krein adler transformations for shape invariant potentials and pseudo virtual states for 11 examples of one dimensional quantum mechanics with shape invariant potentials the darboux crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to krein adler transformations deleting multiple eigenstates with shifted parameters these are based upon infinitely many polynomial wronskian identities of classical orthogonal polynomials i e the hermite laguerre and jacobi polynomials which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape invariant potentials

multi indexed q racah polynomials as the second stage of the project multi indexed orthogonal polynomials we present in the framework of discrete quantum mechanics with real shifts in one dimension the multi indexed q racah polynomials they are obtained from the q racah polynomials by the multiple application of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state vectors in a similar way to the multi indexed laguerre and jacobi polynomials reported earlier the virtual state vectors are the solutions of the matrix schrodinger equation with negative eigenvalues except for one of the two boundary points

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

recurrence relations of the multi indexed orthogonal polynomials v racah and q racah types in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson and askey wilson types in this paper we explore those of the racah and q racah types for the m indexed q racah polynomials we derive 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi indexed q racah polynomials are obtained in appendix b and appendix c we present a proof and some data of the recurrence relations with constant coefficients for the multi indexed wilson and askey wilson polynomials in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson and askey wilson types in this paper we explore those of the racah and q racah types for the m indexed q racah polynomials we derive 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi indexed q racah polynomials are obtained in appendix b and appendix c we present a proof and some data of the recurrence relations with constant coefficients for the multi indexed wilson and askey wilson polynomials

a new recurrence formula for generic exceptional orthogonal polynomials a new recurrence relation for exceptional orthogonal polynomials is proposed which holds for types 1 2 and 3 to provide concrete examples the recurrence relations are then given for xj hermite laguerre and jacobi polynomials in the j 1 2 cases

cherednik hecke and quantum algebras as free frobenius and calabi yau extensions we show how the existence of a pbw basis and a large enough central subalgebra can be used to deduce that an algebra is frobenius we apply this to rational cherednik algebras hecke algebras quantised universal enveloping algebras quantum borels and quantised function algebras in particular we give a positive answer to r rouquier representations of rational cherednik algebras in infinite dimensional aspects of representation theory and applications amer math soc 2005 pp 103 131 stating that the restricted rational cherednik algebra at the value t 0 is symmetric

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

ring theoretical properties of affine cellular algebras abstract as a generalisation of graham and lehrer s cellular algebras affine cellular algebras have been introduced in 12 in order to treat affine hecke algebras of type a and affine versions of diagram algebras like affine temperley lieb algebras in a unifying fashion affine cellular algebras include kleshchev s graded quasihereditary algebras khovanov lauda rouquier algebras and various other classes of algebras in this paper we will study ring theoretical properties of affine cellular algebras we show that any affine cellular algebra a satisfies a polynomial identity furthermore we show that a can be embedded into its asymptotic algebra if the occurring commutative affine k algebras b j are reduced and the determinants of the swich matrices are non zero divisors as a consequence we show that the gelfand kirillov dimension of a is less than or equal to the largest krull dimension of the algebras b j and that equality holds in case all affine cell ideals are idempotent or if the krull dimension of the algebras b j is less than or equal to 1 special emphasis is given to the question when an affine cell ideal is idempotent generated by an idempotent or finitely generated

exceptional laguerre and jacobi polynomials and the corresponding potentials through darboux crum transformations a simple derivation is presented of the four families of infinitely many shapeinvariant hamiltonians corresponding to the exceptional laguerre and jacobi polynomials the darboux crum transformations are applied to connect the well known shape invariant hamiltonians of the radial oscillator and the darboux p oschl teller potential to the shape invariant potentials of odake sasaki dutta and roy derived the two lowest members of the exceptional laguerre polynomials by this method the method is expanded to its full generality and many other ramifications including the aspects of the generalized bochner problem and the bispectral property of the exceptional orthogonal polynomials are discussed

multi indexed wilson and askey wilson polynomials as the third stage of the project multi indexed orthogonal polynomials we present in the framework of discrete quantum mechanics with pure imaginary shifts in one dimension the multi indexed wilson and askey wilson polynomials they are obtained from the original wilson and askey wilson polynomials by multiple applications of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state solutions of types i and ii in a similar way to the multi indexed laguerre jacobi and q racah polynomials reported earlier

affine cellularity of bln algebras abstract we show that the bln algebra which was introduced by mcgerty is affine cellular in the sense of koenig and xi in fact we establish a stronger property namely that the affine cell ideals are generated by idempotents this particularly implies that the global dimension of the bln algebra is finite for affine type a we obtain that the affine q schur algebra u d n n when d n is affine cellular and has finite global dimension

recurrence relations of the multi indexed orthogonal polynomials in paper ii we presented conjectures of the recurrence relations with constant coefficients for the multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types in this paper we present a proof for the laguerre and jacobi cases their bispectral properties are also discussed which gives a method to obtain the coefficients of the recurrence relations explicitly this paper extends to the laguerre and jacobi cases the bispectral techniques recently introduced by gomez ullate et al j approx theory 204 1 2016 e print arxiv 1506 03651 math ca to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of hermite type

affine cellular algebras abstract graham and lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras many classes of finite dimensional algebras including various hecke algebras and diagram algebras have been shown to be cellular and the theory due to graham and lehrer successfully has been applied to these algebras we will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field affine hecke algebras of type a and infinite dimensional diagram algebras like the affine temperley lieb algebras are shown to be examples of our definition the isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties in this way representation theory of non commutative algebras is linked with commutative algebra moreover conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification these conditions are shown to be satisfied for the affine hecke algebra of type a if the quantum parameter is not a root of the poincare polynomial

random matrix derived shrinkage of spectral precision matrices there has been much research on shrinkage methods for real valued covariance matrices and their inverses precision matrices in spectral analysis of p vector valued time series complex valued spectral matrices and precision matrices arise and good shrinkage methods are often required most notably when the estimated complex valued spectral matrix is singular as an improvement on the ledoit wolf lw type of spectral matrix estimator we use random matrix theory to derive a rao blackwell estimator for a spectral matrix its inverse being a rao blackwellized estimator for the spectral precision matrix a random matrix method has previously been proposed for complex valued precision matrices it was implemented by very costly simulations we formulate a fast completely analytic approach moreover we derive a way of selecting an important parameter using predictive risk methodology we show that both the rao blackwell estimator and the random matrix estimator of the precision matrix can substantially outperform the inverse of the lw estimator in a time series setting our new methodology is applied to eeg derived time series data where it is seen to work well and deliver substantial improvements for precision matrix estimation

recurrence relations of the multi indexed orthogonal polynomials iv closure relations and creation annihilation operators we consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types corresponding to the recurrence relations with constant coefficients for the m indexed orthogonal polynomials it is expected that the systems satisfy the generalized closure relations in fact we can verify this statement for small m examples the generalized closure relation gives the exact heisenberg operator solution of a certain operator from which the creation and annihilation operators of the system are obtained

rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials we prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free and therefore can be obtained by applying a finite number of state deleting darboux transformations on the harmonic oscillator equivalently every exceptional orthogonal polynomial system of hermite type can be obtained by applying a darboux crum transformation to the classical hermite polynomials exceptional hermite polynomial systems only exist for even codimension 2m and they are indexed by the partitions of m we provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness exceptional hermite polynomials satisfy a 2l 3 recurrence relation where l is the length of the partition explicit expressions for such recurrence relations are given

affine cellularity of affine hecke algebras of rank two we show that affine hecke algebras of rank two with generic parameters are affine cellular in the sense of koenig xi

multi indexed meixner and little q jacobi laguerre polynomials as the fourth stage of the project multi indexed orthogonal polynomials we present the multi indexed meixner and little q jacobi laguerre polynomials in the framework of discrete quantum mechanics with real shifts defined on the semi infinite lattice in one dimension they are obtained in a similar way to the multi indexed laguerre and jacobi polynomials reported earlier from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state vectors the virtual state vectors are the solutions of the matrix schrodinger equation on all the lattice points having negative energies and infinite norm this is in good contrast to the q racah systems defined on a finite lattice in which the virtual state vectors satisfy the matrix schrodinger equation except for one of the two boundary points

solvable discrete quantum mechanics q orthogonal polynomials with q 1 and quantum dilogarithm several kinds of q orthogonal polynomials with q 1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems their orthogonality weight functions consist of quantum dilogarithm functions which are a natural extension of the euler gamma functions and the q gamma functions q shifted factorials the dimensions of the orthogonal spaces are finite these q orthogonal polynomials are expressed in terms of the askey wilson polynomials and their certain limit forms

unified theory of exactly and quasiexactly solvable discrete quantum mechanics i formalism we present a simple recipe to construct exactly and quasiexactly solvable hamiltonians in one dimensional discrete quantum mechanics in which the schrodinger equation is a difference equation it reproduces all the known ones whose eigenfunctions consist of the askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable the recipe also predicts several new ones an essential role is played by the sinusoidal coordinate which generates the closure relation and the askey wilson algebra together with the hamiltonian the relationship between the closure relation and the askey wilson algebra is clarified

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences

exceptional orthogonal polynomials exactly solvable potentials and supersymmetry we construct two new exactly solvable potentials giving rise to bound state solutions to the schrodinger equation which can be written in terms of the recently introduced laguerre or jacobi type x1 exceptional orthogonal polynomials these potentials extending either the radial oscillator or the scarf i potential by the addition of some rational terms turn out to be translationally shape invariant as their standard counterparts and isospectral to them