transformation groups for soliton equations iv a new hierarchy of soliton equations of kp type abstract a new approach to soliton equations based on functions or hirota s dependent variables vertex operators and the clifford algebra of free fermions is applied to study a new hierarchy of kadomtsev petviashvili type equations the bkp hierarchy the infinite dimensional orthogonal group acts on the space of bkp functions the sawada kotera equation is obtained as a reduction of bkp its infinitesimal transformations constitute the euclidean lie algebra a2 2

bihamiltonian structure of the two component kadomtsev petviashvili hierarchy of type b we employ a lax pair representation of the two component kadomtsev petviashvili hierarchy of type b and construct its bihamiltonian structure with r matrix techniques

ultradiffusion on the fractal branching koch curves ultradiffusion is defined on a family of fractal branching koch curves by an exact renormalization decimation transformation the anomalous long time behaviour of the autocorrelation function is obtained for the particular hierarchy the authors find a non universal crossover c1 m 1 2m and a universal one c2 1 2 as the effective temperature c is increased where m is the maximum ramification number of the fractal they also find that for large m the dynamic behaviour of ultradiffusion on the fractal is independent of the effective temperature c

bosonic and fermionic realizations of the affine algebra g hat l_n we give an explicit description of all inequivalent heisenberg subalgebras of the affine lie algebra g hat l n mathbb c and the associated vertex operator constructions of the level one integrable highest weight representations of this algebra the construction uses multicomponent fermionic fields and yields a correspondence between bosons elements of the heisenberg subalgebra and fermions

trapezoidal approximations of fuzzy numbers the problem of the trapezoidal approximation of fuzzy numbers is discussed a set of criteria for approximation operators is formulated these constraints can be used for direct operator derivation a new nearest trapezoidal approximation operator preserving expected interval is suggested

simulation of fuzzy random variables this work deals with the simulation of fuzzy random variables which can be used to model various realistic situations where uncertainty is not only present in form of randomness but also in form of imprecision described by means of fuzzy sets utilizing the common arithmetics in the space of all fuzzy sets only induces a conical structure as a consequence it is difficult to directly apply the usual simulation techniques for functional data in order to overcome this difficulty two different approaches based on the concept of support functions are presented the first one makes use of techniques for simulating hilbert space valued random elements and afterwards projects on the cone of all fuzzy sets it is shown by empirical results that the practicability of this approach is limited the second approach imitates the representation of every element of a separable hilbert space in terms of an orthonormal basis directly on the space of fuzzy sets in this way a new approximation of fuzzy sets useful to approximate and simulate fuzzy random variables is developed this second approach is adequate to model various realistic situations

multifractality and the kauffman model the author simulates the spread of an initial damage in the kauffman cellular automata at its critical point on the square lattice he calculates various moments of the probability that a site has been damaged n times and checks for multifractality in the critical exponents for different ensembles specifically he finds no evidence for multifractality when the moments of the probabilities are evaluated with lattice size l but multifractal behaviour occurs when the moments are monitored as a function of time t

a remark on kac wakimoto hierarchies of d type for the kac wakimoto hierarchy constructed from the principal vertex operator realization of the basic representation of the affine lie algebra d 1 n we compute the coefficients of the corresponding hirota bilinear equations and verify the coincidence of these bilinear equations with the ones that are satisfied by givental s total descendant potential of the dn singularity as conjectured by givental and milanov 2005 simple singularities and integrable hierarchies the breadth of symplectic and poisson geometry prog math vol 232 boston birkh user pp 173 201

an algebraic and geometric approach to non bijective quadratic transformations this work is concerned with an extension of the so called levi civita and kustaanheimo stiefel transformations the extension is achieved along two lines firstly the authors examine the latter transformations in other dimensions than the ones originally considered by on the one hand levi civita and on the other hand by kustaanheimo and stiefel secondly they pass from the compact to the non compact case this leads to quadratic non bijective transformations that they refer to as hurwitz or kustaanheimo stiefel like and quasihurwitz or levi civita like transformations the hurwitz and quasihurwitz transformations are introduced and studied in an algebraic framework which relies on the use of eight dimensional cayley dickson algebras an explicit formulation of the hurwitz and quasihurwitz transformations is also given in terms of clifford algebras the hurwitz transformations are investigated from a geometrical viewpoint indeed they are connected to hopf and pseudohopf fibrations finally some differential aspects of the hurwitz and to a lesser extent quasihurwitz transformations are developed in view of future physical applications

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

on kernel formulas and dispersionless hirota equations of the extended dispersionless bkp hierarchy we derive dispersionless hirota equations of the extended dispersionless bkp edbkp hierarchy proposed by takasaki from the method of kernel formula provided by carroll and kodama moreover we verify associativity equations wdvv equations in the edbkp hierarchy from dispersionless hirota equations and give a realization of associative algebra with structure constants expressed in terms of the residue formula

multifractal nature of diffusion on hierarchical structures the authors study the moments of the first passage time t of random walks on hierarchical structures using the exact enumeration method they find that the different moments scale with the size of the system with an infinite hierarchy of exponents characterising a multifractal behaviour

trapezoidal approximations of fuzzy numbers revisited fuzzy number approximation by trapezoidal fuzzy numbers which preserve expected interval is discussed the previously proposed approximation operator is improved so as to always produce a well formed trapezoidal fuzzy number

a new type of approximation for fuzzy intervals this paper suggests a new method to approximate a fuzzy interval u by a sequence of differentiable fuzzy intervals this new approximation method involves the construction of differentiable fuzzy intervals using the sup min convolution of fuzzy sets numerical examples and an algorithm for computational implementation of the method proposed are also given

the approximation of piecewise linear membership functions and łukasiewicz operators in this paper we propose an approximation of piecewise linear membership functions with the help of sigmoid functions and certain arithmetic operations the gradient based tuning of piecewise linear membership functions can be achieved with the proposed efficient approximation because it has simple continuous derivatives with this construction we can even approximate the lukasiewicz operator family which plays an important role in fuzzy logic first of all from the theoretical point of view although in practice in optimization and learning it is rarely used because the lack of good analytical properties e g a continuous gradient the proposed approximation enlarges the applicability of fuzzy methods to the operators and membership functions where the differentiability is desirable

approximation of fuzzy numbers using the convolution method in this study we consider the use of the convolution method for constructing approximations comprising fuzzy number sequences with useful properties for a general fuzzy number we show that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers with finite non differentiable points in previous studies this convolution method was only used for constructing differentiable approximations of continuous fuzzy numbers the possible non differentiable points of which were the two endpoints of the 1 cut the construction of smoothers is a key step in the process for producing approximations we also show that if appropriate smoothers are selected then we can use the convolution method to provide approximations that are differentiable lipschitz and that also preserve the core

topological landau ginzburg theory with a rational potential and the dispersionless kp hierarchy based on the dispersionless kp dkp theory we study a topological landau ginzburg lg theory characterized by a rational potential writing the dkp hierarchy in a general form treating all the primaries in an equal basis we find that the hierarchy naturally includes the dispersionless continuous limit of toda hierarchy and its generalizations having a finite number of primaries several flat solutions of the topological lg theory are obtained in this formulation and are identified with those discussed by dubrovin we explicitly construct gravitational descendants for all the primary fields giving a residue formulae for the 3 point functions of the fields we show that these 3 point functions satisfy the topological recursion relation the string equation is obtained as the generalized hodograph solutions of the dkp hierarchy which show that all the gravitational effects to the constitutive equations 2 point functions can be renormalized into the coupling constants in the small phase space

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

scaling of the first passage time and the survival probability on exact and quasi exact self similar structures the authors calculate the asymptotic behaviour of the moments of the first passage time and survival probability for random walks on an exactly self similar tree and on a quasi self similar comb by applying an exact decimation approach to the master equations for the hierarchical comb a transition from ordinary to anomalous diffusion occurs at r 2 where r is the ratio of teeth length in successive iterations of the structure in the anomalous regime r 2 the positive integer moments of the first passage time tq scale as ltau q with tau q 1 2q 1 ln r ln 2 where l is the linear distance from input to output the asymptotic behaviour of the survival probability is studied using both scaling theory and by a direct solution of the master equations they find that the characteristic time t in the asymptotic exponential decay of the survival probability exp t t scales as t approximately ltau with tau ln r2 ln 2 i e tau is distinct from tau 1 however substantial corrections to this asymptotic form for tau exist and these are needed to account for the recent simulation data of havlin and matan 1988

a new integrable equation in the 2 1 dimension dispersionless limit of the bkp equation a new integrable equation is obtained as the dispersionless limit of the bkp equation it is then studied in the framework of kodama s approach 1988 the new equation also turns out to be completely integrable with an infinite number of conservation laws the hodograph transformation method is applied to obtain exact solutions in various cases

on the lax representation of the 2 component kp and 2d toda hierarchies the lax formulation of the multicomponent kadomtsev petviashvili kp and 2d toda hierarchies involves several implicit constraints we show that at least in the 2 component case it is possible to explicitly solve such constraints and identify a set of free dependent variables for such hierarchies

block algebra in two component bkp and d type drinfeld sokolov hierarchies we construct generalized additional symmetries of a two component bkp hierarchy defined by two pseudo differential lax operators these additional symmetry flows form a block type algebra with some modified or additional terms because of a b type reduction condition of this integrable hierarchy further we show that the d type drinfeld sokolov hierarchy which is a reduction of the two component bkp hierarchy possess a complete block type additional symmetry algebra that d type drinfeld sokolov hierarchy has a similar algebraic structure as the bigraded toda hierarchy which is a differential discrete integrable system

scaling properties of diffusion on comb like structures the authors study the scaling properties of the probability density of random walks on comb like structures they find that in contrast to the first passage time in which several exponents are needed to characterise their distribution the probability density can be described by a single exponent

nonlinear poisson structures and r matrices we introduce quadratic poisson structures on lie groups associated with a class of solutions of the modified yang baxter equation and apply them to the hamiltonian description of lax systems the formal analog of these brackets on associative algebras provides second structures for certain integrable equations in particular the integrals of the toda flow on generic orbits are shown to satisfy recursion relations finally we exhibit a third order poisson bracket for which ther matrix approach is feasible

from additional symmetries to linearization of virasoro symmetries abstract we construct the additional symmetries and derive the adler shiota van moerbeke formula for the two component bkp hierarchy considered as certain reductions of the two component bkp hierarchy the drinfeld sokolov hierarchies of type d are proved to possess symmetries written as the linear action of a series of virasoro operators on the tau function it results in that the drinfeld sokolov hierarchies of type d coincide with dubrovin and zhang s hierarchies associated to the frobenius manifolds for coxeter groups of type d and that every solution of such a hierarchy together with the string equation is annihilated by certain combinations of the virasoro operators and the time derivations of the hierarchy

scaling of the first passage time of biased diffusion on hierarchical comb structures biased diffusion on hierarchical comb structures is studied within an exact renormalisation group scheme the scaling exponents of the moments of the first passage time for random walks are obtained it is found that the scaling properties of the diffusion depend only on the direction of bias in a particular case the presence of bias may give rise to a new multifractality

on the approximation of compact fuzzy sets this paper suggests an approximation for a compact fuzzy set by a sequence of lipschitz fuzzy sets this approximation method involves the construction of lipschitz fuzzy sets using the sup min convolution of a compact fuzzy set with a fuzzy zero considering a similar procedure we present the approximation of a trapezoidal type fuzzy interval by a sequence of differentiable fuzzy intervals we show some important properties of both approximations and we give some numerical examples illustrating the results

convolution of fuzzy sets and applications the purpose of this work is studying the approximation in d metric of upper semi continuous and normal fuzzy sets with compact support on rn by using the convolution f g x sup f x y g y y x between two fuzzy sets where the distance d f g is the supremum of the hausdorff distances of their corresponding level sets in particular by using convolution a density result is proved and some applications in choquet integration of fuzzy numbers are presented

infinite dimensional frobenius manifolds for 2 1 integrable systems we introduce a structure of an infinite dimensional frobenius manifold on a subspace in the space of pairs of functions analytic inside outside the unit circle with simple poles at 0 respectively the dispersionless 2d toda equations are embedded into a bigger integrable hierarchy associated with this frobenius manifold

construction of differentiable membership functions in many applications of fuzzy logic it is of special interest to have a transfer function with good properties regarding differentiability to that end it is desirable to have continuously differentiable membership functions with only few parameters in this paper we propose a class of symmetrical and asymmetrical membership functions of exponential order moreover we present a class of more flexible membership functions by construction