quantum cohomology of flag manifolds and toda lattices we discuss relations of vafa s quantum cohomology with floer s homology theory introduce equivariant quantum cohomology formulate some conjectures about its general properties and on the basis of these conjectures compute quantum cohomology algebras of the flag manifolds the answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a toda lattice

gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

basic hypergeometric series very well poised in u n abstract we introduce and study a u n multiple series generalization of classical very well poised basic hypergeometric series of one variable two new multiple series generalizations of the terminating 6 5 summation theorem are derived one of these extends to a u n generalization of the nonterminating 6 5 summation theorem multiple series generalizations of the summation theorems of kummer and dixon appear as special cases we obtain the corresponding theorems for ordinary series by taking the limit as q 1 of all these results

schur functions good s identity and hypergeometric series well poised in su n abstract a simple direct proof is given of a fundamental identity involving schur functions which contains as special cases the identity responsible for good s proof of the dyson conjecture and the summation theorem of biedenharn and louck that appears frequently in dealing with the explicit matrix elements which arise in the unitary groups by using the weyl character formula a general identity is obtained which implies our result involving schur functions when a root system of type a n 1 is considered as a further application of our general identity explicit analogs of good s identity are given corresponding to the root systems of types b n c n and d n in addition methods to obtain q analogs of all of these results are briefly described

hunting for the fractal dimension of the kauffman model abstract the kauffman model is a random mixture of all possible cellular automata on the square lattice with nearest neighbor interactions simulations give a fractal dimension d above 1 8 compatible with the usual scaling law d d for damage spreading

mutations of puzzles and equivariant cohomology of two step flag varieties we introduce a mutation algorithm for puzzles that is a three direction analogue of the classical jeu de taquin algorithm for semistandard tableaux we apply this algorithm to prove our conjectured puzzle formula for the equivariant schubert structure constants of two step flag varieties this formula gives an expression for the structure constants that is positive in the sense of graham thanks to the equivariant version of the quantum equals classical result our formula specializes to a littlewood richardson rule for the equivariant quantum cohomology of grassmannians

diffusion on a hypersphere application to the wright fisher model the eigenfunction expansion by gegenbauer polynomials for the diffusion on a hypersphere is transformed into the diffusion for the wright fisher model with a particular mutation rate we use the ito calculus considering stochastic differential equations the expansion gives a simple interpretation of the griffiths eigenfunction expansion for the wright fisher model our representation is useful to simulate the wright fisher model as well as brownian motion on a hypersphere

initiation of damage in the kauffman model an alternate method is presented for initiating damage to the kauffman model a numerical study suggests that this modification increases the efficiency of generating damage clusters without changing their fractal properties

a q analog of the 5f4 1 summation theorem for hypergeometric series well poised in su n abstract as an application of a general q difference equation for basic hypergeometric series well poised in su n an elementary proof is given of a q analog of holman s su n generalization of the terminating 5 f 4 1 summation theorem this provides an su n generalization of the terminating 6 5 summation theorem for classical basic hypergeometric series

equivariant quantum schubert polynomials abstract we establish an equivariant quantum giambelli formula for partial flag varieties the answer is given in terms of a specialization of universal double schubert polynomials along the way we give new proofs of the presentation of the equivariant quantum cohomology ring as well as graham positivity of the structure constants in equivariant quantum schubert calculus

on the denominator function for canonical su 3 tensor operators the definition of a canonical unit su 3 tensor operator is given in terms of its characteristic null space as determined by group theoretic properties of the intertwining number this definition is shown to imply the canonical splitting conditions used in earlier work for the explicit and unique up to phases construction of all su 3 wcg coefficients wigner clebsch gordan using this construction an explicit su 3 invariant denominator function characterizing completely the canonically defined wcg coefficients is obtained it is shown that this denominator function squared is a product of linear factors which may be obtained explicitly from the characteristic null space times a ratio of polynomials these polynomials denoted gtq are defined over three shift parameters and three barycentric coordinates the properties of these polynomials hence of the corresponding invariant denominator function are developed in detail these include a derivation of their degree symmetries and zeros the

inhomogeneous basis set of symmetric polynomials defined by tableaux abstract a basis set of inhomogeneous symmetric polynomials denoted by tlambda z where z z1 zn and lambda lambda1 lambdan is a partition is defined in terms of young weyl standard tableaux or equivalently in terms of gel fand weyl patterns a number of significant properties of these polynomials are given together with outlines of proofs and compared with properties of the well known basis set of schur functions which are homogeneous polynomials the basis of the ring of symmetric polynomials defined here is shown to be natural for the expansion of inhomogeneous symmetric functions constructed from rising factorials

emergent properties in random complex automata abstract studies of large randomly assembled binary boolean node automata have demonstrated that such systems can spontaneously exhibit enormously ordered dynamical behavior an important approach to characterizing these behaviors has consisted of studying ensembles of automata ensemble specifications have been based on 1 choice of boolean function regulating each node in the automaton 2 random or biased mappings of the 2 n automata states into themselves 3 the numbers of inputs k per node this article briefly reviews these approaches reports new results on the expected behavior of threshold automata and automata with a particular class of biased mappings in addition it discusses emergent highly ordered dynamical behavior in random automata rich in a specific class of canalizing boolean functions due to the crystallization of powerful subautomata called forcing structures the ordered behaviors may be of importance in biological evolution in physics and in the design of adaptive automata

quantum cohomology of flag manifolds g b and quantum toda lattices let g be a connected semi simple complex lie group b its borel subgroup t a maximal complex torus contained in b and lie t its lie algebra this setup gives rise to two constructions the generalized nonperiodic toda lattice 28 29 and the flag manifold g b the toda lattice for g b t is the dynamical system on the cotangent bundle t lie t endowed with the canonical holomorphic symplectic form and the holomorphic hamiltonian function we consider in this paper

generalizations of cauchy s summation theorem for schur functions cauchy s summation theorem for schur functions is generalized and a number of related results are given the result is applied to a com binatorial problem involving products of pairs of permuations by appeal to properties of the group algebra of the symmetric group in 3 this summation theorem is used to derive a generalization of gessel s 3 result on counting 3 rowed latin rectangles with respect to the number of cycles in each row considered as permuations we also derive proposition 3 2 an explicit expression for a useful character sum associated with the symmetric group 4 contains a number of results for symmetric functions that are equivalent to theorem 2 1 these include a generalization of cauchy s summation theorem for

the kauffman model on small world topology abstract we apply kauffman s automata on small world networks to study the crossover between the short range and the infinite range case we perform accurate calculations on square lattices to obtain both critical exponents and fractal dimensions particularly we find an increase of the damage propagation and a decrease in the fractal dimensions when adding long range connections

gromov witten invariants on grassmannians we prove that any three point genus zero gromov witten invariant on a type a grassmannian is equal to a classical intersection number on a two step flag variety we also give symplectic and orthogonal analogues of this result in these cases the two step flag variety is replaced by a sub maximal isotropic grassmannian our theorems are applied in type a to formulate a conjectural quantum littlewood richardson rule and in the other classical lie types to obtain new proofs of the main structure theorems for the quantum cohomology of lagrangian and orthogonal grassmannians

a new symmetry for biedenharn s g functions and classical hypergeometric series abstract we show that the general bisymmetric polynomials m g n q 1 n 1 m are a limiting case of the bisymmetric invariant polynomials n g n q 1 n 1 n which characterize u n tensor operators p q q 0 0 by taking suitable limits of a pair of difference equations for n g n q we then deduce transposition symmetry for m g n q from the same symmetry for n g n q as an application of transposition symmetry for m g n q we derive an elegant new contiguous relation for classical well poised hypergeometric series and also prove an identity between these series and multiple hypergeometric series well poised in su n

equivariant pieri rules for isotropic grassmannians we give a pieri rule for the torus equivariant cohomology of submaximal grassmannians of lie types b c and d to the authors best knowledge our rule is the first manifestly positive formula beyond the equivariant chevalley formula we also give a simple proof of the equivariant pieri rule for the ordinary type a grassmannian

affine approach to quantum schubert calculus this paper presents a formula for products of schubert classes in the quantum cohomology ring of the grassmannian we introduce a generalization of schur symmetric polynomials for shapes that are naturally embedded in a torus then we show that the coefficients in the expansion of these toric schur polynomials in terms of the regular schur polynomials are exactly the 3 point gromov witten invariants which are the structure constants of the quantum cohomology ring this construction implies three symmetries of the gromov witten invariants of the grassmannian with respect to the groups s3 z nz 2 and z 2z the last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q our construction gives a solution to a problem posed by fulton and woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two schubert classes the curious duality switches the smallest such power of q with the highest power we also discuss the affine nil temperley lieb algebra that gives a model for the quantum cohomology

a q analog of hypergeometric series well poised in tsu n and invariant g functions abstract we introduce natural q analogs of hypergeometric series well poised in su n the related hypergeometric series in u n and invariant g functions we prove that both the su n multiple q series and the invariant g functions satisfy general q difference equations both the su n and u n q series are new multivariable generalizations of classical basic hypergeometric series of one variable we prove an identity which expresses our u n multiple q series as a finite sum of finite products of classical basic hypergeometric series these u n q series also satisfy an elegant reduction formula which is analogous to the inclusion lemma for classical invariant g functions

double schubert polynomials for the classical groups abstract for each infinite series of the classical lie groups of type b c or d we construct a family of polynomials parametrized by the elements of the corresponding weyl group of infinite rank these polynomials represent the schubert classes in the equivariant cohomology of the appropriate flag variety they satisfy a stability property and are a natural extension of the single schubert polynomials of billey and haiman which represent non equivariant schubert classes they are also positive in a certain sense and when indexed by maximal grassmannian elements or by the longest element in a finite weyl group these polynomials can be expressed in terms of the factorial analogues of schur s q or p functions defined earlier by ivanov

equivariant littlewood richardson skew tableaux we give a positive equivariant littlewood richardson rule also discovered independently by molev our proof generalizes a proof by stembridge of the classical littlewood richardson rule we describe a weight preserving bijection between our indexing tableaux and trapezoid puzzles which restricts to a bijection between positive indexing tableaux and knutson tao puzzles

a new symmetry related to su n for classical basic hypergeometric series abstract a direct proof is given of an elegant new contiguous relation for classical well poised basic hypergeometric series which preserves the well poised condition the proof involves elementary series manipulations and does not depend upon the transposition symmetry of the general bisymmetric polynomials m g n q 1 n 1 m which was used to establish the ordinary or q 1 case of the identity the new continguous relation can be considered a generalization of the 6 5 summation theorem

quantum k theory of grassmannians we show that equivariant k theoretic 3 point gromov witten invariants of genus zero on a grassmann variety are equal to triple intersec tions computed in the equivariant k theory of a two step flag manifold thus generalizing an earlier result of buch kresch and tamvakis in the process we show that the gromov witten variety of curves passing through 3 general points is irreducible and rational our applications include pieri and giambelli formulas for the quantum k theory ring of a grassmannian which determine the multiplication in this ring we also compute the dual schubert basis for this ring and show that its structure constants satisfy s3 symmetry our for mula for gromov witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of chaput manivel and perrin

a new tableau representation for supersymmetric schur functions abstract we give a new tableau definition for supersymmetric skew schur functions and obtain a number of properties of these functions as easy corollaries

on the denominator function for canonical su 3 tensor operators ii explicit polynomial form the canonical resolution of the multiplicity problem for tensor operators in su 3 is equivalent to the map the denominator mapping from the set of all su 3 unit tensor operators to su 3 invariant functions the denominator functions the denominator function vanishes precisely on that characteristic null space that specifies each operator uniquely since for su 3 the characteristic null spaces are known to be simply ordered each denominator function can be expressed up to explicitly known multiplicative factors as a ratio of two successive polynomials in the set gtq t 0 1 q 1 q 0 1 by obtaining explicitly the set of all polynomials gtq this paper completes the construction of all su 3 denominator functions

schubert calculus and equivariant cohomology of grassmannians abstract we give a description of equivariant cohomology of grassmannians that places the theory into a general framework for cohomology theories of grassmannians as a result we obtain a formalism for equivariant cohomology where the basic results of equivariant schubert calculus the basis theorem pieri s formula and giambelli s formula can be obtained from the corresponding results of the general framework by a change of basis in order to show that our formalism reflects the geometry of grassmannians we relate our theory to the treatment of equivariant cohomology of grassmannians by a knutson and t tao

a connection between the percolation transition and the onset of chaos in the kauffman model it is demonstrated numerically that the percolation transition of the unstable sites and the onset of chaos in the kauffman model 1969 happens for the same value of the bias on the randomly chosen rules in the two dimensional triangular lattice and in the three and four dimensional hypercubic lattices the percolation thresholds for the stable sites are different for the three and presumably also four dimensional lattices on the triangular lattice the differences between the thresholds are too small to be resolved the fractal dimension of the damage is calculated and also the spreading time on the four dimensional hypercubic lattice

the factorial schur function the application of the divided difference of a function to the inhomogeneous symmetric functions factorial schur functions of biedenharn and louck is shown to lead to new relations and simplified proofs of their properties these results include determinantal definitions and the factorial jacobi trudi identities with extensions to skew versions similar properties of a second class of symmetric functions depending on an arbitrary parameter and of importance for generalized hypergeometric functions and series are shown also to be derivable from the divided difference notion slightly extended

an umbral calculus for polynomials characterizing u n tensor operators after the change of variables i i i and xi i 1 i i 1 we show that the invariant polynomials g n q i xi i 1 characterizing u n tensor operators p q q 0 0 become an integral linear combination of schur functions s in the symbol where denotes the difference of the two sets of variables 1 n and 1 n we obtain a similar result for the yet more general bisymmetric polynomials m g n q 1 n 1 m making use of properties of skew schur functions s and s we put together an umbral calculus for m g n q that is working entirely with polynomials we uniquely determine m g n q from m g n q 1 and combinatorial rules involving ferrers diagrams i e partitions provided that n 1 q this restriction does not interfere with writing the general case of m g n q as a linear combination of s as an application we deduce conjugation symmetry for n g n q from transposition symmetry by showing that these two symmetries are equivalent

factorial type schur functions orthogonal rational functions and discrete dressing chains a relationship between orthogonal rational functions and discrete integrable systems is studied by an approach based on schur type symmetric functions a system of orthogonal rational functions is constructed using a multiparameter deformation of the schur functions spectral equations for the orthogonal rational functions are derived by using properties of the schur type symmetric functions the compatibility condition of the spectral equations induces a discrete dressing chain which is a toda type discrete integrable system describing dressing transformations for orthogonal rational functions

weight 2 zeros of 3j coefficients and the pell equation all weight 2 zeros of the wigner 3j coefficients may be obtained from the quadratic diophantine equation known as pell s equation these zeros may then be classified by the orbits of a discrete infinite order subgroup of the lorentz group so 1 1 this is carried out by transforming the polynomial part of a weight 2 3j coefficient to pellian form and obtaining the fundamental zeros numerically the relation of this polynomial to a family of binary quadratic forms is also given together with a discussion of the invariance group

littlewood richardson polynomials abstract we introduce a family of rings of symmetric functions depending on an infinite sequence of parameters a distinguished basis of such a ring is comprised by analogues of the schur functions the corresponding structure coefficients are polynomials in the parameters which we call the littlewood richardson polynomials we give a combinatorial rule for their calculation by modifying an earlier result of b sagan and the author the new rule provides a formula for these polynomials which is positive in the sense of w graham we apply this formula for the calculation of the product of equivariant schubert classes on grassmannians which implies a stability property of the structure coefficients the first manifestly positive formula for such an expansion was given by a knutson and t tao by using combinatorics of puzzles while the stability property was not apparent from that formula we also use the littlewood richardson polynomials to describe the multiplication rule in the algebra of the casimir elements for the general linear lie algebra in the basis of the quantum immanants constructed by a okounkov and g olshanski

dynamical properties of the two and three dimensional ising models by damage spreading the method of damage spreading is used to measure the dynamic exponent z for both the two and three dimensional ising models with heat bath dynamics the authors also measure a new exponent alpha describing the rate at which a damage cloud grows they report values of z 2 24 or 0 04 and alpha 0 77 or 0 11 in two dimensions and z 2 202 or 0 03 and alpha 0 92 or 0 10 in three dimensions

two dimensional cellular automata a largely phenomenological study of two dimensional cellular automata is reported qualitative classes of behavior similar to those in one dimensional cellular automata are found growth from simple seeds in two dimensiona cellular automata can produce patterns with complicated boundaries characterized by a variety of growth dimensions evolution from disordered states can give domains with boundaries that execute effectively continuous motions some global properties of cellular automata can be described by entropies and lyapunov exponents others are undecidable

a littlewood richardson rule for factorial schur functions we give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial schur functions it is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial schur function applications to capelli operators and quantum immanants are also given

on equivariant quantum schubert calculus for g p we show a z2 filtered algebraic structure and a quantum to classical principle on the torus equivariant quantum cohomology of a complete flag variety of general lie type generalizing earlier works of leung and the second author we also provide various applications on equivariant quantum schubert calculus including an equivariant quantum pieri rule for partial flag variety fln1 nk n 1 of lie type a

quantum schubert polynomials where in is the ideal generated by symmetric polynomials in x1 xn without constant term another geometric description of the cohomology ring of the flag manifold is based on the decomposition of fln into schubert cells these are even dimensional cells indexed by the elements w of the symmetric group sn the corresponding cohomology classes oa called schubert classes form an additive basis in h fln 2 to relate the two descriptions one would like to determine which elements of 2 xl xn in correspond to the schubert classes under the isomorphism 1 1 this was first done in 2 see also 8 for a general case of an arbitrary complex semisimple lie group later lascoux and schiitzenberger 22 came up with a combinatorial version of this theory for the type a by introducing remarkable polynomial representatives of the schubert classes oa called schubert polynomials and denoted gw recently motivated by ideas that came from the string theory 31 30 mathematicians defined for any kahler algebraic manifold x the small quantum cohomology ring qh x 2 which is a certain deformation of the classical cohomology ring see e g 28 19 14 and references therein the additive structure of qh x 2 is essentially the same as that of ordinary cohomology in particular qh fln z is canonically isomorphic as an abelian group to the tensor product h fln 2 0 z ql qn 1 where the qi are formal variables deformation parameters the multiplicative structure of the quantum cohomology is however

classification of square lattice cellular automata abstract all 65 536 automata on the square lattice with nearest neighbor interactions ignoring the central spin were simulated we determined which percentage of them ended up in a fixed point with all spins parallel in other fixed points in up down oscillations with all spins parallel and in other oscillations of period two nine tenths of the rules did not fall into these classes we also checked for chaos i e how does the damage spread if one line of spins is reversed for 62 percent of the possible rules it spread over the lattice 31 percent had the damage remaining localized but nonzero and in 7 percent of all rules the damage healed out after some time these simulations were done with a speed of up to 120 updates per microsecond and cray xmp processor

equivariant quantum schubert calculus abstract we study the t equivariant quantum cohomology of the grassmannian we prove the vanishing of a certain class of equivariant quantum littlewood richardson coefficients which implies an equivariant quantum pieri rule as in the equivariant case this implies an algorithm to compute the equivariant quantum littlewood richardson coefficients

a pieri type formula for h t sln c b the singular cohomology of the grassmann variety of k planes in cn has a basis s indexed by partitions the classical pieri formula is an explicit rule for determining the coefficients in the expansion of the cup products1m s c1m s where 1m is a column of length m and s1m is the mth chern class of the tautological bundle lascoux and schutzenberger c r acad sci paris294 1982 447 450 formulated a generalization of pieri s formula to the cohomology of the flag variety sln c b and briefly indicated an algebraic proof manivel cours specialises 3 1998 provides details of this proof a geometric proof was given by sottile ann inst fourier grenoble 46 1996 89 110 in this paper we state and prove a generalization of this pieri type formula for the t equivariant cohomology of the flag variety we use the algebraic description of the t equivariant cohomology of the flag variety due to kostant and kumar adv math 62 1986 187 237 and arabia bull soc math france117 1989 129 165 and our new formula exposes an equality of certain structure constants in this algebra our proof is an induction based on the original idea in lascoux and schutzenberger

stochastic processes with orthogonal polynomial eigenfunctions markov processes which are reversible with either gamma normal poisson or negative binomial stationary distributions in the meixner class and have orthogonal polynomial eigenfunctions are characterized as being processes subordinated to well known diffusion processes for the gamma and normal and birth and death processes for the poisson and negative binomial a characterization of markov processes with beta stationary distributions and jacobi polynomial eigenvalues is also discussed

quantum integrability and generalised quantum schubert calculus abstract we introduce and study a new mathematical structure in the generalised quantum cohomology theory for grassmannians namely we relate the schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six vertex model our approach offers a new perspective on already established and well studied special cases for example equivariant k theory and in addition allows us to formulate a conjecture on the so far unknown case of quantum equivariant k theory

littlewood richardson coefficients for reflection groups abstract in this paper we explicitly compute all littlewood richardson coefficients for semisimple and kac moody groups g that is the structure constants also known as the schubert structure constants of the cohomology algebra h g p c where p is a parabolic subgroup of g these coefficients are of importance in enumerative geometry algebraic combinatorics and representation theory our formula for the littlewood richardson coefficients is purely combinatorial and is given in terms of the cartan matrix and the weyl group of g however if some off diagonal entries of the cartan matrix are 0 or 1 the formula may contain negative summands on the other hand if the cartan matrix satisfies a i j a j i 4 for all i j then each summand in our formula is nonnegative that implies nonnegativity of all littlewood richardson coefficients we extend this and other results to the structure coefficients of the t equivariant cohomology of flag varieties g p and bott samelson varieties i g

positivity in equivariant quantum schubert calculus a conjecture of d peterson proved by w graham states that the structure constants of the t equivariant cohomology of a homogeneous space g p satisfy a certain positivity property in this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology 1 introduction it is well known that the integral cohomology of the homogeneous space x g p for g a connected semisimple complex lie group and p a parabolic subgroup satisfies a positivity property its structure constants are nonnegative integers equal to the number of intersection points of three schubert varieties in general position whose codimensions add up to the dimension of x recently graham gr has proved a conjecture of peterson p asserting that h x the t equivariant cohomology of x where t c r is a maximal torus in g enjoys a more general positivity property

multivalley structure in kauffman s model analogy with spin glasses kauffman s model describes a random network of automata the authors calculations indicate that the multivalley structure of the basins of attraction in kauffman s model is very similar to that of infinite range spin glasses the similarity with spin glasses is tested quantitatively by computing the probability that two initial configurations fall into the same valley

classification of networks of automata by dynamical mean field theory dynamical mean field theory is used to classify the 65536 different networks of binary automata on a square lattice with nearest neighbour interactions application of mean field theory gives 700 different mean field classes which fall in seven classes of different asymptotic dynamics characterised by fixed points and 2 cycles

schubert classes in the equivariant cohomology of the lagrangian grassmannian abstract let lg n denote the lagrangian grassmannian parametrizing maximal isotropic lagrangian subspaces of a fixed symplectic vector space of dimension 2n for each strict partition 1 k with 1 n there is a schubert variety x let t denote a maximal torus of the symplectic group acting on lg n consider the t equivariant cohomology of lg n and the t equivariant fundamental class of x the main result of the present paper is an explicit formula for the restriction of the class to any torus fixed point the formula is written in terms of factorial analogue of the schur q function introduced by ivanov as a corollary to the restriction formula we obtain an equivariant version of the giambelli type formula for lg n as another consequence of the main result we obtained a presentation of the ring h t lg n

search for multifractality in damage spreading for kauffman cellular automata the authors check how an initial disturbance called the damage spreads through a square lattice of kauffman cellular automata at their critical point they determine the moments of the probability that a site has been damaged n times and check for multifractality in the fractal dimensions or critical exponents of these moments of the damage probability

comment on a genetic application of square lattice kauffman models biological genes are argued to have an infinite range of interaction in agreement with the original kauffman model and in disagreement with recent modifications which put them on a lattice with nearest neighbor interaction

upper critical dimension of kauffman cellular automata random boolean networks on nearest neighbour d dimensional lattices are argued to belong to the universality class of directed percolation with quenched disorder in d 1 dimensions hansen s computer simulations for d 4 are thus interpreted as being at the upper critical dimension

equivariant quantum cohomology of the odd symplectic grassmannian the odd symplectic grassmannian mathrm ig mathrm ig k 2n 1 parametrizes k dimensional subspaces of mathbb c 2n 1 which are isotropic with respect to a general necessarily degenerate symplectic form the odd symplectic group acts on mathrm ig with two orbits and mathrm ig is itself a smooth schubert variety in the submaximal isotropic grassmannian mathrm ig k 2n 2 we use the technique of curve neighborhoods to prove a chevalley formula in the equivariant quantum cohomology of mathrm ig i e a formula to multiply a schubert class by the schubert divisor class this generalizes a formula of pech in the case k 2 and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring

giambelli formulae for the equivariant quantum cohomology of the grassmannian we find presentations by generators and relations for the equivariant quantum cohomology of the grassmannian for these presentations we also find determinantal formulae for the equivariant quantum schubert classes to prove this we use the theory of factorial schur functions and a characterization of the equivariant quantum cohomology ring

fractal dimensions in three dimensional kauffman cellular automata the kauffman random networks of automata are studied on a simple cubic lattice by computer simulations each automaton follows random rules depending on its six neighbours and fixed in time a transition between the frozen and the chaotic phase is observed and the fractal dimension of the asymptotic actual damage at the phase transition is measured

the strange man in random networks of automata abstract we have performed computer simulations of kauffman s automata on several graphs such as the regular square lattice and invasion percolation clusters in order to investigate phase transitions radial distributions of the mean total damage dynamical exponent and propagation speeds of the damage when one adds a damaging agent nicknamed strange man despite the increase in the damaging efficiency we have not observed any appreciable change of the transition threshold to chaos neither for the short range nor for the small world case on the square lattices when the strange man is added in comparison to when small initial damages are inserted in the system particularly we have checked the damage spreading when some connections are removed on the square lattice and when one considers special invasion percolation clusters high boundary saturation clusters it is seen that the propagation speed in these systems is quite sensible to the degree of dilution on the square lattice and to the degree of saturation on invasion percolation clusters