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

universal space time codes from demultiplexed trellis codes in broadcast scenarios or in the absence of accurate channel probability distribution information code design for consistent channel by channel performance rather than average performance over a channel distribution may be desirable root and varaiya s compound channel theorem for linear gaussian channels promises the existence of universal codes that operate reliably whenever the channel mutual information mi is above the transmitted rate this paper presents two dimensional trellis codes that provide such universal performance over the compound linear vector gaussian channel when demultiplexed over two three and four transmit antennas the presented trellis codes found by exhaustive search guarantee consistent performance on every matrix channel that supports the information transmission rate with an mi gap that is similar to the capacity gap of a well designed additive white gaussian noise awgn specific code on the awgn channel as a result of their channel by channel consistency the universal trellis codes presented here also deliver comparable or in some cases superior frame error rate and bit error rate performance under quasi static rayleigh fading as compared with trellis codes of similar complexity that are designed specifically for the quasi static rayleigh fading scenario

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

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

log canonical singularities are du bois a recurring difficulty in the minimal model program mmp is that while log terminal singularities are quite well behaved for instance they are rational log canonical singularities are much more complicated they need not even be cohenmacaulay the aim of this paper is to prove that as conjectured in kol92 1 13 log canonical singularities are du bois the concept of du bois singularities abbreviated as db was introduced by steenbrink in ste83 as a weakening of rationality it is not clear how to define du bois singularities in positive characteristic so we work over a field of characteristic 0 throughout the paper the precise definition is rather involved see 1 10 but our main applications rely only on the following consequence

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

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

duality achievable rates and sum rate capacity of gaussian mimo broadcast channels we consider a multiuser multiple input multiple output mimo gaussian broadcast channel bc where the transmitter and receivers have multiple antennas since the mimo bc is in general a nondegraded bc its capacity region remains an unsolved problem we establish a duality between what is termed the dirty paper achievable region the caire shamai see proc ieee int symp information theory washington dc june 2001 p 322 achievable region for the mimo bc and the capacity region of the mimo multiple access channel mac which is easy to compute using this duality we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the mimo bc we also show that the dirty paper achievable region achieves the sum rate capacity of the mimo bc by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the mimo bc

on the design and maximum likelihood decoding of space time trellis codes in this letter we present a simple generalization of the maximum ratio combining principle for space time coded systems this result leads to a maximum likelihood decoder implementation that does not depend on the number of receive antennas and avoids the loss in performance incurred in the decoders proposed by tarokh and lo 1998 and biglieri et al the insights offered by this decoding rule allow for a simple and elegant proof for the space time code design criterion in systems with large number of receive antennas we further present an upper bound on probability of error that captures the dependence of space time code design on the number of receive antennas finally we present a computationally efficient approach for constructing space time trellis codes that exhibit satisfactory performance in systems with variable number of receive antennas

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

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

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

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

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

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

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

trellis codes for periodic erasures this paper describes techniques for the design and analysis of trellis codes that provide reliable communication over every channel in a specified set of possible channels where each channel is characterized by additive white gaussian noise with a distinct periodic variation in signal to noise ratio an important practical application for such trellis codes is the periodic erasure channel produced by partial band interference dispersed by a block interleaver we present trellis codes that provide reliable communication over all periodic erasure patterns of a given period for which the number of unerased coded bits per period is at least equal to the number of information bits per period

the indices of log canonical singularities let p x be a three dimensional log canonical pair such that has only standard coefficients and that p is a center of log canonical singularities for x then we get an effective bound of the indices of these pairs and actually determine all the possible indices furthermore under certain assumptions including the log minimal model program an effective bound is also obtained in dimension n 4

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

existence of log canonical closures let f x u be a projective morphism of normal varieties and x a dlt pair we prove that if there is an open set u 0 u such that x u u 0 has a good minimal model over u 0 and the images of all the non klt centers intersect u 0 then x has a good minimal model over u as consequences we show the existence of log canonical compactifications for open log canonical pairs and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness

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

extension theorems non vanishing and the existence of good minimal models we prove an extension theorem for effective purely log terminal pairs x s b of non negative kodaira dimension kappa k x s b ge 0 the main new ingredient is a refinement of the ohsawa takegoshi l 2 extension theorem involving singular hermitian metrics

mimo ofdm for wireless communications signal detection with enhanced channel estimation multiple transmit and receive antennas can be used to form multiple input multiple output mimo channels to increase the capacity by a factor of the minimum number of transmit and receive antennas in this paper orthogonal frequency division multiplexing ofdm for mimo channels mimo ofdm is considered for wideband transmission to mitigate intersymbol interference and enhance system capacity the mimo ofdm system uses two independent space time codes for two sets of two transmit antennas at the receiver the independent space time codes are decoded using prewhitening followed by minimum euclidean distance decoding based on successive interference cancellation computer simulation shows that for four input and four output systems transmitting data at 4 mb s over a 1 25 mhz channel the required signal to noise ratios snrs for 10 and 1 word error rates wer are 10 5 db and 13 8 db respectively when each codeword contains 500 information bits and the channel s doppler frequency is 40 hz corresponding normalized frequency 0 9 increasing the number of the receive antennas improves the system performance when the number or receive antennas is increased from four to eight the required snrs for 10 and 1 wer are reduced to 4 db and 6 db respectively therefore mimo ofdm is a promising technique for highly spectrally efficient wideband transmission

universal serially concatenated trellis coded modulation for space time channels this paper presents serially concatenated trellis coded modulations sctcms that perform consistently close to the available mutual information for periodic erasure channel pec periodic fading channel pfc and the 2 times 2 compound matrix channel we use both the maximum likelihood decoding criteria and iterative decoding criteria to design universal sctcms for the pec and the pfc for the space time channel by demultiplexing the symbols across the antennas the proposed universal sctcms for the period 2 pfc deliver consistent performance over the eigenvalue skew of the matrix channel within the family of channels having the same eigenvalue skew a time varying linear transformation tvlt is used to mitigate the performance variation over different eigenvectors the proposed space time sctcms of 1 0 2 0 and 3 0 bits per transmission require excess mutual information in the ranges 0 11 0 15 0 23 0 26 and 0 35 0 53 bits per antenna respectively because of their consistent performance over all channels the proposed codes will have good frame error rate fer performance over any quasi static fading distribution in particular the codes provide competitive fer performance in quasi static rayleigh fading

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

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

universal space time trellis codes this article gives practical examples of space time trellis codes performing as predicted by root and varaiya s 1968 compound channel theorem specifically 32 state and 64 state 2 spl times 2 space time trellis codes are presented that provide a bit error rate ber of 10 sup 5 on all 2 spl times 2 matrix channels with an excess mutual information mi within 8 of the excess mi required by standard trellis codes of the same complexity operating only in additive white gaussian noise awgn not surprisingly the universal space time trellis coded modulations st tcms provide average bit and frame error rates in quasi static rayleigh fading qrf that are comparable to those achieved by st tcms designed specifically for qrf as well as standard tcms followed by the alamouti 1998 space time block code however all of these other schemes require more excess mi in the worst case and some have a significantly wider variation in the required excess mi the article also compares the universal and quasi static rayleigh fading design approaches analytically and bounds the worst case distance of a trellis code on a 2 spl times 2 channel using the distances of the code on singular and unitary channels this bound is extended to the more general n sub t spl times n sub r scenario

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

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

asymptotically log fano varieties abstract motivated by the study of fano type varieties we define a new class of log pairs that we call asymptotically log fano varieties and strongly asymptotically log fano varieties we study their properties in dimension two under an additional assumption of log smoothness and give a complete classification of two dimensional strongly asymptotically log smooth log fano varieties based on this classification we formulate an asymptotic logarithmic version of calabi s conjecture for del pezzo surfaces for the existence of kahler einstein edge metrics in this regime we make some initial progress towards its proof by demonstrating some existence and non existence results among them a generalization of matsushima s result on the reductivity of the automorphism group of the pair and results on log canonical thresholds of pairs one by product of this study is a new conjectural picture for the small angle regime and limit which reveals a rich structure in the asymptotic regime of which a folklore conjecture concerning the case of a fano manifold with an anticanonical divisor is a special case

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

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

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