ising models julia sets and similarity of the maximal entropy measures we study the phase transition of ising models on diamondlike hierarchical lattices following an idea of lee and yang one can make an analytic continuation of free energy of this model to the complex temperature plane it is known that the migdal kadanoff renormalization group of this model is a rational endomorphism denoted byf of and that the singularities of the free energy lie on the julia setj f the aim of this paper is to prove that the free energy can be represented as the logarithmic potential of the maximal entropy measure onj f moreover using this representation we can show a close relationship between the critical exponent and local similarity of this measure

electrical circuits on mesoscopic sierpinski gaskets a deterministic network of impedances on the sierpinski gasket fractal is considered with particular regard to low generation or mesoscopic systems we study the fractal set of the resonances of the circuit and present calculations on frequency dependent systems the condition of an intermediate size can lead to an oscillatory scaling of the total impedance that disappears in the asymptotic limit several examples are given

twists of quantum borel algebras abstract we classify drinfeld twists for the quantum borel subalgebra u q b in the frobenius lusztig kernel u q g where g is a simple lie algebra over c and q an odd root of unity more specifically we show that alternating forms on the character group of the group of grouplikes for u q b generate all twists for u q b under a certain algebraic group action this implies a simple classification of finite dimensional hopf algebras whose categories of representations are tensor equivalent to that of u q b we also show that cocycle twists for the corresponding de concini kac algebra are in bijection with alternating forms on the aforementioned character group

liftings of nichols algebras of diagonal type iii cartan type g2 abstract we complete the classification of hopf algebras whose infinitesimal braiding is a principal yetter drinfeld realization of a braided vector space of cartan type g 2 over a cosemisimple hopf algebra we develop a general formula for a class of liftings in which the quantum serre relations hold we give a detailed explanation of the procedure for finding the relations based on the recent work of andruskiewitsch angiono and rossi bertone

the asymmetric clock model on a cayley tree the asymmetric clock model is studied on a cayley tree and commensurate and incommensurate phases are found the transition lines are obtained from stability conditions and characteristic points in the phase diagram are analysed by numerical iteration a critical endpoint is found for the case of three states per site and a lifshitz point for the case of four states per site

a new role of green s function in interpolation and sampling theory abstract one of the most fundamental theorems in interpolation and sampling theory is the whittaker shannon kotel nikov sampling theorem which gives an algorithm for reconstructing functions that are fourier transforms of functions with compact supports from their sampled values at a discrete set of points functions that are integral transforms of types other than the fourier one may also be reconstructed from their sampled values at a discrete set of points by using a generalization of the whittaker shannon kotel nikov sampling theorem known as kramer s sampling theorem the main assumption in kramer s sampling theorem is that the kernel of the integral transform to be reconstructed has to arise from a self adjoint boundary value problem whose eigenvalues are all simple and the eigenfunctions are all generated by one single function in this paper we derive a kramer type sampling theorem for a larger class of integral transforms than that considered by kramer s theorem but under less stringent assumptions for example the kernel of any of these integral transforms to be reconstructed from its sampled values may arise from a non self adjoint boundary value problem with repeated eigenvalues and eigenfunctions that are not necessarily generated by one single function we only require that the eigenvalues be simple poles of the associated green s function our technique is based on the use of green s functions of non self adjoint boundary value problems to reconstruct the interpolating functions

on lagrange interpolations and kramer s sampling theorem associated with self adjoint boundary value problems abstract it is known that kramer s sampling theorem and lagrange type interpolation generalize the celebrated whittaker shannon kotel nikov sampling theorem in two different directions however no direct connection between these two directions seems to be known in this article we show that kramer s sampling theorem gives nothing more than the lagrange type interpolations provided that the kernel function associated with kramer s theorem arises from a self adjoint boundary value problem with sample eigenvalues

rank 4 finite dimensional nichols algebras of diagonal type in positive characteristic abstract nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed hopf algebras by lifting method of andruskiewitsch and schneider arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property in the present paper all rank 4 nichols algebras of diagonal type with a finite arithmetic root system over fields of arbitrary characteristic are classified our proof uses the classification of the finite arithmetic root systems of rank 4

on kramer s sampling theorem associated with general strum liouville problems and lagrange interpolation kramer s sampling theorem which is a generalization of shannon s sampling theorem and its relationship with lagrange interpolation is studied kramer s sampling theorem is extended to the case where the interval a b is infinite and it is shown that any function that has a sampling expansion in the scope of kramer s theorem also has a lagrange type interpolation expansion provided that the kernel associated with kramer s theorem arises from a second order sturm liouville boundary value problem whether it is regular or singular this new technique not only reproduces many of the known sampling expansions such as those for the cosine the finite hankel and the continuous legendre transforms but also generates new ones such as those for the continuous jacobi laguerre and hermite transforms

the discrete kramer sampling theorem and indeterminate moment problems in this paper we propose candidates to be the kernel appearing in the discrete kramer sampling theorem these kernels arise either from orthonormal polynomials associated with indeterminate hamburger or stieltjes moment problems or from the second kind orthogonal polynomials associated with the former ones the sampling points are given by the zeros of the denominator in the nevanlinna parametrization of the n extremal measures explicit formulae are given associated with some cases where the nevanlinna parametrization is known explicitly

a rescaling approach to two and three dimensional lattice spectra an exact real space rescaling transformation is used to calculate the spectral properties of tight binding hamiltonians with an arbitrary range of interaction of two and three dimensional lattices the intrinsic problem of the proliferation of initial short range interactions under rescaling is shown to be tractable numerically this method provides a direct way of obtaining the local green functions and local density of states singularities of the spectral functions are associated with fixed points or limit cycles of the rescaling transformation band edge fixed points and exponents have been found for the square triangular and cubic lattices in two and three dimensions there is rapid convergence to accurate numerical results for the green function outside the band while good results have been obtained for the in band e g density of states spectra in two dimensions by suitable choices of cluster size and attenuation factor insight into the relationship between these factors follows from an investigation of finite cluster approximations to the exact transformation it is found that the zeros of the green functions in the complex plane are julia sets which have a fractal structure for a given cluster size the attenuation factors must lie outside these zeros in order for the procedure to be convergent

hierarchical ensembles random fractals flow fractals and the renormalisation group hierarchical ensembles possess exactly renormalisable partition functions examples studied in this work are iterative geometric constructions which at criticality give random self similar fractals the ensembles include novel non self similar constructions termed flow fractals here the simple criticality of these ensembles is analysed and the self similar fractals are found to be homogeneous a multiple weight ensemble the ising snowflake is introduced the transfer matrix of fractals is defined via the renormalisation group a discussion and interpretation is given of the eigenvalues of the transfer matrix fractals for the ising snowflake

on nichols algebras over basic hopf algebras this is a contribution to the classification of finite dimensional hopf algebras over an algebraically closed field mathbb k of characteristic 0 concretely we show that a finite dimensional hopf algebra whose hopf coradical is basic is a lifting of a nichols algebra of a semisimple yetter drinfeld module and we explain how to classify nichols algebras of this kind we provide along the way new examples of nichols algebras and hopf algebras with finite gelfand kirillov dimension

chaotic properties of multipoint correlation functions of an ising model with long range interactions on the sierpinski gasket lattice a class of multispin correlation functions of an ising model with ferromagnetic nearest neighbor interactionsk and constant distance independent long range interactionsq1 q l 1 2 on the sierpinski gasket lattice is considered using an exact method for calculating thermodynamic functions of hierarchically constructed ising systems it is shown that for a set of values ofq and for almost all values ofk somemk spin correlation functions wheremk 3 k 3 withk 1 2 n andn 1 2 being the order of lattice construction change chaotically asn k and therebymk increase to infinity accordingly in the thermodynamic limit these correlation functions prove to be nonanalytic for appropriate values ofq andk sincemk point correlation functions withk being finite i e correlation functions involving finite numbers of spins remain analytic asn tends to infinity there is a smooth crossover between analytic properties of correlation functions of the two types

ac properties of 3d sierpinski gaskets rigorous results the problem of the decimation of a network of impedances on the three dimensional sierpinski gasket is solved the exact map is given and its asymptotic behaviours are studied the most significant invariant subspaces of and the associated submaps are considered this also allows us to address the problem of small size phenomena such as oscillating asymptotic behaviour on this kind of fractal the set of the resonances of the system and the frequency dependence of the total impedance are studied both in the thermodynamic limit and in mesoscopic systems

asymptotics of the susceptibility for the ising model on the hierarchical lattices using the gks inequality griffiths and kaufman showed in 12 that the susceptibility in the ising model on the hierarchical lattices is infinite fort tc in the present work we give an analytic calculation of the dominant singularity of the susceptibility whenh 0 fort tc moreover we analytically find the spontaneous magnetization fort tc and prove that fort tc the susceptibility is finite

numerical experiments on yang lee zeros it is argued that since any real space renormalisation r is a transformation under which the partition function is invariant those r which are based upon various real space approximation schemes can in principle be extended into the complex field of the parameter space in this extension the julia set of r now extended to include r which are algebraic is an approximation to the locus of the yang lee zeros finite size scaling using onsager s solution of the two dimensional ising model is used to construct r which are highly accurate in real temperatures in the critical region these are extended into the complex temperature plane and results are displayed for both the anisotropic and isotropic cases

renormalisation groups with periodic and aperiodic orbits motivated by recent results for some frustrated hierarchical models the authors analyse the consequences of renormalisation group transformations with periodic and aperiodic orbits they find that the free energy has in general an infinite number of singularities this occurs already when the transformation has a stable periodic orbit they present an idealised example and a new class of hierarchical potts models to which the theory applies

one dimensional mapping modulated phases and lyapunov exponent for the antiferromagnetic q state potts and multi site exchange interaction ising models abstract we describe the bifurcation structure period doubling and chaos for the antiferromagnetic q state potts model on the bethe lattice and three site interaction ising model on husimi one in a magnetic field by using the recursion relation technique a chaotic behavior of the magnetic susceptibility for the models is observed at low temperatures the resulting one dimensional rational mapping has a positive lyapunov exponent in the region of the chaotic regime for the antiferromagnetic q state potts q 2 and three site interaction ising models we discuss modulated phases for the antiferromagnetic q state potts q 2 and q 2 and three site interaction ising model at low temperatures the q state potts model q 2 has only one modulated phase with 1 2 pinching corresponding to the 2 cycle the q state potts q 2 and three site interaction ising models have an infinite number of modulated phases with different pinching numbers we construct the first modulated phase after the first bifurcation point

ising model in an external field on a hierarchical lattice the two dimensional renormalization map of the diamond hierarchical ising model in an external field is given and pictures of the distribution of zeros of the partition function in the complex plane of temperature for varying values of coupling constant and external field are shown critical exponents of the model are found and results are different from those of the ising model on a two or three dimensional regular lattice

on analytic sampling theory let h h be a complex separable hilbert space with orthonormal basis x n n 1 and let be a domain in c the field of complex numbers suppose k is a h valued function defined on for each x h define f x z k z x h and let h denote the collection of all such functions f x in this paper we endow h with a structure of a reproducing kernel hilbert space furthermore we show that each element in h is analytic on if and only if k is analytic on or equivalently if and only if k z x n is analytic for each n n and k h is bounded on all compact subsets of in this setting an abstract version of the analytic kramer theorem is exhibited some examples considering different h spaces are given to illustrate these new results

sampling of vector valued transforms associated with green s matrix of dirac systems recently a sampling theorem associated with regular one dimensional dirac systems has been developed integral transforms whose kernels are solutions of the systems have been reconstructed from their values at the eigenvalues in the present article the situation is considered where the kernels are replaced by green s matrix of the problem thus a vector valued sampling theorem is established for vector valued transforms a family of examples where the systems have variable coefficients and a vector valued kramer type sampling lemma are given at the end of the paper

ferromagnetic phase transitions of inhomogeneous systems modelled by square ising models with diamond type bond decorations abstract the two dimensional ising model defined on square lattices with diamond type bond decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems the model is studied analytically under the bond renormalization scheme for an n level decorated lattice the long range ordering occurs at the critical temperature given by the fitting function k b t c j n 1 6410 0 6281 exp 0 5857 n and the local ordering inside n level decorated bonds occurs at the temperature given by the fitting function k b t m j n 1 6410 0 8063 exp 0 7144 n the critical amplitude a sin g n of the logarithmic singularity in specific heat characterizes the width of the critical region and it varies with the decoration level n as a sin g n 0 2473 exp 0 3018 n obtained by fitting the numerical results the cross over from a finite decorated system to an infinite decorated system is not a smooth continuation for the case of infinite decorations the critical specific heat becomes a cusp with the height c n 0 639852 the results are compared with those obtained in the cell decorated ising model

discrete sturm liouville problems jacobi matrices and lagrange interpolation series abstract the close relationship between discrete sturm liouville problems belonging to the so called limit circle case the indeterminate hamburger moment problem and the search of self adjoint extensions of the associated semi infinite jacobi matrix is well known in this paper all these important topics are also related with associated sampling expansions involving analytic lagrange type interpolation series

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

on the spectral characterization of entire operators with deficiency indices 1 1 abstract for entire operators and entire operators in the generalized sense we provide characterizations based on the spectra of their selfadjoint extensions in order to obtain these spectral characterizations we discuss the representation of a simple regular closed symmetric operator with deficiency indices 1 1 as a multiplication operator in a certain de branges space

on the density of partition function temperature zeros the density of partition function temperature zeros for the two dimensional spatially anisotropic ising model in the absence of a magnetic field satisfies a linear homogeneous partial differential equation a generalised form of this equation for a non zero specific heat exponent can be derived by applying non linear scaling to the expression for the specific heat as an integral over the density of zeros

meblin transforms associated with julia sets and physical applications we introduce the mellin transform of the balanced invariant measure associated to the julia set generated by a rational transformation we show that its analytic continuation is a meromorphic function the poles of which are on a semi infinite periodic lattice this allows one to have an understanding of the behavior of the measure near a repulsive fixed point trace identities corresponding to the fact that the analytically continued mellin transform vanishes at negative integers are derived for the polynomial case the quadratic map is first analyzed in detail and the analytic properties of the inverse of the green s function are exhibited of interest is the appearance of a dense set of spikes at dyadic points when the julia set is disconnected these results are used to study the residues of the mellin transform a certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems the excitation spectrum of which displays unusual oscillations the appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion

remarks on disorder and aperiodicity in a model for interacting polymers abstract we present a comparative study of the effects of random and aperiodically distributed interactions on the critical behavior of a model for two interacting polymers on a diamond hierarchical lattice the problem is formulated in terms of exact renormalization group rg recursion relations in the disordered case it is possible to develop a perturbative treatment in order to obtain the fixed points of the moments associated with the random distribution of interactions fully uncorrelated disorder may become relevant driving the system away from a homogeneous fixed point layered disorder may lead to a breakdown of the perturbative treatment in the case of aperiodic interactions we also show some examples of relevance and irrelevance of geometric fluctuations and further investigate the models by resorting to an independent transfer matrix tm analysis which fully corroborates the scaling results

on lagrange interpolation and kramer type sampling theorems associated with sturm liouville problems this article is devoted to a connection between kramer s sampling theorem and sampling expansions generated by lagrange interpolation it is shown that any function that has a sampling expansion in the scope of kramer s theorem also has a lagrange type interpolation expansion provided that the kernel associated with kramer s theorem arises from a second order sturm liouville boundary value problem this new approach which for a variety of regular and singular sturm liouville problems leads to associated sampling theorems recovers not only many known sampling expansions but also gives new ways to calculate the corresponding sampling functions new sampling series are included

bounded rank one perturbations in sampling theory sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points in this paper we derive an orthogonal sampling theory and associated lagrange interpolation formulae from a family of bounded rank one perturbations of a self adjoint operator that has only discrete spectrum of multiplicity one

a classification result on prime hopf algebras of gk dimension one abstract in this paper we classify all prime hopf algebras h of gk dimension one satisfying the following two conditions 1 h has a 1 dimensional representation of order pi deg h and 2 the invariant components of h with respect to this 1 dimensional representation are domains see section 2 for related definitions as consequences 1 a number of new hopf algebras of gk dimension one are found and some of them are not pointed 2 we give a partial answer to a question posed in 12 and 3 the question 1 1 is solved

orthogonal sampling formulas a unified approach this paper intends to serve as an educational introduction to sampling theory basically sampling theory deals with the reconstruction of functions signals through their values samples on an appropriate sequence of points by means of sampling expansions involving these values in order to obtain such sampling expansions in a unified way we propose an inductive procedure leading to various orthogonal formulas this procedure which we illustrate with a number of examples closely parallels the theory of orthonormal bases in a hilbert space all intermediate steps will be described in detail so that the presentation is self contained the required mathematical background is a basic knowledge of hilbert space theory finally despite the introductory level some hints are given on more advanced problems in sampling theory which we motivate through the examples

multiplication operators with deficiency indices p p and sampling formulas in reproducing kernel hilbert spaces of entire vector valued functions abstract a number of recent papers have established connections between reproducing kernel hilbert spaces h of entire functions de branges spaces sampling formulas and a class of symmetric operators with deficiency indices 1 1 in this paper analogous connections between reproducing kernel hilbert spaces of entire vector valued functions de branges spaces of entire vector valued functions sampling formulas and symmetric operators with deficiency indices p p are obtained enroute an analog of l de branges characterization of the reproducing kernel hilbert spaces of entire functions that are now called de branges spaces is obtained for the p 1 vector valued case a special class of these de branges spaces of p 1 vector valued entire functions is identified as a functional model for m g krein s class of entire operators with deficiency indices p p

pointed hopf algebras over non abelian groups with decomposable braidings i abstract we describe all finite dimensional pointed hopf algebras whose infinitesimal braiding is a fixed yetter drinfeld module decomposed as the sum of two simple objects a point and the one of transpositions of the symmetric group in three letters we give a presentation by generators and relations of the corresponding nichols algebra and show that andruskiewitsch schneider conjecture holds for this kind of pointed hopf algebras

finite dimensional pointed or copointed hopf algebras over affine racks abstract we study the pointed or copointed liftings of nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal yd realization of these racks in the copointed case we complete the classification for the six affine racks whose nichols algebra is known to be of finite dimension in the pointed case our method allows us to finish four of them in all of the cases the hopf algebras obtained turn out to be cocycle deformations of their associated graded hopf algebras all of them are new examples of finite dimensional copointed or pointed hopf algebras over non abelian groups

pointed hopf actions on fields ii abstract this is a continuation of the authors study of finite dimensional pointed hopf algebras h which act inner faithfully on commutative domains as mentioned in part i of this work the study boils down to the case where h acts inner faithfully on a field these hopf algebras are referred to as galois theoretical in this work we provide classification results for finite dimensional pointed galois theoretical hopf algebras h of finite cartan type namely we determine when such h of type a 1 r and some h of rank two possess the galois theoretical property moreover we provide necessary and sufficient conditions for reshetikhin twists of small quantum groups to be galois theoretical

sampling expansions associated with kamke problems the present paper is devoted to the derivation of sampling expansions for entire functions which are represented as integral transforms where a differential operator is acting on the kernels the situation generalizes the results obtained in sampling theory associated with boundary value problems to the case when the differential equation has the form n y lambda p y where n and p are two differential expressions of orders n and p respectively n p and lambda is the eigenvalue parameter both self adjoint and non self adjoint cases will be considered with examples in which the boundary conditions are strongly regular

rank one perturbations at infinite coupling abstract we discuss rank one perturbations a a r a 0 self adjoint let d x be the spectral measure defined by a z 1 d x x z we prove there is a measure d which is the weak limit of 1 2 d x as if is cyclic for a then a the strong resolvent limit of a is unitarily equivalent to multiplication by x on l 2 r d this generalizes results known for boundary condition dependence of sturm liouville operators on half lines to the abstract rank one case

the class of n entire operators we introduce a classification of simple regular closed symmetric operators with deficiency indices 1 1 according to a geometric criterion that extends the classical notions of entire operators and entire operators in the generalized sense due to m g krein we show that these classes of operators have several distinctive properties some of them related to the spectra of their canonical self adjoint extensions in particular we provide necessary and sufficient conditions on the spectra of two canonical self adjoint extensions of an operator for it to belong to one of our classes our discussion is based on some recent results in the theory of de branges spaces

finite time singularity in the dynamics of the world population economic and financial indices contrary to common belief both the earth s human population and its economic output have grown faster than exponential i e in a super malthusian mode for most of the known history these growth rates are compatible with a spontaneous singularity occurring at the same critical time 2052 10 signaling an abrupt transition to a new regime the degree of abruptness can be infered from the fact that the maximum of the world population growth rate was reached in 1970 i e about 80 years before the predicted singular time corresponding to approximately 4 of the studied time interval over which the acceleration is documented this rounding off of the finite time singularity is probably due to a combination of well known finite size effects and friction and suggests that we have already entered the transition region to a new regime as theoretical support a multivariate analysis coupling population capital r d and technology shows that a dramatic acceleration in the population growth during most of the timespan can occur even though the isolated dynamics do not exhibit it possible scenarios for the cross over and the new regime are discussed

applications of krein s theory of regular symmetric operators to sampling theory the classical kramer sampling theorem establishes general conditions that allow the reconstruction of functions by mean of orthogonal sampling formulae one major task in sampling theory is to find concrete non trivial realizations of this theorem in this paper we provide a new approach to this subject on the basis of krein s theory of representation of simple regular symmetric operators having deficiency indices 1 1 we show that the resulting sampling formulae have the form of lagrange interpolation series we also characterize the space of function reconstructible by our sampling formulae our construction allows a rigorous treatment of certain ideas proposed recently in quantum gravity

sampling theorem associated with a dirac operator and the hartley transform abstract kramer s sampling theorem indicates that under certain conditions sampling theorems associated with boundary value problems involving n th order self adjoint differential operators can be obtained in this paper we extend this result and derive a sampling theorem associated with boundary value problems involving a one dimensional system of dirac operators as a special case we obtain a sampling theorem for the hartley transform of a bandlimited function

covariant information density cutoff in curved space time in information theory the link between continuous information and discrete information is established through well known sampling theorems sampling theory explains for example how frequency filtered music signals are reconstructible perfectly from discrete samples in this letter sampling theory is generalized to pseudo riemannian manifolds this provides a new set of mathematical tools for the study of space time at the planck scale theories formulated on a differentiable space time manifold can be equivalent to lattice theories there is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity

hierarchical lattice with competing interactions an example of a nonlinear map an ising model with competing nearest and more than nearest neighbour interactions on a hierarchical lattice is solved by decimation in the context of nonlinear mappings the renormalisation group trajectories are analysed as the competition parameter is varied commensurate and incommensurate phases are identified

critical properties of an aperiodic model for interacting polymers we investigate the effects of aperiodic interactions on the critical behaviour of an interacting two polymer model on hierarchical lattices equivalent to the migadal kadanoff approximation for the model on bravais lattices via renormalization group and transfer matrix calculations the exact renormalization group recursion relations always present a symmetric fixed point associated with the critical behaviour of the underlying uniform model if the aperiodic interactions defined by substitution rules lead to relevant geometric fluctuations this fixed point becomes fully unstable giving rise to novel attractors of different nature we present an explicit example in which this new attractor is a two cycle attractor with critical indices different from the uniform model in the case of the four letter rudin shapiro substitution rule we find a surprising closed curve whose points are attractors of period two associated with a marginal operator nevertheless a scaling analysis indicates that this attractor may lead to a new critical universality class in order to provide an independent confirmation of the scaling results we turn to a direct thermodynamic calculation of the specific heat exponent the thermodynamic free energy is obtained from a transfer matrix formalism which had been previously introduced for spin systems and is now extended to the two polymer model with aperiodic interactions

mapping between hierarchical lattices by renormalisation and duality the renormalisation group scheme of migdal 1976 and kadanoff 1976 is applied to a ferromagnetic potts model on two hierarchical lattices in the two cases the renormalisation transformation is split into two elementary processes parallel bonds or series bonds and each lattice can be mapped onto the other one it is then easy to deduce the characteristic parameters critical exponents repulsive fixed points etc of one lattice from those of the other lattice moreover the classical geometrical duality offers a third possibility of mapping and the relationship between the renormalisation and the duality methods is presented in the peculiar case of a two dimensional lattice

calculating dynamic structure factors with the real space renormalisation group the authors develop exact methods based on the real space renormalisation group for calculating dynamic structure factors of heisenberg ferromagnets the method is applied to a pure chain a sierpinski gasket and a quasi one dimensional model of a dilute chain the result for the sierpinski gasket exhibits a rich structure because of the requirement of discrete scale invariance by the use of the distribution scaling the method is generalised to treat a randomly diluted chain

oversampling and aliasing in de branges spaces arising from bessel operators abstract we show that a class of de branges spaces generated by means of generalized fourier transforms associated with perturbed bessel differential equations has the properties of oversampling and aliasing

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

a new real space renormalisation method and its julia set the authors introduce a new real space renormalisation method which is valid not only on the real axis but also everywhere in the complex plane of temperatures the method requires the knowledge of the partition function on small lattices and can in principle be improved systematically they test this method on the two dimensional ising model they find that it gives a satisfactory picture for the set of zeros of the partition function when restricted to the real axis the estimates for the renormalisation transformation the critical point and the thermal exponent are also very good

renormalization group analysis of the 2000 2002 anti bubble in the us s p500 index explanation of the hierarchy of five crashes and prediction abstract we propose a straightforward extension of our previously proposed log periodic power law model of the anti bubble regime of the usa stock market since the summer of 2000 in terms of the renormalization group framework to model critical points using a previous work by gluzman and sornette phys rev e 65 2003 036142 on the classification of the class of weierstrass like functions we show that the five crashes that occurred since august 2000 can be accurately modeled by this approach in a fully consistent way with no additional parameters our theory suggests an overall consistent organization of the investors forming a collective network which interact to form the pessimistic bearish anti bubble regime with intermittent acceleration of the positive feedbacks of pessimistic sentiment leading to these crashes we develop retrospective predictions that confirm the existence of significant arbitrage opportunities for a trader using our model finally we offer a prediction for the unknown future of the us s p500 index extending over 2003 and 2004 that refines the previous prediction of sornette and zhou quant finance 2 2002 468

anomalous diffusion on fractal lattices with site disorder like random walks on euclidean lattices random walks on fractal lattices are modified by a waiting time site disorder when the first moment of the waiting time distribution diverges it is shown that for lattices which support recurrent walks spectral dimension smaller than two the inverse of the diffusion exponent in the presence of disorder is increased by the difference between the waiting time fractal dimension and the usual fractal dimension this hyperscaling relation is derived for sierpinski gaskets in arbitrary dimension with scale dependent waiting times this provides qualitative insight into this problem and the exponent relation derived should also hold for statistically self similar structure such as percolation clusters for lattices whose usual spectral dimension is larger than two a mean field result holds

sampling theorems associated with fourth and higher order self adjoint eigenvalue problems abstract consider a regular self adjoint eigenvalue problem of n th order and the associated whittaker shannon kramer sampling series the samples being taken at the eigenvalues and the sinc function being replaced by the generalized fourier coefficients of the kernel arising from the eigenvalue problem with respect to the complete set of orthogonal eigenfunctions the major aim of this paper is to show that each such sampling expansion can be written as a lagrange interpolation series provided the kernel satisfies suitable conditions two concrete fourth order eigenvalue problems are examined a regular one as an application of the general result and a singular one the problem whether the two resulting sampling series are really new or can be reduced to already known results is in part open