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

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

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

the cells of the affine weyl group c n in a certain quasi split case abstract the affine weyl group c n s can be realized as the fixed point set of the affine weyl group a 2 n 1 s under a certain group automorphism with s s let l be the length function of a 2 n 1 we study the cells of the weighted coxeter group c n l the main results of the paper are to give an explicit description for all the cells of c n l corresponding to the partitions k 1 2 n k and h 21 2 n h 2 for all 1 k 2 n and 2 h 2 n 2 and also for all the cells of c 3 l

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

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

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

the laurent polynomials my ws in the hecke algebra with unequal parameters abstract let w s be a coxeter system and h the associated hecke algebra with unequal parameters the laurent polynomials m y w s and p y w for y w w and s s play an important role in the representations of h we study the properties of m y w s and p y w the relations among them as well as with the left right and two sided cells of w

some cells in the weighted coxeter group c n l 2n 1 abstract the fixed point set of the affine weyl group a 2 n 1 s under a certain group automorphism with s s can be considered as the affine weyl group c n s then we study the cells of the weighted coxeter group c n l 2 n 1 with l 2 n 1 the length function of a 2 n 1 we give an explicit description for all the cells of c n l 2 n 1 corresponding to the partitions k 1 2 n 2 k and h 2 n 2 h for any 1 k 2 n 2 and n 1 h 2 n and also for all the cells of c 3 l 7

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

on convexity of fuzzy sets and fuzzy relations abstract various kinds of set theoretic operations produce new fuzzy sets and fuzzy relations from old ones what are the convexities of these sets and relations this is the problem discussed herein

a proof of lusztig s conjectures for affine type g2 with arbitrary parameters we prove lusztig s conjectures bf p1 bf p15 for the affine weyl group of type tilde g 2 for all choices of parameters our approach to compute lusztig s mathbf a function is based on the notion of a balanced system of cell representations for the hecke algebra we show that for arbitrary coxeter type the existence of balanced system of cell representations is sufficient to compute the mathbf a function and we explicitly construct such a system in type tilde g 2 for arbitrary parameters we then investigate the connection between kazhdan lusztig cells and the plancherel theorem in type tilde g 2 allowing us to prove bf p1 and determine the set of duflo involutions from there the proof of the remaining conjectures follows very naturally essentially from the combinatorics of weyl characters of types g 2 and a 1 along with some explicit computations for the finite cells

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

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

the weighted universal coxeter group and some related conjectures of lusztig abstract lusztig proposed 15 conjectures for any weighted coxeter groups in this study we explicitly describe kazhdan lusztig cells distinguished involutions and the function a for any weighted universal coxeter group we also verify these conjectures for this group

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

computing kazhdan lusztig cells for unequal parameters abstract following lusztig we consider a coxeter group w together with a weight function l this gives rise to the pre order relation l and the corresponding partition of w into left cells we introduce an equivalence relation on weight functions such that in particular l is constant on equivalent classes we shall work this out explicitly for w of type f 4 and check that several of lusztig s conjectures concerning left cells with unequal parameters hold in this case even for those parameters which do not admit a geometric interpretation the proofs involve some explicit computations using chevie

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