geometric characteristics of planar quintic pythagorean hodograph curves abstract this study examines necessary and sufficient conditions for a planar quintic bezier curve to be a pythagorean hodograph ph curve quintic ph curves can be categorized into two classes according to the representation of their derivatives while the first class has been studied by farouki 1994 to be a family of regular curves already a more succinct proof by introducing auxiliary control points is provided in this paper geometric characteristics of the second class of quintic ph curves are also studied the key technique to simplify the discussion is to represent a planar bezier curve with a complex polynomial in bernstein form benefiting from such complex expression algebraic characteristics of quintic ph curves can be described by nonlinear complex systems with respect to control points by treating these systems with geometric methods conditions for a quintic planar curve to be a ph curve can be described in terms of geometric constraints on its control polygon furthermore we provide methods for the construction of the second class of quintic ph curves in particular parameter values of cusps can be explicitly determined in advance for irregular curves

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

characterization and construction of helical polynomial space curves helical space curves are characterized by the property that their unit tangents maintain a constant inclination with respect to a fixed line the axis of the helix equivalently a helix exhibits a circular tangent indicatrix and constant curvature torsion ratio if a polynomial space curve is helical it must be a pythagorean hodograph ph curve the quaternion representation of spatial ph curves is used to characterize and construct helical curves whereas all spatial ph cubics are helical the helical ph quintics form a proper subset of all ph quintics two types of ph quintic helix are identified i the monotone helical ph quintics in which a scalar quadratic factors out of the hodograph and the tangent exhibits a consistent sense of rotation about the axis and ii general helical ph quintics which possess irreducible hodographs and may suffer reversals in the sense of tangent rotation first order hermite interpolation is considered for both helical ph quintic types the helicity property offers a means of fixing the residual degrees of freedom in the general ph quintic hermite interpolation problem and yields interpolants with desirable shape features

a unified theory of full order and low order observers based on singular system theory the standard construction of the minimal order state observer for a linear time invariant plant is shown to be a degenerate form of the ordinary full order observer the naive state estimation technique of successively differentiating the plant output is also shown to be a limiting case of the same full order structure various results concerning the set of all degenerate forms of the full order observer are presented in particular conditions are established under which shifting observer eigenvalues far to the left in the complex plane implies proximity to a successively differentiating system in addition it is shown that such a scheme can never yield satisfactory estimation singular perturbation and singular system theory are used throughout the analysis

factorization theorems and the structure of operators on hilbert space soit 7b h un espace de hilbert complexe et soit 7b l 7b h l algebre des operateurs lineaires bornes agissant sur 7b h on demontre des theoremes de factorisation on etudie la structure des operateurs sur 7b h

hermite interpolation by pythagorean hodograph quintics the pythagorean hodograph ph curves are polynomial parametric curves x t y t whose hodograph derivative components satisfy the pythagorean condition x 2 t y 2 t 2 t for some polynomial t thus unlike polynomial curves in general ph curves have arc lengths and offset curves that admit exact rational representations the lowest order ph curves that are sufficiently flexible for general interpolation approximation problems are the quintics while the ph quintics are capable of matching arbitrary first order hermite data the solution procedure is not straightforward and furthermore does not yield a unique result there are always four distinct interpolants of which only one in general has acceptable shape characteristics we show that formulating ph quintics as complex valued functions of a real parameter leads to a compact hermite interpolation algorithm and facilitates an identification of the good interpolant in terms of minimizing the absolute rotation number this algorithm establishes the ph quintics as a viable medium for the design or approximation of free form curves and allows a one for one substitution of ph quintics in lieu of the widely used ordinary cubics

identification of two classes of planar septic pythagorean hodograph curves abstract polynomial pythagorean hodograph ph curves have a polynomial arc length function and rational offsets which distinguish ph curves from general polynomial parametric curves however these algebraic properties can hardly be used directly for the identification of ph curves in order to determine whether or not a given septic planar polynomial curve is a ph curve zheng et al 2016 studied a class of septic curves geometric properties and proposed an efficient algorithm in this paper we further complete their work on bezier control polygons of septic ph curves we point out that there are three classes of septic ph curves according to different factorizations of their derivatives except the first class which has been studied geometric characteristics of the other two classes are proposed by introducing auxiliary points the results are in terms of angles and lengths of legs of their bezier control polygons moreover we give feasible methods for the construction of auxiliary points including various degenerate cases

hypercyclic operators that commute with the bergman backward shift the backward shift b on the bergman space of the unit disc is known to be hypercyclic meaning it has a dense orbit here we ask which operators that commute with b inherit its hypercyclicity we show that the problem reduces to the study of operators of the form 9p b where so is a holomorphic self map of the unit disc that multiplies the dirichlet space into itself and that the question of hypercyclicity for such an operator depends on how freely 9p z is allowed to approach the unit circle as lz s 1

pythagorean hodograph preserving mappings we study the scaled pythagorean hodograph ph preserving mappings these mappings make offset rational isothermal surfaces and map ph curves to ph curves we present a method to produce a great number of the scaled ph preserving mappings for an application of the ph preserving mappings we solve the hermite interpolation problem for ph curves in the space

c1 hermite interpolation with spatial pythagorean hodograph cubic biarcs in this paper the c 1 hermite interpolation problem by spatial pythagorean hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described each ph cubic segment interpolates c 1 data at one point and they are then joined together with a c 1 continuity at some unknown common point sharing some unknown tangent vector biarcs are expressed in a closed form with three shape parameters two of them are selected based on asymptotic approximation order while the remaining one can be computed by minimizing the length of the biarc or by minimizing the elastic bending energy the final interpolating spline curve is globally c 1 continuous it can be constructed locally and it exists for arbitrary hermite data configurations

cyclicity of bicyclic operators and completeness of translates we study cyclicity of operators on a separable banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions a simple consequence of our main result is that a bicyclic unitary operator on a banach space with separable dual is cyclic our results also imply that if s a n n in mathbb z longmapsto a n 1 n in mathbb z is the shift operator acting on the weighted space of sequences ell omega 2 mathbb z if the weight satisfies some regularity conditions and n 1 for nonnegative n then s is cyclic if rm lim n rightarrow infty rm log omega n sqrt n 0 on the other hand one can see that s is not cyclic if the series sum n geq 1 rm log omega n n 2 diverges we show that the question of herrero whether either s or s is cyclic on ell omega 2 mathbb z admits a positive answer when the series sum n in mathbb z rm log s n n 2 1 is convergent we also prove completeness results for translates in certain banach spaces of functions on mathbb r

minkowski pythagorean hodograph preserving mappings we state and prove the sufficient and necessary condition for a mapping to be a scaled mph preserving mapping which preserves the mph property of a curve with rescaling the speed by a rational function in r 2 1 and show how to produce polynomial scaled mph preserving mappings from given generating polynomials we introduce s cubic mph preserving mappings of the first kind and their corresponding surfaces we show that these mappings can be used to solve interpolation problems for c 1 hermite data sets with admissible velocity vectors on their corresponding surfaces

singular observer based compensators and the deterministic separation principle the concept of observer based compensators obcs and the deterministic separation principle have long been cornerstones of modern control theory in our paper we extend these ideas via singular system theory to encompass a wider variety of rational compensators we show that the separation principle generalizes in a natural way to obcs of all orders including the static case we pay particular attention to the problems of properness and conjugate symmetry our intention is to provide an introductory account of this framework and to suggest possible avenues of further development

geometric hermite interpolation with tschirnhausen cubics explicit formulae are found that give the unique tschirnhausen cubic that solves a geometric hermite interpolation problem that solution is used to create a planar g1 spline by joining segments of tschirnhausen cubics if the geometric hermite data is from a smooth function the tschirnhausen cubic approximates the smooth function the error in the approximation of a short segment of length h can be expressed as a power series in h the error is o h4 and the coefficient of the leading term is found

on the parametrization of all regularly implementing and stabilizing controllers in this paper we deal with problems of controller parametrization in the context of behavioral systems given a full plant behavior a subbehavior of the manifest plant behavior is called regularly implementable if it can be achieved as the controlled behavior resulting from the interconnection of the full plant behavior with a suitable controller behavior in such a way that the controller does not impose restrictions that are already present in the plant we establish a parametrization of all controllers that regularly implement a given behavior we also obtain a parametrization of all stabilizing controllers

on the structure of contraction operators ii soit 7b h un espace de hilbert complexe separable de dimension infinie et 7b l 7b h l algebre de tous les operateurs lineaires bornes sur 7b h soit d le disque unite ouvert dans c d et n est l ensemble des entiers positifs il existe r 6 tel que chaque contraction t dans 7b l 7b h avec c t soit a un sous espace hyperinvariant non trivial soit appartient a la classe a 1 r

c 2 hermite interpolation by pythagorean hodograph space curves we solve the problem of c 2 hermite interpolation by pythagorean hodograph ph space curves more precisely for any set of c 2 space boundary data two points with associated first and second derivatives we construct a four dimensional family of ph interpolants of degree 9 and introduce a geometrically invariant parameterization of this family this parameterization is used to identify a particular solution which has the following properties first it preserves planarity i e the interpolant to planar data is a planar ph curve second it has the best possible approximation order 6 third it is symmetric in the sense that the interpolant of the reversed set of boundary data is simply the reversed original interpolant this particular ph interpolant is exploited for designing algorithms for converting possibly piecewise analytical curves into a piecewise ph curve of degree 9 which is globally c 2 and for simple rational approximation of pipe surfaces with a piecewise analytical spine curve the algorithms are presented along with an analysis of their error and approximation order

pythagorean hodograph curves in euclidean spaces of dimension greater than 3 a polynomial pythagorean hodograph ph curve r t x 1 t x n t in r n is characterized by the property that its derivative components satisfy the pythagorean condition x 1 2 t x n 2 t s 2 t for some polynomial s t ensuring that the arc length s t s t dt is simply a polynomial in the curve parameter t ph curves have thus far been extensively studied in r 2 and r 3 by means of the complex number and the quaternion or hopf map representations and the basic theory and algorithms for their practical construction and analysis are currently well developed however the case of ph curves in r n for n 3 remains largely unexplored due to difficulties with the characterization of pythagorean n 1 tuples when n 3 invoking recent results from number theory we characterize the structure of ph curves in dimensions n 5 and n 9 and investigate some of their properties

c1 hermite interpolation with ph curves by boundary data modification we show how to create a new class of infinitely many ph hermite interpolants for a given ordinary c 1 hermite data set in a way that ameliorates the rapid shape transitions in ph quintic interpolants that occurs when boundary data has poor geometric compatibility we first interpolate a hermite data set using two asymmetric ph cubics each of which satisfies the data at one boundary then we selectively apply one of three shape improvement techniques to the interpolant so as to eliminate rapid shape transitions and produce c 1 ph curves with lower bending energy at the expense of a small increase in arc length

hermite interpolation by pythagorean hodograph curves of degree seven polynomial pythagorean hodograph ph curves form a remarkable subclass of polynomial parametric curves they are distinguished by having a polynomial arc length function and rational offsets parallel curves many related references can be found in the article by farouki and neff on c 1 hermite interpolation with ph quintics we extend the c 1 hermite interpolation scheme by taking additional curvature information at the segment boundaries into account as a result we obtain a new construction of curvature continuous polynomial ph spline curves we discuss hermite interpolation of g 2 c 1 boundary data points first derivatives and curvatures with ph curves of degree 7 it is shown that up to eight possible solutions can be found by computing the roots of two quartic polynomials with the help of the canonical taylor expansion of planar curves we analyze the existence and shape of the solutions more precisely for hermite data which are taken from an analytical curve we study the behaviour of the solutions for decreasing stepsize it is shown that a regular solution is guaranteed to exist for sufficiently small stepsize provided that certain technical assumptions are satisfied moreover this solution matches the shape of the original curve the approximation order is 6 as a consequence any given curve which is assumed to be g 2 curvature continuous and to consist of analytical segments can approximately be converted into polynomial ph form the latter assumption is automatically satisfied by the standard curve representations of computer aided geometric design such as bezier or b spline curves the conversion procedure acts locally without any need for solving a global system of equations it produces g 2 polynomial ph spline curves of degree 7

cyclicity in lp spaces and zero sets of the fourier transforms abstract we study the cyclicity of vectors u in l p z it is known that a vector u is cyclic in l 2 z if and only if the zero set z u of its fourier transform u has lebesgue measure zero and log u l 1 t where t is the unit circle here we show that unlike l 2 z there is no characterization of the cyclicity of u in l p z 1 p 2 in terms of z u and the divergence of the integral t log u moreover we give both necessary conditions and sufficient conditions for u to be cyclic in l p z 1 p 2

singular inner functions and biinvariant subspaces for dissymetric weighted shifts abstract we consider dissymetric weights on z i e nonincreasing unbounded weights z 1 such that n 1 for n 0 and such that lim n n 1 n 1 where n sup p z n p p a dissymetric weight is said to be quasianalytic if n log n n 2 denote by the unit circle and by d the open unit disc set l 2 f l 2 f n z f n 2 2 n 1 2 we identify the usual hardy space h 2 d to the set f l 2 f n 0 n s f e it e it f e it the usual shift on l 2 if u is a singular inner function denote by e u the closure of span s n u n z in l 2 we show that if is a dissymetric weight such that log n n n then e u is a proper subspace of l 2 for every singular inner function u also if is any dissymetric weight then there exists some singular inner function u such that e u is a proper subspace of l 2 so that s always has nontrivial biinvariant subspaces no nontrivial biinvariant subspaces of s were known so far in the quasianalytic case

the existence of translation invariant subspaces of symmetric self adjoint sequence spaces on z we prove that if x is a reflexive translation invariant banach space of complex sequences on z that contains all finitely supported sequences in which the coordinate functionals are continuous and for every sequence c n in the space the sequences c n and c n are also in the space then x has a nontrivial translation invariant subspace this provides in particular a positive solution to the translation invariant subspace problem for weighted lp spaces on z with even weights for 1 p the proof is based on an intermediate result that asserts that if a is an operator on a reflexive real banach space of dimension greater than one and there exist non zero vectors u in the space and v in the dual space such that anu v n 0 is a moment sequence of a finite positive borel measure on a bounded interval on the real line then a has a nontrivial invariant subspace

construction of ph splines based on h bezier curves abstract there is considerable interest in the properties of ph curves in geometric modeling and cagd because ph curves can be computed at speeds comparable to polynomial curves and used to calculate curve arc lengths and isometric lines the purpose of this paper is to develop a general approximation of the h bezier curve based on ph curves we call the resulting approximations phh bezier curves for convenience first a necessary and sufficient condition for a cubic plane h bezier curve to be a ph curve is obtained second based on the h bezier curve the control polygon of the cubic phh bezier curve is constructed from an orderly triangle the cubic phh bezier curve is then introduced and a new algorithm for their construction is proposed according to geometric modeling the error analysis between h bezier curve and phh bezier curve is estimated finally the proposed algorithm is verified experimentally it is demonstrated that cubic phh bezier curves can accurately approximate h bezier curves but that the selection of the middle two control points of the phh bezier curve has a profound impact on the quality of the approximation error analyses demonstrate that when the middle two control points of the phh bezier curve are mixed with the corresponding original control points a good approximation is achieved the new algorithm may thus have considerable potential for use in geometry modeling

on interpolation by planar cubic g 2 pythagorean hodograph spline curves in this paper the geometric interpolation of planar data points and boundary tangent directions by a cubic g 2 pythagorean hodograph ph spline curve is studied it is shown that such an interpolant exists under some natural assumptions on the data the construction of the spline is based upon the solution of a tridiagonal system of nonlinear equations the asymptotic approximation order 4 is confirmed