two dimensional percolation logarithmic corrections to the critical behaviour from series expansions analyses several extant series for the mean cluster size the zeroth moment of the pair connectedness and the percolation probability for bond and site percolation on two dimensional lattices with a view to detecting possible logarithmic corrections the logarithmic correction exponent z is found to be in the range 0 or z or 0 15 and our analysis also provides some new information about the critical exponents gamma and beta

rigorous results for the number of convex polygons on the square and honeycomb lattices the generating functions for the number of convex polygons on the square and honeycomb lattices are derived rigorously these functions were found by guttmann and enting 1988 their calculation is based on the series expansions up to the 64th order and is nonrigorous the asymptotic form of the mean squared radius of gyration of n step convex polygons on the square lattice is determined and the critical exponent v is 1

a priori probabilities of separable quantum states zyczkowski horodecki sanpera and lewenstein zhsl recently proposed a natural measure on the n dimensional quantum systems but expressed surprise when it led them to conclude that for n 2 2 disentangled separable systems are more probable 0 632 0 002 in nature than entangled ones we contend however that zhsl s rejected intuition has in fact a sound theoretical basis and that the a priori probability of disentangled 2 2 systems should more properly be viewed as considerably less than 0 5 we arrive at this conclusion in two quite distinct ways the first based on classical and the second quantum considerations both approaches however replace in whole or part the zhsl product measure by ones based on the volume elements of monotone metrics which in the classical case amounts to adopting the jeffreys prior of bayesian theory only the quantum theoretic analysis which yields the smallest probabilities of disentanglement uses the minimum number of parameters possible that is n2 1 as opposed to n2 n 1 although this over parametrization as recently indicated by byrd should be avoidable however despite substantial computation we are not able to obtain precise estimates of these probabilities and the need for additional possibly supercomputer analyses is indicated particularly so for higher dimensional quantum systems such as the 2 3 ones which we also study here

volumes of conditioned bipartite state spaces we analyze the metric properties of conditioned quantum state spaces these spaces are the convex sets of density matrices that when partially traced over m degrees of freedom respectively yield the given n n density matrix for the case n 2 the volume of equipped with the hilbert schmidt measure can be conjectured to be a simple polynomial of the radius of in the bloch ball remarkably for we find numerically that the probability to find a separable state in is independent of except for pure for the same holds for the probability to find a state with a positive partial transpose in these results are proven analytically for the case of the family of 4 4 x states and thoroughly numerically investigated for the general case the important implications of these findings for the clarification of open problems in quantum theory are pointed out and discussed

polygons on the honeycomb lattice the authors enumerated the number of self avoiding polygons on the honeycomb lattice to 82 steps analysis of the resulting series gives for the connective constant mu 1 8477594 or 6 10 6 in agreement with nienhuis value 1984 of mu 2 2 12 1 84775906 the critical exponent alpha 0 4999 or 10 4 unbiased or alpha 0 49998 or 2 10 0 5 biased a non physical singularity on the negative real axis is accurately located and some curious results concerning the confluent singularity are obtained

canonical coset parametrization and the bures metric of the three level quantum systems an explicit parametrization for the state space of an n level density matrix is given the parametrization is based on the canonical coset decomposition of unitary matrices we also compute explicitly the bures metric tensor over the state space of two and three level quantum systems

a new two variable generating function for convex self avoiding polygons the width of a self avoiding polygon on the square lattice is defined as the minimal horizontal or vertical distance between two of its parallel edges if the polygons are convex this distance is internal the perimeter generating functions for such convex polygons whose widths exceed a threshold can be given explicitly from these expressions a two variable width and perimeter generating function can be constructed the corresponding phase diagram shows two types of critical behaviour which meet at a tricritical point

high temperature expansions for classical spin models iii series and phase diagrams for the square lattice abstract the concepts of duality and broken symmetry are discussed briefly it is shown how generalized colouring polynomials can be used to calculate the terms in a high temperature expansion effectively these results and those from two previous papers are used to obtain high temperature series for a great variety of models these are given explicitly for all models with two boltzmann factors and for the ashkin teller model many phase diagrams obtained from the series by pade analysis are presented and their salient features discussed a principle suggested by these phase diagrams is applied to the dilute potts model

number of almost convex polygons on the square lattice the generating function for the number nc n of almost convex polygons on the square lattice with concavity index c 1 and perimeter n is derived rigorously the asymptotic behaviour of nc n for large n is determined and this result confirms a conjecture by enting et al 1992

high temperature expansions for classical spin models ii the square and hypercubic lattices and the potts model abstract all graphs and their contributions needed for a high temperature expansion of the partition function of a completely permissible spin model up to order 16 for the square lattice and up to order 11 for the hypercubic lattice are listed for the potts model the corresponding series are derived and analyzed by means of pade approximants in the square lattice case the phase transition of the m state potts model is reproduced to within 1 for m 2

differential geometry on su 3 with applications to three state systems the left and right invariant vector fields are calculated in an euler angle type parametrization for the group manifold of su 3 referred to here as euler coordinates the corresponding left and right invariant one forms are then calculated this enables the calculation of the invariant volume element or haar measure these are then used to describe the density matrix of a pure state and geometric phases for three state systems

the q state potts ferromagnet on d dimensional hypercubic lattices we use a high temperature star graph expansion to compute the free energy and the susceptibility of a q state potts model for arbitrary q on d dimensional hypercubic lattices the series are to order o 10 in the expansion variable nu ek 1 ek q 1 we show how to compute the expectation value of any operator on a finite graph for arbitrary number q of states of the potts model

a word counting technique for the solution of stochastic equations a solution method for finding a stationary state of stochastic equations describing one dimensional nonequilibrium systems with random sequential update is proposed it is proved that if such a system with periodic boundary condition allows for a solution with finite range correlations then this solution can be written as a product of cluster functions the solution can then be found by setting the coefficients of the linearly independent word counts occurring for a ring to zero these sets of linearly independent word counts are described generally the extension of these results to an open linear chain is indicated general solutions for 1 and 2 clusters are derived completely solved examples include i a monomer dimer model ii the general pair reaction diffusion system iii an extension of the asymmetric exclusion process to two different types of particles on a ring and on a chain and iv the general exclusion process with dependence on the states of the neighbour sites many more examples are discussed more briefly in all cases with particle conservation the results are corroborated by simulations

highly symmetric cellular automata and their symmetry breaking patterns abstract a highly symmetric cellular automaton in one dimension is defined by two properties i its local rule is a completely symmetric function of the n cells of its surrounding and ii it is invariant with respect to a regular abelian group acting on its m letter alphabet which can then be identified with this group the motivation for studying such systems is provided by the unsolved problem of partial symmetry breaking in spin models with such an abelian symmetry it is shown that the symmetries i and ii do not contradict each other if and only if n and m have no common factor as examples the four letter alphabets which may be identified either with the cyclic group c 4 or with the klein group k 4 s 2 s 2 are studied for the standard n 3 surrounding it is shown that these automata show complicated patterns of broken symmetries these give information on the corresponding spin models the chiral 4 state clock model and the symmetric and general ashkin teller models respectively

the number of convex polygons on the square and honeycomb lattices a subset of the set of self avoiding polygons sap embeddable on the square lattice which display the property of convexity is defined an algorithm for their enumeration is developed and from the available series coefficients the exact generating function is found the singularity structure appears similar to that of the unsolved sap problem but with different critical exponents and critical points the enumeration of convex polygons allows the extension of the existing series for the square lattice sap by one term for the honeycomb lattice similar results have been obtained despite a less natural definition of convexity

groupoids associated with endomorphisms to a compact hausdorff space which covers itself we associate an r discrete locally compact hausdorff groupoid its c algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the cuntz algebra op we consider examples related to coverings of the circle and of a heisenberg 3 manifold given a compact hausdorff space x and a continuous surjective map a x x we want to associate to the pair x a a c algebra c x a which in the case a is a homeomorphism coincides with the well known crossed product c x x z in this paper we consider the case of a covering a x x and c x a will be defined via a groupoid that generalizes the construction of the cuntz groupoid op considered by renault in re in that case x is a unilateral sequence space and a is the unilateral shift we start with some definitions and notation that make sense in greater generality definition given a set x and a map a x x let i x n y e xx zx x i 3k 1 0 n i k akx aly with the set of composable pairs i 2 x n y w m z e jx j w y the multiplication and inversion are given by x n y y m z x m n z and x n y 1 y n x the range and source maps are r x n y x 0 x s x n y y 0 y with these operations if becomes a groupoid with the unit space identified with x the equivalence relation associated to if will be denoted by r where r x y e x x x i 3k 1 0 akx ly and the isotropy group bundle is i x n x exxzxx ik 1 0 n l k akx alx received by the editors march 25 1993 and in revised form january 21 1994 originally communicated to the proceedings of the ams by palle e t jorgensen 1991 mathematics subject classification primary 46l05 46l55 46l80 research supported in part by the national science foundation 1995 american mathematical society 0002 9947 95 1 00 25 per page

universal amplitude ratio γ γ for two dimensional percolation the amplitude ratio of the susceptibility or second size moment for two dimensional percolation is calculated by two series methods and also by monte carlo simulation the first series method is an approach based upon integrating approximations to the scaling function the second series method directly uses low and high density series expansions of the susceptibility going to unprecedented orders for both bond and site percolation on the square lattice putting all methods together we find a consistent value ensuremath gamma ensuremath ensuremath gamma 162 5 ifmmode pm else textpm fi 2 a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220

quantum heisenberg manifolds as twisted groupoid c algebras abstract the quantum heisenberg manifolds are noncommutative manifolds constructed by m rieffel as strict deformation quantizations of heisenberg manifolds and have been studied by various authors rieffel constructed the quantum heisenberg manifolds as the generalized fixed point algebras of certain crossed product c algebras and they also can be realized as crossed products of c t 2 by hilbert c bimodules in the sense of abadie et al in this paper we describe how the quantum heisenberg manifolds can also be realized as twisted groupoid c algebras in the final section for a transformation groupoid x z with x compact and e p x a locally trivial principal g bundle where g is a compact abelian group we form a groupoid that is a locally trivial principal g bundle over x z which reduces to the twist groupoid of the second author when g t

a parametrization of bipartite systems based on su 4 euler angles in this paper we give an explicit parametrization for all two qubit density matrices this is important for calculations involving entanglement and many other types of quantum information processing to accomplish this we present a generalized euler angle parametrization for su 4 and all possible two qubit density matrices the important group theoretical properties of such a description are then manifest we thus obtain the correct haar hurwitz measure and volume element for su 4 which follows from this parametrization in addition we study the role of this parametrization in the peres horodecki criteria for separability and its corresponding usefulness in calculating entangled two qubit states as represented through the parametrization

hilbert schmidt volume of the set of mixed quantum states we compute the volume of the convex n2 1 dimensional set n of density matrices of size n with respect to the hilbert schmidt measure the hyper area of the boundary of this set is also found and its ratio to the volume provides information about the structure of n similar investigations are also performed for the smaller set of all real density matrices as an intermediate step we analyse volumes of the unitary and orthogonal groups and of the flag manifolds

number of spanning clusters at the high dimensional percolation thresholds a scaling theory is used to derive the dependence of the average number k of spanning clusters at threshold on the lattice size l this number should become independent of l for dimensions d 6 depend on the boundary conditions and the results there may vary between l d 6 and l0 while simulations in six dimensions are consistent with this prediction after including corrections of order ln ln l in five dimensions the average number of spanning clusters still increases as ln l even up to l 201 however the histogram p k of the spanning cluster multiplicity does scale as a function of kx l with x l 1 const l indicating that for sufficiently large l the average k will approach a finite value a fit of the five dimensional multiplicity data with a constant plus a simple linear correction to scaling reproduces the data very well numerical simulations for d 6 and for d 4 are also presented

the numbers of line defect self avoiding polygons and related subgraphs of the square lattice abstract those self avoiding polygons of the square lattice which have a perimeter larger by 2d than the perimeter of their bounding rectangle and for which this is due to a line defect are defined also classes of spiked convex self avoiding polygons are introduced a large number of generating functions for these types of graphs can be calculated exactly all imply an asymptotic behaviour for the number of such graphs with perimeter p as p 2 2 p for p

precise determination of the bond percolation thresholds and finite size scaling corrections for the sc fcc and bcc lattices extensive monte carlo simulations were performed to study bond percolation on the simple cubic s c face centered cubic f c c and body centered cubic b c c lattices using an epidemic kind of approach these simulations provide very precise values of the critical thresholds for each of the lattices pc s c 0 2488126 0 0000005 pc f c c 0 1201635 0 0000010 and pc b c c 0 1802875 0 0000010 for p close to pc the results follow the expected finite size and scaling behavior with values for the fisher exponent 2 189 0 002 the finite size correction exponent 0 64 0 02 and the scaling function exponent 0 445 0 01 confirmed to be universal

area weighted moments of convex polygons on the square lattice when constructing series expansions for the zero field partition function of the q state potts model on the square lattice using the finite lattice method the series can be extended by one term by adding a correction of q 1 times the number of convex polygons of an appropriate order by virtue of duality such corrections apply for both high and low temperature series for low temperature series corrections to the field dependent partition function are of the same order in the temperature variable as the zero field correction the general corrections are given by area weighted moments of the number of convex polygons the authors obtain conjectured closed form expressions for the two generating functions that give corrections to the first two field derivatives of the low temperature partition function

the yang mills functional and laplace s equation on quantum heisenberg manifolds in this paper we discuss the yang mills functional and a certain family of its critical points on quantum heisenberg manifolds using noncommutative geometrical methods developed by a connes and m rieffel in our main result we construct a certain family of connections on a projective module over a quantum heisenberg manifold that gives rise to critical points of the yang mills functional moreover we show that there is a relationship between this particular family of critical points of the yang mills functional and laplace s equation on multiplication type skew symmetric elements of quantum heisenberg manifolds recall that laplacian is the leading term for the coupled set of equations making up the yang mills equation

the ferromagnetic q state potts model on three dimensional lattices a study for real values of q abstract we study the phase diagram of the ferromagnetic q state potts model on the various three dimensional lattices for integer and non integer values of q 1 our approach is based on a thermodynamically self consistent ornstein zernike approximation for the two point correlation functions we calculate the transition temperatures and when the transition is first order the jump discontinuities in the magnetization and the internal energy as well as the coordinates of the critical endpoint in external field our predictions are in very good agreement with the best available estimates from the numerical study of the region of weak first order transition we estimate the critical value qc for which the transition changes from second to first order the q 1 limit that describes the bond percolation problem is also investigated

a generalized pancharatnam geometric phase formula for three level quantum systems we describe a recently developed generalization of the poincare sphere method to represent pure states of a three level quantum system in a convenient geometrical manner the construction depends on the properties of the group su 3 and its generators in the defining representation and uses geometrical objects and operations in an eight dimensional real euclidean space this construction is then used to develop a generalization of the well known pancharatnam geometric phase formula for evolution of a three level system along a geodesic triangle in state space

on the hamiltonian reduction of geodesic motion on su 3 to su 3 su 2 the reduced hamiltonian system on t su 3 su 2 is derived from a riemannian geodesic motion on the su 3 group manifold parametrized by the generalized euler angles and endowed with a bi invariant metric our calculations show that the metric defined by the derived reduced hamiltonian flow on the orbit space su 3 su 2 s5 is not isometric or even geodesically equivalent to the standard riemannian metric on the five sphere s5 embedded into r6

geometry on the quantum heisenberg manifold abstract a class of c algebras called quantum heisenberg manifolds were introduced by rieffel in comm math phys 122 1989 531 as strict deformation quantization of heisenberg manifolds using the ergodic action of heisenberg group we construct a family of spectral triples it is shown that associated kasparov modules are homotopic we also show that they induce cohomologous elements in entire cyclic cohomology the space of connes derham forms have been explicitly calculated then we characterize torsionless unitary connections and show that there does not exist a connection that is simultaneously torsionless and unitary explicit examples of connections are produced with negative scalar curvature this part illustrates computations involving some of the concepts introduced in frohlich et al comm math phys 203 1999 119 for which to the best of our knowledge no infinite dimensional example is known other that the noncommutative torus

yang mills connections on quantum heisenberg manifolds abstract we investigate critical points and minimizers of the yang mills functional ym on quantum heisenberg manifolds d c where the yang mills functional is defined on the set of all compatible linear connections on finitely generated projective modules over d c a compatible linear connection which is both a critical point and minimizer of ym is called a yang mills connection in this paper we investigate yang mills connections with constant curvature we are interested in yang mills connections on the following classes of modules over d c i abadie s module of trace 2 and its submodules ii modules of trace 2 iii tensor product modules of the form p e c where e c is morita equivalent to d c and p is a projection in e c we present a characterization of critical points and minimizers of ym and provide a class of new yang mills connections with constant curvature on over d c via concrete examples in particular we show that every yang mills connection on over d c with constant curvature should have a certain form of the curvature such as x y x z 0 and y z i id e also we show that these yang mills connections with constant curvature do not provide global minima but only local minima we do this by constructing a set of compatible connections that are not critical points but their values are smaller than those of yang mills connections with constant curvature our other results include i an example of a compatible linear connection with constant curvature on d c such that the corresponding connection on an isomorphic projective module does not have constant curvature and ii the construction of a compatible linear connection with constant curvature which neither attains its minimum nor is a critical point of ym on d c consequently the critical points and minimizers of ym depend crucially on the geometric structure of d c and of the projective modules over d c furthermore we construct the grassmannian connection on the projective modules with trace 2 over d c and compute its corresponding curvature finally we construct tensor product connections on p e c whose coupling constant is 2 and characterize the critical points of ym for this projective module

exact solution of the convex polygon perimeter and area generating function an explicit expression is derived for the three variable generating function p x y z sigma m or 1 sigma n or 1 sigma r or 1 x2ny2mzrcn m r where cn m r is the number of convex polygons with horizontal width n vertical height m and area r

susceptibility amplitudes for the three and four state potts models abstract the finite lattice method of series expansions is used to calculate a number of extended series for susceptibilities of the three and four state potts models on the square lattice we analyse all these series and estimate their critical amplitudes we resolve an uncertainty in amplitude ratios for high to low temperature susceptibilities and thus provide confirmation of field theoretic predictions

self avoiding walks neighbour avoiding walks and trails on semiregular lattices we study self avoiding and neighbour avoiding walks and lattice trails on two semiregular lattices the lattice and the lattice for the lattice we find the exact connective constant for both self avoiding walks neighbour avoiding walks and trails for the lattice we generate long series which permit the accurate estimation of the connective constant for self avoiding walks and trails

self avoiding polygons on the square lattice we have developed an improved algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 90 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 529 27 1 biased and the critical exponent 0 500 0005 10 unbiased the critical point is indistinguishable from a root of the polynomial 581x4 7x2 13 0 an asymptotic expansion for the coefficients is given for all n there is strong evidence for the absence of any non analytic correction to scaling exponent

generating functions for classes of self avoiding polygons on the square lattice abstract a transfer matrix technique is presented which allows the explicit calculation of the generating functions for those self avoiding polygons on the square lattice which have a perimeter equal to or two larger than the perimeter of their bounding rectangles

yang mills on quantum heisenberg manifolds in the noncommutative geometry program of connes there are two variations of the concept of the yang mills action functional we show that for the quantum heisenberg manifolds for generic parameter values they agree

on the moduli space of a quantum heisenberg manifold we investigate the yang mills problem on a quantum heisenberg manifold d c in the setting of the non commutative differential geometry this problem was already studied by kang 2010 in 6 for a specific module over d c and kang obtained a family of connections which are critical points of the yang mills functional on but it turned out that they are neither minima nor maxima in this article we construct a connection 0 on and show that it is a minimum of the yang mills functional on the module moreover we give a certain family of minima including 0 and show that the moduli space for is non trivial

interaction algebras of spin models and their relevance for disorder solutions the interaction algebra of a discrete classical spin model with transitive symmetry group is generated by the s independent matrices corresponding to its set of minimal edge transitive graphs if this algebra is non abelian its centre is at least the potts model interaction algebra for a spin model on the square lattice with four different interactions the existence of an abelian interaction algebra implies the existence of four 3s dimensional disorder solutions in the non abelian case there are at least four 2s 2 dimensional disorder solutions as examples the potts and ashkin teller models abelian and the f 6 model non abelian are discussed the possibility of an expansion around a disorder solution in terms of a class of subgraphs of the shadow lattice is indicated inversion relations for spin models on the checker board lattice are derived using an explicit form for the inverse of the transfer matrix

berry s phase for compact lie groups the methods of kahler geometry are applied to generalize the results of berry obtained for su 2 namely the existence of a geometrical part in the adiabatic phase to any compact lie group we obtain explicit expressions for berry s geometric phases berry s connections and berry s curvatures in terms of parameters of the corresponding lie algebra valued hamiltonian it is demonstrated that the parameter space of the hamiltonian in the general theory is essentially a homogeneous kahler manifold the fundamental kahler potentials of this manifold completely determine berry s phase a general approach is exemplified by the lie algebra hamiltonians corresponding to su 2 and su 3 evolution groups

universality of the excess number of clusters and the crossing probability function in three dimensional percolation extensive monte carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three dimensional percolation on the simple cubic s c face centred cubic f c c and body centred cubic b c c lattices systems l l l0 with l0 l were studied for both bond s c f c c b c c and site f c c percolation the excess number of clusters q b per unit length was confirmed to be a universal quantity with a value q b 0 412 likewise the critical crossing probability in the l0 direction with periodic boundary conditions in thell plane was found to follow a universal exponential decay as a function of r d l 0 l for large r simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f c c and b c c lattices yielding pc f c c d 0 199 2365 0 000 0010 pc b c c d 0 245 9615 0 000 0010 we also report the value pc s c d 0 311 6080 0 000 0004 for site percolation

hall normalization constants for the bures volumes of the n state quantum systems we report the results of certain integrations of quantum theoretic interest relying in this regard upon recently developed parametrizations of boya et al 1998 preprint quant ph 9810084 of the n n density matrices in terms of squared components of the unit n 1 sphere and the n n unitary matrices firstly we express the normalized volume elements of the bures minimal monotone metric for n 2 and 3 thereby obtaining bures prior probability distributions over the two and three state systems then as a first step in extending these results to n 3 we determine that the hall normalization constant cn for the marginal bures prior probablity distribution over the n 1 dimensional simplex of the n eigenvalues of the n n density matrices is for n 4 equal to 71 680 2 since we also find that c3 35 it follows that c4 is simply equal to 211c3 c2 itself is known to equal 2 the constant c5 is also found it too is associated with a remarkably simple decomposition involving the product of the eight consecutive prime numbers from 3 to 23 we also preliminarily investigate several cases n 5 with the use of quasi monte carlo integration we hope that the various analyses reported will prove useful in deriving a general formula which evidence suggests will involve the bernoulli numbers for the hall normalization constant for arbitrary n this would have diverse applications including quantum inference and universal quantum coding

star graph expansions for bond diluted potts models we derive high temperature series expansions for the free energy and the susceptibility of random bond q state potts models on hypercubic lattices using a star graph expansion technique this method enables the exact calculation of quenched disorder averages for arbitrary uncorrelated coupling distributions moreover we can keep the disorder strength p as well as the dimension d as symbolic parameters by applying several series analysis techniques to the new series expansions one can scan large regions of the p d parameter space for any value of q for the bond diluted four state potts model in three dimensions which exhibits a rather strong first order phase transition in the undiluted case we present results for the transition temperature and the effective critical exponent gamma as a function of p as obtained from the analysis of susceptibility series up to order 18 a comparison with recent monte carlo data chatelain et al phys rev e 64 036120 2001 shows signals for the softening to a second order transition at finite disorder strength

a heuristic principle for the phase boundaries of pure and quenched bond diluted spin models results from high temperature series for a class of discrete classical spin models on the square lattice suggest that the phase boundary is at least in part given by a simple heuristic principle it is shown how this principle can be extended to triangular and hexagonal lattices by making use of duality and the star triangle transformation for the potts model the exact phase transition is recovered not only for integer number of states but also in the percolation limit for more complicated models a number of phase diagrams are derived which are presumably exact except in regions where partially broken symmetry phases or massless phases intervene the principle can also be applied to quenched bond diluted spin models the results for the potts model are obtained for all three lattices and shown to be exact in certain limiting cases the m to 1 limit of the wreath product model s m s m which corresponds to double percolation is studied to provide an example for which the principle is completely incorrect

clusters size degree distribution for bond percolation abstract to address some physical properties of percolating systems it can be useful to know the degree distributions in finite clusters along with their size distribution here we show that to achieve this aim for classical bond percolation one can use the q 1 limit of suitably modified q state potts model we consider a version of such model with the additional complex variables and show that its partition function gives the bond percolation s generating function for the size and degree distribution in the q 1 limit for the first time we derive this distribution analytically for bond percolation on bethe lattices and complete graph the possibility to expand the applications of present method to other clusters characteristics and to models of correlated percolation is discussed

exact solution of the row convex polygon perimeter generating function an explicit expression is obtained for the perimeter generating function g y sigma n or 2 any2n for row convex polygons on the square lattice where an is the number of 2n step row convex polygons an asymptotic expression for an approximately a mu nn 3 2 is obtained where mu 3 22 and a are given the authors also show that the generating function is an algebraic function and that it satisfies an inhomogeneous linear differential equation of degree three

generalized euler angle parametrization for su n in a previous paper 1 an euler angle parametrization for su 4 was given here we present the derivation of a generalized euler angle parametrization for su n the formula for the calculation of the haar measure for su n as well as its relation to marinov s volume formula for su n 2 3 will also be derived as an example of this parametrization s usefulness the density matrix parametrization and invariant volume element for a qubit qutrit three qubit and two three state systems also known as two qutrit systems 4 will also be given

high temperature expansions for classical spin models abstract it is shown that the transitivity of the symmetry group of a classical spin model leads to a high temperature expansion of the free energy of such a model in terms of multiply connected graphs for models with a completely permissible symmetry group e g for ising and potts models these graphs fall into equivalence classes such classes are classified according to their cyclomatic index several different methods for obtaining the high temperature series in a practical way are also discussed the extension of these results to models with an infinite symmetry group is indicated

explicit thermostatics of stanley s n vector model on the harmonic chain by fourier analysis thermostatic properties of stanley s classical n vector model on the harmonic chain are studied it is shown that this model belongs to a rather general class of classical spin models in one dimension the members of this class are characterized by spins which are elements of a homogenous space with transformation group g and a g invariant and exchange invariant spin pair interaction for the derivation of basic thermostatic quantities and correlation functions we use the method of abstract fourier analysis this allows to derive rather explicit results for any member of the aforementioned class of spin models we present an exact closed form expression as well as a high and a low temperature expansion for the free energy of stanley s n vector model on the harmonic chain from this basic thermostatic quantities like internal energy entropy and heat capacity are obtained furthermore we present results for expectation values of one spin and two spin functions from the latter it is possible to derive the zero field susceptibility and it also allows for a discussion of magnetostrictive effects

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

on two dimensional percolation we present new series data for both high and low density bond and site percolation on the square lattice the series have been obtained by the finite lattice method and in all cases extend pre existing series an analysis of these series gives refined estimates of critical points critical exponents and amplitudes for bond and site animals and for the percolation probability and mean size exponents

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

unitary irreducible representations of sl 2 c in discrete and continuous su 1 1 bases we derive the matrix elements of generators of unitary irreducible representations of sl 2 c with respect to basis states arising from a decomposition into irreducible representations of su 1 1 this is done with regard to a discrete basis diagonalized by j3 and a continuous basis diagonalized by k1 and for both the discrete and continuous series of su 1 1 for completeness we also treat the more conventional su 2 decomposition as a fifth case the derivation proceeds in a functional differential framework and exploits the fact that state functions and differential operators have a similar structure in all five cases the states are defined explicitly and related to su 1 1 and su 2 matrix elements

an equivariant brauer group and actions of groups onc algebras suppose that g t is a second countable locally compact trans formation group given by a homomorphism l g homeo t and that a is a separable continuous trace c algebra with spectrum t an action g aut a is said to cover l if the induced action of g on t coincides with the original one we prove that the set brg t of morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product a b a c0 t b and we refer to brg t as the equivariant brauer group we give a detailed analysis of the structure of brg t in terms of the moore cohomology of the group g and the integral cohomology of the space t using this we can characterize the stable continuous trace c algebras with spectrum t which admit actions covering l in particular we prove that if g r then every stable continuous trace c algebra admits an essentially unique action covering l thereby substantially improving results of raeburn and rosenberg

an explicit computation of the bures metric over the space of n dimensional density matrices the aim of this paper is to provide a method for explicit computation of the bures metric over the space of n level quantum system based on the coset parametrization of density matrices

the high temperature expansion for the potts model and nowhere zero flows it is pointed out that a graph contributes to the high temperature expansion of the m state potts model if and only if it has a nowhere zero m flow this graph theoretical property is shown to be equivalent to the existence of a proper arrowing of the graph which can be visualized easily by means of arrows put on the edges in such a way that the sum of these arrows is 0 mod m at every vertex by means of this concept it is rather easy to solve the problem for planar graphs completely one result being that all planar graphs contribute for m 4 for nonplanar graphs all of them certainly contribute for m 6 but it is an unproven conjecture that m 5 suffices the classes of graphs contributing for 2 m 6 are characterized

high temperature series expansions for the q state potts model on a hypercubic lattice and critical properties of percolation we present results for the high temperature series expansions of the susceptibility and free energy of the q state potts model on a d dimensional hypercubic lattice zd for arbitrary values of q the series are up to order 20 for dimension d 1 limit of these series we estimate the percolation threshold pc and critical exponent gamma for bond percolation in different dimensions we also extend the 1 d expansion of the critical coupling for arbitrary values of q up to order d 9

scaling corrections site percolation and ising model in three dimensions using finite size scaling techniques we obtain accurate results for critical quantities of the ising model and the site percolation in three dimensions we pay special attention to parametrizing the corrections to scaling which is necessary to bring the systematic errors below the statistical ones

on c algebras associated with c correspondences abstract we study c algebras arising from c correspondences which were introduced by the author we prove the gauge invariant uniqueness theorem and obtain conditions for our c algebras to be nuclear exact or satisfy the universal coefficient theorem we also obtain a 6 term exact sequence of k groups involving the k groups of our c algebras

monte carlo study of the site percolation model in two and three dimensions we investigate the site percolation problem on the square and simple cubic lattices by means of a monte carlo algorithm that in fact simulates systems with size l d 1 x infinity where l specifies the linear system size this algorithm can be regarded either as an extension of the hoshen kopelman method or as a special case of the transfer matrix monte carlo technique various quantities such as the magnetic correlation function are sampled in the finite directions of the above geometry simulations are arranged such that both bulk and surface quantities can be sampled on the square lattice we locate the percolation threshold at p c 0 592 746 5 4 and determine two universal quantities as q gbc 0 930 34 1 and q gsc 0 793 72 3 which are associated with bulk and surface correlations respectively these values agree well with the exact values 2 5 48 and 2 1 3 respectively which follow from conformal invariance on the simple cubic lattice we locate the percolation threshold at p c 0 311 607 7 4 we further determine the bulk thermal and magnetic exponents as y t 1 1437 6 and y h 2 5219 2 respectively and the surface magnetic exponent at the ordinary phase transition as y o hs 1 0248 3

the cancellation theorem for projective modules over irrational rotation c algebras on montre que deux modules projectifs sur a qui sont isomorphes de facon stable sont en fait isomorphes

percolation threshold fisher exponent and shortest path exponent for four and five dimensions we develop a method of constructing percolation clusters that allows us to build very large clusters using very little computer memory by limiting the maximum number of sites for which we maintain state information to a number of the order of the number of sites in the largest chemical shell of the cluster being created the memory required to grow a cluster of mass s is of the order of s ensuremath theta bytes where ensuremath theta ranges from 0 4 for two dimensional 2d lattices to 0 5 for six or higher dimensional lattices we use this method to estimate d mathrm min the exponent relating the minimum path mathcal l to the euclidean distance r for 4d and 5d hypercubic lattices analyzing both site and bond percolation we find d mathrm min 1 607 ifmmode pm else textpm fi 0 005 4d and d mathrm min 1 812 ifmmode pm else textpm fi 0 006 5d in order to determine d mathrm min to high precision and without bias it was necessary to first find precise values for the percolation threshold p c p c 0 196889 ifmmode pm else textpm fi 0 000003 4d and p c 0 14081 ifmmode pm else textpm fi 0 00001 5d for site and p c 0 160130 ifmmode pm else textpm fi 0 000003 4d and p c 0 118174 ifmmode pm else textpm fi 0 000004 5d for bond percolation we also calculate the fisher exponent ensuremath tau determined in the course of calculating the values of p c ensuremath tau 2 313 ifmmode pm else textpm fi 0 003 4d and ensuremath tau 2 412 ifmmode pm else textpm fi 0 004 5d

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

the size and number of rings on the square lattice the authors have calculated the number of self avoiding polygons on the square lattice to 56 steps and the caliper size to 54 steps analysis of the generating function permits estimates of the connective constant mu 2 638 1585 or 10 6 and the critical exponents alpha 0 500 06 or 0 000 06 and v 0 753 or 0 007 the singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent delta 1 5 as predicted by nienhuis 1982 1984 the confluent part however maps into the additive analytic term due to the value of the exponent alpha

critical percolation in high dimensions we present monte carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4 13 dimensions for d 6 they are preliminary for d 6 they are between 20 and 10 4 times more precise than the best previous estimates this was achieved by three ingredients i simple and fast hashing that allowed us to simulate clusters of millions of sites on computers with less than 500 mbytes memory ii a histogram method that allowed us to obtain information for several p values from a single simulation and iii a variance reduction technique that is especially efficient at high dimensions where it reduces error bars by a factor of up to 30 and more based on these data we propose a scaling law for finite cluster size corrections

an on line library of extended high temperature expansions of basic observables for the spin s ising models on two and three dimensional lattices we present an on line library of unprecedented extension for high temperature expansions of basic observables in the ising models of general spin s with nearest neighbor interactions we have tabulated through order 25 the series for the nearest neighbor correlation function the susceptibility and the second correlation moment in two dimensions on the square lattice and in three dimensions on the simple cubic and the body centered cubic lattices the expansion of the second field derivative of the susceptibility is also tabulated through 23 for the same lattices we have thus added several terms from four up to thirteen to the series already published for spin s 1 2 1 3 2 2 5 2 3 7 2 4 5

exact solution of the staircase and row convex polygon perimeter and area generating function an explicit expression is obtained for the perimeter and area generating function g y z sigma n or 2 sigma m or 1 cn mynzm where cn m is the number of row convex polygons with area m and perimeter n a similar expression is obtained for the area perimeter generating function for staircase polygons both expressions contain q series