmm architectures from and above 1 024 bit operand sizes we have obtained area latency efficiency improvements up to 50 9 percent for 1 024 bit 41 9 percent for 2 048 bit 37 8 percent for 4 096 bit and 103 2 percent for 7 680 bit operands furthermore the operating latency is also outperformed with high clock frequency for length 64 transform and above

transfer matrix test of the two cell mean field renormalization group abstract a transfer matrix approach has been worked out to test the predictions of the two cell mean field renormalization group mfrg for the square ising cells with linear size up to l 11 the convergence of the finite size critical couplings towards the exact value both for mfrg and for the cluster mfa is shown and the limit of the ratio yh yt consistent with the corresponding universal value is found applications within mfa are also briefly discussed

on the eight vertex model and related models a new method and some new results abstract we report recent work on classical models in two dimensions and related one dimensional quantum hamiltonians which are exactly solvable by using inversion relations for the transfer matrix of such models we have developed a new method to calculate all its eigenvalues and to determine the eigenstates in the thermodynamic limit for the 8 vertex model this leads to a complete classification of the excited states for the self dual q state potts model we derive the excitations of the diagonal transfer matrix and determine the diagonal correlation length explicitly we also report on the exact solution of the biquadratic spin 1 quantum antiferromagnet

finite size scaling of critical cumulants near the ferromagnetic phase boundary abstract we employ the recently proposed scaling theory for first order transitions to derive a detailed prediction for the large argument behaviour of the critical point scaling function of the reduced fourth order cumulant g l 4 2 2 3 correct up to a term exponentially small in l t

higher dimensional temperley lieb algebras a category which generalizes to higher dimensions many of the features of the temperley lieb category is introduced

finite size effects and phase transition in the three dimensional three state potts model the three state potts model in three dimensions is studied by monte carlo and finite size scaling techniques using a histogram method recently proposed by ferrenberg and swendsen the finite size dependence for the maximum of the specific heat is found to scale with the volume of the system indicating that the phase transition is of first order the value of the latent heat per spin and the correlation length at the transition are estimated

bivariant chern character and longitudinal index in this paper we consider a family of dirac type operators on fibration p b equivariant with respect to an action of an etale groupoid such a family defines an element in the bivariant k theory we compute the action of the bivariant chern character of this element on the image of connes map in the cyclic cohomology a particular case of this result is connes index theorem for etale groupoids a connes noncommutative geometry academic press 1994 in the case of fibrations

static and dynamic critical phenomena of the two dimensional q state potts model theq state potts model on the square lattice is studied by monte carlo simulation forq 3 4 5 6 very good agreement is obtained with exact results of kiharaet al and baxter for energy and free energy at the critical point critical exponent estimates forq 3 are 0 4 0 1 1 45 in rough agreement with high temperature series extrapolation and real space renormalization group methods the transition forq 5 6 is found to be a very weakly first order transition i e pronounced pseudocritical phenomena occur specific heat susceptibility etc nearly diverge at the first order transition temperature dynamics is associated to the model in the same way as for the kinetic ising model and the nonlinear slowing down of the order parameter and of the energy is studied the dynamic exponent is estimated to be zv 1 9 within our accuracy noq dependence is detected the relaxation is found to be consistent with dynamic scaling predictions and dynamic scaling functions associated with the nonlinear relaxation are estimated

the gauge potts model on a generalized bethe lattice the q states gauge potts model is exactly solved on a special infinite dimensional lattice which is a bethe lattice of plaquettes the critical properties are studied the model exhibits a first order transition for any number of states q and coordination number gamma 1 the critical values of coupled constant beta c is shown to be in a good agreement with known results

the temperature exponent of the n component cubic model we have applied transfer matrix and finite size scaling techniques to obtain critical couplings and temperature exponents for n component cubic models on the square lattice in these calculations n enters as a continuously variable parameter the results for the temperature exponent are in agreement with recent predictions

structure of the partition function and transfer matrices for the potts model in a magnetic field on lattice strips we determine the general structure of the partition function of the q state potts model in an external magnetic field z g q v w for arbitrary q temperature variable v and magnetic field variable w on cyclic mobius and free strip graphs g of the square sq triangular tri and honeycomb hc lattices with width ly and arbitrarily great length lx for the cyclic case we prove that the partition function has the form z lambda l y times l x q v w sum d 0 l y tilde c d mathrm tr t z lambda l y d m where denotes the lattice type tilde c d are specified polynomials of degree d in q t z lambda l y d is the corresponding transfer matrix and m lx lx 2 for sq tri hc respectively an analogous formula is given for mobius strips while only t z lambda l y d 0 appears for free strips we exhibit a method for calculating t z lambda l y d for arbitrary ly and give illustrative examples explicit results for arbitrary ly are presented for t z lambda l y d with d ly and d ly 1 we find very simple formulas for the determinant mathrm det t z lambda l y d we also give results for self dual cyclic strips of the square lattice

a first order phase transition in the three dimensional four state antiferromagnetic potts model studies the four state antiferromagnetic potts model with next nearest neighbour ferromagnetic interactions on the simple cubic lattice by the monte carlo method the ratio j2 j1 of the strength of the next nearest neighbour interactions to that of the nearest neighbour interactions is to be 10 the author analyses the finite size effects of the energy the specific heat the fourth order cumulant the magnetization the susceptibility and the effective transition temperatures with the help of the finite size scaling theory the author concludes that the transition of the model is assigned to be of first order

critical behaviour of the 1d q state potts model with long range interactions the critical behaviour of the one dimensional q state potts model with long range interactions decaying with distance r as r 1 sigma has been studied in the wide range of parameters 0 sigma or 1 and 1 16 or q or 64 a transfer matrix has been constructed for a truncated range of interactions for integer and continuous q and finite range scaling has been applied results for the phase diagram and the correlation length critical exponent are presented

hamiltonian group actions on symplectic deligne mumford stacks and toric orbifolds abstract we develop differential and symplectic geometry of differentiable deligne mumford stacks orbifolds including hamiltonian group actions and symplectic reduction as an application we construct new examples of symplectic toric dm stacks

riemannian metrics on lie groupoids we introduce a notion of metric on a lie groupoid compatible with multipli cation and we study its properties we show that many families of lie groupoids admit such metrics including the important class of proper lie groupoids the exponential map of these metrics allows us to establish a linearization theorem for riemannian groupoids obtaining both a simpler proof and a stronger version of the weinstein zung linearization theorem for proper lie groupoids this new notion of metric has a simplicial nature which will be explored in future papers of this series

on an algebraic approach to higher dimensional statistical mechanics we study representations of temperley lieb algebras associated with the transfer matrix formulation of statistical mechanics on arbitrary lattices we first discuss a new hyperfinite algebra the diagram algebra d underset raise0 3em hbox smash scriptscriptstyle n q which is a quotient of the temperley lieb algebra appropriate for potts models in the mean field case and in which the algebras appropriate for all transverse lattice shapesg appear as subalgebras we give the complete structure of this subalgebra in the case hat a n potts model on a cylinder the study of the full temperley lieb algebra of graphg reveals a vast number of infinite sets of inequivalent irreducible representations characterized by one or more complex parameters associated to topological effects such as links we give a complete classification in the hat a n case where the only such effects are loops and twists

quenched bond dilution in two dimensional potts models we report a numerical study of the bond diluted two dimensional potts model using transfer matrix calculations for different numbers of states per spin we show that the critical exponents at the random fixed point are the same as in self dual random bond cases in addition we determine the multifractal spectrum associated with the scaling dimensions of the moments of the spin spin correlation function in the cylinder geometry we show that the behaviour is fully compatible with the one observed in the random bond case confirming the general picture according to which a unique fixed point describes the critical properties of different classes of disorder dilution self dual binary random bond self dual continuous random bond

the o n model on the triangular lattice to find critical points of o n models on the triangular lattice we apply two methods first we investigate the yang baxter equations on the triangular lattice we find only solvable points directly related to those for the square lattice second we construct intersections with the potts model this yields eight branches of critical points parametrized by n we establish the equivalence of these branches with the known critical points of the o n model on the square lattice transfer matrix calculations are performed to obtain the conformal anomaly and the thermal exponent of these branches these results include a numerical analysis of a q 3 potts tricritical point we find analytic relations between potts and o n models as well as among o n models with different values of n and among potts models with different values of q

toric symplectic stacks abstract we give an intrinsic definition of compact toric symplectic stacks and show that they are classified by simple convex polytopes equipped with some additional combinatorial data this generalizes delzant s classification of compact toric symplectic manifolds as an application we show that any compact toric symplectic stack can be deformed to an ineffective toric orbifold

the localized longitudinal index theorem for lie groupoids and the van est map abstract we define the localized index of longitudinal elliptic operators on lie groupoids associated with lie algebroid cohomology classes we derive a topological expression for these numbers using the algebraic index theorem for poisson manifolds on the dual of the lie algebroid underlying the definition and computation of the localized index is an action of the hopf algebroid of jets around the unit space and the characteristic map it induces on lie algebroid cohomology this map can be globalized to differentiable groupoid cohomology giving a definition of the global index that can be computed by localization this correspondence between the global and localized index is given by the van est map for lie groupoids

crossover finite size scaling at first order transitions in a recent paper we developed a method which allows one to control rigorously the finite size behavior in long cylinders near first order phase transitions at low temperature here we apply this method to asymmetric transitions with two competing phases and to theq state potts model as a typical model of a temperature driven transition whereq low temperature phases compete with one high temperature phase we obtain the finite size scaling of the firstn eigenvalues wheren is the number of competing phases of the transfer matrix in a periodic box of volumel l t and as a corollary the finite size scaling of the shape of the order parameter in a hypercubic box t l the infinite cylinder t and the crossover regime from hypercubic to cylindrical scaling for the two phase case n 2 we find that the crossover length l is given by o lw exp lv where is the inverse temperature is the surface tension and w 1 2 if v 1 2 whilew 0 if v 1 2 for the standard ising model we also consider free boundary conditions showing that l exp lv o lv 1 for any dimension v 1 2 for v 1 2 we finally discuss a class of boundary conditions which interpolate between free corresponding to the interpolating parameter g 0 and periodic boundary conditions corresponding to g 1 finding that l o lw exp lv withw 0 forg 0 andw 1 2 for 0 g 1

collapse of a polymer in two dimensions abstract we numerically investigate the influence of self attraction on the critical behaviour of a polymer in two dimensions by means of an analysis of finite size results of transfer matrix calculations the transfer matrix is constructed on the basis of the o n loop model in the limit n 0 it yields finite size results for the magnetic correlation length of systems with a cylindrical geometry a comparison with the predictions of finite size scaling enables us to obtain information about the phase diagram as a function of the chemical potential of the loop segments and the strength of the attractive potential results for the magnetic scaling dimension can be interpreted in terms of known universality classes in particular when the attractive potential is increased we observe the crossover between the polymer critical behaviour of the self avoiding walk type to behaviour described earlier for the theta point

on the role of effective representations of lie groupoids abstract in this paper we undertake the study of the tannaka duality construction for the ordinary representations of a proper lie groupoid on vector bundles we show that for each proper lie groupoid g the canonical homomorphism of g into the reconstructed groupoid t g is surjective although contrary to what happens in the case of groups it may fail to be an isomorphism we obtain necessary and sufficient conditions in order that g may be isomorphic to t g and more generally in order that t g may be a lie groupoid we show that if t g is a lie groupoid the canonical homomorphism g t g is a submersion and the two groupoids have isomorphic categories of representations

ising model with two spin interactions and three spin interactions on a square lattice abstract an ising model with two spin interactions in one direction and four spin interactions on the smallest plaquettes is investigated on the square lattice the part of the two spin interactions is essential to determine the nature of the phase transition of the model the system with nearest neighbour interactions has no phase transition the one with next nearest neighbour interactions has a spontaneous magnetization and belongs to the same class of universality as the eight vertex model the one with third neighbour interactions has no spontaneous magnetization but belongs to the same class of universality as onsager s lattice and so on

nonvanishing vector fields on orbifolds we introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold q motivated by the inertia orbifold the space of multi sectors and the generalized orbifold euler characteristics we construct for each finitely generated group an orbifold called the space of sectors of q the obstruction occurs as the euler satake characteristics of the sectors for an appropriate choice of in the case that q is oriented this obstruction is expressed as a cohomology class the euler satake class we also acquire a complete obstruction in the case that q is compact with boundary and in the case that q is an open suborbifold of a closed orbifold

on the expressive power of univariate equations over sets of natural numbers equations of the form email protected x are considered where the unknown x is a set of natural numbers the expression f x may contain the operations of set addition defined as s t m n email protected s email protected t union intersection as well as ultimately periodic constants an equation with a non periodic solution of exponential growth rate is constructed at the same time it is demonstrated that no sets with super exponential growth rate can be represented it is also shown that restricted classes of these equations cannot represent sets with super linearly growing complements nor sets that are additive bases of order 2 the results have direct implications on the power of unary conjunctive grammars with one nonterminal symbol

critical behaviour and conformal anomaly of the o n model on the square lattice finite size scaling and transfer matrix techniques are used to determine the conformal anomaly and critical exponents of o n models on the square lattice these calculations were performed on five branches of critical points parametrised by n the results for two of the branches agree with the known universal properties of the o n model as derived for the honeycomb lattice finite size convergence is poor for most of the third branch so that in this case the classification in terms of known models is uncertain the authors data show that the fourth branch can be interpreted in terms of a combination of a low temperature o n and a critical ising model finally the numerical results for the o n model on the remaining branch agree accurately with the known results for the low temperature o n 1 model they prove this equivalence by means of an exact mapping

two dimensional scaling in magnetic systems with non integer characteristic lengths we studied the ferromagnetic ising model on two dimensional finite systems with non integer characteristic lengths first we considered very long strips with a finite number i of complete rows and one partially filled row with probability x which is the two dimensional version of a layer by layer growth at fixed temperature when the characteristic length l i x increases the free energy per spin the specific heat and the magnetic susceptibility oscillate attaining relative extremes at integer l the oscillations in the free energy are interpreted as surface corrections related to the oscillations in the mean coordination number finite size scaling relations are not satisfied with continuous l but still hold for fixed x and variable i where the differences of mean widths are integers for fixed x the fits of the free energy give the conformal anomaly with very good accuracy we also studied strips with discretized gaussian distributions of widths with non integer means l and rms deviation in these structures the thermodynamic quantities vary monotonically with continuous l but some methods for calculating critical exponents do not work properly when generalized to continuous l we also obtain with good accuracy in these systems we discuss the possible implications of our results to real systems behaviour

correlation length of the three state potts model in three dimensions a monte carlo study is carried out for the three state potts model in three dimensions the correlation length is shown to remain finite and to be discontinuous at the transition point it is also shown that the transition exhibits genuine first order characteristics in all aspects

excision in cyclic homology and in rational algebraic k theory for a precise definition see 1 below by replacing everywhere k by k q one obtains the corresponding notion in rational algebraic k theory the above definition has an obvious counterpart for cyclic homology and algebras over a fixed field in the case of a general commutative ground ring k some restrictions on the class of allowable extensions seem inevitable due to the well known limitations of cyclic homology considered as a homology functor for algebras not flat over a ground ring an extension of k algebras will be called pure if it is pure as an extension of k modules in the sense of p m cohn 6 cf also appendix a 3 below the class of pure extensions contains e g i extensions which admit a k module splitting ii extensions a e r s with s flat over k in fact one among the several possible characterizations of purity says that an extension is pure if and only if the underlying extension of k modules is an inductive limit of split extensions see theorem a 4 of appendix a below everywhere in this paper the word excision used in the context of cyclic homology will mean excision with respect to the class of pure extensions the second purpose of the present paper is to give a complete characterization of the class of algebras possessing the excision property in cyclic homology before stating the corresponding result we need the following definition let us

effects of interface width scaling and spatial correlations on ising systems with rough boundaries we studied the ferromagnetic ising model on two dimensional systems with rough boundaries and several thickness distributions first we considered very long strips with discretized gaussian distributions of widths with mean 3 l 12 systems with fixed interface width w and with increasing roughness w l with 0 1 were analysed we also considered strips with columns heights correlations where the difference in the heights of neighbouring columns was 0 or 1 with 0 0 5 free energies per spin fl and initial susceptibilities l were calculated using transfer matrix techniques at the critical temperature of the two dimensional ising model the scaling of fl and of the finite size estimates of the ratio of exponents l was analysed in systems with small and slowly increasing w 0 2 correlated or not extrapolations of fl and l to l give accurate estimates of the critical parameters f and in systems with rapidly increasing roughness the deviations of those estimates from the exact values indicate the presence of large corrections in finite size scaling relations the comparison of systems with nearly the same shows that spatial correlations have no systematic effect on corrections to scaling it suggests the reliability of uncorrelated models or related approximate systems for the description of scaling properties of thin magnetic films and other low dimensional structures

faster polynomial multiplication over finite fields polynomials over finite fields play a central role in algorithms for cryptography error correcting codes and computer algebra the complexity of multiplying such polynomials is still a major open problem let p be a prime and let mp n denote the bit complexity of multiplying two polynomials in fp x of degree less than n for n large compared to p we establish the bound mp n o n log n 8log n log p where log n min k n log k log n 1 stands for the iterated logarithm this improves on the previously best known bound mp n o n log n log log n log p which essentially goes back to the 1970s

on the duality relation for correlation functions of the potts model we prove a recent conjecture on the duality relation for correlation functions of the potts model for boundary spins of a planar lattice specifically we deduce the explicit expression for the duality of the n site correlation functions and establish sum rule identities in the form of the mobius inversion of a partially ordered set the strategy of the proof is by first formulating the problem for the more general chiral potts model the extension of our consideration to the many component potts models is also given

parsing boolean grammars over a one letter alphabet using online convolution consider context free grammars generating strings over a one letter alphabet for the membership problem for such grammars stated as given a grammar g and a string a n determine whether a n is generated by g only a naive o g n 2 time algorithm is known this paper develops a new algorithm solving this problem which is based upon fast multiplication of integers works in time g nlog 3n 2 o l o g n and is applicable to context free grammars augmented with boolean operations known as boolean grammars for unambiguous grammars the running time of the algorithm is reduced to g nlog 2n 2 o l o g n the algorithm is based upon a simplification of the online integer multiplication algorithm by fischer and stockmeyer m j fischer l j stockmeyer fast on line integer multiplication journal of computer and system sciences 9 3 1974 317 331

on the integrability of lie subalgebroids abstract let g be a lie groupoid with lie algebroid g it is known that unlike in the case of lie groups not every subalgebroid of g can be integrated by a subgroupoid of g in this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible we also consider the problem of integrability by closed subgroupoids and we give conditions under which the closure of a subgroupoid is again a subgroupoid

the number of link and cluster states the core of the 2d q state potts model due to fortuin and kastelyin the q state potts model has a representation as a sum over random graphs generalizing the potts model to arbitrary q is based on this representation a key element of the random cluster representation is the combinatorial factor which is the number of ways to form distinct clusters consisting of totally edges we have devised a method to calculate from monte carlo simulations

foliation groupoids and their cyclic homology the purpose of this paper is to prove two theorems which concern the position of etale groupoids among general smooth or lie groupoids our motivation comes from the noncommutative geometry and algebraic topology concerning leaf spaces of foliations here one is concerned with invariants of the holonomy groupoid of a foliation such as the cohomology of its classifying space the cyclic homology of its smooth convolution algebra or the ktheory of the cconvolution algebras many results here depend on the fact that such a holonomy groupoid can be reduced to what is called a complete transversal of the foliation giving rise to an equivalent etale groupoid for etale groupoids sometimes called rdiscrete groupoids in the literature the cyclic homology sheaf theory and classifying spaces are each well understood as is the relation between these

faster integer multiplication for more than 35 years the fastest known method for integer multiplication has been the schonhage strassen algorithm running in time o n log n log log n under certain restrictive conditions there is a corresponding omega n log n lower bound all this time the prevailing conjecture has been that the complexity of an optimal integer multiplication algorithm is theta n log n we take a major step towards closing the gap between the upper bound and the conjectured lower bound by presenting an algorithm running in time n log n 2 o log n the running time bound holds for multitape turing machines the same bound is valid for the size of boolean circuits

transfer matrix computation of generalized critical polynomials in percolation percolation thresholds have recently been studied by means of a graph polynomial pb p henceforth referred to as the critical polynomial that may be defined on any periodic lattice the polynomial depends on a finite subgraph b called the basis and the way in which the basis is tiled to form the lattice the unique root of pb p in 0 1 either gives the exact percolation threshold for the lattice or provides an approximation that becomes more accurate with appropriately increasing size of b initially pb p was defined by a contraction deletion identity similar to that satisfied by the tutte polynomial here we give an alternative probabilistic definition of pb p which allows for much more efficient computations by using the transfer matrix than was previously possible with contraction deletion we present bond percolation polynomials for the 4 82 kagome and 3 122 lattices for bases of up to respectively 96 162 and 243 edges much larger than the previous limit of 36 edges using contraction deletion we discuss in detail the role of the symmetries and the embedding of b for the largest bases we obtain the thresholds pc 4 82 0 676 803 329 pc kagome 0 524 404 998 pc 3 122 0 740 420 798 comparable to the best simulation results we also show that the alternative definition of pb p can be applied to study site percolation problems this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

corner exponents in the two dimensional potts model the critical behavior at a corner in two dimensional ising and three state potts models is studied numerically on the square lattice using transfer operator techniques the local critical exponents for the magnetization and the energy density for various opening angles are deduced from finite size scaling results at the critical point for isotropic or anisotropic couplings the scaling dimensions compare quite well with the values expected from conformal invariance provided the opening angle is replaced by an effective one in anisotropic systems

approximation schemes for finite lattices two algorithms are formulated for systematically truncating the set of basis states required in lattice hamiltonian calculations thus allowing larger lattice abort sizes to be treated they are tested for the case of the 1 1 dimensional ising model it is shown that smoothly convergent finite lattice sequences are obtained which can successfully the extrapolated to the bulk limit the authors point out the relevance of the technique for lattice calculations in higher dimensions

convergence and extrapolation in finite size scaling renormalization abstract finite size renormalization technique is studied by examining a test model which is exactly solvable and the problem of lattice animals and self avoiding walks bulk estimates of the leading irrelevant rg eigenvalue are found and compared with independent estimates from finite size data we introduce an extrapolation technique which is applicable to non monotonic sequences of exponent approximants

multiple phase coexistence and the scaling transformation it is shown that a simple extension of the finite size scaling method in the theory of critical phenomena can yield a sequence of approximants to any point subset of the interaction parameter space where multiple phase coexistence is possible the method is illustrated by an application to the three four and five state potts models and even in its lowest order of approximation is able to distinguish the difference in the zero field critical behaviour between the five state and the three and four state models

latent heats from finite size scaling a method of obtaining the latent heat at a first order transition from finite lattice matrix elements is discussed and demonstrated for the case of the 1 1 d potts model the method is shown to provide smoothly convergent finite lattice sequences the structure of the finite lattice hamiltonian eigenvalues is exhibited and a characteristic signal of the first order transition is proposed the finite size scaling behaviour at the transition is also discussed

convergence of finite size scaling renormalisation techniques abstract the rate of convergence of the transfer matrix finite size scaling or phenomenological renormalisation method is studied it is shown both heuristically and numerically that the convergence of estimates for exponents etc is governed asymptotically by the leading irrelevant variable scaling exponent the more rapid apparent convergence rates observed in many practical calculations for two dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms

finite size scaling for first order phase transitions rigorous results abstract recent rigorous results on finite size effects for lattice models are reviewed we consider symmetric and asymmetric first order phase transitions in cubic and cylindrical geometries and discuss the determination of transition points from finite size data

critical behaviour and logarithmic corrections of a quantum model with three spin interaction extended finite size scaling and series expansion methods are used to determine the critical properties of the one dimensional quantum ising model with three spin interaction by supposing the presence of logarithmic corrections the universality class of the model is investigated the results obtained by different methods support the conjecture that the model belongs to the same universality class as the four state potts model

a transfer matrix for the backbone exponent of two dimensional percolation rephrasing the backbone of two dimensional percolation as a monochromatic path crossing problem we investigate the latter by a transfer matrix approach conformal invariance links the backbone dimension db to the highest eigenvalue of the transfer matrix t and we obtain the result db 1 6431 0 0006 for a strip of width l t is roughly of size 23l but we manage to reduce it to l we find that the value of db is stable with respect to inclusion of additional blobs tangent to the backbone in a finite number of points

finite size scaling at first and second order phase transitions of baxter wu model abstract using a finite size phenomenological theory we investigate the behavior of the baxter wu model for both first and second order transitions in order to distinguish between the two kinds of transition we study the finite size scaling behavior of the order parameter and the susceptibility of the model at the critical temperature the known critical exponents of the system govern the scaling behavior while below the critical temperature the scaling exponents are the dimensionality of the system the presented scaling theory for the baxter wu model is an appropriate generalization of the corresponding theory of binder and landau for the ising model binder and landau phys rev b 30 1984 1477 we perform numerical simulations using standard monte carlo techniques and we find that our results are in good agreement with theoretical predictions

surface induced finite size effects for first order phase transitions we consider classical lattice models describing first order phase transitions and study the finite size scaling of the magnetization and susceptibility in order to model the effects of an actual surface in systems such as small magnetic clusters we consider models with free boundary conditions for a field driven transition with two coexisting phases at the infinite volume transition pointh h t we prove that the low temperature finite volume magnetizationmfree l h per site in a cubic volume of sizel d behaves like m free l h frac m m 2 frac m m 2 tanh left frac m m 2 l d h h chi l right o left frac 1 l right whereh x l is the position of the maximum of the finite volume susceptibility andm are the infinite volume magnetizations ath h t 0 andh h t 0 respectively we show thath x l is shifted by an amount proportional to 1 l with respect to the infinite volume transition pointh t provided the surface free energies of the two phases at the transition point are different this should be compared with the shift for periodic boundary conditions which for an asymmetric transition with two coexisting phases is proportional only to 1 l2d one can consider also other definitions of finite volume transition points for example the positionh u l of the maximum of the so called binder cumulantufree l h whileh u l is again shifted by an amount proportional to 1 l with respect to the infinite volume transition pointh t its shift with respect toh l is of the much smaller order 1 l2d we give explicit formulas for the proportionality factors and show that in the leading 1 l2d term the relative shift is the same as that for periodic boundary conditions

monte carlo method for critical systems in infinite volume the planar ising model in this paper we propose a monte carlo method for generating finite domain marginals of critical distributions of statistical models in infinite volume the algorithm corrects the problem of the long range effects of boundaries associated to generating critical distributions on finite lattices it uses the advantage of scale invariance combined with ideas of the renormalization group in order to construct a type of holographic boundary condition that encodes the presence of an infinite volume beyond it we check the quality of the distribution obtained in the case of the planar ising model by comparing various observables with their infinite plane prediction we accurately reproduce planar two three and four point of spin and energy operators we also define a lattice stress energy tensor and numerically obtain the associated conformal ward identities and the ising central charge

a family of methods for the solution of lattice models abstract we discuss a family of methods for evaluating the thermodynamic functions for a class of lattice models that includes the ising model it is shown that the usual transfer matrix method can be considered as a member of the family some typical members of this family are also introduced furthermore we are concerned with the acceleration of these methods for this reason an extra linear extrapolation algorithm which is based on the member of this family related to the transfer matrix method is presented by this algorithm not only can the bulk physical quantities be evaluated more precisely but also an approximation to the boundary thermodynamic quantities can be obtained a new expression for the boundary free energy of the ising model is derived in addition the convergence to the thermodynamic limit is proven for the scaling method presented by chorin some numerical results for the bulk and boundary thermodynamic functions of the ising model are included

efficient computation of the characteristic polynomial of a threshold graph an efficient algorithm is presented to compute the characteristic polynomial of a threshold graph threshold graphs were introduced by chvatal and hammer as well as by henderson and zalcstein in 1977 a threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex which is a vertex adjacent to all the other vertices threshold graphs are special kinds of cographs which themselves are special kinds of graphs of clique width 2 we obtain a running time of o n log 2 n for computing the characteristic polynomial while the previously fastest algorithm ran in quadratic time we improve the running time drastically in the case where there is a small number of alternations between 0 s and 1 s in the sequence defining a threshold graph a simple recurrence equation for the determinant of a weighted threshold graph matrix is presented the characteristic polynomial is computed in o n log 2 n arithmetic operations the algorithm is more efficient if the alternations in the defining sequence are o n as the numbers may be of length n log n the bit complexity is investigated too