colored solutions of the yang baxter equation from representations of uqgl 2 we study the hopf algebra structure and the highest weight representation of a multiparameter version of uqgl 2 the hopf algebra maps of this algebra are explicitly given we show that the multiparameter universal r matrix can be constructed directly as a quantum double intertwiner without using reshetikhin s twisting transformation we find there are two types highest weight representations for this algebra type a corresponds to the qeneric q and type b corresponds to the case that q is a root of unity when applying the representation theory to the multi parameter universal r matrix both standard and nonstandard colored solutions of the yang baxter equation are obtained

exact scaling functions for self avoiding loops and branched polymers it is shown that a recently conjectured form for the critical scaling function for planar self avoiding polygons weighted by their perimeter and area also follows from an exact renormalization group flow into the branched polymer problem combined with the dimensional reduction arguments of parisi and sourlas the result is generalized to higher order multicritical points yielding exact values for all their critical exponents and exact forms for the associated scaling functions

two dimensional lattice vesicles and polygons we consider a lattice model of two dimensional vesicles in which the boundary of the vesicle is the perimeter of a self avoiding polygon embeddable in the square lattice with fixed boundary length m we incorporate an osmotic pressure difference by associating a fugacity with the area enclosed by the polygon we derive rigorous results concerning the behaviour of the associated free energy and the form of the phase diagram by deriving exact values ofthe numbers of polygons with m edges which enclose area n and analysing the resulting series we obtain the free energy the phase boundary and various scaling exponents and amplitudes numerically

solutions of yang baxter equations with colours standard and nonstandard colour extended solutions of ybe associated with g bn cn and dn are derived and the coloured birman wenzl algebra cb wa is established the authors point out that the cb wa still satisfies the redundance conditions of murakami 1989 for nonstandard solution they take the nontrivial colour solution of d2 as an example they find that the colour constraints induce a new solution with q4 1 the implication of this is not clear

fractal and nonfractal shapes in two dimensional vesicles abstract ongoing collaborative work on the statistical mechanics of two dimensional vesicles is reviewed with emphasis on the range of fractal and nonfractal shapes exhibited by the bead and tether model of a closed membrane which also embodies an osmotic pressure difference p and a bending rigidity modulus monte carlo simulations and scaling analyses are described flaccid vesicles with p 0 represent closed self avoiding rings their mean area a scales with the mean square size r g 2 their mean shape is close to elliptical inflated p 0 vesicles become circular deflated p branched polymers the vesicle shapes appear to vary continuously with the scaling combination p n 2 where n is the number of monomers beads and 3 4 for p k b t large relative to n 3 a range of characteristic nonfractal shapes or cytotypes appears

polygonal figures and their form spaces ii angle permutations in equilateral polygons abstract in a closed nonregular equilateral polygonal curve angles cannot be permuted arbitrarily without opening the curve here we investigate the possible angle permutations that leave a polygonal curve closed to this end we introduce angle sequences i e symbol sequences describing the number and arrangement of like angle values in classes of forms we find that there exist universal angle sequences giving closed polygons for arbitrary angle values provided that the angle sum is kept constant at one of certain permissible values the universal angle sequences are periodic they consist of at least two identical series of symbols within a series the symbols can be permuted freely on condition that in all other series the very same permutation is applied all these correlated permutations when applied to the angles of a corresponding polygonal curve leave this curve closed polygons belonging to a given angle sequence form class constitute a submanifold of the manifold m i of equilateral polygons within the form space these submanifolds structure m i in a hierarchical manner so that the form class concept gives a valuable complementary classification scheme for the forms of m i furthermore we find that polygons with constant angle composition the term composition is used here in its chemical sense show a wide variety of possible shapes from quasi circular to wormlike and even self intersecting ones corresponding to different degrees of segregation of like angles within the polygonal curve

convex polyominoes and heaps of segments builds a one to one correspondence between some polyominoes called parallelogram polyominoes and some heaps of segments then the author derives explicit expressions for the generating function of convex polyominoes according to their height width and area the author also enumerates two subsets of convex polyominoes namely the directed and convex polyominoes and the parallelogram polyominoes according to these three parameters generating functions of polyominoes having a fixed width are also studied

type i quantum superalgebras q supertrace and two variable link polynomials a new general eigenvalue formula for the eigenvalues of casimir invariants for the type i quantum superalgebras is applied to the construction of link polynomials associated with any finite dimensional unitary irrep for these algebras this affords a systematic construction of new two variable link polynomials associated with any finite dimensional irrep with a real highest weight for the type i quantum superalgebras in particular infinite families of nonequivalent two variable link polynomials are determined in fully explicit form

the universal r matrix and the yang baxter equation with parameters the authors formulate and demonstrate with some examples a method for obtaining solutions of the yang baxter equation with parameters

inflated vesicles a lattice model abstract a three dimensional vesicle model on the cubic lattice with fixed surface area n subject to an internal pressure increment p 0 is analyzed using a monte carlo method and finite size scaling at the branching point p 0 the conformations are of branched polymer like structures and the mean volume and the mean square radius of gyration vary as vb r2b n in the large inflation scaling regime 100 v 2 3 r 2 n v p 2 and v 6 5 and 1 5 the pincus fisher stretching exponent is 1 3 at the crumpling point xc pcn 100 separating the branched and the stretched regime the conformations are crumpled with v c 2 3 r 2 c n vc and v c 4 5 the transition between stretched and crumpled conformations is continous whereas the transformation between branched and crumpled shapes is discontinuous the various regimes are discussed in terms of simple flory type arguments

coloured frt algebra and its yang baxterization leading to integrable models with non additive r matrices a colour representation of faddeev reshetikhin takhtajan frt algebra which in contrast to the standard case is related to the coloured braid group representation with generic values of q is presented explicit realizations of l or matrices occurring in this coloured variant of frt algebra are also obtained for the uq gl 2 quantized algebra though these realizations are found to depend manifestly on the colour parameters the underlying quantum group structure and associated co product are interestingly free from such dependence this allows us to perform the yang baxterization of the coloured frt algebra successfully which leads to the construction of an ancestor law operator associated with a new non additive type quantum r matrix through different realizations of this lax operator a new class of quantum integrable models representing colour generalizations of the well known models such as the lattice sine gordon model the ablowitz ladik model lattice and the derivative nonlinear schrodinger model etc is generated

new constant and trigonometric 4 4 solutions to the yang baxter equations a classification of triangular constant solutions of the yang baxter equations is performed the formulae for the baxterization of braid group representation are used for finding trigonometric solutions of the yang baxter equations from the constant ones some of the obtained solutions correspond to known solutions but several new ones are obtained as well the calculations confirm that the baxterization formulae for more than two eigenvalues of the r matrix are not universal

the aspherity of random walks the asphericity a of a random walk is defined and calculated for non self avoiding walks the definition of a generalises to self avoiding walks percolating clusters and other fractal objects and thus provides a generalised quantitative measure of the departure from spherical symmetry of the gross shape of these objects

the chow ring of the classifying space bso 2n c we compute the chow ring of the classifying space bso 2n c in the sense of totaro using the fibration gl 2n so 2n bso 2n bgl 2n and a computation of the chow ring of gl 2n so 2n in a previous paper we find this chow ring is generated by chern classes and a characteristic class defined by edidin and graham which maps to 2n 1 times the euler class under the usual class map from the chow ring to ordinary cohomology moreover we show this class represents 1 2n 1 n 1 times the nth chern class of the representation of so 2n whose highest weight vector is twice that of the half spin representation

more about the nonstandard r matrix associated with suq 2 the markov trace is constructed for the general r matrices associated with suq 2 at q a root of unity the relationship between the r matrices derived based on quantum algebra and those obtained by akutsu et al 1991 is clarified

vesicles on a hierarchical lattice an exact renormalization group approach a model of two dimensional vesicles on a sierpinski gasket is introduced and solved by iterative methods the effect of osmotic pressure is obtained by associating a fugacity to each plaquette of the underlying triangular lattice enclosed by the vesicle several results concerning the critical behaviour and phase diagram are discussed in particular it is shown exactly that deflated critical vesicles fall into the universality class of branched polymers

scaling behaviour of two dimensional polygon models exactly solvable two dimensional polygon models counted by perimeter and area are described by q algebraic functional equations we provide techniques to extract the scaling behaviour of these models up to arbitrary order and apply them to some examples these are then used to analyze the unsolved model of self avoiding polygons where we numerically confirm predictions about its scaling function and its first two corrections to scaling

exact solution of the convex polygon perimeter and area generating function an explicit expression is derived for the three variable generating function p x y z sigma m or 1 sigma n or 1 sigma r or 1 x2ny2mzrcn m r where cn m r is the number of convex polygons with horizontal width n vertical height m and area r

characteristic classes and quadric bundles in this paper we construct stiefel whitney and euler classes in chow cohomology for algebraic vector bundles with nondegenerate quadratic form these classes are not in the algebra generated by the chern classes of such bundles and are new characteristic classes in algebraic geometry on complex varieties they correspond to classes with the same name pulled back from the cohomology of the classifying space bso n c the classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups t2 eg we begin by using the geometry of quadric bundles to study chern classes of maximal isotropic subbundles if v x is a vector bundle with quadratic form and if e and f are maximal isotropic subbundles of v then we prove theorem 1 that ci e and ci f are equal mod 2 moreover if the rank of v is 2n then cn e cn f proving a conjecture of fulton we define stiefel whitney and euler classes as chow cohomology classes which pull back to chern classes of maximal isotropic subbundles of the pullback bundle using the above theorem we show theorem 2 that these classes exist and are unique even though v need not have a maximal isotropic subbundle these constructions also make it possible to give schubert presentations

colored quantum universal enveloping algebras we define some new algebraic structures termed colored hopf algebras by combining the coalgebra structures and antipodes of a standard hopf algebra set h corresponding to some parameter set q with the transformations of an algebra isomorphism group g herein called color group such transformations are labeled by some color parameters taking values in a color set c we show that various classes of hopf algebras such as almost cocommutative coboundary quasitriangular and triangular ones can be extended into corresponding colored algebraic structures and that colored quasitriangular hopf algebras in particular are characterized by the existence of a colored universal r matrix satisfying the colored yang baxter equation the present definitions extend those previously introduced by ohtsuki which correspond to some substructures in those cases where the color group is abelian we apply the new concepts to construct colored quantum universal enveloping algebras of both semisimple and nonsemisimple lie algebras considering several examples with fixed or varying parameters as a by product some of the matrix representations of colored universal r matrices derived in the present paper provide new solutions of the colored yang baxter equation which might be of interest in the context of integrable models

examples for the mod p motivic cohomology of classifying spaces let bg be the classifying space of a compact lie group g some examples of computations of the motivic cohomology h eg z p are given by comparing with h bg z p ch eg and bp bg

size of rings in two dimensions the authors report enumeration results for the radius of gyration and caliper size distribution of self avoiding unrooted polygons of up to 28 steps on the square lattice the second moment radius of gyration series is sufficiently smooth to allow verification of the theoretical prediction v rings v walks to 0 2 accuracy

convex polyominoes and algebraic languages uses the so called dsv methodology that links some enumeration problems to the theory of algebraic languages to get a system of q difference equations involving the generating functions of convex polyominoes of convex and directed polyominoes and of parallelogram polyominoes according to their height width and area then the author shows various applications of this system

self avoiding surfaces and vesicles on a lattice abstract recent results concerning the statistics of self avoiding random surfaces sas and vesicles made of elementary lattice plaquettes in d 3 are reviewed in all cases progress follows from introduction of spin or gauge ising vacancies into a suitable gauge model with n component classical vectors as degrees of freedom sas statistics is generated in the n 0 limit topics include in particular the possibility of defining sas models with the geometry of ising spin cluster hulls crossover from sas to deflated vesicles and the role of topology in determining the universality class of sas problems critical properties of sas models in the presence of an adsorbing boundary plane are also discussed

new hierarchy of colored braid group representations novel solutions of the yang baxter relation for a series of multistate vertex models are presented in the braid group limit the obtained family of solutions gives a sequence of colored braid group representations possible generalizations of the multivariable alexander polynomial are discussed

graded solutions of the yang baxter relation and link polynomials from a family of graded solvable models the authors derive representations of the braid group associated with the lie superalgebra gl m mod n and give explicitly a general form of the markov traces on the representations the braid operators thus obtained satisfy the hecke algebra the authors construct composite solvable models and obtain link polynomials from the braid operators for the composite models

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

non birman wenzl algebraic properties and redundancy of exotic enhanced yang baxter operator for spin model in this paper we explicitly construct the markov trace for the general coloured exotic braid group representations bgr with spin j it is verified that the bgr for j 1 is redundant in the sense of murakami 1990 but no birman wenzl algebra

random walks on braid groups brownian bridges complexity and statistics we investigate the limit behaviour of random walks on some non commutative discrete groups related to knot theory namely we study the connection between the limit behaviour of the lyapunov exponent of products of non commutative random matrices generators of the braid group and the asymptotics of powers of the algebraic invariants of randomly generated knots we turn the simplest problems of knot statistics into the context of random walks on hyperbolic groups we also consider the limit distribution of brownian bridges on so called locally non commutative groups

integrable spin chains associated with slq n and slp q n the hopf structure of the central extension of the uq sl n algebra is considered the intertwine matrix induces new integrable chain models we show the relation of these models and the biparametric spin chain slp q n models the cases n 2 are n 3 are discussed and for n 2 we obtain the model of dasgupta and chowdhury 1993 the case n 3 is solved by the nested bethe ansatz method and the dependence of the bethe equations on the second parameter introduced is demonstrated

a new transfer matrix algorithm for exact enumerations self avoiding polygons on the square lattice we present a new and more efficient implementation of transfer matrix methods for exact enumerations of lattice objects the new method is illustrated by an application to the enumeration of self avoiding polygons on the square lattice a detailed comparison with the previous best algorithm shows significant improvement in the running time of the algorithm the new algorithm is used to extend the enumeration of polygons to length 130 from the previous record of 110

scaling function and universal amplitude combinations for self avoiding polygons we analyse new data for self avoiding polygons saps on the square and triangular lattices enumerated by both perimeter and area providing evidence that the scaling function is the logarithm of an airy function the results imply universal amplitude combinations for all area moments and suggest that rooted saps may satisfy a q algebraic functional equation

asymptotic behavior of inflated lattice polygons we study the inflated phase of two dimensional lattice polygons with fixed perimeter n and variable area associating a weight exp pa jb to a polygon with area a and b bends for convex and column convex polygons we calculate the average area for positive values of the pressure for large pressures the area has the asymptotic behaviour langle a rangle a max 1 k j tilde p 2 mathcal o rho tilde p where tilde p pn gg 1 and 1 the constant k j is found to be the same for both types of polygons we argue that self avoiding polygons should exhibit the same asymptotic behavior for self avoiding polygons our predictions are in good agreement with exact enumeration data for j 0 and monte carlo simulations for j 0 we also study polygons where self intersections are allowed verifying numerically that the asymptotic behavior described above continues to hold

quantum harmonic oscillator algebra and link invariants associated with the q deformation of the harmonic oscillator algebra we define an infinite dimensional braid group representation on the hilbert space of the harmonic oscillator and an extended yang baxter system in the sense of turaev the corresponding link invariant is computed in some particular cases and coincides with the inverse of the alexander conway polynomial the r matrix of uq h4 can be interpreted as defining a baxterization of the intertwiners for semicyclic representations of su 2 q at q e2 i n in the n limit finally we define new multicolored braid group representations and study their relation to the multivariable alexander conway polynomial

phase transitions in pressurized semiflexible polymer rings we propose and study a model for the equilibrium statistical mechanics of a pressurized semiflexible polymer ring in two dimensions the hamiltonian has a term which couples to the algebraic or signed area of the ring and a term which accounts for bending semiflexibility the model allows for self intersections using a combination of monte carlo simulations flory type scaling theory mean field approximations and lattice enumeration techniques we obtain a phase diagram in which collapsed and inflated phases are separated by a continuous transition the scaling properties of the averaged area as a function of the number of units of the ring are derived for large pressures the asymptotic behavior of the area is calculated for both continuum and lattice versions of the model for small pressures the area is obtained through a known mapping onto the quantum mechanical problem of an electron moving in a magnetic field the simulation data agree well with the analytic and mean field results

on the asymptotics of the finite perimeter partition function of two dimensional lattice vesicles we derive the dominant asymptotic form and the order of the correction terms of the finite perimeter partition function of self avoiding polygons on the square lattice which are weighted according to their area a as qa in the inflated regime q 1 the approach q 1 of the asymptotic form is examined

the chow ring of the symmetric space gl 2n c so 2n c abstract we show this chow ring is z z we do this by partitioning the space into 2n subvarieties each of which is fibered over gl 2 n 2 c so 2 n 2 c

convergence and extrapolation in finite size scaling renormalization abstract finite size renormalization technique is studied by examining a test model which is exactly solvable and the problem of lattice animals and self avoiding walks bulk estimates of the leading irrelevant rg eigenvalue are found and compared with independent estimates from finite size data we introduce an extrapolation technique which is applicable to non monotonic sequences of exponent approximants

monte carlo study of 3d vesicles abstract a model of vesicles on a cubic lattice is studied by monte carlo methods vesicles have spherical topology and are made of lattice plaquettes an osmotic pressure difference acts between interior and exterior results are given for critical properties in the deflated regime

invariant tangles of coloured non standard solutions of braid relations the invariant tangles for murakami s coloured solution 1990 of braid relations are explicitly calculated in terms of the kauffman saleur fermionic integral 1991 a more general coloured solution r c1 c2 of braid group representation is obtained the authors verify that such r c1 c2 satisfies all the redundant conditions presented by murakami they thus derive invariant alexander link polynomials for the new coloured solution

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

a nonlinear deformed su 2 algebra with a two color quasitriangular hopf structure nonlinear deformations of the enveloping algebra of su 2 involving two arbitrary functions of j0 and generalizing the witten algebra were introduced some time ago by delbecq and quesne in the present paper the problem of endowing some of them with a hopf algebraic structure is addressed by studying in detail a specific example referred to as aq 1 this algebra is shown to possess two series of n 1 dimensional unitary irreducible representations where n 0 1 2 to allow the coupling of any two such representations a generalization of the standard hopf axioms is proposed by proceeding in two steps in the first one a variant and extension of the deforming functional technique is introduced variant because a map between two deformed algebras suq 2 and aq 1 is considered instead of a map between a lie algebra and a deformed one and extension because use is made of a two valued functional whose inverse is singular as a result the hopf structure of suq 2 is carried over to aq 1 ther

effects of surface fluctuations in a two dimensional emulsion abstract we investigate the role of geometrical shape fluctuations for the thermodynamic properties of a polydisperse ensemble of two dimensional droplets the energy of a droplet derives from the hamiltonian for surface bending rigidity with spontaneous curvature interactions between individual droplets are omitted but polydispersity in droplet size and arbitrarily large distortions of the droplet shape are incorporated fully the physical property of self avoidance is approximated by the less stringent topological requirement of unit winding number this topological model shows that some physical observables are considerably more sensitive to the presence of shape fluctuations than others the specific heat in particular is strongly affected by the presence of shape fluctuations this observable is also very sensitive to approximate treatments of the shape fluctuations the periodic gaussian or villain approximation is found to be reliable at all temperatures whereas the gaussian approximation fails except for very low temperatures

the brown peterson cohomology of the classifying spaces of the projective unitary groups pu p and exceptional lie groups for a fixed prime p we compute the brown peterson cohomologies of classifying spaces of pu p and exceptional lie groups by using the adams spectral sequence in particular we see that bp bpu p and k n bpu p are even dimensionally generated

multi colour braid monoid algebras we define multi colour generalizations of braid monoid algebras and present explicit matrix representations which are related to two dimensional exactly solvable lattice models of statistical mechanics in particular we show that the two colour braid monoid algebra describes the yang baxter algebra of the critical dilute a d e models which were recently introduced by warnaar nienhuis and seaton 1992 as well as by roche 1992 these and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure it is conjectured that the baxterization of general multi colour braid monoid algebras will lead to the construction of further solvable lattice models

torsion algebraic cycles and complex cobordism atiyah and hirzebruch gave the first counterexamples to the hodge conjecture with integral coefficients 3 that conjecture predicted that every integral cohomology class of hodge type p p on a smooth projective variety should be the class of an algebraic cycle but atiyah and hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle here we provide a more systematic explanation for their results by showing that the classical cycle map from algebraic cycles modulo algebraic equivalence to integral cohomology factors naturally through a topologically defined ring which is richer than integral cohomology the new ring is based on complex cobordism a well developed topological theory which has been used only rarely in algebraic geometry since hirzebruch used it to prove the riemann roch theorem 17 this factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by atiyah and hirzebruch it goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective as well as nonsurjective the kernel of the classical cycle map is called the griffiths group and the topological proof here that the griffiths group can be nonzero is the first proof of this fact which does not use hodge theory the proof here gives nonzero torsion elements in the griffiths group whereas griffiths s hodge theoretic proof gives nontorsion elements 13 this topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before thus answering some questions posed by colliot thelene and schoen 8 p 14 37 p 13 colliot thelene asked in particular whether the map ch2 x n h4 x z n is injective for all smooth complex projective varieties x here ch x is the group of codimension i algebraic cycles modulo rational equivalence the first examples where colliot thelene s map is not injective were found by kollar and van geemen see 4 p 135 very recently bloch and esnault found examples defined over number fields 7 over nonalgebraically closed fields k there are other examples of smooth projective varieties xk with ch2 xk n hg4t xk z n not injective due to colliot thelene and sansuc as reinterpreted by salberger see 9 and 8 remark 7 6 1 and parimala and suresh 31 these elements of ch2 xk n are not shown to remain nonzero in ch2 xc n however here our topological method