quadri tilings of the plane we introduce quadri tilings and show that they are in bijection with dimer models on a family of graphs r arising from rhombus tilings using two height functions we interpret a sub family of all quadri tilings called triangular quadri tilings as an interface model in dimension 2 2 assigning critical weights to edges of r we prove an explicit expression only depending on the local geometry of the graph r for the minimal free energy per fundamental domain gibbs measure this solves a conjecture of kenyon invent math 150 409 439 2002 we also show that when edges of r are asymptotically far apart the probability of their occurrence only depends on this set of edges finally we give an expression for a gibbs measure on the set of all triangular quadri tilings whose marginals are the above gibbs measures and conjecture it to be that of minimal free energy per fundamental domain

entropy and boundary conditions in random rhombus tilings the tilings of rhombi in two dimensions and of rhomboedra in three dimensions are studied when they are constrained by fixed boundary conditions we establish a link between those conditions and free or periodic boundary ones the entropy is written as a functional integral which is treated via a saddle point method we can exhibit the dominant states of the statistical ensemble of tilings and show that they can display a strong structural inhomogeneity caused by the boundary this inhomogeneity is responsible for a difference of entropy between the studied fixed boundary tilings and free boundary ones this method uses a representation of tilings by membranes embedded in a higher dimensional hypercubic lattice it is illustrated in the case of 60 degree rhombus tilings

markov chain algorithms for planar lattice structures consider the following markov chain whose states are all domino tilings of a 2n 2n chessboard starting from some arbitrary tiling pick a 2 2 window uniformly at random if the four squares appearing in this window are covered by two parallel dominoes rotate the dominoes 90 rm o in place repeat many times this process is used in practice to generate a random tiling and is a widely used tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics analogous markov chains are used to randomly generate other structures on various two dimensional lattices this paper presents techniques which prove for the first time that in many interesting cases a small number of random moves suffice to obtain a uniform distribution

area efficient and fast combined binary decimal floating point fused multiply add unit in this work we present a new 64 bit floating point fused multiply add fma unit that can perform both binary and decimal addition multiplication and fused multiply add operations the presented fma has 6 percent less delay than the fastest stand alone decimal unit and 23 percent less area than both binary and decimal units together these results were achieved by the use of 1 column by column reduction to reduce the partial products in the multiplier tree 2 a new leading zeros detector that produces its output in base 3 to simplify the normalization shifting in the binary datapath 3 the use of a redundant adder to perform the final addition 4 using a new rounding while redundant technique to hide the rounding delay and remove it from the critical path and 5 using a new simple conversion technique from redundant to binary decimal

improved design of high performance parallel decimal multipliers the new generation of high performance decimal floating point units dfus is demanding efficient implementations of parallel decimal multipliers in this paper we describe the architectures of two parallel decimal multipliers the parallel generation of partial products is performed using signed digit radix 10 or radix 5 recodings of the multiplier and a simplified set of multiplicand multiples the reduction of partial products is implemented in a tree structure based on a decimal multioperand carry save addition algorithm that uses unconventional non bcd decimal coded number systems we further detail these techniques and present the new improvements to reduce the latency of the previous designs which include optimized digit recoders for the generation of 2n tuples and 5 tuples decimal carry save adders csas combining different decimal coded operands and carry free adders implemented by special designed bit counters moreover we detail a design methodology that combines all these techniques to obtain efficient reduction trees with different area and delay trade offs for any number of partial products generated evaluation results for 16 digit operands show that the proposed architectures have interesting area delay figures compared to conventional booth radix 4 and radix 8 parallel binary multipliers and outperform the figures of previous alternatives for decimal multiplication

high speed multioperand decimal adders there is increasing interest in hardware support for decimal arithmetic as a result of recent growth in commercial financial and internet based applications consequently new specifications for decimal floating point arithmetic have been added to the draft revision of the ieee 754 standard for floating point arithmetic this paper introduces and analyzes three techniques for performing fast decimal addition on multiple binary coded decimal bcd operands two of the techniques speculate bcd correction values and correct intermediate results while adding the input operands the first speculates over one addition the second speculates over two additions the third technique uses a binary carry save adder tree and produces a binary sum combinational logic is then used to correct the sum and determine the carry into the next more significant digit multioperand adder designs are constructed and synthesized for four to 16 input operands analyses are performed on the synthesis results and the merits of each technique are discussed finally these techniques are compared to several previous techniques for high speed decimal addition

a variational principle for a non integrable model we show the existence of a variational principle for graph homomorphisms from mathbb z m to a d regular tree the technique is based on a discrete kirszbraun theorem and a concentration inequality obtained through the dynamics of the model as another consequence of the concentration inequality we also obtain the existence of a continuum of translation invariant ergodic gradient gibbs measures for graph homomorphisms from mathbb z m to a regular tree the method is sufficiently robust such that it could be applied to other discrete models with a quite general target graphs

phase diagram of a random tiling quasicrystal we study the phase diagram of a two dimensional random tiling model for quasicrystals at proper concentrations the model has 8 fold rotational symmetry landau theory correctly gives most of the qualitative features of the phase diagram which is in turn studied in detail numerically using a transfer matrix approach we find that the system can enter the quasicrystal phase from many other crystalline and incommensurate phases through first order or continuous transitions exact solutions are given in all phases except for the quasicrystal phase and for the phase boundaries between them we calculate numerically the phason elastic constants and entropy density and confirm that the entropy density reaches its maximum at the point where phason strains are zero and the system possesses 8 fold rotational symmetry in addition to the obvious application to quasicrystals this study generalizes certain surface roughening models to two dimensional surfaces in four dimensions

arctic octahedron in three dimensional rhombus tilings and related integer solid partitions three dimensional integer partitions provide a convenient representation of codimension one three dimensional random rhombus tilings calculating the entropy for such a model is a notoriously difficult problem we apply transition matrix monte carlo simulations to evaluate their entropy with high precision we consider both free and fixed boundary tilings our results suggest that the ratio of free and fixed boundary entropies is free fixed 3 2 and can be interpreted as the ratio of the volumes of two simple nested polyhedra this finding supports a conjecture by linde moore and nordahl concerning the arctic octahedron phenomenon in three dimensional random tilings

extremal principle for the steady state selection in driven lattice gases with open boundaries this paper investigates the steady states of one dimensional driven lattice gases with open boundary conditions it shows how the extremal principle proposed recently by popkov and schutz can be modified to apply to more general cases monte carlo simulations are presented for a one dimensional totally asymmetric simple exclusion process with nearest neighbor repulsion under parallel update as an example the simulations enable one to guess the exact phase diagram for this particular lattice gas with deterministic bulk dynamics by fitting the data to analytic formulas which appear to be exact in the thermodynamic limit

solution of the dimer problem on a hexagonal lattice with boundary the dimer problem is solved exactly for a hexagonal lattice with general boundary using a known generating function from the theory of partitions it is shown that the leading term in the entropy depends on the shape of the boundary

multioperand parallel decimal adder a mixed binary and bcd approach decimal arithmetic has been revived in recent years due to the large amount of data in commercial applications we consider the problem of multioperand parallel decimal addition with an approach that uses binary arithmetic suggested by the adoption of binary coded decimal bcd numbers this involves corrections in order to obtain the bcd result or a binary to decimal bd conversion we adopt the latter approach which is particularly efficient for a large number of addends conversion requires a relatively small area and can afford fast operation the bd conversion moreover allows an easy alignment of the sums of adjacent columns we treat the design of bcd digit adders using fast carry free adders and the conversion problem through a known parallel scheme using elementary conversion cells spreadsheets have been developed for adding several bcd digits and for simulating the bd conversion as a design tool

high speed parallel decimal multiplication with redundant internal encodings the decimal multiplication is one of the most important decimal arithmetic operations which have a growing demand in the area of commercial financial and scientific computing in this paper we propose a parallel decimal multiplication algorithm with three components which are a partial product generation a partial product reduction and a final digit set conversion first a redundant number system is applied to recode not only the multiplier but also multiples of the multiplicand in signed digit sd numbers furthermore we present a multioperand sd addition algorithm to reduce the partial product array finally a digit set conversion algorithm with a hybrid prefix network to decrease the number of the logic gates on the critical path is discussed an analysis of the timing delay and an hdl model synthesized under 90 nm technology show that by considering the tradeoff of designs among three components the overall delay of the proposed 16 16 digit multiplier takes about 11 percent less timing delay with 2 percent less area compared to the current fastest design

boundary conditions entropy and the signature of random tilings we investigate the influence of boundary conditions on the results of monte carlo simulations of 2d random tilings looking at the fluctuations in internal space we compare fixed and periodic boundaries we find that fixed boundary conditions will lead to a different random tiling ensemble with reduced finite size entropy density in comparison with periodic boundary conditions as a by product we derive improved estimates for the elastic constants of the octagonal rhombus tiling ensemble finally we introduce a robust tool also suitable for the analysis of experimental data to distinguish between quasiperiodic and random tiling models

inversion relations and disorder solutions on potts models the inversion symmetry and the automorphy group generated by the latter when combined with the spatial symmetries are studied for the anisotropic potts models on the triangular and checkerboard lattices the exact expression of the partition function which is known on some particular disorder subvarieties of the triangular model is checked and a generalisation is proposed for the checkerboard lattice the automorphy group is then used to extend the disorder solutions to the infinity of transformed subvarieties

random tilings of high symmetry ii boundary conditions and numerical studies we perform numerical studies including monte carlo simulations of high rotational symmetry random tilings for computational convenience our tilings obey fixed boundary conditions in regular polygons such tilings are put in correspondence with algorithms for sorting lists in computer science we obtain statistics on path counting and vertex coordination which compare well with predictions of mean field theory and allow estimation of the configurational entropy which tends to the value 0 568 per vertex in the limit of continuous symmetry tilings with phason strain appear to share the same entropy as unstrained tilings as predicted by mean field theory we consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy although these differ for finite rotational symmetry

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

boundary effects in the dimer problem on a non bravais lattice the dimer problem on the hexagonal lattice is solved for various boundary shapes the number of arrangements of dimers in the asymptotic limit of large lattices proves to be very sensitive to the precise form of the lattice boundary if n is the number of lattice points and for large n wn 2 the number of dimer configurations then the molecular freedom w reduces to 1 in the case of standard boundaries parallelogram rectangle and there is in the thermodynamic limit no freedom in placing a dimer at all but in the case of a hexagonal lattice which has been given the shape of a macroscopic honeycomb and which thus exhibits the symmetry properties of the unit cell the molecular freedom is found to tend to w 3 4 3 1 299 for n investigating various other boundary forms the regular hexagon proves to be a convenient surface shape with a maximum molecular freedom which differs markedly from the known value w 1 381 for the hexagonal lattice wrapped on a torus

phase transitions in a cellular automaton model of a highway on ramp we introduce a lattice gas model for the merging of two single lane automobile highways the merging rules for traffic on the two lanes are deterministic but the inflow on both lanes is stochastic analysing the stationary distribution of this stochastic cellular automaton we find a discontinuous phase transition from a free flow phase which depends on the initial state of the road to a jammed phase where all memory of the initial state is lost inside the jammed phase we identify dynamical phase transitions in the approach to stationarity each dynamical phase is characterized by a fixed number of relaxation cycles which is decreasing as one moves deeper into the jammed phase in each cycle step the number of desperate drivers who force their way onto the main road when they reach the end of the on ramp increases until stationarity

s4 symmetry on the checkerboard potts model the large q expansion of the partition function z of the checkerboard q state potts model with a magnetic field is given up to sixth order in q 12 up to this order z is shown to exhibit an unexpected s4 symmetry using various arguments exact results expansion etc the authors suggest the general validity of this new symmetry property of the considered model an exact expression for the magnetisation jump is also proposed and checked on the large q expansion

fast radix 10 multiplication using redundant bcd codes we present the algorithm and architecture of a bcd parallel multiplier that exploits some properties of two different redundant bcd codes to speedup its computation the redundant bcd excess 3 code xs 3 and the overloaded bcd representation odds in addition new techniques are developed to reduce significantly the latency and area of previous representative high performance implementations partial products are generated in parallel using a signed digit radix 10 recoding of the bcd multiplier with the digit set 5 5 and a set of positive multiplicand multiples 0x 1x 2x 3x 4x 5x coded in xs 3 this encoding has several advantages first it is a self complementing code so that a negative multiplicand multiple can be obtained by just inverting the bits of the corresponding positive one also the available redundancy allows a fast and simple generation of multiplicand multiples in a carry free way finally the partial products can be recoded to the odds representation by just adding a constant factor into the partial product reduction tree since the odds uses a similar 4 bit binary encoding as non redundant bcd conventional binary vlsi circuit techniques such as binary carry save adders and compressor trees can be adapted efficiently to perform decimal operations to show the advantages of our architecture we have synthesized a rtl model for 16 times 16 digit and 34 times 34 digit multiplications and performed a comparative survey of the previous most representative designs we show that the proposed decimal multiplier has an area improvement roughly in the range 20 35 percent for similar target delays with respect to the fastest implementation

tiling with dominoes pictures of plane figures without holes graphic basis of the thurston algorithm parallelization uniqueness and decomposition resume le desormais classique algorithme de thurston 1990 pour le pavage des figures planes finies et sans trous par des dominos peut etre retrouve par la condition de hall sur les couplages dans les graphes bipartis et la consideration de plus courts chemins dans un graphe associe a la figure a paver ce point de vue nouveau permet une parallelisation en temps o log n avec n 3 processeurs sur un modele pram c est a dire de facon efficace classe nc ce meme algorithme resout la question de l unicite d un pavage et de facon plus generale donne une decomposition canonique de la figure

totally asymmetric exclusion process with extended objects a model for protein synthesis the process of protein synthesis in biological systems resembles a one dimensional driven lattice gas in which the particles have spatial extent covering more than one lattice site we expand the well studied totally asymmetric exclusion process in which particles typically cover a single lattice site to include cases with extended objects exact solutions can be determined for a uniform closed system we analyze the uniform open system through two approaches first a continuum limit produces a modified diffusion equation for particle density profiles second an extremal principle based on domain wall theory accurately predicts the phase diagram and currents in each phase finally we briefly consider approximate approaches to a nonuniform open system with quenched disorder in the particle hopping rates and compare these approaches with monte carlo simulations

random tilings of high symmetry i mean field theory we study random tiling models in the limit of high rotational symmetry in this limit a mean field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions we base our mean field theory on an iterative tiling construction inspired by the work of de bruijn in addition to the entropy we consider correlation functions phason elasticity and the thermodynamic limit tilings of dimension other than two are considered briefly

fixed boundary octagonal random tilings a combinatorial approach some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied a geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems thus generalizing previous results on the more restricted class of codimension one tilings in particular this method gives access to counting formulas which are directly related to questions of entropy in these statistical systems methods and tools from the field of enumerative combinatorics are used

configurational entropy of codimension one tilings and directed membranes the calculation of random tiling configurational entropy amounts to an enumeration of partitions a geometrical description of the configuration space is given in terms of integral points in a high dimensional space and the entropy is deduced from the integral volume of a convex polytope in some cases the latter volume can be expressed in a compact multiplicative formula and in all cases in terms of binomial series the origin of which is given a geometrical meaning our results mainly concern codimension one tilings but can also be extended to higher codimension tilings we also discuss the link between freeboundary and fixed boundary condition problems