the multiplier of the interval 1 1 for the dunkl transform on the real line abstract we study the boundedness of the multiplier of the interval 1 1 for the dunkl transform of order 1 2 on weighted l p spaces with 1 p in particular we get that it is bounded from l p r x 2 1 d x into itself if and only if 4 1 2 3 p 4 1 2 1

conewise linear systems non zenoness and observability conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation this class of dynamical systems represents a large number of piecewise linear systems most notably linear complementarity systems with the p property and their generalizations to affine variational systems which have many applications in engineering systems and dynamic optimization the challenges of dealing with this type of hybrid system are due to two major characteristics mode switchings are triggered by state evolution and states are constrained in each mode in this paper we first establish the absence of zeno states in such a system based on this fundamental result we then investigate and relate several state observability notions short time and t time or finite time local global observability for the short time observability notions constructive finitely verifiable algebraic both sufficient and necessary conditions are derived due to their long time mode transitional behavior which is very difficult to predict only partial results are obtained for the t time observable states nevertheless we completely resolve the t time local observability for the bimodal conewise linear system for finite t and provide numerical examples to illustrate the difficulty associated with the long time observability

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

the phase transition of the directed polymer on disordered hierarchical lattices the directed polymer on disordered hierarchical lattices is studied using an exact renormalization scheme the phase transition is studied and a hyperscaling relation is derived the fixed distributions of the renormalized energies are obtained numerically the specific heat and associated exponents are computed from the fixed distributions

perturbation expansions of the axial next nearest neighbour ising and asymmetric four state clock models the axial next nearest neighbour ising annni and four state asymmetric clock 4 asc models are investigated by perturbing about points at which they reduce the simple nearest neighbour ising models these limits are a for the annni model i small nearest neighbour interaction j1 in the x direction and ii small next nearest neighbour interaction j2 in the x direction and b for the 4 asc model small asymmetry delta in the limit j1 to 0 the phase boundary is located to first order in j1 and found to be ising like in the limit j2 to 0 the behaviour to second order in j2 is more complicated but is identical to that found to second order in delta for the 4 asc model these results are discussed in the light of other recent work

the kadanoff lower bound variational renormalization group applied to an su 2 lattice spin model the authors apply the variational lower bound renormalization group transformation of kadanoff 1975 to an su 2 lattice spin model in two and three dimensions even in the one hypercube framework of this renormalization group transformation the present model is characterized by having an infinite basis of fundamental operators they investigate whether the lower bound variational renormalization group transformation yields results stable under different truncations of this operator basis their results show that for this particular spin model this is not the case

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

strongly regular differential variational systems a differential variational system is defined by an ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution of a finite dimensional variational inequality containing the state variable of the system this paper addresses two system theoretic topics for such a nontraditional nonsmooth dynamical system namely non zenoness and local observability of a given state satisfying a blanket strong regularity condition for the former topic which is of contemporary interest in the study of hybrid systems we extend the results in our previous paper where we have studied zeno states and switching times in a linear complementarity system lcs as a special case of the differential variational inequality dvi the lcs consists of a linear time invariant ode and a linear complementarity problem the extension to a nonlinear complementarity system ncs with analytic inputs turns out to be non trivial as we need to use the lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state further extension to a differential variational inequality is obtained via its equivalent karush kuhn tucker formulation for the second topic which is classical in system theory we use the non zenoness result and the recent results in a previous paper pertaining to the b differentiability of the solution operator of a nonsmooth ode to obtain a sufficient condition for the short time local observability of a given strongly regular state of an ncs refined sufficient conditions and necessary conditions for local observability of the lcs satisfying the p property are obtained

positive invariance of constrained affine dynamics and its applications to hybrid systems and safety verification motivated by long time dynamic analysis of hybrid systems and safety verification problems this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long time oscillatory dynamic analysis a linear program based algorithm is proposed to verify these conditions and its computational complexity is analyzed the positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems further the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

interpolation by weighted paley wiener spaces associated with the dunkl transform abstract given 1 2 0 and 1 p we study the interpolation problem in the space pw p of entire functions f c c of exponential type for which r f x p x 2 1 d x with nodes of interpolation at s j j z where s j j n is the increasing sequence of all positive roots of the bessel function j 1 z of order 1 and s j s j for all j z we prove that if 4 1 2 3 p 1 p p 2 4 1 2 1 the interpolation problem f pw p and f 1 s j c j for all j z has a unique solution for every sequence c j of complex numbers satisfying j z c j p 1 j 2 1 and that if p p 1 the corresponding interpolation problem may not have a solution and that the solution if exists is unique if and only if p p 2 finally we show that r f x p x 2 1 d x 2 2 j z f s j 1 p 1 j 2 1 with the constant of equivalence depending only on p and holds for all entire functions f of exponential type if and only if p 1 p p 2

a transference theorem for the dunkl transform and its applications for a family of weight functions invariant under a finite reflection group we show how weighted lp multiplier theorems for dunkl transform on the euclidean space rd can be transferred from the corresponding results for h harmonic expansions on the unit sphere sd of rd 1 the result is then applied to establish a hormander type multiplier theorem for the dunkl transform and to show the convergence of the bochner riesz means of the dunkl transform of order above the critical index in weighted lp spaces

the migdal kadanoff approximation optimisation generalisation ising model in external fields and the migdal kadanoff hierarchies the simplest migdal kadanoff approximation is optimised on hypercubic lattices generalisation is made to all regular lattices leading to recursion relations continuous in lattice coordination one variety of these relations shows universality whilst another variety shows duality on planar lattices the ising model in external fields both normal and staggered is investigated phase diagrams and exponents are found under a complete treatment of the ambiguity in the handling of external fields by bond moving finally the underlying exactly solved family of hierarchical lattices is considered it is shown that these lattices exhibit a complete breakdown in the concept of universality

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

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

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

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

condensed julia sets with an application to a fractal lattice model hamiltonian on considere l ensemble de julia pour l application rationnelle complexe z z 2 e z ou et e sont des parametres complexes dans la limite e 0

infinite susceptibility at high temperatures in the migdal kadanoff scheme the migdal kadanoff real space renormalisation group scheme applied to the ising and classical xy and probably other ferromagnetic models yields an infinite susceptibility at all temperatures above the critical temperature for a hypercubic lattice of dimension d or 2

grassman path integral solution for a class of triangular type decorated ising models abstract grassman path integral solution is given for a class of two dimensional triangular type decorated ising models canonical ising lattices rectangular triangular and hexagonal enter into this class as the most simple particular cases as a first step the problem is reformulated in terms of a free fermionic field theory the method is based on the mirror factorization principle for the density matrix traditional transfer matrix or combinatorial considerations are not needed the solution exhibits the characteristics free fermionic structure providing the universal logarithmic singularity in the specific heat the symmetries and the critical point behaviour are investigated within the spin polynomial interpretation of the problem some concrete decorated lattices are treated by illustration attention is given to the choice of rational computational devices

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

mixed norm estimates for the cesaro means associated with dunkl hermite expansions our main goal in this article is to study mixed norm estimates for the cesaro means associated with dunkl hermite expansions on r d these expansions arise when one considers the dunkl hermite operator or dunkl harmonic oscillator h kappa delta kappa vertical bar x vertical bar 2 where delta kappa stands for the dunkl laplacian it is shown that the desired mixed norm estimates are equivalent to vector valued inequalities for a sequence of cesaro means for laguerre expansions with shifted parameter in order to obtain such vector valued inequalities we develop an argument to extend these laguerre operators for complex values of the parameters involved and apply a version of the three lines lemma

controlling a class of nonlinear systems on rectangles in this paper we focus on a particular class of nonlinear affine control systems of the form xdot f x bu where the drift f is a multi affine vector field i e affine in each state component the control distribution b is constant and the control u is constrained to a convex set for such a system we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant we then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time if the control constraints are polyhedral we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle this work is motivated by the need to construct discrete abstractions for continuous and hybrid systems in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs we show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators

hierarchical models and chaotic spin glasses renormalization group studies in position space have led to the discovery of hierarchical models which are exactly solvable exhibiting nonclassical critical behavior at finite temperature position space renormalization group approximations that had been widely and successfully used are in fact alternatively applicable as exact solutions of hierarchical models this realizability guaranteeing important physical requirements for example a hierarchized version of the sierpiriski gasket is presented corresponding to a renormalization group approximation which has quantitatively yielded the multicritical phase diagrams of submonolayers on graphite hierarchical models are now being studied directly as a testing ground for new concepts for example with the introduction of frustration chaotic renormalization group trajectories were obtained for the first time thus strong and weak correlations are randomly intermingled at successive length scales and a new microscopic picture and mechanism for a spin glass emerges an upper critical dimension occurs via a boundary crisis mechanism in cluster hierarchical variants developed to have well behaved susceptibilities

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

phase diagram of the ising model on a cayley tree in the presence of competing interactions and magnetic field we study the phase diagram for the ising model on a cayley tree with competing nearest neighbor interactionsj1 and next nearest neighbor interactionsj2 andj3 in the presence of an external magnetic field to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular cases previous works by vannimenus and by inawashiroet al at vanishing temperature the phase diagram is fully determined for all values and signs ofj2 j1 andj3 j2 in particular we verify that values ofj3 j2 high enough favor the paramagnetic phase at finite temperatures several interesting features evolution of reentrances separation of the modulated region into two disconnected pieces etc are exhibited for typical values ofj2 j1 andj3 j2

on linear systems containing strict inequalities this paper deals with systems of an arbitrary possibly infinite number of both weak and strict linear inequalities we analyze the existence of solutions for such kind of systems and show that the large class of convex sets which admit this kind of linear representations i e the so called evenly convex sets enjoys most of the well known properties of the subclass of the closed convex sets we also show that it is possible to obtain geometrical information on these sets from a given linear representation finally we discuss the theory and methods for those linear optimization problems which contain strict inequalities as constraints

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

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

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

lp fourier multipliers for the dunkl operator on the real line we consider fourier multipliers for lp associated with the dunkl operator on r and establish a version of hormander s multiplier theorem in applying this version we come up with some results regarding the oscillating multipliers partial sum operators and generalized bessel potentials

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

fefferman stein inequalities for the mathbb z _2 d dunkl maximal operator in this article we establish the fefferman stein inequalities for the dunkl maximal operator associated with a finite reflection group generated by the sign changes similar results are also given for a large class of operators related to dunkl s analysis

phase diagram of the one state potts model on the cayley tree the phase diagram of the one state potts model on the closed asymmetric cayley tree with branching ratior 2 is obtained from the bethe peierls map the route to chaos via the period doubling cascade is obtained by considering the antiferromagnetic coupling limit the connection of the potts model with the percolation problem is shown by calculating the order parameter its susceptibility the internal energy and the specific heat as well as their asymptotic behavior at the paramagnetic ferromagnetic critical point due to the type of the lattice and to the polynomial character of the map this is the simplest known example of a mckay berker kirkpatrick spin glass

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

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

robust non zenoness of piecewise affine systems with applications to linear complementarity systems piecewise affine systems pass constitute an important class of nonsmooth switching dynamical systems subject to state dependent mode transitions arising from control and dynamic optimization a fundamental issue in dynamics analysis of switching systems pertains to the possible occurrence of infinitely many switchings in finite time referred to as the zeno behavior there has been a growing interest in characterization of zeno free switching systems different from the recent non zeno analysis of switching systems the present paper studies non zeno properties of pass subject to system parameter and or initial state perturbations inspired by sensitivity and uncertainty analysis of pass specifically by exploiting the geometry of polyhedral subdivisions and dynamical system techniques this paper establishes a uniform bound on the number of mode switchings for a family of lipschitz pass under mild uniform conditions on system parameters and associated polyhedral subdivisions this result is employed t

critical properties of ising models with long range interactions on sierpinski gasket lattices in higher euclidean dimensions abstract ising systems with two spin long range interactions are studied on sierpinski gasket lattices generalized to euclidean dimensions d 3 4 5 6 for each lattice the long range interactions are assigned to edges that are part of the appropriate self similar structure in consequence these systems are infinitely ramified and exhibit critical behavior at finite temperatures an analysis of critical properties of the systems in particular the calculation of critical exponents is carried out by using an exact renormalization group method it is shown that there exists some similarity between critical properties of the long range interacting fractal models under study here and critical properties of short range interacting ising models defined on abstract regular lattices interpolated to noninteger dimensionalities

riesz transform and riesz potentials for dunkl transform analogues of riesz potentials and riesz transforms are defined and studied for the dunkl transform associated with a family of weight functions that are invariant under a reflection group the lp boundedness of these operators is established in certain cases

hormander s multiplier theorem for the dunkl transform abstract for a normalized root system r in r n and a multiplicity function k 0 let n n r k denote by d w x r x k d x the associated measure in r n let f stand for the dunkl transform given a bounded function m on r n we prove that if there is s n such that m satisfies the classical hormander condition with the smoothness s then the multiplier operator t m f f 1 m f f is of weak type 1 1 strong type p p for 1 p and is bounded on a relevant hardy space h 1 to this end we study the dunkl translations and the dunkl convolution operators and prove that if f is sufficiently regular for example its certain schwartz class seminorm is finite then the dunkl convolution operator with the function f is bounded on l p d w for 1 p we also consider boundedness of maximal operators associated with the dunkl convolutions with schwartz class functions

a framework for worst case and stochastic safety verification using barrier certificates this paper presents a methodology for safety verification of continuous and hybrid systems in the worst case and stochastic settings in the worst case setting a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region no explicit computation of reachable sets is required in the construction of barrier certificates which makes it possible to handle nonlinearity uncertainty and constraints directly within this framework in the stochastic setting our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set a bound whose validity is proven by the existence of a barrier certificate for polynomial systems barrier certificates can be constructed using convex optimization and hence the method is computationally tractable some examples are provided to illustrate the use of the method

on fundamental harmonic analysis operators in certain dunkl and bessel settings we consider several harmonic analysis operators in the multi dimensional context of the dunkl laplacian with the underlying group of reflections isomorphic to z2n also negative values of the multiplicity function are admitted our investigations include maximal operators g functions lusin area integrals riesz transforms and multipliers of laplace and laplace stieltjes transform type using the general calderon zygmund theory we prove that these objects are bounded in weighted lp spaces 1 p and from l1 into weak l1

statistical mechanics of z m models on cayley trees abstract the problem of small field phase transitions for z m models on cayley trees is solved in detail phase diagrams for zero field are obtained for m 2 3 4 5 and 6 as special cases potts models are also considered and all phases not only those in zero field are identified the m limit the planar rotator model is also solved completely for the zero field case the relevance of the results especially of the phase diagrams for the problems associated with z m models on real lattices is discussed

on invariant polyhedra of continuous time linear systems this note presents some conditions of existence of positively invariant polyhedra for linear continuous time systems these conditions are first described algebraically then interpreted on the basis of the system eigenstructure then a simple state feedback placement method is described for solving some linear regulation problems under constraints

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

recursion relations for the q state potts model on a closed cayley tree abstract recursion relations which determine the free energy of a q state potts model on a closed cayley tree are derived the fixed points of these relations and their relative stabilities are discussed in detail

observability analysis of conewise linear systems via directional derivative and positive invariance techniques belonging to the broad framework of hybrid systems conewise linear systems clss form a class of lipschitz piecewise linear systems subject to state triggered mode switchings motivated by state estimation of nonsmooth switched systems this paper exploits directional derivative and positive invariance techniques to characterize finite time and long time local observability of a general cls for the former observability notion directional derivative results are developed from the simple switching property and they yield improved observability conditions for the latter notion we focus on the case where a nominal trajectory has finitely many switchings in order to characterize long time behaviors of the cls necessary and sufficient conditions are obtained for the interior of a positively invariant cone by employing these conditions we establish connections between finite time and long time local observability underlying positive invariance properties are unveiled

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

riesz transforms and fractional integration for orthogonal expansions on spheres balls and simplices abstract this paper studies the hardy littlewood sobolev hls inequality and the riesz transforms for fractional integration associated to weighted orthogonal polynomial expansions on spheres balls and simplexes with weights being invariant under a general finite reflection group on r d the sharp index for the validity of the hls inequality is determined and the l p boundedness of the riesz transforms is established in particular our results extend a classical inequality of muckenhoupt and stein on conjugate ultraspherical polynomial expansions our idea is based on a new decomposition of the dunkl laplace beltrami operator on the sphere and some sharp asymptotic estimates of the weighted christoffel functions

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

phase transitions in regularly diluted 2d ising models with a complicated inherent structure of a lattice cell abstract we analyze the critical point and study the nonuniversal peculiarities occuring in the temperature dependence of the specific heat for a sequence of the exactly solvable highly decorated two dimensional triangular type ferromagnetic ising models with a complicated fractal type inherent structure of elementary cell the lattices are indexed by the decoration parameter n 0 1 2 3 the elementary cell for the n lattice is the n th order lattice approximation of the sierpinski gasket fractal at any finite n there is the second order phase transition indicated by the logarithmic singularity in the specific heat the critical temperature decreases as 4 ln 4 n 1 with increasing decoration the specific heat critical amplitude vanishes as 3 n n ln n 2 as n the basic nonuniversal features observable in the specific heat at strong decoration are the anomalous narrowing of the critical interval and the appearance of a rounded peak above the true t c point these effects are closely related and can both be explained by the above t c local ordering in the local lattice inhomogeneities the mechanism of the above t c local ordering is analyzed the models under discussion can as well be interpreted as the regular bond diluted 2d ising ferromagnets with a hierarchy of the nonmagnetic holes at different scales and varying dilution a comparative discussion on the properties of the regularly and randomly diluted 2d ising ferromagnets is given

hierarchical lattices some examples with a comparison of intrinsic dimension and connectivity and ising model exponents the recently recognised class of hierarchical lattices is examined through a number of examples definitions of length intrinsic dimension and connectivity are made and used to discuss variations of exponents calculated on the ising model generalisations of regular lattice results are found for exponents at discontinuity fixed points and in 1 epsilon dimensions

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

exact results for an ising system with long range interactions on the sierpinski gasket lattice thermodynamic properties of an ising model with two point long range interactions on the sierpinski gasket lattice are studied using an exact recursive methodfor calculating the partition function the long range interactions are introduced in a self similar manner reflecting the structure of the lattice and are assumed to be of the form ql q 2 l 1 where q 0 0 are parameters and l 1 2 indicates that ql couples pairs of spins on the distance 2l measured in units of the lattice spacing in the case of constant long range interactions 0 exact analytic expression for the critical temperature and the critical exponent v are derived it is also shown that the recursive method can be applied to analyze the problem of the existence of phase transitions at finite temperatures in the case of 0 i e when the long range coupling decrease with the distance

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

chiral potts model on a cayley tree with complete and incomplete devil s staircase the three state chiral potts model on a cayley tree is analysed in the limit of infinitely large coordination number the fractal dimensionalities of the wavenumber against chiral field curves are computed it is shown that they change from a complete to an incomplete devil s staircase as the temperature is raised commensurate phase boundaries are determined analytically for high and low temperatures and the discommensurations are shown to result from tangent bifurcations

exact solutions for ising model odd number correlations on the honeycomb and triangular lattices abstract investigating honeycomb and triangular simple spin 1 2 ising model ferromagnets exact algebraic systems of linear identities are developed containing the spontaneous magnetization and other odd number correlations where the coefficients depend upon the interaction parameters aided by star triangle type relationships and making supplemental use of only the spontaneous magnetization from the literature it is found that these identity systems are exactly solvable for the odd number multisite correlations the method therefore satisfies closure non hierarchal and linear independence requirements is relatively transparent and approximately eighty odd number localized correlations are obtained containing up to and including nine seven sites for the honeycomb triangular lattice and a simple prescription is given and demonstrated for extracting their critical amplitudes the results also offer examples of correlation degeneracies both essential and accidental types and reveal the existence of linear correlation identities which do not depend explicitly upon the interaction parameters

fisher zeros and julia sets a multifractal analysis abstract we consider the potts model on a hierarchical lattice a multifractal analysis of the julia set associated with the renormalization group transformation which gives the fisher zeros of the partition function in the complex temperature plane is performed the behavior of the singularity spectrum f and the generalized dimension dq as a function of the number of states is studied

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

reachability and control synthesis for piecewise affine hybrid systems on simplices in this paper we consider the synthesis of control laws for piecewise affine hybrid systems on simplices the construction is based on the solution to the control to facet problem at the continuous level and on dynamic programming at the discrete level the construction is given as an explicit algorithm using only linear algebra and reach set computations for automata no numerical integration is required the method is conservative in that it may fail to find a control law where one exists but one cannot hope for a sharp algorithm for control synthesis since reachability for piecewise affine hybrid systems is undecidable

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

a positive radial product formula for the dunkl kernel it is an open conjecture that generalized bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated dunkl operators in this paper a partial result towards this conjecture is proven namely a positive radial product formula for the non symmetric counterpart of the generalized bessel function the dunkl kernel radial here means that one of the factors in the product formula is replaced by its mean over a sphere the key to this product formula is a positivity result for the dunkl type spherical mean operator it can also be interpreted in the sense that the dunkl type generalized translation of radial functions is positivity preserving as an application we construct dunkl type homogeneous markov processes associated with radial probability distributions

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

oscillatory critical amplitudes in hierarchical models and the harris function of branching processes oscillatory critical amplitudes have been repeatedly observed in hierarchical models and in the cases that have been taken into consideration these oscillations are so small to be hardly detectable hierarchical models are tightly related to iteration of maps and in fact very similar phenomena have been repeatedly reported in many fields of mathematics like combinatorial evaluations and discrete branching processes it is precisely in the context of branching processes with bounded off spring that t harris in 1948 first set forth the possibility that the logarithm of the moment generating function of the rescaled population size in the super critical regime does not grow near infinity as a power but it has an oscillatory prefactor the harris function these oscillations have been observed numerically only much later and while the origin is clearly tied to the discrete character of the iteration the amplitude size is not so well understood the purpose of this note is to reconsider the issue for hierarchical models and in what is arguably the most elementary setting the pinning model that actually just boils down to iteration of polynomial maps and notably quadratic maps in this note we show that the oscillatory critical amplitude for pinning models and the harris function coincide moreover we make explicit the link between these oscillatory functions and the geometry of the julia set of the map making thus rigorous and quantitative some ideas set forth in derrida et al commun math phys 94 115 132 1984

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

temporal logic analysis of gene networks under parameter uncertainty the lack of precise numerical information for the values of biological parameters severely limits the development and analysis of models of genetic regulatory networks to deal with this problem we propose a method for the analysis of genetic regulatory networks under parameter uncertainty we consider models based on piecewise multiaffine differential equations dynamical properties expressed in temporal logic and intervals for the values of uncertain parameters the problem is then either to guarantee that the system satisfies the expected properties for every possible parameter value the corresponding parameter set is then called valid or to find valid subsets of a given parameter set the proposed method uses discrete abstractions and model checking and allows for efficient search of the parameter space however the abstraction process creates spurious behaviors in the abstract systems along which time does not progress consequently the verification of liveness properties expressing that something will eventually happen and implicitly assuming progress of time often fails a solution to this second problem is proposed using the notion of transient regions this approach has been implemented in a tool for robust verification of gene networks and applied to the tuning of a synthetic network built in e coli