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

a syntactic approach to scale space based corner description planar curves are described by information about corners integrated over various levels of resolution the detection of corners takes place on a digital representation to compensate for ambiguities arising from sampling problems due to the discreteness results about the local behavior of curvature extrema in continuous scale space are employed

contains and inside relationships within combinatorial pyramids irregular pyramids are made of a stack of successively reduced graphs embedded in the plane such pyramids are used within the segmentation framework to encode a hierarchy of partitions the different graph models used within the irregular pyramid framework encode different types of relationships between regions this paper compares different graph models used within the irregular pyramid framework according to a set of relationships between regions we also define a new algorithm based on a pyramid of combinatorial maps which allows to determine if one region contains the other using only local calculus

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

a characterization of all elliptic algebro geometric solutions of the akns hierarchy here 9 x p x wl w3 denotes the elliptic weierstrass function with fundamental periods 2wl and 2w3 im w3 wl 0 in the special case where wl is real and w3 is purely imaginary the potential q x in 1 1 is real valued and ince s striking result 51 in modern spectral theoretic terminology yields that the spectrum of the unique self adjoint operator associated with the differential expression l2 d2 dx 2 q x in l2 r exhibits finitely many bands and gaps respectively that is

linear operators with invariant polynomial space and graded algebra the irreducible finite dimensional representations of the graded algebras osp j 2 j 1 2 3 are expressed in terms of differential operators some quantum deformations of these algebras are shown to admit similar kinds of representations these are formulated in terms of finite difference operators the results are discussed in the framework of the quasi exactly solvable equations

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

on the sutherland spin model of b n type and its associated spin chain the b n hyperbolic sutherland spin model is expressed in terms of a suitable set of commuting dunkl operators this fact is exploited to derive a complete family of commuting integrals of motion of the model thus establishing its integrability the dunkl operators are shown to possess a common flag of invariant finite dimensional linear spaces of smooth scalar functions this implies that the hamiltonian of the model preserves a corresponding flag of smooth spin functions the discrete spectrum of the restriction of the hamiltonian to this spin flag is explicitly computed by triangularization the integrability of the hyperbolic sutherland spin chain of b n type associated with the dynamical model is proved using polychronakos s freezing trick

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators

an su 2 analogue of the azbel hofstadter hamiltonian motivated by the recent findings of wiegmann zabrodin and faddeev kashaev concerning the appearance of the quantum symmetry in the problem of a bloch electron on a two dimensional magnetic lattice we introduce a modification of the tight binding azbel hofstadter hamiltonian that is a specific spin s euler top that can be considered as its classical analogue the eigenvalue problem for the proposed model in the coherent state representation is described by the s gap lame equation and thus is completely solvable we observe a striking similarity between the shapes of the spectra of the two models for various values of the spin s

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

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

groups of piecewise linear homeomorphisms in this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line we use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them we also identify large simple subgroups in all of them providing examples of finitely presented infinite simple groups

quasi exactly solvable extensions of the lame equation quasi exactly solvable equations are spectral equations which possess a finite number of algebraic eigenvectors for a few of these equations such as the lame equation the number of algebraic solutions is larger than predicted by the general theory a class of quasi exactly solvable equations in one variable is considered and the conditions under which a rich algebraization occurs is discussed families of equations which by themselves are not quasi exactly solvable but can be transformed into systems of coupled quasi exactly solvable equations are discussed these results suggest a scheme for the classification of quasi exactly solvable systems in one variable

picard potentials and hill s equation on a torus hill s equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness of particular importance in the last 20 years is its connection with the kdv hierarchy and hence with integrable systems we show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finitegap potentials a major open problem in the field specifically we show that elliptic finitegap potentials of hill s equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable complementing a classical theorem of picard the intimate connection between picard s theorem and elliptic finite gap solutions of completely integrable systems is established in this paper for the first time in addition we construct the hyperelliptic riemann surface associated with a finitegap potential not necessarily elliptic i e determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain auxiliary operators associated with hill s equation these multiplicities are intimately correlated with the pole structure of the diagonal green s function of the operator h d2 dx2 q x in l2 r our construction is new in the present general complex valued periodic finite gap case before describing our approach in some detail we shall give a brief account of the history of the problem involved this theme dates back to a 1940 paper of ince 43 who studied what is presently called the lam6 ince potential

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

the structure of quasi exactly solvable systems the linear differential operators that preserve the n dimensional vector space whose entries are polynomials of fixed but arbitrary degrees in one variable are considered herein they are relevant for the classification of quasi exactly solvable systems of equations generating elements are explicitly constructed and rules for ordering their products are derived some examples of equations that can be reduced to such systems are discussed

multiple algebraisations of an elliptic calogero sutherland model recently gomez ullate et al phys lett b 511 112 2001 have studied an n particle quantum problem with elliptic function potentials they have shown that the hamiltonian operator preserves a finite dimensional space of functions and as such is quasi exactly solvable qes in this article we show that other types of invariant function spaces exist which are in close relation to the algebraic properties of the elliptic functions accordingly series of new algebraic eigenfunctions can be constructed

families of quasi exactly solvable extensions of the quantum oscillator in curved spaces we introduce two new families of quasi exactly solvable qes extensions of the oscillator in a d dimensional constant curvature space for the first three members of each family we obtain closed form expressions of the energies and wavefunctions for some allowed values of the potential parameters using the bethe ansatz method we prove that the first member of each family has a hidden sl 2 symmetry and is connected with a qes equation of the first or second type respectively one dimensional results are also derived from the d dimensional ones with d 2 thereby getting qes extensions of the mathews lakshmanan nonlinear oscillator

a new proposal for graph based image classification using frequent approximate subgraphs graph based data representations are an important research topic due to the suitability of this kind of data structure to model entities and the complex relations among them in computer vision graphs have been used to model images in order to add some high level information relations to the low level representation of individual parts how to deal with these representations for specific tasks is not easy due to the complexity of the data structure itself in this paper we propose to use a graph mining technique for image classification introducing approximate patterns discovery in the mining process in order to allow certain distortions in the data being modeled we are proposing to combine a powerful graph based image representation adapted to this specific task and frequent approximate subgraph fas mining algorithms in order to classify images in the case of image representation we are proposing to use more robust descriptors than our previous approach in this topic and we also suggest a criterion to select the isomorphism threshold for the graph mining step this proposal is tested in two well known collections to show the improvement with respect to the previous related works highlightswe propose a new framework for image classification which uses frequent approximate subgraph patterns as features we propose to compute automatically the substitution matrices needed in the process instead of using expert knowledge we propose to use a new graph based image representation we propose a criterion for selecting isomorphism threshold for the graph mining process

the symmetric form of the koekoeks laguerre type differential equation abstract koekoek and koekoek 1991 proved that the laguerre type polynomials ln n x n 0 which are orthogonal on 0 with respect to the measure d n defined on 0 by d n x x e x 1 n x dx satisfy a differential equation l2 4 n y x ny x of order 2 4 when is a nonnegative integer in this paper we announce that the differential expression l2 4 n is symmetrizable on 0 with symmetry factor x e x and explicitly give this formally symmetric expression x e xl2 4 n y

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

constructing finitely presented simple groups that contain grigorchuk groups abstract we construct infinite finitely presented simple groups that have subgroups isomorphic to grigorchuk groups we also prove that up to one possible exception all previously known finitely presented simple groups are torsion locally finite

the bc1 quantum elliptic model algebraic forms hidden algebra sl 2 polynomial eigenfunctions the potential of the bc1 quantum elliptic model is a superposition of two weierstrass functions with a doubling of both periods two coupling constants the bc1 elliptic model degenerates to an a1 elliptic model characterized by the lame hamiltonian it is shown that in the space of the bc1 elliptic invariant the potential becomes a rational function while the flat space metric becomes a polynomial the model possesses the hidden sl 2 algebra for arbitrary coupling constants it is equivalent to the sl 2 quantum top in three different magnetic fields it is shown that three one parametric families of coupling constants exist for which a finite number of polynomial eigenfunctions up to a factor occur

type b 3 fold supersymmetry and non polynomial invariant subspaces we obtain the most general type b 3 fold supersymmetry by solving directly the intertwining relation we then show that it is a necessary and sufficient condition for a second order linear differential operator to have three linearly independent local analytic solutions we find that there are eight linearly independent non trivial linear differential operators of this kind as a by product we find new quasi solvable second order operators preserving a monomial or polynomial subspace one in type b two in type c and four in type x2 all of which have been missed in the existing literature in addition we show that type a type b and type c 3 fold supersymmetries are connected continuously via one parameter a few new quasi solvable models are also presented

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

nonlinear pseudo supersymmetry in the framework of fold supersymmetry we recall the importance of recognizing the different mathematical nature of various concepts relating to symmetric quantum theories after clarifying the relation between supersymmetry and pseudo supersymmetry we prove generically that nonlinear pseudo supersymmetry recently proposed by sinha and roy is just a special case of fold supersymmetry in particular we show that all the models constructed by these authors have type a two fold supersymmetry furthermore we prove that an arbitrary one body quantum hamiltonian which admits two local solutions in closed form belongs to type a two fold supersymmetry irrespective of whether or not it is hermitian symmetric pseudo hermitian and so on

quasi exactly solvable spin 1 2 schrodinger operators the algebraic structures underlying quasi exact solvability for spin 1 2 hamiltonians in one dimension are studied in detail necessary and sufficient conditions for a matrix second order differential operator preserving a space of wave functions with polynomial components to be equivalent to a schrodinger operator are found systematic simplifications of these conditions are analyzed and are then applied to the construction of new examples of multi parameter qes spin 1 2 hamiltonians in one dimension

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

realizations of the lie superalgebra q 2 and applications the lie superalgebra q 2 and its class of irreducible representations vp of dimension 2p p being a positive integer are considered the action of the q 2 generators on a basis of vp is given explicitly and from here two realizations of q 2 are determined the q 2 generators are realized as differential operators in one variable x and the basis vectors of vp as 2 arrays of polynomials in x following such realizations it is observed that the hamiltonian of certain physical models can be written in terms of the q 2 generators in particular the models given here as an example are the sphaleron model the moszkowski model and the jaynes cummings model for each of these it is shown how the q 2 realization of the hamiltonian is helpful in determining the spectrum

invariant modules and the reduction of nonlinear partial differential equations to dynamical systems abstract we completely characterize all nonlinear partial differential equations leaving a given finite dimensional vector space of analytic functions invariant existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides a nonlinear counterpart to quasi exactly solvable quantum hamiltonians these results rely on a useful extension of the classical wronskian determinant condition for linear independence of functions in addition new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented

on orthogonal polynomials with respect to a class of differential operators we consider orthogonal polynomials with respect to a linear differential operatorl m k 0m r k z d kdz k where r k k 0 m are complex polynomials such that deg r k

criterion for polynomial solutions to a class of linear differential equations of second order we consider the differential equations y 0 x y s0 x y where 0 x s0 x are c functions we prove i if the differential equation has a polynomial solution of degree n 0 then n nsn 1 n 1sn 0 where n n 1 sn 1 0 n 1andsn s n 1 s0 k 1 n 1 2 conversely ii if n n 1 0 and n 0 then the differential equation has a polynomial solution of degree at most n we show that the classical differential equations of laguerre hermite legendre jacobi chebyshev first and second kinds gegenbauer and the hypergeometric type etc obey this criterion further we find the polynomial solutions for the generalized hermite laguerre legendre and chebyshev differential equations

design and formal proof of a new optimal image segmentation program with hypermaps this article presents the design of a new functional 2d image segmentation algorithm by cell merging in a subdivision its proof of total correctness and the derivation of an optimal imperative program the planar subdivisions are modeled by hypermaps the formal specifications of hypermaps and segmentation are developed in the calculus of inductive constructions the proofs are assisted by the coq system the final program is written in c

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

representation reduction and solution space contraction in quasi exactly solvable systems in quasi exactly solvable problems partial analytic solutions energy spectrum and associated wavefunctions are obtained if some potential parameters are assigned specific values we introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters to obtain a larger solution space one varies the energy over a discrete set the spectrum by simply changing the value of a given integer a unified treatment that includes the standard as well as the new class of quasi exactly solvable problems is presented and a few examples are given the solution space is spanned by discrete square integrable basis functions in which the matrix representation of the hamiltonian is tridiagonal consequently the wave equation gives a three term recursion relation for the expansion coefficients of the wavefunction imposing quasi exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component the finite is real and exactly solvable whereas the infinite is complex and associated with zero norm states consequently the whole physical space contracts to a finite dimensional subspace with normalizable states

nd generalized map pyramids definition representations and basic operations graph pyramids are often used for representing irregular image pyramids for the 2d case combinatorial pyramids have been recently defined in order to explicitly represent more topological information than graph pyramids the main contribution of this work is the definition of pyramids of n dimensional nd generalized maps this extends the previous works to any dimension and generalizes them in order to represent any type of pyramid constructed by using any removal and or contraction operations we give basic algorithms that allow to build an nd generalized pyramid that describes a multi level segmented image a pyramid of nd generalized maps can be implemented in several ways we propose three possible representations and give conversion algorithms

on one dimensional schrodinger problems allowing polynomial solutions we discuss the explicit construction of the schrodinger equations admitting a representation through some family of general polynomials almost all solvable quantum potentials are shown to be generated by this approach some generalization has also been performed in higher dimensional problems

lame equation sl 2 algebra and isospectral deformations it is proven that the n zone lame equation is equivalent to a spectral problem for a quadratic element of a universal enveloping sl 2 algebra the riemannian surface corresponding to the eigenstates in a parametric space forms the 2n 1 sheet surface which splits in four subsurfaces three of them contain the same number of sheets

group theoretical properties and band structure of the lame hamiltonian we study the group theoretical properties of the lame equation and its relation to su 1 1 and su 2 we compute the band structure dispersion relation and transfer matrix and discuss the dynamical symmetry limits

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

extended jaynes cummings models and quasi exact solvability the original jaynes cummings model is described by a hamiltonian which is hermitian and exactly solvable here we extend this model by several types of interactions leading to a non hermitian operator which does not satisfy the physical condition of spacetime reflection symmetry pt symmetry the new hamiltonians are either exactly solvable admitting an entirely real spectrum or quasi exactly solvable with a real algebraic part of their spectrum

failure of the hill determinant method for the sextic anharmonic oscillator an extended analysis of eigenvalues and wavefunctions resulting from the hill determinant method is carried out for the one dimensional sextic anharmonic oscillator it is shown that in spite of a seemingly natural ansatz of the method irrelevant wavefunctions may appear leading to incorrect eigenvalues the domain of applicability of the method is limited and its practical use demands a priori investigation before concrete calculations of spectral characteristics effective variational calculations based on adequate trial functions are performed and comparison of the results of both approaches is presented

quommutator deformations of spl n 1 we analyze the witten woronowicz s type deformations of the lie superalgebras spl n 1 and obtain deformations parametrized by n n 1 2 independent parameters for n 2 and by three parameters for n 2 for some of these algebras finite dimensional representations can be constructed in terms of finite difference operators providing operators that are relevant for the classification of quasiexactly solvable finite difference equations

fisher information of special functions and second order differential equations we investigate a basic question of analytic information theory namely the evaluation of the fisher information and the relative fisher information with respect to a non negative function for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second order differential equations we obtain explicit expressions for these information theoretic properties via the expectation values of the coefficients of the differential equation we illustrate our approach for various nonrelativistic d dimensional wavefunctions and some special functions of physicomathematical interest emphasis is made in the nikiforov uvarov hypergeometric type functions which include and generalize the hermite functions and the gauss and kummer hypergeometric functions among others

formalizing the trading theorem in coq this paper is the second half of a two part series devoted to the formalization of a combinatorial model of geometric varieties generalized maps we study here how to express continuous notions like homeomorphisms in the combinatorial world in order to constructively describe the well known theorem of surface classification according to genus number of boundaries and orientability coefficient we then use the coq specification elaborated in the previous paper to prove the first half of this theorem called the trading theorem

umbral calculus discretization and quantum mechanics on a lattice umbral calculus deals with representations of the canonical commutation relations we present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of lie algebras on a lattice related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus this observation then suggests various generalizations of these examples a special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail

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

quasi exact solvability of the a _ 2 g _ 2 elliptic model algebraic forms sl 3 g 2 hidden algebra and polynomial eigenfunctions the potential of the a2 quantum elliptic model three body calogero moser elliptic model is defined by the pairwise three body interaction through the weierstrass function and has a single coupling constant a change of variables has been found which are a2 elliptic invariants such that the potential becomes a rational function while the flat space metric as well as its associated vector are polynomials in two variables it is shown that the model possesses the hidden algebra the hamiltonian is an element of the universal enveloping algebra for the arbitrary coupling constant thus it is equivalent to the quantum euler arnold top the integral in a form of the third order differential operator with polynomial coefficients is constructed explicitly being also an element of it is shown that there exists a discrete sequence of the coupling constants for which a finite number of polynomial eigenfunctions up to a non singular gauge factor occurs for these values of the coupling constants there exists a particular integral it commutes with the hamiltonian in action on the space of polynomial eigenfunctions and the hamiltonian is invariant with respect to two dimensional projective transformations it is shown that the a2 model has another hidden algebra introduced in rosenbaum et al 1998 int j mod phys a 13 3885 the potential of the g2 quantum elliptic model three body wolfes elliptic model is defined by the pairwise and three body interactions through the weierstrass function and has two coupling constants a change of variables has been found which are g2 elliptic invariants such that the potential becomes a rational function while the flat space metric as well as its associated vector are polynomials in two variables it is shown the model possesses the hidden algebra it is shown that there exists a discrete family of the coupling constants for which a finite number of polynomial eigenfunctions up to a non singular gauge factor occurs for these values of the coupling constants there exists a particular integral

new classes of quasi solvable potentials their exactly solvable limit and related orthogonal polynomials we have generated using an sl 2 r lie algebraic formalism several new classes of quasi solvable elliptic potentials which in the appropriate limit go over to the exactly solvable forms we have obtained exact solutions of the corresponding spectral problem for some real values of the potential parameters we have also given explicit expressions of the families of associated orthogonal polynomials in the energy variable

associated lame and various other new classes of elliptic potentials from sl 2 r and related orthogonal polynomials using representations of sl 2 r generators which yield associated lame hamiltonians we obtain new classes of elliptic potentials we explicitly calculate eigenstates and spectra for these potentials and construct the associated orthogonal polynomials we show that in the proper limit these potentials reduce to well known exactly solvable potentials

hierarchical image analysis using irregular tessellations a novel multiresolution image analysis technique based on hierarchies of irregular tessellations generated in parallel by independent stochastic processes is presented like traditional image pyramids these hierarchies are constructed in a number of steps on the order of log image size steps however the structure of a hierarchy is adapted to the image content and artifacts of rigid resolution reduction are avoided two applications of these techniques are presented connected component analysis of labeled images and segmentation of gray level images in labeled images every connected component is reduced to a separate root with the adjacency relations among the components also extracted in gray level images the output is a segmentation of the image into a small number of classes as well as the adjacency graph of the classes

quasi exactly solvable lie superalgebras of differential operators in this paper we study lie superalgebras of matrix valued first order differential operators on the complex line we first completely classify all such superalgebras of finite dimension among the finite dimensional superalgebras whose odd subspace is non trivial we find those admitting a finite dimensional invariant module of smooth vector valued functions and classify all the resulting finite dimensional modules the latter lie superalgebras and their modules are the building blocks in the construction of quasi exactly solvable quantum mechanical models for spin particles in one dimension

quasi exact solvability resonances and trivial monodromy in ordinary differential equations a correspondence between the sextic anharmonic oscillator and a pair of third order ordinary differential equations is used to investigate the phenomenon of quasi exact solvability for eigenvalue problems involving differential operators with order greater than 2 in particular links with bender dunne polynomials and resonances between independent solutions are observed for certain second order cases and extended to the higher order problems this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

the embedding of certain linear and abelian groups in finitely presented simple groups in 2 g higman showed than any countable locally finite group can be embedded in a finitely presented simple group in this paper it is shown that any linear group gl n z over the integers and any countable abelian group whose torsion factor group is a subgroup of an x module oieimi where i is a countable set m is an infinite cyclic module and x is the subset of rationals whose denominators are powers of the integer p can be embedded in a finitely presented simple group i am grateful to j s wilson who pointed out to me that it is not necessary to assume that the torsion free x module is finitely generated both of the above results are proved using the construction procedure described in 3 the first result also shows that some new finitely presented simple groups can be constructed using the procedures in 3 for gl 3 z cannot be embedded in any of the previously known finitely presented simple groups see 2 the definitions techniques and notation of 3 will be freely used in this work it will be assumed that the reader is familiar with the first two sections the defining relations a b c and d and the fourth section of 3 but a knowledge of the rest of 3 is not essential it will be useful to have recalled parts of the basic construction that were only mentioned in 3 but will be used heavily in what follows from a homomorphism 0 from a free group f to a wreath product f wr s we obtain the following diagram see 3

new quasi exactly solvable periodic potentials we present two new classes of quasi exactly solvable periodic potentials it is shown that their solutions are given in terms of a known special function when the parameters in the two periodic potentials satisfy a certain condition some of the bloch waves and associated energies can be found exactly and in closed form at either the center or the edge of the first brillouin zone two fundamental sets of solutions have been constructed to calculate all the bloch waves and energy bands

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

quasiexactly solvable 2 2 matrix equations the conditions under which systems of two differential eigenvalue equations are quasiexactly solvable are investigated these systems reveal a rich set of algebraic structures some of them are explicitly described an example of a quasiexactly solvable system which provides a direct counterpart of the lame equation is studied

quasi exactly solvable quartic bose hamiltonians we consider hamiltonians which are even polynomials of the fourth order with respect to bose operators we find subspaces preserved by the action of the hamiltonian these subspaces being finite dimensional include nonetheless states with an infinite number of quasi particles corresponding to the original bose operators the basis functions look rather simple in the coherent state representation and are expressed in terms of the degenerate hypergeometric function with respect to the complex variable labelling the representation in some particular degenerate cases they turn up to the power factor into trigonometric or hyperbolic functions bessel functions or combinations of the exponent and hermite polynomials we find explicitly the relationship between coefficients at different powers of bose operators that ensure quasi exact solvability of hamiltonians

exact results for the potentials v ax2s bx4s 2 the condition for a double well potential v ax2s bx4s 2 s 1 2 3 to have a negative energy branch is derived for an arbitrary geometry of the well the total number of negative levels is determined some general features of the spectrum are pointed out the results are based on a sequence of exactly obtainable zero energy solutions

on some solutions to generalized spheroidal wave equations and applications expansions in series of coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations gswes are analysed and written together in pairs each pair consists of a solution in series of hypergeometric functions and another in series of coulomb wavefunctions and has the same recurrence relations for the series coefficients but the solutions inside it present different radii of convergence expansions without a phase parameter are derived by truncating the series with a phase parameter for the whittaker hill equation solutions are found by treating that equation as a particular case of gswe while for the confluent gswe solutions with and without a phase parameter are given as pairs of series of coulomb wavefunctions amongst the applications there are equations for the time dependence of dirac test fields in some nonflat friedmann robertson walker spacetimes the radial schrodinger equation for an electron in the field of two coulombian centres and the schrodinger equation for the razavy type quasi exactly solvable potentials for these problems it is possible to find wavefunctions in terms of infinite series regular and convergent over the entire range of the independent variable by matching expansions belonging to one or more of the above pairs the infinite series solutions for the razavy type potentials in addition to the polynomial ones suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi classical methods of approximation

polyhedra genus theorem and euler formula a hypermap formalized intuitionistic proof this article presents formalized intuitionistic proofs for the polyhedra genus theorem the euler formula and a sufficient condition of planarity they are based on a hypermap model for polyhedra and on formal specifications in the calculus of inductive constructions first a type of free maps is inductively defined from three atomic constructors next a hierarchy of types defined by invariants with operations constrained by preconditions is built on the free maps hypermaps orientated combinatorial maps and a central notion of quasi hypermaps besides the proofs of their properties are established until the genus theorem and the euler formula mainly using a simple induction principle based on the free map term algebra finally a constructive sufficient condition for polyhedra to be planar is set and proved the whole process is assisted by the interactive coq proof system

hidden algebras of the super calogero and sutherland models we propose to parametrize the configuration space of one dimensional quantum systems of n identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates it is shown that in this parametrization the hamiltonians of the an bcn bn cn and dn calogero and sutherland models as well as their supersymmetric generalizations can be expressed for arbitrary values of the coupling constants as quadratic polynomials in the generators of a borel subalgebra of the lie algebra gl n 1 or the lie superalgebra gl n 1 n for the supersymmetric case these algebras are realized by first order differential operators this fact establishes the exact solvability of the models according to the general definition given by turbiner and implies that the calogero and jack sutherland polynomials as well as their supersymmetric generalizations are related to finite dimensional irreducible representations of the lie algebra gl n 1 and the lie superalgebra gl n 1 n

heun equations and quasi exact solubility we consider the heun equations in the context of the quasi exactly solvable spectral problems and establish the conditions for this class of equations to admit algebraic solutions the schrodinger operators that can be associated with heun equations are constructed explicitly in some cases we present their supersymmetric partners we study a heun like system of equations which is relevant for the classical solution of the two dimensional abelian higgs model

the relation between polynomial deformations of sl 2 r and quasi exact solvability a general method based on the polynomial deformations of the lie algebra sl 2 r is proposed in order to exhibit the quasi exact solvability of specific hamiltonians implied by quantum physical models this method using the finite dimensional representations and differential realizations of such deformations is illustrated on the sextic oscillator as well as on second harmonic generation

polynomial families and schrodinger equation one example for nonhypergeometric type of correspondence we discuss the explicit construction of the schrodinger equations admitting representation through some family of general nonorthogonal polynomials the specific choice of the third order polynomial coefficient functions that lead to quasi solvable families of schrodinger potentials is considered in detail

tridiagonalization and the heun equation it is shown that the tridiagonalization of the hypergeometric operator l yields the generic heun operator m the algebra generated by the operators l m and z l m is quadratic and a one parameter generalization of the racah algebra the racah heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators l and m an interpretation in terms of the racah problem for su 1 1 and separation of variables in a superintegrable system are discussed

an median graphs properties algorithms and applications in object prototype learning and similar tasks median computation is an important technique for capturing the essential information of a given set of patterns we extend the median concept to the domain of graphs in terms of graph distance we introduce the novel concepts of set median and generalized median of a set of graphs we study properties of both types of median graphs for the more complex task of computing generalized median graphs a genetic search algorithm is developed experiments conducted on randomly generated graphs demonstrate the advantage of generalized median graphs compared to set median graphs and the ability of our genetic algorithm to find approximate generalized median graphs in reasonable time application examples with both synthetic and nonsynthetic data are shown to illustrate the practical usefulness of the concept of median graphs

maximal dimension of invariant subspaces admitted by nonlinear vector differential operators in this paper the dimension of invariant subspaces admitted by m component nonlinear vector differential operators is estimated it is shown that if the m component nonlinear vector differential operators of order k preserves the invariant subspace wn11 wnmm where wnqq is the space generated by solutions of linear ordinary differential equations of order nq q 1 m then max n1 nm 2mk 1 to illustrate the approach examples of nonlinear vector differential operators admitting invariant subspace wn11 wn22 with max n1 n2 9 and three component nonlinear vector differential operators admitting invariant subspace wn11 wn22 wn33 with max n1 n2 n3 13 are presented

quasi exactly solvable models in nonlinear optics we study a large class of models with an arbitrary finite number of degrees of freedom described by hamiltonians which are polynomial in bosonic creation and annihilation operators and including as particular cases nth harmonic generation and photon cascades for each model we construct a complete set of commuting integrals of motion of the hamiltonian fully characterize the common eigenspaces of the integrals of motion and show that the action of the hamiltonian on these common eigenspaces can be represented by a quasi exactly solvable reduced hamiltonian whose expression in terms of the usual generators of 2 is computed explicitly

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

reply to comment on the darboux transformation and algebraic deformations of shape invariant potentials we reply to the comment on our recent paper made by drs sinha and roy we agree that the backward darboux transformation method used in our paper is equivalent to the approach based on ces potentials but we stress that the emphasis and the results of our paper are different

solvability of the hamiltonians related to exceptional root spaces rational case solvability of the rational quantum integrable systems related to exceptional root spaces g2 f4 is re examined and for e6 7 8 is established in the framework of a unified approach it is shown that hamiltonians take algebraic form being written in certain weyl invariant variables it is demonstrated that for each hamiltonian the finite dimensional invariant subspaces are made from polynomials and they form an infinite flag a notion of minimal flag is introduced and minimal flag for each hamiltonian is found corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants the hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite dimensional but finitely generated lie algebra of differential operators taken in a finite dimensional representation