automorphisms of the nottingham group abstract let k be a finite field of characteristic p 5 the nottingham group n over k is the group of normalised automorphisms of the local field k t in this paper we determine the automorphism group of n in particular we show that every automorphism of the nottingham group is standard we also give a complete description of the conjugacy classes of elements of order p in n

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 fast elementary algorithm for computing the determinant of toeplitz matrices in recent years a number of fast algorithms for computing the determinant of a toeplitz matrix were developed the fastest algorithm we know so far is of order k 2 log n k 3 where n is the number of rows of the toeplitz matrix and k is the bandwidth size this is possible because such a determinant can be expressed as the determinant of certain parts of the n th power of a related k i k companion matrix in this paper we give a new elementary proof of this fact and provide various examples we give symbolic formulas for the determinants of toeplitz matrices in terms of the eigenvalues of the corresponding companion matrices when k is small

a new family of maximal curves over a finite field it has been known for a long time that the deligne lusztig curves associated to the algebraic groups of type 2a 2 2b 2 and 2g 2 defined over the finite field mathbb f n all have the maximum number of mathbb f n rational points allowed by the weil explicit formulas and that these curves are mathbb f q 2 maximal curves over infinitely many algebraic extensions mathbb f q 2 of mathbb f n serre showed that an mathbb f q 2 rational curve which is mathbb f q 2 covered by an mathbb f q 2 maximal curve is also mathbb f q 2 maximal this has posed the problem of the existence of mathbb f q 2 maximal curves other than the deligne lusztig curves and their mathbb f q 2 subcovers see for instance garcia on curves with many rational points over finite fields in finite fields with applications to coding theory cryptography and related areas pp 152 163 springer berlin 2002 and garcia and stichtenoth a maximal curve which is not a galois subcover of the hermitan curve bull braz math soc n s 37 139 152 2006 in this paper a positive answer to this problem is obtained for every q n 3 with n p r 2 p 2 prime we give a simple explicit construction of an mathbb f q 2 maximal curve mathcal x that is not mathbb f q 2 covered by any mathbb f q 2 maximal deligne lusztig curve furthermore the mathbb f q 2 automorphism group aut mathcal x has size n 3 n 3 1 n 2 1 n 2 n 1 interestingly mathcal x has a very large mathbb f q 2 automorphism group with respect to its genus g frac 1 2 n 3 1 n 2 2 1

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

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

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

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

a robust optimal model matching control a deductive design approach is proposed to develop an optimal feedback control while preserving quantitative robustness and noise rejection properties the design procedure involves the minimization of a suitably selected performance index to reflect a required model matching following objective gradient projection schemes applied to the performance index and feasible constraints are used to accomplish the design tradeoff a mechanical manipulator control system example is employed as a vehicle to illustrate the overall design selection and optimization uncertainty of the system description is included as a feature of the example

explicit inverse of a tridiagonal k toeplitz matrix we obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k toeplitz matrix a the proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of chebyshev polynomials of the second kind we also compute the characteristic polynomial of a which enables us to state some conditions for the existence of a 1 our results also extend known results for the case when the residue mod k of the order of a is equal to 0 or k 1 numer math 10 1967 pp 153 161

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

entries of continuous functions of large hermitian tridiagonal 2 toeplitz matrices abstract in this paper we present an eigenvalue decomposition for any real symmetric tridiagonal 2 toeplitz matrix of odd order where the eigenvector matrix is orthogonal using this decomposition we study the entries of continuous functions of large real symmetric tridiagonal 2 toeplitz matrices furthermore we also study in the present paper the entries of continuous functions of large hermitian toeplitz matrices with at most three non zero diagonals

eigenvalues of a certain class of tridiagonal matrices we make a short comment on a paper involving the minimal eigenvalues of a specialized version of 2 toeplitz tridiagonal matrices

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

inverses of k toeplitz matrices with applications to resonator arrays with multiple receivers abstract we find closed form algebraic formulas for the elements of the inverses of tridiagonal 2 and 3 toeplitz matrices which are symmetric and have constant upper and lower diagonals these matrices appear respectively as the impedance matrices of resonator arrays in which a receiver is placed over every 2 or 3 resonators consequently our formulas allow to compute the currents of a wireless power transfer system in closed form allowing for a simple exact and symbolic analysis thereof small numbers are chosen for illustrative purposes but the elementary linear algebra techniques used can be extended to k toeplitz matrices of this special form with k arbitrary hence resonator arrays with a receiver placed over every k resonators can be analysed in the same way

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

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

on a constant diagonals matrix in this paper we consider a constant diagonals matrix the matrix was discussed in witula and slota r witula d slota on computing the determinants and inverses of some special type of tridiagonal and constant diagonals matrices appl math comput 189 1 2007 514 527 the authors gave some results on determinant and the inverse of the matrix for some special cases we give lu factorization and then compute determinant of the matrix we determine eigenvalues of the matrix further we obtain some relationships between permanent of the matrix and terms of a certain recurrence relation

locally toeplitz sequences spectral properties and applications abstract the notion of locally toeplitz sequence of matrices is introduced which extends the notion of toeplitz sequence of matrices the singular value distribution and in the hermitian case the eigenvalue distribution is completely characterized for locally toeplitz sequences and their sums obtaining weighted szeg formulas which extend well known results due to tyrtyshnikov concerning toeplitz matrices indeed any toeplitz sequence t n f where f is bounded is proved to be a locally toeplitz sequence moreover sufficient conditions are given for the product of two locally toeplitz sequences to be also locally toeplitz by combining these theoretic results we are able to explicitly compute the asymptotic spectral distribution of a large class of matrices arising in the applications including the algebra generated by toeplitz sequences and virtually all matrices resulting from the discretization of a unidimensional differential operator with non constant coefficients finally a large number of examples is discussed

galois structure of zariski cohomology for weakly ramified covers of curves we compute equivariant euler characteristics of locally free sheaves on curves thereby generalizing several results of kani and nakajima for instance we extend kani s computation of the galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case

spectral properties of certain tridiagonal matrices abstract we study spectral properties of irreducible tridiagonal k toeplitz matrices and certain matrices which arise as perturbations of them

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

galois structure of de rham cohomology of tame covers of schemes a class on in the grothendieck group ko z g of all finitely generated projective z g modules the class group cl z g of z g is defined to be the quotient of ko z g by the subgroup generated by the class of z g let on stab be the image of on in cl z g in 34 taylor proved fr6hlich s conjecture that on stab is equal to another invariant wn k in cl z g which cassou nogues had defined by means of the root numbers of symplectic representations of the galois group g the root number of a representation v of g is the constant which appears in the functional equation of the artin lfunction of v the connection between galois structure invariants and artin l functions has been a basic theme in research on galois structure for further discussion see 12 and 4

generalized locally toeplitz sequences spectral analysis and applications to discretized partial differential equations abstract starting from the finite difference discretization of an elliptic second order pde as i j 1 d x i a i j x x j u x b x over a bounded domain we introduce the notion of generalized locally toeplitz sequence of matrices the singular value distribution the eigenvalue distribution in the hermitian case is studied and characterized for generalized locally toeplitz sequences in terms of weighted multidimensional szeg formulas this extends preceding results attributed to tilli which concern the unilevel case the application of this theoretic analysis to the numerical solution of pdes is finally discussed

inversion of k tridiagonal matrices with toeplitz structure in this paper we consider an inverse problem with the k tridiagonal toeplitz matrices a theoretical result is obtained that under certain assumptions the explicit inverse of a k tridiagonal toeplitz matrix can be derived immediately two numerical examples are given to demonstrate the validity of our results

explicit inverses of some tridiagonal matrices abstract we give explicit inverses of tridiagonal 2 toeplitz and 3 toeplitz matrices which generalize some well known results concerning the inverse of a tridiagonal toeplitz matrix

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

the determination of companion matrices characterizing toeplitz and r toeplitz matrices abstract it is shown that the unknown elements in the companion matrix used to characterize a nonsingular toeplitz matrix are easily found from the elements of the inverse of the toeplitz matrix similar formulae are obtained for the two companion like matrices characterizing an r toeplitz matrix this case includes the result for a block toeplitz matrix

eigenstructure assignment in multivariable linear systems abstract a new design method of regulators using state feedback is developed this method not only assigns eigenvalues to the closed loop system but also assigns parts of the closed loop eigenvector structure illustration of the method in the case of a two link manipulator is included a new robustness measure is introduced

eigenvalues assignment by a particular lyapunov type equation this paper proposes a particular lyapunov type equation and uses its solution to synthesize a state feedback control such that the eigenvalues of the closed loop multiple inputs system can be assigned to the left half plane with arbitrary stability degree

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

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

cohomology of g sheaves in positive characteristic let g be a finite group and x a noetherian g scheme defined on an algebraically closed field k whose characteristic divides the order of g we define a refinement of the equivariant k theory of x devoted to give a better account of the information related to modular representation theory the construction relies in an essential way on the work of m auslander in modular representation theory and the use of sheaves of rings with several objects the main applications of this modular k theory are in dimension one where we show how it allows to extend the work of s nakajima

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

the eigenproblem of a tridiagonal 2 toeplitz matrix abstract the characteristic polynomial of a tridiagonal 2 toeplitz matrix is shown to be closely connected to polynomials which satisfy the three point chebyshev recurrence relationship this is an extension of the well known result for a tridiagonal toeplitz matrix when the order of the matrix is odd the eigenvalues are found explicitly in terms of the chebyshev zeros the eigenvectors are found in terms of the polynomials satisfying the three point recurrence relationship

equivariant epsilon constant conjectures for weakly ramified extensions we study the local epsilon constant conjecture as formulated by breuning in breuning j london math soc 70 2 289 306 2004 this conjecture fits into the general framework of the equivariant tamagawa number conjecture etnc and should be interpreted as a consequence of the expected compatibility of the etnc with the functional equation of artin l functions let k mathbb q p be unramified under some mild technical assumption we prove breuning s conjecture for weakly ramified abelian extensions n k with cyclic ramification group as a consequence of breuning s local global principle we obtain the validity of the global epsilon constant conjecture as formulated in bley and burns proc lond math soc 87 3 545 590 2003 and of chinburg s omega 2 conjecture as stated in chinburg ann math 121 2 351 376 1985 for certain infinite families f e of weakly and wildly ramified extensions of number fields

explicit expression for powers of tridiagonal 2 toeplitz matrix of odd order abstract in this paper we give the eigenvalue decomposition for odd order tridiagonal 2 toeplitz matrix and derive the explicit expression for integer powers of such matrix

robustness measure bounds for optimal model matching control designs the theorems introduced by r v patel et al 1977 for multivariable linear quadratic state feedback design in the presence of perturbations are extended to optimal model matching control system designs a robustness measure bound is introduced for optimal model matching control systems bounds are obtained for allowable nonlinear time varying perturbations such that the resulting closed loop system remain stable bounds on a special but important class of perturbations in which the perturbation is linear are also derived

robust eigenstructure assignment for multi input systems abstract a new eigenstructure assignment scheme for multi input systems is proposed it is well known that for multi input systems the feedback gain is not unique and so cannot be used to obtain a desired set of closed poles in this paper the set of all such gains is parametrized by a matrix partition scheme it is shown that this non uniqueness of the feedback gains can be used to improve the robustness and noise rejection properties of the closed system when it is subject to possible uncertainties an illustrative example is included

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

characterisation and properties of r toeplitz matrices abstract an r toeplitz rt matrix can be regarded as a block toeplitz matrix with order some multiple of r from which a bottom border and a right hand border have been removed it is shown that some known properties of toeplitz matrices extend to this new class in particular a necessary and sufficient condition is given in terms of a linear matrix equation for a nonsingular matrix to have rt form from which it follows that the elements of the inverse are completely determined by those in its first and last r rows and columns in addition some results are presented for cases when the inverse of an rt matrix has certain banded forms

a cohomological treatise of hkg covers with applications to the nottingham group abstract we characterize harbater katz gabber curves in terms of a family of cohomology classes satisfying a compatibility condition our construction is applied to the description of finite subgroups of the nottingham group

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

orthogonal polynomials on the unit circle via a polynomial mapping on the real line let f n n 0 be a sequence of monic orthogonal polynomials on the unit circle opuc with respect to a symmetric and finite positive borel measure email protected on 0 email protected and let 1 a 0 a 1 a 2 be the associated sequence of verblunsky coefficients in this paper we study the sequence email protected n n 0 of monic opuc whose sequence of verblunsky coefficients is 1 b 1 b 2 b n 1 a 0 b n 1 b 2 b 1 a 1 b 1 b 2 b n 1 a 2 b n 1 b 2 b 1 a 3 where b 1 b 2 b n 1 are n 1 fixed real numbers such that b email protected 1 1 for all j 1 2 n 1 so that email protected n n 0 is also orthogonal with respect to a symmetric and finite positive borel measure email protected on the unit circle we show that the sequences of monic orthogonal polynomials on the real line oprl corresponding to f n n 0 and email protected n n 0 by szego s transformation are related by some polynomial mapping giving rise to a one to one correspondence between the monic opuc email protected n n 0 on the unit circle and a pair of monic oprl on a subset of the interval 1 1 in particular we prove email protected q email protected n 1 q email protected q n q email protected q n q q n q supported on a subset of the union of 2n intervals contained in 0 email protected such that any two of these intervals have at most one common point and where up to an affine change in the variable z n 1 and email protected n are algebraic polynomials in email protected of degrees n 1 and n respectively defined only in terms of a 0 b 1 b n 1 this measure induces a measure on the unit circle supported on the union of 2n arcs pairwise symmetric with respect to the real axis the restriction to symmetric measures or real verblunsky coefficients is needed in order that szego s transformation may be applicable

on some tridiagonal k toeplitz matrices algebraic and analytical aspects applications in this paper we use the analytic theory for 2 and 3 toeplitz matrices to obtain the explicit expressions for the eigenvalues eigenvectors and the spectral measure associated to the corresponding infinite matrices as an application we consider two solvable models related with the so called chain model some numerical experiments are also included

spectra of certain jacobi operators we discuss the spectra and in particular the essential spectra of some bounded self adjoint jacobi operators associated with orthogonal polynomial sequences obtained via polynomial mappings

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