complex temperature singularities in potts models on the square lattice we report some results on the complex temperature ct singularities of q state potts models on the square lattice we concentrate on the problematic region mathrm re a l0 where a e k in which ct zeros of the partition function are sensitive to finite lattice artifacts from analyses of low temperature series expansions for 3l ql 8 we establish the existence in this region of complex conjugate ct singularities at which the magnetization and susceptibility diverge from calculations of zeros of the partition function we obtain evidence consistent with the inference that these singularities occur at endpoints a e a e of arcs protruding into the complex temperature extension of the ferromagnetic phase exponents for these singularities are determined e g for q 3 we find ensuremath beta e ensuremath 0 125 1 consistent with ensuremath beta e ensuremath frac 1 8 by duality these results also imply associated arcs extending into the ct extension of the symmetric paramagnetic phase analytic expressions are suggested for the positions of some of these singularities e g for q 5 our finding is consistent with the exact value a e a e 2 ensuremath 1 ensuremath mp i further discussions of complex temperature phase diagrams are given

a new representation for at s 2 3 a new representation of a t s 2 3 of a having prescribed range space t and null space s is derived using this representation the well known generalized inverses such as the moore penrose inverse the group inverse the drazin inverse the bott duffin inverse and the generalized bott duffin inverse are computed three numerical examples are worked out to demonstrate the computation of the moore penrose inverse the drazin inverse and the 2 3 inverse to show the efficacy of our approach

ground state entropy of the potts antiferromagnet on strips of the square lattice abstract we present exact solutions for the zero temperature partition function chromatic polynomial p and the ground state degeneracy per site w exponent of the ground state entropy for the q state potts antiferromagnet on strips of the square lattice of width ly vertices and arbitrarily great length lx vertices the specific solutions are for a ly 4 fbcy pbcx cyclic b ly 4 fbcy tpbcx mobius c ly 5 6 pbcy fbcx cylindrical and d l y 5 fbc y fbc x open where fbc pbc and tpbc denote free periodic and twisted periodic boundary conditions respectively in the lx limit of each strip we discuss the analytic structure of w in the complex q plane the respective w functions are evaluated numerically for various values of q several inferences are presented for the chromatic polynomials and analytic structure of w for lattice strips with arbitrarily great ly the absence of a nonpathological lx limit for real nonintegral q in the interval 0 0 for strips of the square triangular lattice is discussed

chromatic polynomials for j h i strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p for n vertex strip graphs of the form j l 1 m h i where j and i are various subgraphs on the left and right ends of the strip whose bulk is comprised of m fold repetitions of a subgraph h the strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction this extends our earlier calculations for strip graphs of the form l 1 m h i we use a generating function method from these results we compute the asymptotic limiting function w lim n p 1 n for q z this has physical significance as the ground state degeneracy per site exponent of the ground state entropy of the q state potts antiferromagnet on the given strip in the complex q plane w is an analytic function except on a certain continuous locus b in contrast to the l 1 m h i strip graphs where b i is independent of i and ii consists of arcs and possible line segments that do not enclose any regions in the q plane we find that for some j l 1 m h i strip graphs b i does depend on i and j and ii can enclose regions in the q plane our study elucidates the effects of different end subgraphs i and j and of boundary conditions on the infinite length limit of the strip graphs

transfer matrices and partition function zeros for antiferromagnetic potts models iii triangular lattice chromatic polynomial we study the chromatic polynomial pg q for m n triangular lattice strips of widths m 12p 9f with periodic or free transverse boundary conditions respectively and arbitrary lengths n with free longitudinal boundary conditions the chromatic polynomial gives the zero temperature limit of the partition function for the q state potts antiferromagnet we compute the transfer matrix for such strips in the fortuin kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q plane in the limit n we recompute the limiting curve obtained by baxter in the thermodynamic limit m n and find new interesting features with possible physical consequences finally we analyze the isolated limiting points and their relation with the beraha numbers

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

innovation based on gaussian elimination to compute generalized inverse at s 2 abstract in this paper we execute elementary row and column operations on the partitioned matrix g a g g g 0 into i s 0 0 0 0 0 a t s 2 to compute generalized inverse a t s 2 of a given complex matrix a where g is a matrix such that r g t and n g s the total number of multiplications and divisions operations is t m n s 2 m n 2 4 m s 1 2 n s m s n s m n s and the upper bound of t m n s is less than 6 m n 2 3 2 n 3 1 2 n 2 when n m a numerical example is shown to illustrate that this method is correct

families of graphs with w r g q functions that are nonanalytic at 1 q 0 denoting p g q as the chromatic polynomial for coloring an n vertex graph g with q colors and considering the limiting function w g q lim n to infty p g q 1 n a fundamental question in graph theory is the following is w r g q q 1 w g q analytic or not at the origin of the 1 q plane where the complex generalization of q is assumed this question is also relevant in statistical mechanics because w g q exp s 0 k b where s 0 is the ground state entropy of the q state potts antiferromagnet on the lattice graph g and the analyticity of w r g q at 1 q 0 is necessary for the large q series expansions of w r g q although w r is analytic at 1 q 0 for many g there are some g for which it is not for these w r has no large q series expansion it is important to understand the reason for this nonanalyticity here we give a general condition that determines whether or not a particular w r g q is analytic at 1 q 0 and explains the nonanalyticity where it occurs we also construct infinite families of graphs with w r functions that are non analytic at 1 q 0 and investigate the properties of these functions our results are consistent with the conjecture that a sufficient condition for w r g q to be analytic at 1 q 0 is that g is a regular lattice graph lambda this is known not to be a necessary condition

thin fisher zeros various authors have suggested that the loci of partition function zeros can profitably be regarded as phase boundaries in the complex temperature or field planes we obtain the fisher zeros for ising and potts models on non planar thin regular random graphs using this approach and note that the locus of fisher zeros on a bethe lattice is identical to the corresponding random graph since the number of states q appears as a parameter in the potts solution the limiting locus of chromatic zeros is also accessible

on integral representation of the generalized inverse at s 2 we present a general integral representation for the generalized inverse a t s 2 which extends earlier result on the moore penrose inverse weighted moore penrose inverse and drazin inverse

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

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

zeros of jones polynomials for families of knots and links abstract we calculate jones polynomials v l t for several families of alternating knots and links by computing the tutte polynomials t g x y for the associated graphs g and then obtaining v l t as a special case of the tutte polynomial for each of these families we determine the zeros of the jones polynomial including the accumulation set in the limit of infinitely many crossings a discussion is also given of the calculation of jones polynomials for non alternating links

chromatic polynomials for families of strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p gs m q and from these the asymptotic limiting functions w g s q lim n p g s q 1 n for families of n vertex graphs gs m comprised of m repeated subgraphs h adjoined to an initial graph i these calculations of w gs q for infinitely long strips of varying widths yield important insights into properties of w q for two dimensional lattices in turn these results connect with statistical mechanics since w q is the ground state degeneracy of the q state potts model on the lattice for our calculations we develop and use a generating function method which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant w gs q function in the limit n from this we obtain the exact continuous locus of points b where w gs q is nonanalytic in the complex q plane this locus is shown to consist of arcs which do not separate the q plane into disconnected regions zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities b we find that as the width of the infinitely long strips is increased the arcs comprising b elongate and move toward each other which enables one to understand the origin of closed regions that result for the infinite 2d lattice

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

flow polynomials and their asymptotic limits for lattice strip graphs we present exact calculations of flow polynomials f g q for lattice strips of various fixed widths l y 4 and arbitrarily great lengths l x with several different boundary conditions square honeycomb and triangular lattice strips are considered we introduce the notion of flows per face fl in the infinite length limit we study the zeros of f g q in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite length limit for the various families of strips the function fl is nonanalytic on this locus the loci are found to be noncompact for many strip graphs with periodic or twisted periodic longitudinal boundary conditions and compact for strips with free longitudinal boundary conditions we also find the interesting feature that aside from the trivial case l y 1 the maximal point q cf where crosses the real axis is universal on cyclic and mobius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf 3 for the infinite square lattice

ground state entropy of potts antiferromagnets on homeomorphic families of strip graphs abstract we present exact calculations of the zero temperature partition function and ground state degeneracy per site w for the q state potts antiferromagnet on a variety of homeomorphic families of planar strip graphs g ch k1 k2 k m where k1 k2 and k describe the homeomorphic structure and m denotes the length of the strip several different ways of taking the total number of vertices to infinity by sending i m with k1 k2 and k fixed ii k1 and or k2 with m and k fixed and iii k with m and p k1 k2 fixed are studied and the respective loci of points b where w is nonanalytic in the complex q plane are determined the b s for limit i are comprised of arcs which do not enclose regions in the q plane and for many values of p and k include support for re q b for limits ii and iii is the unit circle q 1 1

end graph effects on chromatic polynomials for strip graphs of lattices and their asymptotic limits abstract we report exact calculations of the ground state degeneracy per site exponent of the ground state entropy w g q of the q state potts antiferromagnet on infinitely long strips with specified end graphs for free boundary conditions in the longitudinal direction and free and periodic boundary conditions in the transverse direction this is equivalent to calculating the chromatic polynomials and their asymptotic limits for these graphs making the generalization from q z to q c we determine the full locus b on which w is nonanalytic in the complex q plane we report the first example for this class of strip graphs in which b encloses regions even for planar end graphs the bulk of the specific strip graph that exhibits this property is a part of the 3 3 4 2 archimedean lattice

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

the reverse order law for 1 3 4 inverse of the product of two matrices the relationship between 1 3 4 inverses of ab and the product of 1 3 4 inverses of a and b have been studied in this paper the necessary and sufficient conditions for b 1 3 4 a 1 3 4 ab 1 3 4 b 1 3 4 a 1 3 4 ab 1 3 4 and b 1 3 4 a 1 3 4 ab 1 3 4 are given

lower bounds and series for the ground state entropy of the potts antiferromagnet on archimedean lattices and their duals we prove a general rigorous lower bound for w lambda q exp s 0 lambda q k b the exponent of the ground state entropy of the q state potts antiferromagnet on an arbitrary archimedean lattice lambda we calculate large q series expansions for the exact w r lambda q q 1 w lambda q and compare these with our lower bounds on this function on the various archimedean lattices it is shown that the lower bounds coincide with a number of terms in the large q expansions and hence serve not just as bounds but also as very good approximations to the respective exact functions w r lambda q for large q on the various lattices lambda plots of w r lambda q are given and the general dependence on lattice coordination number is noted lower bounds and series are also presented for the duals of archimedean lattices as part of the study the chromatic number is determined for all archimedean lattices and their duals finally we report calculations of chromatic zeros for several lattices these provide further support for our earlier conjecture that a sufficient condition for w r lambda q to be analytic at 1 q 0 is that lambda is a regular lattice

a finite algorithm for computing the weighted moore penrose inverse a mn this paper presents a finite algorithm for computing the weighted moore penrose inverse of a rectangular matrix this algorithm may be viewed as an extension of decell s algorithm

comments on some recent results concerning 2 3 and 2 4 generalized inverses abstract we give a comment on some recent results concerning the representations of generalized 2 3 and 2 4 inverses shorter proofs of some previous results are presented

complex q zeros of the partition function of the potts model with long range interactions abstract the zeros of the partition function of the ferromagnetic q state potts model with long range interactions in the complex q plane are studied in the mean field case while preliminary numerical results are reported for the finite 1 d chains with power law decaying interactions in both cases at any fixed temperature the zeros lie on the arc shaped contours which cross the positive real axis at the value for which the given temperature is transition temperature for finite number of spins the positive real axis is free of zeros which approach to it in the thermodynamic limit the convergence exponent of the zero closest to the positive real q axis is found to have the same value as the temperature critical exponent 1

transfer matrices for the zero temperature potts antiferromagnet on cyclic and mobius lattice strips abstract we present transfer matrices for the zero temperature partition function of the q state potts antiferromagnet equivalently the chromatic polynomial on cyclic and mobius strips of the square triangular and honeycomb lattices of width l y and arbitrarily great length l x we relate these results to our earlier exact solutions for square lattice strips with l y 3 4 5 triangular lattice strips with l y 2 3 4 and honeycomb lattice strips with l y 2 3 and periodic or twisted periodic boundary conditions we give a general expression for the chromatic polynomial of a mobius strip of a lattice and exact results for a subset of honeycomb lattice transfer matrices both of which are valid for arbitrary strip width l y new results are presented for the l y 5 strip of the triangular lattice and the l y 4 and l y 5 strips of the honeycomb lattice using these results and taking the infinite length limit l x we determine the continuous accumulation locus of the zeros of the above partition function in the complex q plane including the maximal real point of nonanalyticity of the degeneracy per site w as a function of q

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact chromatic polynomials for toroidal chains of complete graphs abstract we present exact calculations of the partition function of the zero temperature potts antiferromagnet equivalently the chromatic polynomial for graphs of arbitrarily great length composed of repeated complete subgraphs kb with b 5 6 which have periodic or twisted periodic boundary condition in the longitudinal direction in the lx limit the continuous accumulation set of the chromatic zeros b is determined we give some results for arbitrary b including the extrema of the eigenvalues with coefficients of degree b 1 and the explicit forms of some classes of eigenvalues we prove that the maximal point where b crosses the real axis qc satisfies the inequality qc b for 2 b the minimum value of q at which b crosses the real q axis is q 0 and we make a conjecture concerning the structure of the chromatic polynomial for klein bottle strips

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

the representation of generalized inverse at s 2 3 and its applications this paper presents the explicit expression for matrix right symmetry factor with prescribed rang and null space moreover the explicit expression for generalized inverse a t s 2 3 which is a 2 3 inverse of a having the prescribed rang t and null space s is derived as an application two numerical examples are given

successive matrix squaring algorithm for parallel computing the weighted generalized inverse amn we derive a successive matrix squaring sms algorithm to approximate the weighted generalized inverse which can be expressed in the form of successive squaring of a composite matrix t given an m by n matrix a with m n we show that the weighted generalized inverse of a can be computed in parallel time ranging from o log n to o log 2n provided that there are enough processors to support matrix multiplication in time o log n

ground state degeneracy of potts antiferromagnets cases with noncompact w boundaries having multiple points at we present exact calculations of the zero temperature partition function and ground state degeneracy per site for the q state potts antiferromagnet on a number of families of graphs for which the boundary of regions of analyticity of w in the complex q plane is noncompact and has the properties that i in the plane the point z 0 is a multiple point on and ii includes support for these families are generated by the method of homeomorphic expansion our results give further insight into the conditions for the validity of large q series expansions for the reduced function

the representation and computation of generalized inverse at s 2 this paper presents a novel representation for the generalized inverse a t s 2 based on this we give an algorithm to compute this generalized inverse as an application we use gauss jordan elimination to compute the weighted moore penrose inverse a m n and the drazin inverse a d

t 0 partition functions for potts antiferromagnets on lattice strips with fully periodic boundary conditions abstract we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet equivalently the chromatic polynomial for families of arbitrarily long strip graphs of the square and triangular lattices with width ly 4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation i e of torus or klein bottle type these boundary conditions have the advantage of removing edge effects in the limit of infinite length we calculate the exponent of the entropy w q and determine the continuous locus b where it is singular we also give results for toroidal strips involving crossing subgraphs these make possible a unified treatment of torus and klein bottle boundary conditions and enable us to prove that for a given strip the locus b is the same for these boundary conditions

computing 2 4 and 2 3 inverses by using the sherman morrison formula a finite recursive procedure for computing 2 4 generalized inverses and the analogous recursive procedure for computing 2 3 generalized inverses of a given complex matrix are presented the starting points of both introduced methods are general representations of these classes of generalized inverses these representations are formed using certain matrix products which include the moore penrose inverse or the usual inverse of a symmetric matrix product and the sherman morrison formula for the inverse of a symmetric rank one matrix modification the computational complexity of the methods is analyzed defined algorithms are tested on randomly generated matrices as well as on test matrices from the matrix computation toolbox

matrix left symmetry factor and its applications in generalized inverses a 2 4 t s in this paper we propose some definitions about matrix symmetry factors and discuss the construction of matrix left symmetry factor in detail the explicit expression of the generalized inverses at s 2 4 which is a 2 4 inverses of a having the prescribed rang t and null space s is derived furthermore using the expression at s 2 4 we get the corresponding results for six important generalized inverses

full rank representations of 2 4 2 3 inverses and successive matrix squaring algorithm abstract we present the full rank representations of 2 4 and 2 3 inverses with given rank as well as with prescribed range and null space as particular cases of the full rank representation of outer inverses as a consequence two applications of the successive matrix squaring sms algorithm from p s stanimirovic d s cvetkovic ilic successive matrix squaring algorithm for computing outer inverses appl math comput 203 2008 19 29 are defined using the full rank representations of 2 4 and 2 3 inverses the first application is used to approximate 2 4 inverses the second application after appropriate modifications of the sms iterative procedure computes 2 3 inverses of a given matrix presented numerical examples clarify the purpose of the introduced methods

the generalized bott duffin inverse and its applications abstract the generalized bott duffin inverse a l of a with respect to a subspace l is defined and by means of it a unified treatment of a class of linear systems and extreme value problems with equality constraints is given

transfer matrices and partition function zeros for antiferromagnetic potts models i general theory and square lattice chromatic polynomial we study the chromatic polynomials zero temperature antiferromagnetic potts model partition functions p g q for m n rectangular subsets of the square lattice with m 8 free or periodic transverse boundary conditions and n arbitrary free longitudinal boundary conditions using a transfer matrix in the fortuin kasteleyn representation in particular we extract the limiting curves of partition function zeros when n which arise from the crossing in modulus of dominant eigenvalues beraha kahane weiss theorem we also provide evidence that the beraha numbers b2 b3 b4 b5 are limiting points of partition function zeros as n whenever the strip width m is 7 periodic transverse b c or 8 free transverse b c along the way we prove that a noninteger beraha number except perhaps b10 cannot be a chromatic root of any graph

families of graphs with chromatic zeros lying on circles we define an infinite set of families of graphs which we call p wheels and denote wh p n that generalize the wheel p 1 and biwheel p 2 graphs the chromatic polynomial for wh p n is calculated and remarkably simple properties of the chromatic zeros are found i the real zeros occur at q 0 1 p 1 for n p even and q 0 1 p 2 for n p odd and ii the complex zeros all lie equally spaced on the unit circle q p 1 1 in the complex q plane in the n to infty limit the zeros on this circle merge to form a boundary curve separating two regions where the limiting function w wh p q is analytic viz the exterior and interior of the above circle connections with statistical mechanics are noted