large 2 groups of automorphisms of algebraic curves over a field of characteristic 2 abstract let s be a 2 subgroup of the k automorphism group aut x of an algebraic curve x of genus g x defined over an algebraically closed field k of characteristic 2 it is known that s may be quite large compared to the classical hurwitz bound 84 g x 1 however if s fixes no point then the size of s is smaller than or equal to 4 g x 1 in this paper we investigate algebraic curves x with a 2 subgroup s of aut x having the following properties i s 8 and s 2 g x 1 ii s fixes no point on x theorem 1 2 shows that x is a general curve and that either s 4 g x 1 or s 2 g x 2 or for every involution u z s the quotient curve x u inherits the above properties that is it has genus 2 and its automorphism group s u still has properties i and ii in the first two cases s is completely determined we also give examples illustrating our results in particular for every g 2 h 1 9 we exhibit a general bielliptic curve x of genus g whose k automorphism group has a dihedral 2 subgroup s of order 4 g 1 that fixes no point in x

a laplace dunkl equation on s 2 and the bannai ito algebra the dirac dunkl operator on the two sphere associated to the mathbb z 2 3 reflection group is considered its symmetries are found and are shown to generate the bannai ito algebra representations of the bannai ito algebra are constructed using ladder operators eigenfunctions of the spherical dirac dunkl operator are obtained using a cauchy kovalevskaia extension theorem these eigenfunctions which correspond to dunkl monogenics are seen to support finite dimensional irreducible representations of the bannai ito algebra

the heun racah and heun bannai ito algebras the heun racah and heun bannai ito algebras are introduced specializations of these algebras are seen to be realized by the operators obtained by applying the algebraic heun construct to the bispectral operators of the racah and bannai ito polynomials the study supplements the results on the heun askey wilson algebra and completes the description of the heun algebras associated with the polynomial families at the top of the askey scheme its q analog and the bannai ito one

regular sparse crossbar concentrators a bipartite concentrator is a single stage sparse crossbar switching device that can connect any m of its n spl ges m inputs to its m outputs possibly without the ability to distinguish their order fat and slim crossbars were introduced recently to show that bipartite concentrators can be constructed with a minimum number of crosspoints for any number of inputs and outputs we generalize these graphs to obtain bipartite concentrators with nearly a fixed fanout without altering their n m 1 m crosspoint complexity we also present an o log n time algorithm to route arbitrary concentration assignments on this new family of fat and slim crossbars

a missing family of classical orthogonal polynomials we study a family of classical orthogonal polynomials which satisfy apart from a three term recurrence relation an eigenvalue problem with a differential operator of dunkl type these polynomials can be obtained from the little q jacobi polynomials in the limit q 1 we also show that these polynomials provide a nontrivial realization of the askey wilson algebra for q 1

convolution identities for dunkl orthogonal polynomials from the osp 1 2 lie superalgebra new convolution identities for orthogonal polynomials belonging to the q 1 analog of the askey scheme are obtained specialization of the chihara polynomials will play a central role as the eigenfunctions of a special element of the lie superalgebra osp 1 2 in the positive discrete series representation using the clebsch gordan coefficients a convolution identity for the specialized chihara the dual 1 hahn and the big 1 jacobi polynomials are found using the racah coefficients a convolution identity for the big 1 jacobi and the bannai ito polynomials is found finally these results are applied to construct a bilinear generating function for the big 1 jacobi polynomials new convolution identities for orthogonal polynomials belonging to the q 1 analog of the askey scheme are obtained specialization of the chihara polynomials will play a central role as the eigenfunctions of a special element of the lie superalgebra osp 1 2 in the positive discrete series representation using the clebsch gordan coefficients a convolution identity for the specialized chihara the dual 1 hahn and the big 1 jacobi polynomials are found using the racah coefficients a convolution identity for the big 1 jacobi and the bannai ito polynomials is found finally these results are applied to construct a bilinear generating function for the big 1 jacobi polynomials

some 4 point hurwitz numbers in positive characteristic author s bouw irene i osserman brian abstract in this paper we compute the number of covers of curves with given branch behavior in characteristic p for one class of examples with four branch points and degree p our techniques involve related computations in the case of three branch points and allow us to conclude in many cases that for a particular choice of degeneration all the covers we consider degenerate to separable admissible covers starting from a good understanding of the complex case the proof is centered on the theory of stable reduction of galois covers

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

the hahn superalgebra and supersymmetric dunkl oscillator models a supersymmetric extension of the hahn algebra is introduced this quadratic superalgebra which we call the hahn superalgebra is constructed using the realization provided by the dunkl oscillator model in the plane whose hamiltonian involves reflection operators in this realization the reflections act as grading operators and the odd generators are part of the schwinger dunkl algebra which is a two parameter extension of the bosonic construction the even part of the algebra is built from bilinears in the odd generators and satisfies the hahn algebra supplemented with involutions a family of supersymmetric dunkl oscillator models in n dimensions is also considered the hamiltonians of these supersymmetric models differ from ordinary dunkl oscillators by pure reflection terms in two dimensions the supersymmetric dunkl oscillator is seen to have the even part of the hahn superalgebra as the invariance algebra

the ramification sequence for a fixed point of an automorphism of a curve and the weierstrass gap sequence for nonsingular projective curves defined over algebraically closed fields of positive characteristic the dependence of the ramification filtration of decomposition groups of the automorphism group with weierstrass semigroups attached at wild ramification points is studied a faithful representation of the p part of the decomposition group at each wild ramified point to a riemann roch space is defined

the algebra of dual 1 hahn polynomials and the clebsch gordan problem of sl 1 2 the algebra h of the dual 1 hahn polynomials is derived and shown to arise in the clebsch gordan problem of sl 1 2 the dual 1 hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q 1 limit of the dual q hahn polynomials the hopf algebra sl 1 2 has four generators including an involution it is also a q 1 limit of the quantum algebra slq 2 and furthermore the dynamical algebra of the parabose oscillator the algebra h a two parameter generalization of u 2 with an involution as additional generator is first derived from the recurrence relation of the 1 hahn polynomials it is then shown that h can be realized in terms of the generators of two added sl 1 2 algebras so that the clebsch gordan coefficients of sl 1 2 are dual 1 hahn polynomials an irreducible representation of h involving five diagonal matrices and connected to the difference equation of the dual 1 hahn polynomials is constructed

a superintegrable model with reflections on s n 1 and the higher rank bannai ito algebra a quantum superintegrable model with reflections on the n 1 sphere is presented its symmetry algebra is identified with the higher rank generalization of the bannai ito algebra it is shown that the hamiltonian of the system can be constructed from the tensor product of n representations of the superalgebra osp 1 2 and that the superintegrability is naturally understood in that setting the separated solutions are obtained through the fischer decomposition and a cauchy kovalevskaia extension theorem

a higher rank racah algebra and the mathbb z _2 n laplace dunkl operator a higher rank generalization of the rank one racah algebra is obtained as the symmetry algebra of the laplace dunkl operator associated to the mathbb z 2 n root system this algebra is also the invariance algebra of the generic superintegrable model on the n sphere bases of dunkl harmonics are constructed explicitly using a cauchy kovalevskaia theorem these bases consist of joint eigenfunctions of labelling abelian subalgebras of the higher rank racah algebra a method to obtain expressions for both the connection coefficients between these bases and the action of the symmetries on these bases is presented

automorphisms of harbater katz gabber curves let k be a perfect field of characteristic p 0 and let g be a finite group we consider the pointed g curves over k associated by harbater katz and gabber to faithful actions of g on k t over k we use such hkg g curves to classify the automorphisms of k t of p power order that can be expressed by particularly explicit formulas namely those mapping t to a power series lying in a mathbb z p mathbb z artin schreier extension of k t in addition we give necessary and sufficient criteria to decide when an hkg g curve with an action of a larger finite group j is also an hkg j curve

relevements des revetements de courbes faiblement ramifies let x be a smooth projective curve over a perfect field of characteristic p 0 and g a finite group of automorphism of x let x g be the characteristic of the versal equivariant deformation ring r x g of x g when the ramification is weak i e all second ramification groups are trivial we prove that x g 0 p and we compute r x g

eigenproblems for tridiagonal 2 toeplitz matrices and quadratic polynomial mappings abstract given a system of monic orthogonal polynomials mops p n x n 0 we characterize all the sequences of monic orthogonal polynomials q n x n 0 such that q 1 x x b q 2 n x p n 2 x n 0 1 2 where 2 is a fixed polynomial of degree exactly 2 and b is a fixed complex number with an appropriate choice of the mops p n x n 0 our results enables us to solve the eigenproblem of a tridiagonal 2 toeplitz matrix giving an alternative proof to a recent result by m j c gover we also find the relations between the jacobi matrices corresponding to the mops p n x n 0 and q n x n 0 finally we show that if p n x n 0 is a semiclassical orthogonal polynomial sequence then so is q n x n 0 and in particular we analyze the classical case in detail

bivariate bannai ito polynomials a two variable extension of bannai ito polynomials is presented they are obtained via q 1 limits of the bivariate q racah and askey wilson orthogonal polynomials introduced by gasper and rahman their orthogonality relation is obtained these new polynomials are also shown to be multispectral two dunkl shift operators are seen to be diagonalized by the bivariate bannai ito polynomials and 3 and 9 term recurrence relations are provided a two variable extension of bannai ito polynomials is presented they are obtained via q 1 limits of the bivariate q racah and askey wilson orthogonal polynomials introduced by gasper and rahman their orthogonality relation is obtained these new polynomials are also shown to be multispectral two dunkl shift operators are seen to be diagonalized by the bivariate bannai ito polynomials and 3 and 9 term recurrence relations are provided

the q onsager algebra and multivariable q special functions two sets of mutually commuting q difference operators x i and y j i j 1 n such that x i and y i generate a homomorphic image of the q onsager algebra for each i are introduced the common polynomial eigenfunctions of each set are found to be entangled product of elementary pochhammer functions in n variables and n 3 parameters under certain conditions on the parameters they form two dual bases of polynomials in n variables the action of each operator with respect to its dual basis is block tridiagonal the overlap coefficients between the two dual bases are expressed as entangled products of q racah polynomials and satisfy an orthogonality relation the overlap coefficients between either one of these bases and the multivariable monomial basis are also considered one obtains in this case entangled products of dual q krawtchouk polynomials finally the split basis in which the two families of operators act as block bidiagonal matrices is also provided

cyclic extensions and the local lifting problem the local oort conjecture states that if is cyclic and k is an algebraically closed eld of characteristic p then all extensions of k t should lift to characteristic zero we prove a critical case of this conjecture in particular we show that the conjecture is always true when vp j j 3 and is true for arbitrarily highly p divisible cyclic groups when a certain condition on the higher ramication ltration is satised

large p groups of automorphisms of algebraic curves in characteristic p abstract let s be a p subgroup of the k automorphism group aut x of an algebraic curve x of genus g 2 and p rank defined over an algebraically closed field k of characteristic p 3 nakajima 27 proved that if 2 then s p p 2 g 1 if equality holds x is a nakajima extremal curve we prove that if s p 2 p 2 p 1 g 1 then one of the following cases occurs i 0 and the extension k x k x s completely ramifies at a unique place and does not ramify elsewhere ii s p and x is an ordinary curve of genus g p 1 iii x is an ordinary nakajima extremal curve and k x is an unramified galois extension of a function field of a curve given in ii iii x is an ordinary nakajima extremal curve and k x is an unramified galois extension of a function field of a curve given in ii there are exactly p 1 subgroups m of s such that k x k x m is such a galois extension moreover if some of them is an abelian extension then s has maximal nilpotency class the full k automorphism group of any nakajima extremal curve is determined and several infinite families of nakajima extremal curves are constructed by using their pro p fundamental groups

algebraic curves with many automorphisms abstract let x be a projective geometrically irreducible non singular algebraic curve of genus g 2 defined over an algebraically closed field k of odd characteristic p let aut x be the group of all automorphisms of x which fix k element wise it is known that if aut x 8 g 3 then the p rank equivalently the hasse witt invariant of x is zero this raises the problem of determining the minimum value function f g such that whenever aut x f g then x has zero p rank for even g we prove that f g 900 g 2 the odd genus case appears to be much more difficult although for any genus g 2 if aut x has a solvable subgroup g such that g 252 g 2 then x has zero p rank and g fixes a point of x our proofs use the hurwitz genus formula and the deuring shafarevich formula together with a few deep results from group theory characterizing finite simple groups whose sylow 2 subgroups have a cyclic subgroup of index 2 we also point out some connections with the abhyankar conjecture and the katz gabber covers

generalized artin mumford curves over finite fields let fqfq be the finite field of order q phq ph with p 2p 2 prime and h 1h 1 and let fq be a subfield of fqfq from any two q linearized polynomials l1 l2 f q t of degree q we construct an ordinary curve x l1 l2 x l1 l2 of genus g q 1 2g q 1 2 which is a generalized artin schreier cover of the projective line p1p1 the automorphism group of x l1 l2 x l1 l2 over the algebraic closure f q of fqfq contains a semidirect product of an elementary abelian p group of order q2q2 by a cyclic group of order q 1 we show that for l1 l2l1 l2 is the full automorphism group aut x l1 l2 aut x l1 l2 over f q for l1 l2l1 l2 there exists an extra involution and aut x l1 l1 aut x l1 l1 with a dihedral group of order 2 q 1 containing two different choices of the pair l1 l2 l1 l2 may produce birationally isomorphic curves even for l1 l2l1 l2 we prove that any curve of genus q 1 2 q 1 2 whose f q automorphism group contains an elementary abelian subgroup of order q2q2 is birationally equivalent to x l1 l2 x l1 l2 for some separable q linearized polynomials l1 l2l1 l2 of degree q we produce an analogous characterization in the special case l1 l2l1 l2 this extends a result on the artin mumford curves due to arakelian and korchmaros 1

the oort conjecture on lifting covers of curves we show that the conjecture of oort on lifting covers of curves is true the main ingredients in the proof are a deformation argument in characteristic p and a special case of a very recent result by obus wewers a kind of boundedness result is given as well

resolution of singularities and modular galois theory i shall sketch a brief history of the desingularization problem from riemann thru zariski to hironaka including the part i played in it and the work on galois theory which this led me to and how that caused me to search out many group theory gurus i shall also formulate several conjectures and suggest numerous thesis problems

mumford curves and mumford groups in positive characteristic a mumford group is a discontinuous subgroup gamma of pgl 2 k where k denotes a non archimedean valued field such that the quotient by gamma is a curve of genus 0 as abstract group gamma is an amalgam of a finite tree of finite groups for k of positive characteristic the large collection of amalgams having two or three branch points is classified using these data mumford curves with a large group of automorphisms are discovered a long combinatorial proof involving the classification of the finite simple groups is needed for establishing an upper bound for the order of the group of automorphisms of a mumford curve orbifolds in the category of rigid spaces are introduced for the projective line the relations with mumford groups and singular stratified bundles are studied this paper is a sequel to 26 part of it clarifies corrects and extends work of g cornelissen f kato and k kontogeorgis c 2018 elsevier inc all rights reserved

generating functions for the mathfrak osp 1 2 clebsch gordan coefficients generating functions for clebsch gordan coefficients of are derived these coefficients are expressed as limits of the dual q hahn polynomials the generating functions are obtained using two different approaches respectively based on the coherent state representation and the position representation of

the stack of potts curves and its fibre at a prime of wild ramification in this note we study the modular properties of a family of cyclic coverings of p 1 of degree n in all odd characteristics we compute the moduli space of the corresponding algebraic stack over z 1 2 as well as the picard groups over algebraically closed fields we put special emphasis on the study of the fibre of the stack at a prime of wild ramification in particular we show that the moduli space has good reduction at such a prime 2004 elsevier inc all rights reserved

dunkl shift operators and bannai ito polynomials abstract we consider the most general dunkl shift operator l with the following properties i l is of first order in the shift operator and involves reflections ii l preserves the space of polynomials of a given degree iii l is potentially self adjoint we show that under these conditions the operator l has eigenfunctions which coincide with the bannai ito polynomials we construct a polynomial basis which is lower triangular and two diagonal with respect to the action of the operator l this allows to express the bi polynomials explicitly we also present an anti commutator aw 3 algebra corresponding to this operator from the representations of this algebra we derive the structure and recurrence relations of the bi polynomials we introduce new orthogonal polynomials referred to as the complementary bi polynomials as an alternative q 1 limit of the askey wilson polynomials these complementary bi polynomials lead to a new explicit expression for the bi polynomials in terms of the ordinary wilson polynomials

rational spectral transformations and orthogonal polynomials abstract we show that generic rational transformations of the stieltjes function with polynomial coefficients st can be presented as a finite superposition of four fundamental elementary transforms christoffel transform ct geronimus transform gt and forward and backward associated transformations a t a t it is shown that the laguerre hahn polynomials lhp on arbitrary nonuniform lattice are covariant with respect to st i e st of a lhp yields another lhp whereas the semi classical polynomials are covariant with respect to a subclass of linear st some applications of these results to the theory of the semi classical polynomials are considered

artin characters hurwitz trees and the lifting problem we study finite groups of automorphisms of the p adic open disk in particular we generalize results of green matignon and henrio from cyclic groups of order p to arbitrary finite groups as an application we produce a counterexample to a question of chinburg guralnick and harbater concerning the local lifting problem for generalized quaternion groups

galois groups of unramified covers of projective curves in characteristicp abstract we show several results concerning the finite groups that occur as galois groups of unramified covers of projective curves in characteristic p in particular we prove that every finite group with g generators occurs over some curve of genus g this implies for example that every finite simple group occurs in genus 2 by similar methods we obtain several other families of groups which occur in genus 2 in addition we show that if a group g occurs over some curve of genus g then it must occur over almost all curves of genus g or greater the results are obtained using formal patching

algebraic heun operator and band time limiting we introduce the algebraic heun operator associated to any bispectral pair of operators this operator can be presented as a generic bilinear combination of these bispectral operators we show that the resulting operators are natural generalizations of the ordinary heun operator this leads to a simple construction of the operators commuting with the projection operators in problems of band time limiting and it gives a way to adapt a construction first used by perline in a purely finite setup to general situations we also extend this algebraic construction to cover some purely finite cases which lie beyond this scheme in particular for the anti krawtchouk polynomials in the latter case the commuting operator is represented by a pentadiagonal matrix which can be constructed explicitly as a polynomial of degree 4 of the pair of bispectral operators

a combinatorial li yau inequality and rational points on curves we present a method to control gonality of nonarchimedean curves based on graph theory let k denote a complete nonarchimedean valued field we first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps namely one should minimize over all so called finite harmonic graph morphisms to trees that originate from any refinement of the dual graph of the stable model of the curve next comes our main result we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the laplacian and some volume of the original graph this can be seen as a substitute for graphs of the li yau inequality from differential geometry although we also prove that the strict analogue of the original inequality fails for general graphs finally we apply the results to give a lower bound for the gonality of arbitrary drinfeld modular curves over finite fields and for general congruence subgroups vargamma of vargamma 1 that is linear in the index vargamma 1 vargamma with a constant that only depends on the residue field degree and the degree of the chosen infinite place this is a function field analogue of a theorem of abramovich for classical modular curves we present applications to uniform boundedness of torsion of rank two drinfeld modules that improve upon existing results and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of papikian

a new family of exceptional polynomials in characteristic two we produce a new family of polynomials f x over fields k of characteristic 2 which are exceptional in the sense that f x f y has no absolutely irreducible factors in k x y except for scalar multiples of x y when k is finite this condition is equivalent to saying that the map f induces a bijection on an infinite algebraic extension of k our polynomials have degree 2 e 1 2 e 1 where e 1 is odd we also prove that this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic

connectedness of the moduli space of artin schreier curves of fixed genus abstract we study the moduli space as g of artin schreier curves of genus g over an algebraically closed field k of positive characteristic p the moduli space is partitioned into strata which are irreducible each stratum parameterizes artin schreier curves whose ramification divisors have the same coefficients we construct deformations of these curves to study the relations between those strata as an application when p 3 we prove that as g is connected for all g when p 3 it turns out that as g is connected for a sufficiently large value of g in the course of our work we answer a question of pries and zhu about how a combinatorial graph determines the geometry of as g

finite dimensional irreducible modules of the universal askey wilson algebra since the introduction of askey wilson algebras by zhedanov in 1991 the classification of the finite dimensional irreducible modules of askey wilson algebras remains open a universal analog triangle q of the askey wilson algebras was recently studied in this paper we consider a family of infinite dimensional triangle q modules by the universal property of these triangle q modules we classify the finite dimensional irreducible triangle q modules when q is not a root of unity

modular leonard triples abstract let k denote a field and let v denote a vector space over k of finite positive dimension an ordered triple a a a of linear operators on v is said to be a leonard triple whenever for each b a a a there exists a basis of v with respect to which the matrix representing b is diagonal and the matrices representing the other two operators are irreducible tridiagonal a leonard triple a a a is said to be modular whenever for each b a a a there exists an antiautomorphism of end v which fixes b and swaps the other two operators we classify the modular leonard triples up to isomorphism

a limit q 1 for the big q jacobi polynomials we study a new family of classical orthogonal polynomials here called big 1 jacobi polynomials which satisfy apart from a 3 term recur rence relation an eigenvalue problem with differential operators of dunkl type these polynomials can be obtained from the big q jacobi polynomials in the limit q 1 an explicit expression of these polynomials in terms of gauss hypergeometric functions is found the big 1 jacobi polynomials are orthogo nal on the union of two symmetric intervals of the real axis we show that the big 1 jacobi polynomials can be obtained from the bannai ito polynomials when the orthogonality support is extended to an infinite number of points we further indicate that these polynomials provide a nontrivial realization of the askey wilson algebra for q 1 the novelty lies in the fact that l is a differential difference operator of special type namely l is a linear operator which is of first order in the derivative operator x and contains also the reflection operator r which acts as rf x f x roughly speaking one can say that l belongs to the class of dunkl operators 8 which contain both the operators x and r nevertheless the operator l differs from the standard dunkl operators in a fundamental way indeed l preserves the linear space of polynomials of any given maximal degree this basic property allows to construct a complete system of polynomials pn x n 0 1 2 as eigenfunctions of the operator l guided by the q 1 limit of the little q jacobi polynomials we derived in 20 an explicit expression of the polynomials pn x in terms of gauss hypergeometric functions we also found explicitly the recurrence coefficients and s that the polynomials pn x are orthogonal on the interval 1 1 with a weight function related to the weight function of the generalized jacobi polynomials 7 we also proved that they admit the dunkl classical property 5 and further demonstrated that the operator l together with the multiplication operator x form a special case of the askey wilson algebra aw 3 21 corresponding to the parameter q 1

global oort groups abstract we study the oort groups for a prime p i e finite groups g such that every g galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a g cover of curves in characteristic 0 we prove that all oort groups lie in a particular class of finite groups that we characterize with equality of classes under a conjecture about local liftings we prove this equality unconditionally if the order of g is not divisible by 2 p 2 we also treat the local lifting problem and relate it to the global problem

darboux transformation and perturbation of linear functionals abstract let l be a quasi definite linear functional defined on the linear space of polynomials with real coefficients in the literature three canonical transformations of this functional are studied xl l c x and 1 x l c x where x denotes the linear functional x xk k 0 and k 0 is the kronecker symbol let us consider the sequence of monic polynomials orthogonal with respect to l this sequence satisfies a three term recurrence relation whose coefficients are the entries of the so called monic jacobi matrix in this paper we show how to find the monic jacobi matrix associated with the three canonical perturbations in terms of the monic jacobi matrix associated with l the main tools are darboux transformations in the case that the lu factorization of the monic jacobi matrix associated with xl does not exist and darboux transformation does not work we show how to obtain the monic jacobi matrix associated with x 2 l as a limit case we also study perturbations of the functional l that are obtained by combining the canonical cases finally we present explicit algebraic relations between the polynomials orthogonal with respect to l and orthogonal with respect to the perturbed functionals

the dunkl oscillator in the plane ii representations of the symmetry algebra the superintegrability wavefunctions and overlap coefficients of the dunkl oscillator model in the plane were considered in the first part here finite dimensional representations of the symmetry algebra of the system called the schwinger dunkl algebra sd 2 are investigated the algebra sd 2 has six generators including two involutions and a central element and can be seen as a deformation of the lie algebra mathfrak u 2 two of the symmetry generators j 3 and j 2 are respectively associated to the separation of variables in cartesian and polar coordinates using the parabosonic creation annihilation operators two bases for the representations of sd 2 the cartesian and circular bases are constructed in the cartesian basis the operator j 3 is diagonal and the operator j 2 acts in a tridiagonal fashion in the circular basis the operator j 2 is block upper triangular with all blocks 2 2 and the operator j 3 acts in a tridiagonal fashion the expansion coefficients between the two bases are given by the krawtchouk polynomials in the general case the eigenvectors of j 2 in the circular basis are generated by the heun polynomials and their components are expressed in terms of the para krawtchouk polynomials in the fully isotropic case the eigenvectors of j 2 are generated by little 1 jacobi or ordinary jacobi polynomials the basis in which the operator j 2 is diagonal is considered in this basis the defining relations of the schwinger dunkl algebra imply that j 3 acts in a block tridiagonal fashion with all blocks 2 2 the matrix elements of j 3 in this basis are given explicitly

large p group actions with a p elementary abelian derived group let k be an algebraically closed field of characteristic p 0 and c a connected nonsingular projective curve over k with genus g 2 let c g be a big action i e a pair c g where g is a p subgroup of the k automorphism group of c such that g g 2pp 1 the aim of this paper is to describe the big actions whose derived group g is p elementary abelian in particular we obtain a structure theorem for the functions parametrizing the artin schreier cover c c g using artin schreier duality we shift to a group theoretic point of view to characterize relevant cases then we display universal families and discuss the corresponding deformation space for p 5

the local lifting problem for dihedral groups let g dp be the dihedral group of order 2p where p is an odd prime let k be an algebraically closed field of characteristic p we show that any action of g on the ring k y can be lifted to an action on r y where r is some complete discrete valuation ring with residue field k and fraction field of characteristic 0

large odd prime power order automorphism groups of algebraic curves in any characteristic abstract let x be a projective geometrically irreducible nonsingular algebraic curve of genus g 2 defined over an algebraically closed field k of odd characteristic p 0 and let aut x be the group of all automorphisms of x which fix k element wise for any a subgroup g of aut x whose order is a power of an odd prime d other than p the bound proven by zomorrodian for riemann surfaces is g 9 g 1 where the extremal case can only be obtained for d 3 and g 10 we prove zomorrodian s result for any k the essential part of our paper is devoted to extremal 3 zomorrodian curves x two cases are distinguished according as the quotient curve x z for a central subgroup z of aut x of order 3 is either elliptic or not for elliptic type extremal 3 zomorrodian curves x we completely determine the two possibilities for the abstract structure of g using deeper results on finite 3 groups we also show infinite families of extremal 3 zomorrodian curves for both types of elliptic or non elliptic our paper does not adapt methods from the theory of riemann surfaces nevertheless it sheds a new light on the connection between riemann surfaces and their automorphism groups

further nice equations for nice groups nice sextinomial equations are given for unramified coverings of the affine line in nonzero characteristic p with p 2m q and 2m q as galois groups where m 3 is any integer and q 1 is any power of p 2

deformations formelles des revetements sauvagement ramifies de courbes algebriques in this paper we study formal moduli for wildly ramified galois covering we prove a local global principle we then focus on the infinitesimal deformations of the z pz covers we explicitly compute a deformation of an automorphism of order p which implies a universal obstruction for p 2 by deforming artin schreier equations we obtain a lower bound on the dimension of the local versal deformation ring at last by comparing the global versal deformation ring to the complete local ring in a point of a moduli space we determine the dimensions of the global and local versal deformation rings

dunkl supersymmetric orthogonal functions associated with classical orthogonal polynomials we consider the eigenvalue problem associated with the dunkl type differential operator in which the reflection operator r is involved l dx r v x v x v x in the context of supersymmetric quantum mechanical models by solving this eigenvalue prob lem with the help of known exactly solvable potentials we construct several classes of func tions satisfying certain orthogonality relations we call them the dunkl supersymmetric dunkl susy orthogonal functions these functions can be expressed in terms of the classical orthog onal polynomials cops the key feature of these functions is that they appear by pairs i e qn x and qn x are both the eigenfunctions of l a general formulation of the dunkl susy orthogonal polynomials is also presented

leonard triples of q racah type and their pseudo intertwiners abstract let f denote a field and let v denote a vector space over f with finite positive dimension pick a nonzero q f such that q 4 1 and let a b c denote a leonard triple on v that has q racah type we show that there exist invertible w w w in end v such that i a commutes with w and w 1 b w c ii b commutes with w and w 1 c w a iii c commutes with w and w 1 a w b moreover each of w w w is unique up to multiplication by a nonzero scalar in f we show that the three elements w w w w w w mutually commute and their product is a scalar multiple of the identity a number of related results are obtained we call w w w the pseudo intertwiners for a b c

algebraic curves with a large non tame automorphism group fixing no point let k be an algebraically closed field of characteristic p 0 and let x be a curve over k of genus g 2 assume that the automorphism group aut x of x over k fixes no point of x the following result is proven if there is a point p on x whose stabilizer in aut x contains a p subgroup of order greater than gp p 1 then x is birationally equivalent over k to one of the irreducible plane curves ii iii iv v listed in the introduction

the structure relation for askey wilson polynomials an explicit structure relation for askey wilson polynomials is given this involves a divided q difference operator which is skew symmetric with respect to the askey wilson inner product and which sends polynomials of degree n to polynomials of degree n 1 by specialization of parameters and by taking limits similar structure relations as well as lowering and raising relations can be obtained for other families in the q askey scheme and the askey scheme this is explicitly discussed for jacobi polynomials continuous q jacobi polynomials continuous q ultraspherical polynomials and for big q jacobi polynomials an already known structure relation for this last family can be obtained from the new structure relation by using the three term recurrence relation and the second order q difference formula the results are also put in the framework of a more general theory their relationship with earlier work by zhedanov and bangerezako is discussed there is also a connection with the string equation in discrete matrix models and with the sklyanin algebra

revisiting the askey wilson algebra with the universal r matrix of uq su 2 a description of the embedding of a centrally extended askey wilson algebra aw 3 in uq sl2 3 is given in terms of the universal r matrix of uq sl2 the generators of the centralizer of uq sl2 in its three fold tensor product are naturally expressed through conjugations of casimir elements with r they are seen as the images of the generators of aw 3 under the embedding map by showing that they obey the aw 3 relations this is achieved by introducing a natural coaction also constructed with the help of the r matrix

jordan algebras and orthogonal polynomials we illustrate how jordan algebras can provide a framework for the interpretation of certain classes of orthogonal polynomials the big 1 jacobi polynomials are eigenfunctions of a first order operator of dunkl type we consider an algebra that has this operator up to constants as one of its three generators and whose defining relations are given in terms of anticommutators it is a special case of the askey wilson algebra aw 3 we show how the structure and recurrence relations of the big 1 jacobi polynomials are obtained from the representations of this algebra we also present ladder operators for these polynomials and point out that the big 1 jacobi polynomials satisfy the hahn property with respect to a generalized dunkl operator

deformation data belyi maps and the local lifting problem we prove existence and nonexistence results for certain differential forms in positive characteristic called good deformation data some of these results are obtained by reduction modulo p of belyi maps as an application we solve the local lifting problem for groups with sylow p subgroup of order p

p ranks and automorphism groups of algebraic curves let x be an irreducible complete nonsingular curve of genus g over an algebraically closed field k of positive characteristic p if 9 2 the automorphism group aut x of x is known to be a finite group and moreover its order is bounded from above by a polynomial in g of degree four stichtenoth in this paper we consider the frank zy of x and investigate relations between zy and aut x we show that zy affects the order of a sylow psubgroup of aut x 3 and that an inequality iaut x 84 9 1 9 holds for an ordinary i e zy g curve x 4

centralizers of the superalgebra mathfrak osp 1 2 o s p 1 2 the brauer algebra as a quotient of the bannai ito algebra

lower bounds on crosspoints in concentrators lower bounds on the required number of crosspoints in concentrators a class of interconnection networks are given the lower bounds are obtained from a straightforward necessary condition on the number of crosspoints in sparse crossbar full capacity concentrators because this condition must be satisfied by all full capacity concentrators embedded in more general concentrators the general necessary condition is established several sparse crossbar designs that contain the minimum number of crosspoints are presented an extension of the results to a more general class of interconnection networks is described

the z 2 n dirac dunkl operator and a higher rank bannai ito algebra abstract the kernel of the z 2 n dirac dunkl operator is examined the symmetry algebra a n of the associated dirac dunkl equation on s n 1 is determined and is seen to correspond to a higher rank generalization of the bannai ito algebra a basis for the polynomial null solutions of the dirac dunkl operator is constructed the basis elements are joint eigenfunctions of a maximal commutative subalgebra of a n and are given explicitly in terms of jacobi polynomials the symmetry algebra is shown to act irreducibly on this basis via raising lowering operators a scalar realization of a n is proposed

the higher rank q deformed bannai ito and askey wilson algebra the q deformed bannai ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra mathfrak osp q 1 vert 2 it turned out to be isomorphic to the askey wilson algebra in the present paper these results will be extended to higher rank the rank n 2 q bannai ito algebra mathcal a n q which by the established isomorphism also yields a higher rank version of the askey wilson algebra is constructed in the n fold tensor product of mathfrak osp q 1 vert 2 an explicit realization in terms of q shift operators and reflections is proposed which will be called the mathbb z 2 n q dirac dunkl model the algebra mathcal a n q is shown to arise as the symmetry algebra of the constructed mathbb z 2 n q dirac dunkl operator and to act irreducibly on modules of its polynomial null solutions an explicit basis for these modules is obtained using a q deformed mathbf ck extension and fischer decomposition

self routing quantum sparse crossbar packet concentrators quantum switching networks are derived from conventional switching networks by replacing the classical switches by quantum switches we give the quantum circuit design and routing of an n m network called a quantum concentrator that can direct quantum bit packets in arbitrary quantum states from any of its k inputs to some of its k outputs where 1 k m n our designs are based on sparse crossbars which are rectangular grids of 2 2 crosspoints sparse crossbar concentrator structures with theoretically minimum crosspoint complexities for any values of n and m are well known but no self routing algorithms have been reported for such concentrators we transform two such families of optimal concentrators called fat slim and banded sparse crossbars into quantum networks and provide self routing algorithms for these families of concentrators in this process we extend the notion of packet concentration to a quantum network and design self routing quantum crosspoints from quantum gates we address issues critical to quantum operation like reversibility and localized self routing and give a rigorous proof that quantum fat slim and banded sparse crossbar concentrators are self routable the self routing algorithms described in the paper can be used for both quantum and classical sparse crossbar concentrators by the linearity property of all quantum systems