generalizations of chebyshev polynomials and polynomial mappings in this paper we show how polynomial mappings of degree from a union of disjoint intervals onto generate a countable number of special cases of generalizations of chebyshev polynomials we also derive a new expression for these generalized chebyshev polynomials for any genus from which the coefficients of can be found explicitly in terms of the branch points and the recurrence coefficients we find that this representation is useful for specializing to polynomial mapping cases for small where we will have explicit expressions for the recurrence coefficients in terms of the branch points we study in detail certain special cases of the polynomials for small degree mappings and prove a theorem concerning the location of the zeroes of the polynomials we also derive an explicit expression for the discriminant for the genus 1 case of our chebyshev polynomials that is valid for any configuration of the branch point

motor algebra approach for visually guided robotics abstract this paper presents the formulation of the 3d kinematics in the geometric algebra framework we show that this approach is extremely useful for solving problems in the field of visual guided robotics in this algebraic system the 3d euclidean motion of points lines and planes can be advantageously represented using the algebra of motors the computational complexity of the direct and indirect kinematics and other problems concerning robot manipulators depends on its degrees of freedom as well on its geometric characteristics our approach makes possible a direct algebraic formulation of the concrete problem in such a way that it reflects the underlying geometric structure this is achieved by switching to a description of parts of the problem based on motor representations of points lines and planes where necessary the first robotics task this paper deals with is the formulation and computation of closed form solutions of the direct and indirect kinematics of standard robot manipulators and a simple example of a grasping task the flexible method presented here is new and it widens the current standard point or line representation based approaches for the treatment of problems related to robot manipulators the second challenging task presented in this paper is the solution of the hand eye calibration problem when cameras are attached to robot arms the solution of this problem in the motor algebra framework turns to be linear both tasks are crucial for visual guided robotics that is why we are strongly motivated to present them to show how useful it is to apply motor algebra for solving geometric problems in visual guided robotics

christoffel functions and tura n determinants on several intervals abstract we consider orthogonal polynomials on the real line with an underlying asymptotic periodic structure for the coefficients in the three term recurrence relation these polynomials live on a set e which consists of several intervals we give a survey of some techniques for obtaining asymptotic results of the orthogonal polynomials on the oscillatory region e from these results we obtain a useful technique for approximating the weight function on e by means of christoffel functions and shifted turan determinants

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

extreme points and support points of subordination families this paper surveys the main results and open questions in the theory of extreme points and support points of subordination families of analytic functions an effort has been made to make the list of references as complete as possible

on extreme points of subordination families with a convex majorant abstract let z c z b 0 denote the set of functions analytic in and satisfying z 0 0 suppose f is analytic and univalent in and maps it onto a convex domain d other than a strip a wedge or a half plane let s f fo b 0 let es f denote the set of extreme points of s f d j hallenbeck and t h macgregor asked in 3 whether the set es f ever lies strictly between the minimal fo b 0 lim r 1 re i 1 a e and the maxima fc is an extreme point of b 0 sets we prove that this always happens when the boundary of d contains a line segment let d f be any convex domain and let f be subordinate to f we show that if f is an extreme point of s f then the closure of the boundary values f of f f lim r 1 f re i a e has a nonempty intersection with the set of extreme points of the boundary of d we also completely determine extreme points of s f when the boundary of d consists of finitely many arcs wih positive curvature

relative growth of linear iterations and orthogonal polynomials on several intervals assume a c2 2 and y0 c2 let yn yn a y0 be defined by thelinear iteration yn ayn 1 n 1 2 then for every p 0 and for every norm in c2 there is a constant c depending only on p and such that sup a 1 y0 0 yl a y0 p lk 0 yk a y0 p cl l 1 2 where a denotes the spectral radius of a applications to orthogonal polynomials pn with respect to a positive borel measure on several intervals are studied in particular the growth of orthogonal polynomials relative to their sums satisfies limn supx supp pn x p n 1k 0 pk x p 0 provided that the three term recurrence coefficients of pn are asymptotically periodic

characterizations of generalized hermite and sieved ultraspherical polynomials a new characterization of the generalized hermite polynomials and of the orthogonal polynomials with respect to the measure ixy 1 x2 1 2dx is derived which is based on a reversing property of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of laguerre and jacobi polynomials a similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind these results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order

the absolute line quadric and camera autocalibration we introduce a geometrical object providing the same information as the absolute conic the absolute line quadric alq after the introduction of the necessary exterior algebra and grassmannian geometry tools we analyze the grassmannian of lines of p from both the projective and euclidean points of view the exterior algebra setting allows then to introduce the alq as a quadric arising very naturally from the dual absolute quadric we fully characterize the alq and provide clean relationships to solve the inverse problem i e recovering the euclidean structure of space from the alq finally we show how the alq turns out to be particularly suitable to address the euclidean autocalibration of a set of cameras with square pixels and otherwise varying intrinsic parameters providing new linear and non linear algorithms for this problem we also provide experimental results showing the good performance of the techniques

sieved ultraspherical polynomials the continuous q ultraspherical polynomials contain a number of important examples as limiting or special cases one of these arose in allaway s ph d thesis in a previous paper we solved a characterization problem essentially equivalent to allaway s and showed that these polynomials arose from the q ultraspherical polynomials when q approached a root of unity a second class of such polynomials is found and the recurrence relation and orthogonality relation are found for each of these polynomials the orthogonality is interesting because the weight function has a finite number of zeros in 1 1 generating functions and other formulas are also found

a basic class of symmetric orthogonal functions with six free parameters in a previous paper we introduced a basic class of symmetric orthogonal functions bcsof by an extended theorem for sturm liouville problems with symmetric solutions we showed that the foresaid class satisfies the differential equation x 2 px 2 q f n x x rx 2 s f n x l nx 2 1 1 n c 2 f n x 0 where l n n q 1 1 1 n 2 r n 1 q 1 1 1 n 2 p email protected s q 1 q and contains four important sub classes of symmetric orthogonal functions moreover for q 1 it is reduced to a basic class of symmetric orthogonal polynomials bcsop which respectively generates the generalized ultraspherical polynomials generalized hermite polynomials and two other sequences of finite symmetric orthogonal polynomials in this paper again by using the extended theorem we introduce a further basic class of symmetric orthogonal functions with six parameters and obtain its standard properties we show that the new class satisfies the equation x 2 px 2 q f n x x rx 2 s f n x a nx 2 1 nc d f n x 0 in which c d are two free parameters and a n denotes eigenvalues corresponding to the defined class we then introduce four orthogonal sub classes of the foresaid class and study their properties in detail since the introduced class is a generalization of bcsof for email protected 2 the four mentioned sub classes naturally generalize the generalized ultraspherical polynomials generalized hermite polynomials and two sequences of finite classical symmetric orthogonal polynomials again

an electrostatic interpretation of the zeros of sieved ultraspherical polynomials in an earlier work castillo et al j math anal appl 455 1801 1821 2017 it was proved that the semiclassical class of orthogonal polynomials is stable under polynomial transformations in this work we use this fact to derive in a unified way old and new properties concerning the sieved ultraspherical polynomials of the first and second kind in particular we derive ordinary differential equations for these polynomials as an application we use the differential equation for sieved ultraspherical polynomials of the first kind to deduce that the zeros of these polynomials mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field

canonical bases for real representations of clifford algebras abstract the well known classification of the clifford algebras cl r s leads to canonical forms of complex and real representations which are essentially unique by virtue of the wedderburn theorem for s 1 representations of cl r s on r 2 n are obtained from representations on r n by adding two new generators while in passing from a representation of cl p 0 on r n to a representation of cl r 0 on r 2 n the number of generators that can be added is either 1 2 or 4 according as the clifford algebra represented on r n is of real complex or quaternionic type we have expressed canonical forms of these representations in terms of the complex and quaternionic structures in the half dimension and we obtained algorithms for transforming any given representation of cl r s to a canonical form our algorithm for the transformation of the representations of cl 8 d c 0 c 7 to canonical forms is based on finding an abelian subalgebra of cl 8 d c 0 and its invariant subspace computer programs for determining explicitly the change of basis matrix for the transformation to canonical forms are given for lower dimensions the construction of the change of basis matrices uniquely up to the commutant provides a constructive proof of the uniqueness properties of the representations and may have applications in computer graphics and robotics

45 years of orthogonal polynomials a view from the wings this is an informal and rather personal report on what a bit player observed happening on stage in this ongoing drama we call orthogonal polynomials

cyclic tridiagonal pairs higher order onsager algebras and orthogonal polynomials abstract the concept of cyclic tridiagonal pairs is introduced and explicit examples are given for a fairly general class of cyclic tridiagonal pairs with cyclicity n we associate a pair of divided polynomials the properties of this pair generalize the ones of tridiagonal pairs of racah type the algebra generated by the pair of divided polynomials is identified as a higher order generalization of the onsager algebra it can be viewed as a subalgebra of the q onsager algebra for a proper specialization at q the primitive 2 n th root of unity orthogonal polynomials beyond the leonard duality are revisited in light of this framework in particular certain second order dunkl shift operators provide a realization of the divided polynomials at n 2 or q i

support points of families of analytic functions described by subordination we determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination the families we study include the functions with a positive real part the typically real functions and the functions which are subordinate to a given majorant if the majorant f is univalent then each support point has the form f 0 where p is a finite blaschke product and o 0 o this completely characterizes the set of support points when f is convex the set of support points is found for some specific majorants including f z 1 z l z p where p 1 let k and st denote the set of normalized convex and starlike mappings respectively we find the support points of the families k and st defined by the property of being subordinate to some member of k or st respectively introduction let a z e c i z i 1 and let e denote the set of functions analytic in a then 6 is a locally convex linear topological space with respect to the topology given by uniform convergence on compact subsets of la by a continuous linear functional on e we mean a complex valued functional defined on 6f that is linear and continuous in other words if j is such a functional then j af bg aj f bj g whenever a and b belong to c and f and g belong to b also if fn e 6b n 1 2 and fn f then j fn j f each continuous linear funcn tional j on ef is given by a sequence bn n 0 1 which satisfies limn n i n i 1 and

support points of families of starlike mappings abstract the support points are obtained for the class of those analytic functions starlike of order m fold symmetric that are real on 1 1 we also determine the set of support points for the m fold symmetric analytic functions which are starlike of order

on harmonic typically real mappings abstract in this paper we prove that a univalent orientation preserving harmonic mapping f h g defined on the unit disk u with the normalization f 0 0 h 0 g 0 0 is a typically real mapping if f u is a starlike domain with respect to the origin or f u is convex in one direction

the support points of several classes of analytic functions with fixed coefficients let b denote the set of functions z that are analytic in the unit disk d and satisfy z 1 z 0 z 1 let t denote the set of functions f z that are analytic in d normalized by f 0 0 and f 0 1 and satisfy that f z is real if and only if z is real z 1 in this article we investigate the support points of the subclasses of b p and t of functions with fixed coefficients

zeros and orthogonality of the askey wilson polynomials for q a root of unity we study some properties of the askey wilson polynomials awp when q is a primitive nth root of unity for general four parameter awp zeros of the nth polynomial and the orthogonality measure are found explicitly special subclasses of the awp e g the continuous q jacobi and big q jacobi polynomials are considered in detail a set of discrete weight functions positive on a real interval is described some new trigonometric identities related to the awp are obtained normalization conditions of some polynomials are expressed in terms of the gauss sums

keypoints based surface representation for 3d modeling and 3d object recognition the three dimensional 3d modeling and recognition of 3d objects have been traditionally performed using local features to represent the underlying 3d surface extraction of features requires cropping of several local surface patches around detected keypoints although an important step the extraction and representation of such local patches adds to the computational complexity of the algorithms this paper proposes a novel keypoints based surface representation ksr technique the proposed technique has the following two characteristics 1 it does not rely on the computation of features on a small surface patch cropped around a detected keypoint rather it exploits the geometrical relationship between the detected 3d keypoints for local surface representation 2 ksr is computationally efficient requiring only seconds to process 3d models with over 50 000 points with a matlab implementation experimental results on the uwa and stanford 3d models dataset suggest that it can accurately perform pairwise and multiview range image registration 3d modeling ksr was also tested for 3d object recognition with occluded scenes recognition results on the uwa dataset show that the proposed technique outperforms existing methods including 3d tensor vd lsd keypoint depth based feature spherical harmonics and spin image with a recognition rate of 95 9 the proposed approach also achieves a recognition rate of 93 5 on the challenging ca fascori dataset compared to 92 5 achieved by game theoretic the proposed method is computationally efficient compared to state of the art local feature methods highlightswe propose a novel technique called keypoint based surface representation ksr the proposed technique does not require local features around detected keypoints ksr exploits geometrical relationships between keypoints for surface representation ksr is tested on 3 popular datasets for 3d modeling and 3d object recognition ksr achieves superior 3d modeling and recognition results

on sieved orthogonal polynomials iii orthogonality on several intervals on introduit deux generalisations des polynomes de chelryshev dont le spectre continu est x 2 c 1 c t k x 2 c 1 c ou c est un parametre positif on definit les fonctions poids on obtient les fonctions generatrices et les formules explicites

some properties of two subclasses of k fold symmetric functions associated with srivastava attiya operator abstract in the present paper by using the srivastava attiya operator we define two new subclasses of k fold symmetric analytic functions some interesting properties of these subclasses such as integral representations extreme points close convex hulls and subordinations are obtained which generalize and refine some previous results

on semiclassical orthogonal polynomials via polynomial mappings abstract we consider orthogonal polynomials via polynomial mappings in the framework of the semiclassical class we prove that this class is stable under polynomial transformations several consequences of this fact are deduced as an application we analyze in detail cubic transformations for semiclassical orthogonal polynomials of class at most 2 recovering and extending some results proved recently for class 1 and producing new examples of semiclassical orthogonal polynomials of class 2 in particular we show how to obtain integral representations for the regular functionals with respect to which these new semiclassical families are orthogonal

electrostatic models for zeros of polynomials old new and some open problems we give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well known families of polynomials and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity the leading role is played by the differential equation satisfied by these polynomials some new developments applications and open problems are presented

inverse images of polynomial mappings and polynomials orthogonal on them let j be a polynomial with complex coefficients first we study the inverse images of the real and imaginary axes under a polynomial mapping j in detail then for an arbitrary polynomial and a sequence pn of orthogonal polynomials the orthogonality behaviour of the sequence of polynomials pn j n n is investigated in particular necessary and sufficient conditions are given such that pn j n n is a subsequence of polynomials orthogonal with respect to a positive measure supported on a compact subset of the real line

sieved orthogonal polynomials vii generalized polynomial mappings systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second order difference equations are studied in detail applications are given to the theory of the recently discovered sieved orthogonal polynomials the connection with polynomial mappings is examined an example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials

univalent harmonic functions several families of complex valued univalent harmonic functions are studied from the point of view of geometric function theory one class consists of mappings of a simply connected domain onto an infinite horizontal strip with a normalization at the origin extreme points and support points are determined as well as sharp estimates for fourier coefficients and distortion theorems next mappings in izi 1 are considered that leave infinity fixed some coefficient estimates distortion theorems and covering properties are obtained for such mappings with real boundary values many extremal problems are solved explicitly