on the measure of totally real algebraic integers ii for a certain measure s2 a defined for totally real algebraic integers a 0 1 we find the four smallest values of q a the methods used involve linear programming and the results are verified using interval arithmetic

bandpass correlator analysis for general input assumptions corresp for general assumptions concerning bandpass signal and noise inputs previous analyses of analog cross correlators have yielded very complicated expressions for the probability density function pdf of the correlator output the inherent computational difficulties therefore limited the application of these pdf s almost exclusively to special cases of equal noise powers and noise spectrums symmetric about the signal s frequency in this correspondence it is shown that for general bandpass inputs the correlator output can always be expressed as the difference of two scaled independent noncentral chi squared variables if a certain matrix transformation of variables is employed the characteristic function method is then used to derive the pdf which is as readily computed as for the case of special assumptions

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

link invariant of the izergin korepin model the link invariant associated with the izergin korepin 19 vertex model is deduced using the method of statistical mechanics it is shown that the izergin korepin model leads to an invariant which is precisely the 3 state akutsu wadati polynomial previously known only for 2 and 3 braid knots we give a table of the invariant for all knots and links up to seven crossings

electrical characteristics of a class of n order triangular network abstract this paper investigates the electrical properties of a class of n order triangular resistor network model the authors establish by constructing equivalent model the second order nonlinear difference equation model we propose by using equivalent transformation method the general and specific methods of solution for the nonlinear difference equation and at once obtain an accurate equivalent resistance formula of n order triangular resistor network and a series of special conclusions are analyzed and compared finally we study the oscillation and resonance characteristics of the triangular lc network under different conditions where the method used is variable substitution by using the equivalent resistance results obtained this study reveals the electrical characteristics of triangular lc network and establishes a new theoretical foundation for related disciplines

generating functions and duality for non crossing walks on a plane graph the generating function for the number of non crossing walk configurations of n walks between the roots of a two rooted directed plane graph is introduced this is shown to be a rational function and the structure and symmetry property of its numerator are discussed the walk configurations correspond to flows and the equivalent dual generating function for potentials is investigated independently also equivalences are drawn with the partially order sets that can be constructed from the walk configurations finally the general results developed here are applied to the directed square and honeycomb lattices this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

resistance between two nodes of a ring network abstract the resistance between two nodes in some resistor networks has been studied extensively by mathematicians and physicists given m positive integers m 1 m 2 m n let g m i 1 n be the resistor network with node set v v 1 v 2 v n and with a unit resistor between arbitrary two nodes u v i v v i 1 for i 1 2 n where v i v j 0 if i j and v i m i v n 1 v 1 gervacio 2016 introduces a modified method to compute the resistance between two nodes based on this method in this paper we use the elimination and substitution principles in electrical circuit to obtain the resistance between arbitrary two nodes of g m i 1 n

critical frontier of the potts and percolation models on triangular type and kagome type lattices ii numerical analysis in the preceding paper one of us f y wu considered the potts model and bond and site percolation on two general classes of two dimensional lattices the triangular type and kagome type lattices and obtained closed form expressions for the critical frontier with applications to various lattice models for the triangular type lattices wu s result is exact and for the kagome type lattices wu s expression is under a homogeneity assumption the purpose of the present paper is twofold first an essential step in wu s analysis is the derivation of lattice dependent constants a b c for various lattice models a process which can be tedious we present here a derivation of these constants for subnet networks using a computer algorithm second by means of a finite size scaling analysis based on numerical transfer matrix calculations we deduce critical properties and critical thresholds of various models and assess the accuracy of the homogeneity assumption specifically we analyze the q state potts model and the bond percolation on the 3 12 and kagome type subnet lattices n ifmmode times else texttimes fi n n ifmmode times else texttimes fi n n ensuremath le 4 for which the exact solution is not known our numerical determination of critical properties such as conformal anomaly and magnetic correlation length verifies that the universality principle holds to calibrate the accuracy of the finite size procedure we apply the same numerical analysis to models for which the exact critical frontiers are known the comparison of numerical and exact results shows that our numerical values are correct within errors of our finite size analysis which correspond to 7 or 8 significant digits this in turn infers that the homogeneity assumption determines critical frontiers with an accuracy of 5 decimal places or higher finally we also obtained the exact percolation thresholds for site percolation on kagome type subnet lattices 1 ifmmode times else texttimes fi 1 n ifmmode times else texttimes fi n for 1 ensuremath le n ensuremath le 6

spanning trees on graphs and lattices in d dimensions the problem of enumerating spanning trees on graphs and lattices is considered we obtain bounds on the number of spanning trees nst and establish inequalities relating the numbers of spanning trees of different graphs or lattices a general formulation is presented for the enumeration of spanning trees on lattices in d 2 dimensions and is applied to the hypercubic body centred cubic face centred cubic and specific planar lattices including the kagome diced 4 8 8 bathroom tile union jack and 3 12 12 lattices this leads to closed form expressions for nst for these lattices of finite sizes we prove a theorem concerning the classes of graphs and lattices with the property that nst exp nz as the number of vertices n where z is a finite non zero constant this includes the bulk limit of lattices in any spatial dimension and also sections of lattices whose lengths in some dimensions go to infinity while others are finite we evaluate z exactly for the lattices we consider and discuss the dependence of z on d and the lattice coordination number we also establish a relation connecting z to the free energy of the critical ising model for planar lattices

a generalization of the schur siegel smyth trace problem abstract let be a totally positive algebraic integer and define its absolute trace to be tr deg the trace of divided by the degree of elementary considerations show that the absolute trace is always at least one while it is plausible that for any 0 the absolute trace is at least 2 with only finitely many exceptions this is known as the schur siegel smyth trace problem our aim in this paper is to show that the schur siegel smyth trace problem can be considered as a special case of a more general problem

distribution of algebraic numbers schur studied limits of the arithmetic means an of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk if the leading coefficients are bounded schur proved that lim supn an 1 e 2 we show that an 0 and estimate the rate of convergence by generalizing the erd s turan theorem on the distribution of zeros as an application we show that integer polynomials have some unexpected restrictions of growth on the unit disk schur also studied problems on means of algebraic numbers on the real line when all conjugate algebraic numbers are positive the problem of finding the sharp lower bound for lim infn an was developed further by siegel and others we provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line potential theoretic methods allow us to consider distribution of algebraic numbers in or near general sets in the complex plane we introduce the generalized mahler measure and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one the quantitative aspects of this equidistribution are also analyzed in terms of the generalized mahler measure 1 schur s problems on means of algebraic numbers let e be a subset of the complex plane c consider the set of polynomials zn e of the exact degree n with integer coefficients and all zeros in e we denote the subset of zn e with simple zeros by zn e given m 0 we write pn anz zn e m if an m and pn zn e respectively pn zn e m if an m and pn zn e schur 45 4 8 studied the limit behavior of the arithmetic means of zeros for polynomials from zn e m as n where m 0 is an arbitrary fixed number his results may be summarized in the following statements let r 0 where r is the real line theorem a schur 45 satz ix given a polynomial pn z an n k 1 z k n define the arithmetic mean of squares of its zeros by sn n k 1 2 k n n if pn zn r m is any sequence of polynomials with degrees n then lim inf n sn e 1 6487 1 1 theorem b schur 45 satz xi for a polynomial pn z an n k 1 z k n define the arithmetic mean of its zeros by an n k 1 k n n if pn zn r m is any sequence of polynomials with degrees n then lim inf n an e 1 6487 1 2 1991 mathematics subject classification primary 11c08 secondary 11r04 26c10 30c15

on the absolute mahler measure of polynomials having all zeros in a sector let be an algebraic integer of degree d not 0 or a root of unity all of whose conjugates i are confined to a sector arg z we compute the greatest lower bound c of the absolute mahler measure i 1 d max 1 i 1 d of for belonging to nine subintervals of 0 2 3 in particular we show that c 2 1 12933793 from which it follows that any integer 1 and e i 3 all of whose conjugates have positive real part has absolute mahler measure at least c 2 this value is achieved for satisfying 1 0 2 where 0 1 3247 is the smallest pisot number the real root of 0 3 0 1

simple asymmetric exclusion model and lattice paths bijections and involutions we study the combinatorics of the change of basis of three representations of the stationary state algebra of the two parameter simple asymmetric exclusion process each of the representations considered correspond to a different set of weighted lattice paths which when summed over give the stationary state probability distribution we show that all three sets of paths are combinatorially related via sequences of bijections and sign reversing involutions this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

finite size corrections and scaling for the triangular lattice dimer model with periodic boundary conditions we analyze the partition function of the dimer model on m x n triangular lattice wrapped on the torus obtained by fendley moessner and sondhi phys rev b 66 214513 2002 based on such an expression we then extend the algorithm of ivashkevich izmailian and hu j phys a 35 5543 2002 to derive the exact asymptotic expansion of the first and second derivatives of the logarithm of the partition function at the critical point and find that the aspect ratio dependence of finite size corrections and the finite size scaling functions are sensitive to the parity of the number of lattice sites along the lattice axis

on integer chebyshev polynomials we are concerned with the problem of minimizing the supremum norm on o 1 of a nonzero polynomial of degree at most n with integer coefficients we use the structure of such polynomials to derive an efficient algorithm for computing them we give a table of these polynomials for degree up to 75 and use a value from this table to answer an open problem due to p borwein and t erdelyi and improve a lower bound due to flammang et al

synchronization in weighted uncorrelated complex networks in a noisy environment optimization and connections with transport efficiency motivated by synchronization problems in noisy environments we study the edwards wilkinson process on weighted uncorrelated scale free networks we consider a specific form of the weights where the strength and the associated cost of a link is proportional to kikj beta with ki and kj being the degrees of the nodes connected by the link subject to the constraint that the total edge cost is fixed we find that in the mean field approximation on uncorrelated scale free graphs synchronization is optimal at beta 1 numerical results based on exact numerical diagonalization of the corresponding network laplacian confirm the mean field results with small corrections to the optimal value of beta employing our recent connections between the edwards wilkinson process and resistor networks and some well known connections between random walks and resistor networks we also pursue a naturally related problem of optimizing performance in queue limited communication networks utilizing local weighted routing schemes

two point resistances in a family of self similar x y flower networks abstract the computation of resistance between two nodes in networks is a fundamental problem in both electric theory and graph theory in this paper first a recursive algorithm for computing resistance between any two nodes in a family of self similar x y flower networks is given the x y flower networks display rich behavior as observed in a large variety of real systems then as explanations using the algorithm explicit expressions for some resistances in 1 3 flower networks and 2 2 flower networks are obtained

packing dimers on 2p 1 x 2q 1 lattices we use computational method to investigate the number of ways to pack dimers on m x n lattices where both m and n are odd in this case there is always a single vacancy in the lattices we show that the dimer configuration numbers on 2k 1 x 2k 1 odd square lattices have some remarkable number theoretical properties in parallel to those of close packed dimers on 2k x 2k even square lattices for which an exact solution exists furthermore we demonstrate that there is an unambiguous logarithmic term in the finite size correction of free energy of odd by odd lattice strips with any width n or 1 this logarithmic term determines the distinct behavior of the free energy of odd square lattices these findings reveal a deep and previously unexplored connection between statistical physics models and number theory and indicate the possibility that the monomer dimer problem might be solvable

on the linear independence measure of logarithms of rational numbers in this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and in particular the linear independence measure of 1 log 2 log 3 log 5 and of 1 log 2 log 3 log 5 log 7 we also give a method to search for polynomials of smallest norm on a real interval a b which may be suitable for computing or improving the linear independence measure of logarithms of rational numbers

iterated stochastic measurements we describe a measurement device principle based on discrete iterations of bayesian updating of system state probability distributions although purely classical by nature these measurements are accompanied with a progressive collapse of the system state probability distribution during each complete system measurement this measurement scheme finds applications in analysing repeated non demolition indirect quantum measurements we also analyse the continuous time limit of these processes either in the brownian diffusive limit or in the poissonian jumpy limit in the quantum mechanical framework this continuous time limit leads to belavkin s equations which describe quantum systems under continuous measurements this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

the critical manifolds of inhomogeneous bond percolation on bow tie and checkerboard lattices we give a conditional derivation of the inhomogeneous critical percolation manifold of the bow tie lattice with five different probabilities a problem that does not appear at first to fall into any known solvable class although our argument is mathematically rigorous only on a region of the manifold we conjecture that the formula is correct over its entire domain and we provide a non rigorous argument for this that employs the negative probability regime of the triangular lattice critical surface we discuss how the rigorous portion of our result substantially broadens the range of lattices in the solvable class to include certain inhomogeneous and asymmetric bow tie lattices and that if it could be put on a firm foundation the negative probability portion of our method would extend this class to many further systems including f y wu s checkerboard formula for the square lattice we conclude by showing that this latter problem can in fact be proved using a recent result of grimmett and manolescu for isoradial graphs lending strong evidence in favor of our other conjectured results this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

recursion transform method for computing resistance of the complex resistor network with three arbitrary boundaries we develop a general recursion transform r t method for a two dimensional resistor network with a zero resistor boundary as applications of the r t method we consider a significant example to illuminate the usefulness for calculating resistance of a rectangular m ifmmode times else texttimes fi n resistor network with a null resistor and three arbitrary boundaries a problem never solved before since green s function techniques and laplacian matrix approaches are invalid in this case looking for the exact calculation of the resistance of a binary resistor network is important but difficult in the case of an arbitrary boundary since the boundary is like a wall or trap which affects the behavior of finite network in this paper we obtain several general formulas of resistance between any two nodes in a nonregular m ifmmode times else texttimes fi n resistor network in both finite and infinite cases in particular 12 special cases are given by reducing one of the general formulas to understand its applications and meanings and an integral identity is found when we compare the equivalent resistance of two different structures of the same problem in a resistor network

dimer model on a triangular lattice we analyze the partition function of the dimer model on an mathcal m ifmmode times else texttimes fi mathcal n triangular lattice wrapped on a torus obtained by fendley moessner and sondhi phys rev b 66 214513 2002 from a finite size analysis we have found that the dimer model on such a lattice can be described by a conformal field theory having a central charge c ensuremath 2 the shift exponent for the specific heat is found to depend on the parity of the number of lattice sites mathcal n along a given lattice axis e g for odd mathcal n we obtain the shift exponent ensuremath lambda 1 while for even mathcal n it is infinite ensuremath lambda ensuremath infty in the former case therefore the finite size specific heat pseudocritical point is size dependent while in the latter case it coincides with the critical point of the thermodynamic limit

hitting and commute times in large random neighborhood graphs in machine learning a popular tool to analyze the structure of graphs is the hitting time and the commute distance resistance distance for two vertices u and v the hitting time huv is the expected time it takes a random walk to travel from u to v the commute distance is its symmetrized version cuv huv hvu in our paper we study the behavior of hitting times and commute distances when the number n of vertices in the graph tends to infinity we focus on random geometric graphs e graphs knn graphs and gaussian similarity graphs but our results also extend to graphs with a given expected degree distribution or erdos renyi graphs with planted partitions we prove that in these graph families the suitably rescaled hitting time huv converges to 1 dv and the rescaled commute time to 1 du 1 dv where du and dv denote the degrees of vertices u and v in these cases hitting and commute times do not provide information about the structure of the graph and their use is discouraged in many machine learning applications

resistance distances in composite graphs the resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor let g h g h g h g h g h gh and g h be the join product composition direct product strong product corona and rooted product of two graphs g and h respectively in this paper formulae for resistance distances of these composite graphs are given in terms of parameters of the parent graphs and some properties are established explicit formulae are obtained for resistance distances of some classes of graphs including rook graphs diagonal mesh graphs and generalized double graphs

the smallest perron numbers a perron number is a real algebraic integer of degree d 2 whose conjugates are i such that max 2 i d i in this paper we compute the smallest perron numbers of degree d 24 and verify that they all satisfy the lind boyd conjecture moreover the smallest perron numbers of degree 17 and 23 give the smallest house for these degrees the computations use a family of explicit auxiliary functions these functions depend on generalizations of the integer transfinite diameter of some compact sets in c

two point resistance of a resistor network embedded on a globe we consider the problem of two point resistance in an m ensuremath 1 ifmmode times else texttimes fi n resistor network embedded on a globe a geometry topologically equivalent to an m ifmmode times else texttimes fi n cobweb with its boundary collapsed into one single point we deduce a concise formula for the resistance between any two nodes on the globe using a method of direct summation pioneered by one of us z z tan l zhou and j h yang j phys a math theor 46 195202 2013 this method is contrasted with the laplacian matrix approach formulated also by one of us f y wu j phys a math gen 37 6653 2004 which is difficult to apply to the geometry of a globe our analysis gives the result in the form of a single summation

the integer chebyshev problem the main problem of this chapter is to find a polynomial in z n of minimal supremum norm on an interval this is p1 and it is of a slightly different flavour than most of the other problems in this book in that there is no restriction on the size of the coefficients we now state p1 with greater precision

on the calculation of the generalized q function via parl s method corresp parl has previously given error bounds suitable for calculating marcum s q function however it is shown that additional terms in the error bound are necessary for accurate calculations of the generalized q function q m alpha beta the values of m alpha and beta that result in successful algorithm completion are examined numerical results are presented for calculations of the noncentral chi square distribution and the probability of detection for a classical multiobservation detection problem

the trace of totally positive algebraic integers for all totally positive algebraic numbers except a finite number of explicit exceptions the following inequality holds 1 d 1 d max 1 780022 1 66 1 where d is the degree of a and 0 1 d its conjugates this improves previous results of smyth flammang and rhin

resolution of singularities and modular galois theory i shall sketch a brief history of the desingularization problem from riemann thru zariski to hironaka including the part i played in it and the work on galois theory which this led me to and how that caused me to search out many group theory gurus i shall also formulate several conjectures and suggest numerous thesis problems

the completely packed o n loop model on the square lattice we present an investigation of the completely packed o n loop model on the square lattice by means of the transfer matrix method and finite size scaling we investigate the model for a number of n values covering a wide range this model is known to be equivalent with the q state potts model with q n 2 but here we also investigate the range n 2 the system is in an ordered phase where the majority of the loops cover part of the elementary faces of the lattice in one of two checkerboard patterns that are in phase coexistence furthermore we find that the numerical results for the free energy density are in agreement with the expressions obtained from the exact analysis of the equivalent six vertex model this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

dimers on two types of lattices on the klein bottle the problem of enumerating close packed dimers or perfect matchings on two types of lattices the so called 8 8 4 and 8 8 6 lattices embedded on the klein bottle is considered and we obtain the explicit expression of the number of close packed dimers and entropy our results imply that 8 8 4 lattices have the same entropy under three different boundary conditions cylindrical toroidal and klein bottle and 8 8 6 lattices have the same property this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

algebraic integers whose conjugates all lie in an ellipse we find all 15909 algebraic integers whose conjugates all lie in an ellipse with two of them nonreal while the others lie in the real interval 1 2 this problem has applications to finding certain subgroups of sl 2 c we use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of c this gives good bounds for the coefficients of the minimal polynomial of a

theory of impedance networks the two point impedance and lc resonances we present a formulation of the determination of the impedance between any two nodes in an impedance network an impedance network is described by its laplacian matrix l which has generally complex matrix elements we show that by solving the equation lu u with orthonormal vectors ua the effective impedance between nodes p and q of the network is zpq u p u q 2 where the summation is over all not identically equal to zero and u p is the pth component of u for networks consisting of inductances l and capacitances c the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of this curious result suggests the possibility of practical applications to resonant circuits our formulation is illustrated by explicit examples

the equivalent resistance of a 3 n cobweb network and its conjecture of an m n cobweb network we investigate the equivalent resistance of a 3 n cobweb network the difference equations of the model are constructed by network analysis and their general solution is obtained by matrix transformations it is found that the equivalent resistance can be expressed by the trigonometric function of k 7 which decreases with the increase of the order n by analyzing and comparing the equivalent resistances of the 1 n 2 n and 3 n cobweb networks a conjecture on the equivalent resistance of the m n cobweb network is proposed

properties and applications of laplacian spectra for koch networks for a network knowledge of its laplacian eigenvalues is central to understanding its structure and dynamics in this paper we study the laplacian spectra of a family of koch networks with scale free and small world properties we derive some recursive relations between the laplacian characteristic polynomials of koch networks and their subgraphs at different iterations based on the obtained recurrence relations we determine explicitly the product of all nonzero laplacian eigenvalues as well as the sum of the reciprocals of these eigenvalues then using these results we further evaluate the number of spanning trees kirchhoff index global mean first passage time and average path length of the family of koch networks finally we determine the number of spanning forests under certain conditions we expect that our method can be adapted to other types of self similar networks

approximate expressions for the output signal to noise ratios for two types of cross correlators with correlated bandpass inputs thus far analytical expressions have not been developed for the output signal to noise ratio of a polarity coincidence correlator when the noise is correlated between the input channels approximate expressions are developed here for the output signal to noise ratios for two types of cross correlator systems one being the polarity coincidence correlator with a hard limiter in each input channel and the other variation of this correlator having a hard limiter in only one input channel the approximation scheme is based upon the known output signal to noise characteristics of a limiter and that of the standard analog correlator with no limiters the derived expressions are applicable for the case of deterministic sine waves plus narrow band gaussian noise in each input channel where the noise between channels may be correlated for the case of uncorrelated noise between input channels good agreement is found between the approximate formulas and known exact results finally asymptotic expressions are developed for certain limiting cases of input signal to noise ratios having extreme values in one or both input channels

bivariate factorizations connecting dickson polynomials and galois theory in his ph d thesis of 1897 dickson introduced certain permutation polynomials whose galois groups are essentially the dihedral groups these are now called dickson polynomials of the first kind to distinguish them from their variations introduced by schur in 1923 which are now called dickson polynomials of the second kind in the last few decades there have been extensive investigations of both of these types which are related to the classical chebyshev polynomials we give new bivariate factorizations involving both types of dickson polynomials these factorizations demonstrate certain isomorphisms between dihedral groups and orthogonal groups and lead to the construction of explicit equations with orthogonal groups as galois groups

asymptotic expansion for the resistance between two maximally separated nodes on an m by n resistor network we analyze the exact formulae for the resistance between two arbitrary notes in a rectangular network of resistors under free periodic and cylindrical boundary conditions obtained by wu j phys a 37 6653 2004 based on such expression we then apply the algorithm of ivashkevich izmailian and hu j phys a 35 5543 2002 to derive the exact asymptotic expansions of the

finite size effects for anisotropic 2d ising model with various boundary conditions we analyze the exact partition function of the anisotropic ising model on finite m n rectangular lattices under four different boundary conditions periodic periodic pp periodic antiperiodic pa antiperiodic periodic ap and antiperiodic antiperiodic aa obtained by kaufman 1949 phys rev 76 1232 wu and hu 2002 j phys a math gen 35 5189 and kastening 2002 phys rev e 66 057103 we express the partition functions in terms of the partition functions z j k with 0 0 1 2 0 0 1 2 and 1 2 1 2 j is an interaction coupling and k is an anisotropy parameter based on such expressions we then extend the algorithm of ivashkevich et al 2002 j phys a math gen 35 5543 to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above our result is f fbulk p 0fp k s p 1 where f is the free energy of the system fbulk is the free energy of the bulk s mn is the area of the lattice and m n is the aspect ratio all coefficients in this expansion are expressed through analytical functions we have introduced the effective aspect ratio eff sinh 2jc and show that for pp and aa boundary conditions all finite size correction terms are invariant under the transformation eff 1 eff this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

two new points in the spectrum of the absolute mahler measure of totally positive algebraic integers for totally positive algebraic integers 0 1 of degree d we consider the set l of values of m 1 d where m is the mahler measure of c j smyth has found the four smallest values of l and conjectured that the fifth point is 2 cos 2 60 2 we prove that this is so and moreover we give the sixth point of l

exact evaluation of the resistance in an infinite face centered cubic network the equivalent resistance between the origin and the lattice site 2n 0 0 in an infinite face centered cubic fcc network consisting from identical resistors each of resistance r has been evaluated analytically and numerically the asymptotic behavior of the equivalent resistance has been also investigated finally some numerical values for the equivalent resistance are presented

theory of resistor networks the two point resistance the resistance between two arbitrary nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the laplacian matrix associated with the network explicit formulae for two point resistances are deduced for regular lattices in one two and three dimensions under various boundary conditions including that of a mobius strip and a klein bottle the emphasis is on lattices of finite sizes we also deduce summation and product identities which can be used to analyse large size expansions in two and higher dimensions

resistance based performance analysis of the consensus algorithm over geometric graphs the performance of the linear consensus algorithm is studied by using a linear quadratic lq cost the objective is to understand how the communication topology influences this algorithm this is achieved by exploiting the analogy between markov chains and electrical resistive networks indeed this allows us to uncover the relation between the lq performance cost and the average effective resistance of a suitable electrical network and moreover to show that if the communication graph fulfills some local properties then its behavior can be approximated by that of a grid which is a graph whose associated lq cost is well known

trace of totally positive algebraic integers and integer transfinite diameter the diameter of a totally real algebraic integer of degree d with conjugates 1 2 d is diam d 1 for all positive integers k and n diam 2 cos 2k n is less than 4 r m robinson has computed modulo integer translations all the other totally real algebraic integers with diam 4 for d 8 we have done the computations for all d 15 we use a large family of explicit auxiliary functions related to generalized integer transfinite diameter of real intervals they give good bounds for the coefficients of the minimal polynomial of for d 15 we prove a recent conjecture of capparelli et al mathematics subject classification 2000 primary 11c08 11y40

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

integer transfinite diameter and polynomials with small mahler measure in this work we show how suitable generalizations of the integer transfinite diameter of some compact sets in c give very good bounds for coefficients of polynomials with small mahler measure by this way we give the list of all monic irreducible primitive polynomials of z x of degree at most 36 with mahler measure less than 1 324 and of degree 38 and 40 with mahler measure less than 1 31

resistance distance in potting networks abstract on a resistor network the resistance distance is the effective resistance r u v between two nodes u v of the network in this paper we study a family of self similar and symmetric networks named potting networks and obtain a recursion formula to calculate resistance distances

a trace bound for positive definite connected integer symmetric matrices abstract let a be a connected integer symmetric matrix i e a a i j m n z for some n a a t and the underlying graph vertices corresponding to rows with vertex i joined to vertex j if a i j 0 is connected we show that if all the eigenvalues of a are strictly positive then tr a 2 n 1 there are two striking corollaries first the analogue of the schur siegel smyth trace problem is solved for characteristic polynomials of connected integer symmetric matrices second we find new examples of totally real separable irreducible monic integer polynomials that are not minimal polynomials of integer symmetric matrices

the green function for the radial schramm loewner evolution we prove the existence of the green function for radial sle for 8 unlike the chordal case where an explicit formula for the green function is known for all values of 8 we give an explicit formula only for 4 for other values of we give a formula in terms of an expectation with respect to sle conditioned to go through a point

average resistance of toroidal graphs abstract the average effective resistance of a graph is a relevant performance index in many applications including distributed estimation and control of network systems in this paper we study how the average resistance depends on the graph topology and specifically on the dimension of the graph we concentrate on d dimensional toroidal grids and we exploit the connection between resistance and laplacian eigenvalues our analysis provides tight estimates of the average resistance which are key to study its asymptotic behavior when the number of nodes grows to infinity in dimension two the average resistance diverges in this case we are able to capture its rate of growth when the sides of the grid grow at different rates in higher dimensions the average resistance is bounded uniformly in the number of nodes in this case we conjecture that its value is of order 1 d for large d we prove this fact for hypercubes and when the side lengths go to infinity

resistances between two nodes of a path network abstract considerable progress has recently been made in the development of techniques to exactly determine two point resistances in networks of various topologies given n positive integers m 1 m 2 m n let p m i 1 n be the network with node set v v 1 v 2 v n where v i v j if i j and v i m i and with edge set e u v u v i v v i 1 i 1 2 n 1 in this paper using triangle star transformation and the principle of substitution we express arbitrary two nodes resistances of a path network in terms of the number of their nodes

integrability as a consequence of discrete holomorphicity loop models it has recently been established that imposing the condition of discrete holomorphicity on a lattice parafermionic observable leads to the critical boltzmann weights in a number of lattice models remarkably the solutions of these linear equations also solve the yang baxter equations we extend this analysis for the zn model by explicitly considering the condition of discrete holomorphicity on two and three adjacent rhombi for two rhombi this leads to a quadratic equation in the boltzmann weights and for three rhombi a cubic equation the two rhombus equation implies the inversion relations the star triangle relation follows from the three rhombus equation we also show that these weights are self dual as a consequence of discrete holomorphicity this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

new expressions for the pdf and cdf of the filtered output of an analog cross correlator corresp a general expression is derived for the probability density function pdf associated with the filtered output of an analog cross correlator the inputs of which consist of deterministic signals having the same frequency plus correlated stationary gaussian noise processes the correlator is composed of input bandpass filters a multiplier and a zonal iowpass filter the noise components in each channel have distinct variances and obey no spectral symmetry condition using the derived expression for the pdf the cumulative distribution function cdf under the same general conditions is calculated all previous known special cases are subsequently deduced from the general expressions given here the general results derived here are more suitable for numerical computations than comparable previous results found in the literature due to the fact that expressions here do not involve multiple infinite series lastly asymptotic formulas are developed for both the pdf and cdf which may be helpful in certain calculations

further nice equations for nice groups nice sextinomial equations are given for unramified coverings of the affine line in nonzero characteristic p with p 2m q and 2m q as galois groups where m 3 is any integer and q 1 is any power of p 2

logarithmic corrections in the free energy of monomer dimer model on plane lattices with free boundaries using exact computations we study the classical hard core monomer dimer models on m x n plane lattice strips with free boundaries for an arbitrary number v of monomers or vacancies we found a logarithmic correction term in the finite size correction of the free energy per lattice site the coefficient of the logarithmic correction term depends on the number of monomers present v and the parity of the width n of the lattice strip the coefficient equals to v when n is odd and v 2 when n is even the results are generalizations of the previous results for a single monomer in an otherwise fully packed lattice of dimers we also study the finite size correction in the low dimer density limit where the number of dimers d is fixed in this case the coefficient of the logarithmic correction term equals to d for both odd and even n

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

calculating two point resistances in distance regular resistor networks an algorithm for the calculation of the resistance between two arbitrary nodes in an arbitrary distance regular resistor network is provided where the calculation is based on stratification introduced in jafarizadeh and salimi 2006 j phys a math gen 39 1 29 and the stieltjes transform of the spectral distribution stieltjes function associated with the network it is shown that the resistances between a node and all nodes belonging to the same stratum with respect to the r i belonging to the ith stratum with respect to the a re the same also the analytical formulae for two point resistances r i i 1 2 3 are given in terms of the size of the network and corresponding intersection numbers in particular the two point resistances in a strongly regular network are given in terms of its parameters v moreover the lower and upper bounds for two point resistances in strongly regular networks are discussed

loop models on random maps via nested loops the case of domain symmetry breaking and application to the potts model we use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors which can be assigned different local weights hence allowing for an explicit z2 domain symmetry breaking each loop receives a non local weight n as well as a local bending energy which controls loop turns by a standard cluster construction that we review the q n2 potts model on general random maps is mapped to a particular instance of this problem with domain non symmetric weights we derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color and solve it by extending well known complex analytic techniques in the case where loops are fully packed we analyze in detail the phase diagram of the model and derive exact equations for the position of its non generic critical points in particular we underline that the critical potts model on general random maps is not self dual whenever q 1 in a model with domain symmetric weights we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

potts model and graph theory elementary exposition is given of some recent developments in studies of graphtheoretic aspects of the potts model topics discussed include graphical expansions of the potts partition function and correlation functions and their relationships with the chromatic dichromatic and flow polynomials occurring in graph theory it is also shown that the potts model realization of these classic graph theoretic problems provides alternate and direct proofs of properties established heretofore only in the context of formal graph theory

finite size effects for the ising model on helical tori we analyze the exact partition function of the ising model on a square lattice under helical boundary conditions obtained by liaw et al phys rev e 73 055101 r 2006 based on such an expression we then extend the algorithm of ivashkevich izmailian and hu j phys a 35 5543 2002 to derive an exact asymptotic expansion of the logarithm of the partition function and its first to fourth derivatives at the critical point from such results we find that the shift exponent for the specific heat is ensuremath lambda 1 for all values of the helicity factor d we also find that finite size corrections for the free energy the internal energy and the specific heat of the model in a crucial way depend on the helicity factor of the lattice

computing totally positive algebraic integers of small trace we construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials many new examples of such polynomials of minimal absolute trace for given degree are found the computations are pushed to degrees that previously were unattainable and one consequence is that the new examples form the majority of all those known as an application we produce a new bound for the schur siegel smyth trace problem

comparison of generalized q function algorithms corresp several methods have been proposed for calculating the generalized q function q m alpha beta but only isolated comparisons of the various techniques have been reported the accuracy range and execution time of prominent methods for computing the q function are compared and limitations of the algorithms are identified

a triangular gap of side 2 in a sea of dimers in a 60 angle we consider a triangular gap of side 2 in a 60 angle on the triangular lattice whose sides are zigzag lines we study the interaction of the gap with the corner as the rest of the angle being completely filled with lozenges we show that the resulting correlation is governed by the product of the distances between the gap and its five images in the sides of the angle this provides a new aspect of the parallel between the correlation of gaps in dimer packings and electrostatics developed by the first author in previous work this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

resistance between two nodes in general position on an m n fan network the resistance between two nodes in general position on a fan network with n radial lines and m transverse lines is determined also a similar result of izmailian kenna and wu j phys a math theor 47 035003 2014 for an m ifmmode times else texttimes fi n cobweb network is reproduced but the method used here is significantly different it avoids the use of the kirchhoff matrix requires the solution of just one instead of two eigenvalue problems and results directly in only a single summation further the current distribution is given explicitly as a byproduct of the method the method is the same as that used by tan zhou and yang j phys a math theor 46 195202 2013 to find the cobweb resistance between center and perimeter for 1 ensuremath le m ensuremath le 3 and general n proof of their conjecture for general m is discussed

a theoretical and experimental comparison of output signal to noise ratios of three types of cross correlators the output signal to noise ratio as a function of input signal to noise ratios is compared for three different types of cross correlators 1 the standard analog cross correlator consisting of bandpass filters multiplier and low pass filter 2 the polarity coincidence correlator pcc which incorporates hard limiters in each analog input channel and 3 a modified type of pcc wherein only one of the analog channels utilizes a hard limiter some of the previous theoretical work on the pcc is extended to include new expressions for the output signal to noise ratio from which computations can more readily be made in addition discrete circuits were fabricated that closely approximate idealized elements so as to experimentally verify the theoretical results in almost all cases tested the experimental results were in close agreement with the math models