a new algebraization of the lameequation we develop a new way of writing the lamehamiltonian in lie algebraic form this yields in a natural way an explicit formula for both the lamepolynomials and the classical non meromorphic lame functions in terms of chebyshev polynomials and of a certain family of weakly orthogonal polynomials

quasi exactly solvable models with an inhomogeneous magnetic field let group generators having finite dimensional representation be realized as hermitian linear differential operators without inhomogeneous terms as takes place for example for the so n group then corresponding group hamiltonians containing terms linear in generators along with quadratic ones give rise to quasi exactly solvable models with a magnetic field in a curved space in particular for the so 4 group hamiltonian with isotropic quadratic part the manifold within which a quantum particle moves has the geometry of the einstein universe

quantal problems with partial algebraization of the spectrum we discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly solvable problems e g harmonic oscillator and all others the problems belonging to this class are distinguished by the fact that a part of the eigenvalues and eigenfunctions can be found algebraically but not the whole spectrum the reason explaining the existence of the quasi exactly solvable problems is a hidden dynamical symmetry present in the hamiltonian for one dimensional motion this hidden symmetry issl 2 r it is shown that other groups lead to a partial algebraization in multidimensional quantal problems in particular sl 2 r sl 2 r so 3 andsl 3 r are relevant to two dimensional motion inducing a class of quasi exactly solvable two dimensional hamiltonians typically they correspond to systems in a curved space but sometimes the curvature turns out to be zero graded algebras open the possibility of constructing quasi exactlysolvable hamiltonians acting on multicomponent wave functions for example with a non minimal superextension ofsl 2 r we get a hamiltonian describing the motion of a spinor particle

fractals in physics squig clusters diffusions fractal measures and the unicity of fractal dimensionality the three topics discussed in this paper are largely independent part 1 fractal squig clusters are introduced and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality physics on these new geometric shapes should prove tractable as background the author s theories of squig intervals and squig trees are reviewed and restated in more versatile form part 2 the notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality scaling noises are touched upon a common formalism is presented for three forms of anomalous diffusion the ant in the fractal labyrinth fractional brownian motion and levy stable motion the fractal dimensionalities common to diverse shapes generated by diffusion are given in table i as functions of the latent dimensionalities of the support of the motion and of the diffusion itself part 3 it is argued that every fractal point set has a unique fractal dimensionality but it is pointed out that many fractals involve diverse combinations of many fractal point sets such is in particular the case for fractal measures and for fractal graphs often called hierarchical lattices the fractal measures that the author had introduced in the early 1970s are described including new developments

exact enumeration approach to fractal properties of the percolation backbone and 1 σ expansion an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold the authors use this approach to calculate exactly the first low density expansion of lbb p for arbitrary system dimensionality d where lbb p is the mean of backbone bonds and p is the bond occupation probability standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d these disagree with the sierpinski gasket model of the backbone they also calculate the first low density expansions of lmin p and lred p which are respectively the mean number of bonds in the minimum path between i and j and the mean number of singly connected red bonds

moment of inertia of doubly connecting bonds in two dimensional bond percolation by applying a theorem from conformal field theory to bond percolation in two dimensions the author obtains an equation for the second moment i of bonds which doubly connect a percolation cluster i a 5 square root 3 27 pi 2 p pc 2 the equation is exact as the occupancy probability p approaches the critical probability pc from either side a is a calculable lattice dependent number which for a square lattice equals 1 2

on the families of orthogonal polynomials associated to the razavy potential we show that there are two different families of weakly orthogonal polynomials associated to the quasi exactly solvable razavy potential v x cos 2x m 2 0 m one of these families encompasses the four sets of orthogonal polynomials recently found by khare and mandal while the other one is new these results are extended to the related periodic potential u x cos 2x m 2 for which we also construct two different families of weakly orthogonal polynomials we prove that either of these two families yields the ground state when m is odd and the lowest lying gaps in the energy spectrum of the latter periodic potential up to and including the m 1 th gap and having the same parity as m 1 moreover we show that the algebraic eigenfunctions obtained in this way are the well known finite solutions of the whittaker hill or hill s three term periodic differential equation thus the foregoing results provide a lie algebraic justification of the fact that the whittaker hill equation unlike for instance mathieu s equation admits finite solutions

a regular model for cluster numbers and structure just below percolation threshold a two dimensional deterministic fractal model is proposed to imitate the geometric texture just below percolation threshold the model is constructed for bond percolation via a rule of bond occupation on the square lattice it show the typical percolation behaviour as a function of a parameter p p is the bond concentration the model describes the approach towards the threshold below criticality the critical bond concentration pc and the correlation length exponent nu are found the scaling property of the cluster size distribution is studied below the threshold the critical concentration and critical exponents are shown to agree with those derived from the regular model proposed for the geometric texture above the threshold the scaling relations for the critical exponents are shown to be satisfied

exact enumeration of self avoiding walks on critical percolation clusters in 2 7 dimensions we study self avoiding walks on critical percolation clusters by means of a recently developed exact enumeration method which can handle walks of several thousand steps we had previously presented results for the two and three dimensional cases here we take a wider perspective and vary the system s dimensions up to d 7 beyond the supposed upper critical dimension of these results may serve to check analytical predictions and help understand how the medium s fractal structure impacts on the walks scaling behavior

percolation and spreading of damage in a simplified kauffman model abstract a simplified version of the kauffman cellular automaton is introduced as in the usual kauffman model there is a transition between a frozen phase and a chaotic phase where damage may spread we associate the onset of chaos in this model with a percolation transition of certain rules occurring in the model it seems to be in a different universality class from the usual kauffman cellular automaton

a note on coniglio s lemma it is proved that coniglio s 1981 1982 lemma holds for any pair of points in any graph in the low density series expansion of the cut edge weighted pair connectedness

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

position space renormalisation group approach to fractal dimensions of an infinite cluster its backbone and cutting bonds for percolation a position space renormalisation group method is presented to study the cluster structure of an infinite cluster at the percolation threshold fractal dimensions of an infinite cluster its backbone and cutting bonds are determined by the small cell method for two dimensional bond percolation it is shown that the fractal dimension of its cutting bonds agrees with the inverse of the connectedness length exponent at the fixed point

quasi exactly solvable lie algebras of differential operators in two complex variables the authors completely classify all quasi exactly solvable lie algebras of first order differential operators in two complex variables applications to quasi exactly solvable quantum problems are indicated

quasi exact solvability and the direct approach to invariant subspaces we propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of this approach is used here to construct new exactly solvable and quasi exactly solvable quantum hamiltonians on the line which are not lie algebraic it is also applied to generate potentials with multiple algebraic sectors we discuss two illustrative examples of these two applications we show that the generalized lame potential possesses four algebraic sectors and describe a quasi exactly solvable deformation of the morse potential which is not lie algebraic

the starry sky model of the percolation phase transition iii a new equilibrium equation and two kinds of the special elastic nodes abstract the equilibrium lame equation in two dimensional and three dimensional spaces can be presented as conductivity in magnetic field equation with lorenz force expression this allows us to apply the perturbation theory and generalize the lame equation and obtain a new equilibrium equation with all responses of compression and tension it means that compression in one direction leads to tension in the perpendicular direction and vice versa if the boundaries are fixed the solution in recurrent integral form for contributions to all orders of poisson ratio is then obtained zero order solution of lame equilibrium equation expresses the poisson ratio as ve p pc k k 0 3 and creates the special elastic nodes with maximum local deformation the first order solution gives mean values of effective lateral displacements or forces when lateral boundaries are fixed and expresses the bulk modulus as ke p pc 2t g where g 0 6 is a hall coefficient critical exponent it creates the special active elastic nodes where the whole sample microrotates around a new universality class of elasticity problem belongs to the hall coefficient universality class therefore the elasticity backbone coincides with the hall coefficient backbone with the fractal dimension d f b 2 25

cumulant renormalisation group and its application to the incipient infinite cluster in percolation the authors introduce the cumulant method into the context of position space renormalisation group the advantage of this method over previous procedures is that one need not systematically evaluate all 2n states of each kadanoff cell where n is the number of elements sites or bonds in the cell they illustrate this method by calculating for bond percolation with an eight bond cell the critical exponents characterizing the following quantities i the mean number of backbone bonds ii the mean number of bonds in the minimum and maximum paths along the backbone iii the mean number of singly connected red bonds and iv the mean number of bonds in all the self avoiding walks connecting the two extreme points they derive previously obtained results of the conductivity and resistivity

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

a new percolation statistic with unusual properties in bond percolation on lattice with p pc there are unoccupied bonds which if occupied would immediately join the backbone in general when such a bond is occupied several tag end bonds may join the backbone as well the author defines ni to be the number of bonds that would join the backbone if such a bond i were occupied and n to be the average of ni taken over all such bonds on the lattice the author derives a formula for n and finds that for all lattices near pc n gamma bpc p pc 1 where gamma b is the exponent for the divergence of the backbone fraction identical formulae apply to site percolation

new quasi exactly solvable hamiltonians in two dimensions quasi exactly solvable schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods such operators lie in the enveloping algebra of a finite dimensional lie algebra of first order differential operators the hidden symmetry algebra in this paper we develop some general techniques for constructing quasi exactly solvable operators our methods are applied to provide a wide variety of new explicit two dimensional examples on both flat and curved spaces of quasi exactly solvable hamiltonians corresponding to both semisimple and more general classes of lie algebras

logarithmic corrections for the percolative properties of the four state potts model clusters in the potts model are connected sets of nearest neighbour sites for which the spin variables are in the same state droplets can be obtained by adding bonds with probability p 1 exp k between the sites in a cluster where k is the potts model s inverse temperature when q to 4 renormalization group rg fixed points describing clusters and droplets will coalesce leading to logarithmic corrections we calculate the precise form of these corrections used in a differential rg method our predictions are then tested using extensive monte carlo calculations theory and the simulations are found to be in excellent agreement

segment distribution in self similar patterns and backbone structure of percolation clusters a distribution function of segments in self similar patterns is investigated and determined from the fractal dimensionality d of the pattern and that d of the segments on the basis of scaling assumptions the critical behaviour of the segment distribution is studied critical exponents are introduced and expressed in terms of d and d the general theory is applied to link distribution in the backbone of an infinite percolation cluster critical exponents for link distribution are also introduced and evaluated explicitly

on algebraic classification of hermitian quasi exactly solvable matrix schrodinger operators on line we construct six multi parameter families of hermitian quasi exactly solvable matrix schrodinger operators in one variable the method for finding these operators relies heavily upon a special representation of the lie algebra o 2 2 sl 2 sl 2 whose representation space contains an invariant finite dimensional subspace furthermore we select those quasi exactly solvable matrix models that have square integrable eigenfunctions on these models are in direct analogy with the quasi exactly solvable scalar schrodinger operators obtained by turbiner and ushveridze

on algebraic classification of quasi exactly solvable matrix models we suggest a generalization of the lie algebraic approach for constructing quasi exactly solvable one dimensional schrodinger equations this generalization is based on representations of lie algebras by first order matrix differential operators we have classified inequivalent representations of the lie algebras of dimensions up to three by first order matrix differential operators in one variable next we describe invariant finite dimensional subspaces of the representation spaces of the one two dimensional lie algebras and of the algebra these results enable us to construct multiparameter families of first and second order quasi exactly solvable models in particular we have obtained two classes of quasi exactly solvable matrix schrodinger equations

transition between strong and weak disorder regimes for the optimal path abstract we study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path l opt in a disordered erd s renyi er random network and scale free sf network each link i is associated with a weight i exp ar i where r i is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we find that for any finite a there is a crossover network size n a such that for n n a the scaling behavior of l opt is in the strong disorder regime while for n n a the scaling behavior is in the weak disorder regime we derive the scaling relation between n a and a with the help of simulations and also present an analytic derivation of the relation

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

new algebraic quantum many body problems we develop a systematic procedure for constructing quantum many body problems whose spectrum can be partially or totally computed by purely algebraic means the exactly solvable models include rational and hyperbolic potentials related to root systems in some cases with an additional external field the quasi exactly solvable models can be considered as deformations of the previous ones which share their algebraic character

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

dynamic scaling on fractals with sublattices dynamic scaling techniques are developed for antiferromagnets and multiatomic lattices the methods are applied to antiferromagnets and triatomic lattices on sierpinski gasket fractals and to ferro and antiferromagnets on berker hierarchical lattices relationships between the frequencies of ferro and antiferromagnets on the same fractal and of mono and multiatomic fractal systems are given in particular the dynamic exponents za and zf of antiferromagnetic and ferromagnetic sierpinski gaskets or berker lattice fractals are shown to satisfy za 1 2zf

on polynomial solutions of differential equations a general method of obtaining linear differential equations having polynomial solutions is proposed the method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some lie algebra in the projectivized representation possessing an invariant subspace and the spectral problem for a certain linear differential operator with variable coefficients it is shown in general that polynomial solutions of partial differential equations occur in the case of lie superalgebras there are polynomial solutions of some matrix differential equations quantum algebras give rise to polynomial solutions of finite difference equations particularly known classical orthogonal polynomials will appear when considering sl 2 r acting on rp1 as examples some polynomials connected to projectivized representations of sl2 r sl2 r q osp 2 2 and so3 are briefly discussed

high field vs low field magneto transport in a percolating medium recent theoretical and numerical work on high field magneto transport in a percolating medium is described and compared to earlier work on weak field magneto transport in such systems while the weak field behavior is well described by the simple nodes links picture which ignores blobs and loops on a scale smaller than the percolation correlation length p the strong field behavior is extremely sensitive to those features the critical behavior at strong magnetic fields h near the percolation threshold is governed by competition between the usual h 0 fixed point and a new h fixed point which of those fixed points dominates the behavior is determined by the relative sizes of two characteristic lenghts the percolation correlation length p and a new magnetic field dependent length h

lie algebras of differential operators and lie algebraic potentials abstract an explicit characterisation of all second order differential operators on the line which can be written as bilinear combinations of the generators of a finite dimensional lie algebra of first order differential operators is found solving a problem arising in the lie algebraic approach to scattering theory and molecular dynamics one dimensional potentials corresponding to these lie algebras are explicitly classified which include the harmonic oscillator morse one soliton poschl teller mathieu lame confluent hypergeometric and bessel potentials

series expansion analysis of the backbone properties of two dimensional percolation clusters low density series expansions for the backbone properties of two dimensional bond percolation clusters are derived and analysed expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice earlier series work was confined to three properties of the square lattice and was to order the fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest monte carlo results the union intersection and length of the longest self avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for on the other hand the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection the triangular and square lattices are found to have sigificantly different dimensions the fractal dimension of the shortest path also appears to be non universal and we find for the square lattice and for the triangular lattice critical amplitude ratios are considered and found to be in agreement with theoretical inequalities

possibly exact fractal dimensions from conformal invariance based on an equivalence between the q state potts model and conformally invariant field theories the author speculates that all fractal dimensionalities of two dimensional percolation are given by the formula d 100 x2 48 where x is an integer the same formula should also apply to self avoiding walks and lattice animals this approach is extended to cluster weighted percolation his conjecture seems to agree with previous exact and numerical results except for the backbone exponent and possibly for lattice animals

exponent inequalities for the bulk conductivity of a hierarchical model the bulk conductivity p of the bond lattice in d is considered where the bonds have conductivity 1 with probabilityp or e 0 with probability 1 p various representations of the derivatives of p are developed these representations are used to analyze the behavior of p for e 0 near the percolation thresholdp c when the conducting backbone is assumed to have a hierarchical node link blob nlb structure this model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds exact asymptotics of frac d 2 sigma dp 2 for the nlb model are proven under some technical assumptions the proof employs a novel technique whereby frac d 2 sigma dp 2 for the nlb model with e 0 andp nearp c is computed using perturbation theory for p for two and three component resistor lattices aroundp 1 with a sequence of e s converging to 1 as one goes deeper in the hierarchy these asymptotics establish convexity of p for the nlb model nearp c and that its critical exponentt obeys the inequalities 1 t 2 ford 2 3 while 2 t 3 ford 4 the upper boundt 2 ind 3 which is realizable in the nlb class virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model

transport on percolation clusters with power law distributed bond strengths the simplest transport problem namely finding the maximum flow of current or maxflow is investigated on critical percolation clusters in two and three dimensions using a combination of extremal statistics arguments and exact numerical computations for power law distributed bond strengths of the type p ensuremath sigma ensuremath sim ensuremath sigma ensuremath ensuremath alpha assuming that only cutting bonds determine the flow the maxflow critical exponent v is found to be v ensuremath alpha d ensuremath 1 ensuremath nu 1 1 ensuremath ensuremath alpha this prediction is confirmed with excellent accuracy using large scale numerical simulation in two and three dimensions however in the region of anomalous bond capacity distributions 0l ensuremath alpha l 1 we demonstrate that due to cluster structure fluctuations it is not the cutting bonds but the blobs that set the transport properties of the backbone this blob dominance avoids a crossover to a regime where structural details the distribution of the number of red or cutting bonds would set the scaling the restored scaling exponents however still follow the simplistic red bond estimate this is argued to be due to the existence of a hierarchy of so called minimum cut configurations for which cutting bonds form the lowest level and whose transport properties scale all in the same way we point out the relevance of our findings to other scalar transport problems i e conductivity

a class of graded lie algebras of vector fields and first order differential operators finite dimensional lie algebras of polynomial vector fields on rn that contain the elements xi and xi xi for i 1 n were studied to any lie algebra l of this class an n valued n n matrix a and a set of special elements s 1 n are associated it is proven that the pair a s necessarily satisfies two properties conversely to any pair a s satisfying those two properties is associated a lie algebra l a s such that l a s is maximal in the class of all l with matrix a and special elements s for the lie algebras l a s the possible extensions to first order differential operators and its modules of c functions are discussed 1994 american institute of physics

on the construction of quasi exactly solvable schrodinger operators on homogeneous spaces the closure conditions for a quasi exactly solvable operator is the requirement that a second order lie algebraic differential operator be equivalent up to scale change to a schrodinger operator on curved space the present work begins with an invariant characterization of the closure conditions and after a series of steps gives a reformulation of the closure conditions in terms of finite dimensional representations of the underlying lie algebra these techniques are used to give a complete solution to the closure condition for two different planar realizations of sl 2 along the way two theorems about homogeneous solutions to the closure conditions are introduced and proved also introduced is the class of abelian solutions to the closure conditions such solutions appear on flat spaces and give rise to a schrodinger operator with zero potential the concluding remarks highlight remaining questions and give references to research that address said questions

percolation fractals and anomalous diffusion both the infinite cluster and its backbone are self similar at the percolation threshold pc this self similarity also holds at concentrationsp nearpc for length scalesl which are smaller than the percolation connectedness length forl the number of bonds on the infinite cluster scales asld where the fractal dimensionalityd is equal to d v geometrical fractal models which imitate the backbone and on which physical models are exactly solvable are presented above six dimensions one has d 4 and an additional scaling length must be included the effects of the geometrical structure of the backbone on magnetic spin correlations and on diffusion at percolation are also discussed

algebraic approach to the scattering matrix a purely algebraic procedure is presented for calculating recursion relations for s matrices belonging to problems associated with the group su 1 1 the procedure involves supplementing a recently introduced group theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior our formulation makes use of the euclidean group which contains the symmetry transformations for a free particle the coulomb and poeschl teller potentials are discussed to illustrate our methods

exact exponent relations for random resistor diode networks investigates the percolation properties of a random network of non directed bonds resistors and arbitrarily oriented directed bonds diodes for the square lattice there is a multicritical line which connects the isotropic percolation threshold with a network fully occupied by randomly oriented diodes random manhattan along this line symmetry and invariance properties are used to demonstrate that the correlation length exponents are constant furthermore a lattice independent relation is proved which shows that at the isotropic percolation threshold the correlation length diverges with the same exponent when the transition is approached by varying either the resistor concentration or the concentration of randomly oriented diodes

scaling of optimal path lengths distribution in complex networks we study the distribution of optimal path lengths in random graphs with random weights associated with each link disorder with each link i we associate a weight i expari where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we suggest in an analogy with the average length of the optimal path that the distribution of optimal path lengths has a universal form that is controlled by the expression 1 pc a where is the optimal path length in strong disorder a and pc is the percolation threshold this relation is supported by numerical simulations for erd s renyi and scale free graphs we explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

group classification of heat conductivity equations with a nonlinear source we suggest a systematic procedure for classifying partial differential equations pdes invariant with respect to low dimensional lie algebras this procedure is a proper synthesis of the infinitesimal lie method the technique of equivalence transformations and the theory of classification of abstract low dimensional lie algebras as an application we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms we have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry it is shown that there are 3 7 28 and 12 inequivalent classes of pdes of the type considered that are invariant under the one two three and four dimensional lie algebras correspondingly furthermore we prove that any pde belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation this classification is compared with existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach furthermore a number of new invariant equations are constructed which have rich symmetry properties and therefore may be used for mathematical modelling of say nonlinear heat transfer processes

algebraic exact solvability of trigonometric type hamiltonians associated to root systems in this article we study and settle several structural questions concerning the exact solvability of the olshanetsky perelomov quantum hamiltonians corresponding to an arbitrary root system we show that these operators can be written as linear combinations of certain basic operators admitting infinite flags of invariant subspaces namely the laplacian and the logarithmic gradient of invariant factors of the weyl denominator the coefficients of the constituent linear combination become the coupling constants of the final model we also demonstrate the l2 completeness of the eigenfunctions obtained by this procedure and describe a straightforward recursive procedure based on the freudenthal multiplicity formula for constructing the eigenfunctions explicitly

fractals in two dimensions and conformal invariance abstract the exact fractal properties of critical geometrical systems in two dimensions are briefly reviewed they are generically described by an infinite discrete spectrum of scaling dimensions associated with underlying conformal operators applications involve polymers percolation brownian motion and ising clusters a seemingly essential distinction between critical phenomena spectra and multifractal ones is given

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

the fractal dimension of cluster perimeters generated by a kinetic walk the fractal dimension dg of cluster perimeters generated by a recently proposed butterfly growth walk is considered in the long range limit of the walk on a percolation cluster dg appears to be equal to the fractal dimension of the singly connected bonds dg 1 nu the new relation for chemical dimension dl is proposed dl df df dg in the short range limit the butterfly walk on a euclidean lattice appears to be in the same universality class as a random walk the dynamic aspect of the growth walk is discussed and the continuously tunable spectral dimension is obtained both short and long range limits of this diffusion process are different from a random diffusion on percolation

a self avoiding walk on random strips to describe a polymer in a random medium the author considers a self avoiding walk saw on a lattice whose bonds are randomly favourable with probability 1 p or unfavourable with probability p the size of a saw of n steps is calculated when the lattice is a strip by performing products of random matrices when the weight of unfavourable bonds tends to 0 there exist critical concentrations pc where the size of the saw undergoes first order transitions the origin of these transitions is explained as well as the fact that the limits p to 0 or p to 1 are discontinuous possible consequences for higher dimensional systems are discussed

the resistive susceptibility of random diode insulator networks the authors consider regular crystal lattices in which the bonds are either diodes probability p or insulators the mean number of backbone bonds lbb p when averaged over lattice points is found to be related to the mean size of clusters s p by lbb p zps p where z is the number of bonds directed away from any lattice site thus lbb diverges at pc with the mean size exponent gamma the resistive susceptibility chi r p of harris and fisch 1977 1978 is expanded graphically as a power series in p and it is found that term by term chi r p psi r p s p 2 where psi r p is obtained from chi r p by ignoring contributions from nodal graphs the above results are valid for bond and site dilution in any dimension for bond dilution on the square lattice the authors have determined psi r p to p17 and pade analysis of the resulting series for chi r p shows that it diverges with exponent gamma r 3 654 or 0 017 using a scaling relation the exponent t for the conductivity of the infinite cluster is estimated to be t 0 75 or 0 02

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

a transfer matrix for the backbone exponent of two dimensional percolation rephrasing the backbone of two dimensional percolation as a monochromatic path crossing problem we investigate the latter by a transfer matrix approach conformal invariance links the backbone dimension db to the highest eigenvalue of the transfer matrix t and we obtain the result db 1 6431 0 0006 for a strip of width l t is roughly of size 23l but we manage to reduce it to l we find that the value of db is stable with respect to inclusion of additional blobs tangent to the backbone in a finite number of points

field theoretic approaches to biconnectedness in percolating systems two field theoretic formulations for the percolation problem are presented from which the critical exponents describing the backbone of the infinite cluster at the percolation threshold are obtained at high spatial dimension d the order parameter exponent for the backbone beta 2 is given by beta 2 2 beta psi 2 nu where beta is the critical exponent for the density of the infinite cluster and psi 2 is a new crossover exponent in mean field theory psi 2 0 and for d 6 epsilon psi 2 2 epsilon 2 49 o epsilon 3 presumably psi 2 d is a smooth function of d for d d where numerical and theoretical work indicates that d is about 3 the result indicates that the fractal dimensionality of the backbone is given in terms of the percolation exponents as gamma nu psi 2

scaling theory for the statistics of self avoiding walks on random lattices the authors study self avoiding walks saw on randomly diluted quenched lattices with direct configurational averaging over the moments of the saw distribution function a scaling function representation of rn the average end to end distance of n step walks is studied here both for saw on a the infinite percolation cluster and b any cluster they have shown that at the percolation threshold nu p nu p 1 beta p 2 nu p where beta p and nu p are the percolation order parameter and correlation length exponents respectively the authors also propose a scaling function representation for the total number of n step saw configuration gn approximately mu nngamma 1 for infinite cluster averaging which gives gamma p gamma d nu p nu for all cluster averaging gamma will remain unchanged

fractals their transfer matrices and their eigen dimensional sequences this paper defines and studies a general algorithm for constructing new families of fractals in euclidean space this algorithm involves a sequence of linear interscale transformations that proceed from large to small scales the authors find that the fractals obtained in this fashion decompose in intrinsic fashion into linear combinations of a variable number of addend fractals the addends relative weights and fractal dimensionalities are obtained explicitly through an interscale matrix which they call the transfer matrix of the fractal tmf the authors first demonstrate by a series of examples then prove rigorously that the eigenvalues of our tmfs are real and positive and that the fractal dimensions of the addend fractals are the logarithms of the eigenvalues of their tmf they say that these dimensions form the overall fractal s eigendimensional sequence the eigenvalues of their tmf are integers in the non random variants of the construction but are non integer in the random variants a geometrical interpretation of the eigenvalues and the eigenvectors is given their tmf have other striking and very special properties that deserve additional attention