an operator without invariant subspaces on a nuclear frechet space let x be a locally convex topological complex vector space throughout this paper the term operator on x will mean a continuous linear transformation mapping x into itself and the term invariant subspace for an operator t on x will mean a closed subspace m of x such that m f o m p x and tm c m it is still an open problem whether or not every operator on an infinite dimensional separable hilbert space has an invariant subspace on the other hand p enflo 8 constructed an operator without invariant subspaces on a nonreflexive banach space in this paper we construct an operator t on an infinite dimensional nuclear frechet space f such that t has no invariant subspaces and t2 has an invariant subspace moreover the space f is a nuclear countably hilbert space in the sense of gelfand and vilenkin 12 that is f is the intersection of a decreasing sequence of hilbert spaces hn n 1 2 such that for every n the imbedding of hn 1 into hn is a nuclear mapping and the topology on f is defined by the sequence of hn norms the elements of f are complex sequences c ci j 0 and the operator t is the associated backward shift that is

a family of non stationary subdivision schemes reproducing exponential polynomials we present a family of non stationary subdivision schemes taking the exponential quasi spline as the basic limit function reaching up to interpolatory refinable functions the exponential quasi spline generalizes the exponential b spline the associated non stationary subdivision scheme can reproduce the desired number of exponential polynomials due to the built in parameter permitting a wide range of tension control our scheme can avoid the undesirable artifact generated by usual non stationary interpolatory schemes on the irregularly spaced region of control points under a natural condition on the rate of convergence of the shape parameters all the subdivision schemes in the family have the same smoothness and approximation order as those of the asymptotically equivalent stationary schemes we present numerical examples to verify that our subdivision schemes have high flexibility and versatility in design

construction of binary four and five point non stationary subdivision schemes from hyperbolic b splines binary non stationary four and five point subdivision schemes have been developed using hyperbolic b spline basis functions the proposed subdivision schemes generate pleasing curves near the control polygon as compared to the existing non stationary subdivision schemes developed using trigonometric b splines basis functions the main speciality of the hyperbolic subdivision schemes is that they can regenerate hyperbolas and parabolas quite efficiently

reproduction of exponential polynomials by multivariate non stationary subdivision schemes with a general dilation matrix we study scalar multivariate non stationary subdivision schemes with a general integer dilation matrix we characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols these algebraic conditions provide a useful theoretical tool for checking the reproduction properties of existing schemes and for constructing new schemes with desired reproduction capabilities and other enhanced properties we illustrate our results with several examples

component groups of unipotent centralizers in good characteristic abstract let g be a connected reductive group over an algebraically closed field of good characteristic for u g unipotent we describe the conjugacy classes in the component group a u of the centralizer of u our results extend work of the second author done for simple adjoint g over the complex numbers when g is simple and adjoint the previous work of the second author makes our description combinatorial and explicit moreover it turns out that knowledge of the conjugacy classes suffices to determine the group structure of a u thus we obtain the result previously known through case checking that the structure of the component group a u is independent of good characteristic

generation of fractal curves and surfaces using ternary 4 point interpolatory subdivision scheme in this paper the generation of fractal curves and surfaces along with their properties using ternary 4 point interpolatory subdivision scheme with one parameter are analyzed the relationship between the tension parameter and the fractal behavior of the limiting curve is demonstrated through different examples the specific range of the tension parameter has also been depicted which provides a clear way to generate fractal curves since the fractal scheme introduces in the paper have more number of control points therefore it gives more degree of freedom to control the shape of the fractal curve

a family of non uniform subdivision schemes with variable parameters for curve design in this paper we present non uniform subdivision schemes with variable parameter sequences a locally different tension parameter is set at each edge of the initial control polygon to control locally the shape of the resulting curve such that the scheme becomes non uniform due to the variable parameters the scheme can reproduce locally different analytic curves such as conics lissajous trigonometric and catenary curves hence blending curves including such analytic components can be successfully generated we discuss the convergence and smoothness of the proposed non uniform schemes and present some numerical results to demonstrate their advantages in geometric modeling furthermore as an application we propose a chamfering algorithm which can be used in designing automobile and mechanical products

shape controlled interpolatory ternary subdivision ternary subdivision schemes compare favorably with their binary analogues because they are able to generate limit functions with the same or higher smoothness but smaller support in this work we consider the two issues of local tension control and conics reproduction in univariate interpolating ternary refinements we show that both these features can be included in a unique interpolating 4 point subdivision method by means of non stationary insertion rules that do not affect the improved smoothness and locality of ternary schemes this is realized by exploiting local shape parameters associated with the initial polyline edges

convergence of nonstationary cascade algorithms a nonstationary multiresolution of l 2 mathbb r s is generated by a sequence of scaling functions phi k in l 2 mathbb r s k in mathbb z we consider phi k that is the solution of the nonstationary refinement equations phi k m sum j h k 1 j phi k 1 m cdot j k in mathbb z where h k is finitely supported for each k and m is a dilation matrix we study various forms of convergence in l 2 mathbb r s of the corresponding nonstationary cascade algorithm phi k n m sum j h k 1 j phi k 1 n 1 m cdot j as k or n tends to infty it is assumed that there is a stationary refinement equation at infty with filter sequence h and that sum k h k j h j infty the results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h

hyperbolic forms of ternary non stationary subdivision schemes originated from hyperbolic b splines in this work ternary non stationary subdivision schemes based on hyperbolic b spline basis functions have been presented the proposed hyperbolic ternary three point and four point subdivision schemes give pleasing as well as consistent curves with the control polygons as compared to the existing non stationary subdivision schemes obtained from trigonometric b splines the main speciality of the hyperbolic schemes is that they can reproduce hyperbolas and parabolas quite efficiently which has been demonstrated with the help of curvature plots

the invariant subspace problem for non archimedean kothe spaces abstract in this study a proof is given that if a non archimedean kothe space which is generated by an infinite matrix b b n k k n n such that b n k b n 1 k for k n n and for each k l n m n exists such that b n k 1 b n l k for n m then a continuous operator t exists that has no nontrivial closed invariant subspaces

a non stationary binary three point approximating subdivision scheme in this paper we present a non stationary binary three point approximating subdivision scheme which can generate a wide variety of c3 continuous limit curves with different initial control parameter 0 the continuity of the scheme is proved by using the conclusion in dyn and levin 1995 5 the advantage of the scheme is illustrated by a number of examples

invariant subspaces for non normable frechet spaces abstract a frechet space x satisfies the hereditary invariant subspace resp subset property if for every closed infinite dimensional subspace m in x each continuous operator on m possesses a non trivial invariant subspace resp subset in this paper we exhibit a family of non normable separable infinite dimensional frechet spaces satisfying the hereditary invariant subspace property and we show that many non normable frechet spaces do not satisfy this property we also state sufficient conditions for the existence of a continuous operator without non trivial invariant subset and deduce among other examples that there exists a continuous operator without non trivial invariant subset on the space of entire functions h c

non uniform interpolatory subdivision surface abstract this paper presents an interpolatory subdivision scheme with non uniform parametrization for arbitrary polygon meshes with arbitrary manifold topology this is the first attempt to generalize the non uniform four point interpolatory curve subdivision to surface with extraordinary points the scheme is constructed from the inspiration of the relation between the non uniform four point interpolatory subdivision scheme and the non uniform b spline refinement rule numerical examples and comparisons with the uniform interpolatory subdivision schemes indicate that the quality of the limit surface can be improved by using non uniform parameter values for non uniform initial data

on four point penalized lagrange subdivision schemes this paper is devoted to the definition and analysis of new subdivision schemes called penalized lagrange their construction is based on an original reformulation for the construction of the coefficients of the mask associated to the classical 4 points lagrange interpolatory subdivision scheme these coefficients can be formally interpreted as the solution of a linear system similar to the one resulting from the constrained minimization problem in kriging theory which is commonly used for reconstruction in geostatistical studies in such a framework the introduction in the formulation of a so called error variance can be viewed as a penalization of the oscillations of the coefficients following this idea we propose to penalize the 4 points lagrange system this penalization transforms the interpolatory schemes into approximating ones with specific properties suitable for the subdivision of locally noisy or strongly oscillating data according to a so called penalization vector a family of schemes can be generated a full theoretical study is first performed to analyze this new type of non stationary subdivision schemes then in the framework of position dependant penalization vector several numerical tests are provided to point out the efficiency of these schemes compared to standard approaches

kw sections for vinbergʼs θ groups of exceptional type let k be an algebraically closed field of characteristic not equal to 2 or 3 let g be an almost simple algebraic group of type f4 g2 or d4 and let theta be an automorphism of g of finite order coprime to the characteristic in this paper we consider the theta group in the sense of vinberg associated to these choices we classify the positive rank automorphisms via kac diagrams and we describe the little weyl group in each case as a result we show that all groups in types g2 f4 and d4 have kw sections confirming a conjecture of popov in these cases

fractal properties of the generalized chaikin corner cutting subdivision scheme in this paper we analyze the fractal properties of the generalized chaikin corner cutting subdivision scheme with two arbitrary parameters on the basis of the discussion of limit points on the limit curve the fractal range of the scheme is obtained and illustrated the results obtained suggest a clear direction for generating fractal curves by using this scheme two fractal examples and their complexity evaluation results obtained by using the box dimension are given

generalized daubechies wavelet families we present a generalization of the orthonormal daubechies wavelets and of their related biorthogonal flavors cohen daubechies feauveau 9 7 our fundamental constraint is that the scaling functions should reproduce a predefined set of exponential polynomials this allows one to tune the corresponding wavelet transform to a specific class of signals thereby ensuring good approximation and sparsity properties the main difference with the classical construction of daubechies is that the multiresolution spaces are derived from scale dependent generating functions however from an algorithmic standpoint mallat s fast wavelet transform algorithm can still be applied the only adaptation consists in using scale dependent filter banks finite support ensures the same computational efficiency as in the classical case we characterize the scaling and wavelet filters construct them and show several examples of the associated functions we prove that these functions are square integrable and that they converge to their classical counterparts of the corresponding order

binary 3 point and 4 point non stationary subdivision schemes using hyperbolic function in this paper binary 3 point and 4 point non stationary subdivision schemes are presented using hyperbolic function as basis function asymptotic equivalence method is used to investigate the continuity of the proposed schemes comparison between the proposed schemes their stationary counterparts and some existing non stationary schemes has been depicted through examples it can be observed that the proposed schemes have the capability to reproduce conics particularly parabola

involutions of reductive lie algebras in positive characteristic abstract let g be a reductive group over a field k of characteristic 2 let g lie g let be an involutive automorphism of g and let g k p be the associated symmetric space decomposition for the case of a ground field of characteristic zero the action of the isotropy group g on p is well understood since the well known paper of kostant and rallis b kostant s rallis orbits and representations associated with symmetric spaces amer j math 93 1971 753 809 such a theory in positive characteristic has proved more difficult to develop here we use an approach based on some tools from geometric invariant theory to establish corresponding results in good positive characteristic among other results we prove that the variety n of nilpotent elements of p has a dense open orbit and that the same is true for every fibre of the quotient map p p g however we show that the corresponding statement for g conjectured by richardson is not true we provide a new mostly calculation free proof of the number of irreducible components of n extending a result of sekiguchi for k c finally we apply a theorem of skryabin to describe the infinitesimal invariants k p k

a non stationary combined subdivision scheme generating exponential polynomials abstract in this paper a non stationary combined subdivision scheme is presented which can unify several existing non stationary approximating and interpolatory subdivision schemes this scheme is obtained by generalizing the connection between the approximating and interpolatory schemes in the stationary case first formalized by maillot stam using a push back operator to the non stationary case for such a combined scheme we investigate its c l convergence and exponential polynomial generation reproduction property and get that it can reach c 4 degree of smoothness and generate reproduce certain exponential polynomials with suitable choices of the parameters besides we give a more generalized combined scheme for the purpose of generating and reproducing more general exponential polynomials the performance of our new schemes is illustrated by several numerical examples

improved binary four point subdivision scheme and new corner cutting scheme a binary 4 point approximating subdivision scheme presented by siddiqi and ahmad 2006 9 has been improved by introducing a global tension parameter an improved form of the scheme generates a family of c n n 1 2 3 4 limiting curves for certain range of global tension parameter and a c 5 limiting curve for a fixed value also a new corner cutting subdivision scheme has been proposed which generates a limiting curve of c 1 continuity the laurent polynomial method has been used to investigate the order of the derivative continuity of both subdivision schemes the performance of the subdivision schemes has been exposed by considering examples and compared with the existing schemes

a combined approximating and interpolating ternary 4 point subdivision scheme abstract in this paper a new combined approximating and interpolating ternary 4 point subdivision scheme with multiple parameters is proposed a set of nice properties such as support continuity and polynomial generation are briefly discussed the new combined scheme not only contains a lot of classical ternary schemes as special cases but also generates brand new ternary schemes compared to other approximating subdivision schemes limit curves generated by the new scheme are more consistent with the corresponding control polygons and keep detail features better examples are given to show the effectiveness of the scheme furthermore fractal property is analyzed and fractal curves are also given

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

the invariant subspace problem for the space of smooth functions on the real line abstract we construct a continuous linear operator acting on the space of smooth functions on the real line without non trivial invariant subspaces this is a first example of such an operator acting on a frechet space without a continuous norm the construction is based on the ideas due to c read who constructed a continuous operator without non trivial invariant subspaces on the banach space l 1

analysis of asymptotically equivalent binary subdivision schemes non stationary uniform subdivision schemes are presented including some important examples as the up function and exponential splines the multivariate analysis of subdivision schemes is extended to a class of non stationary schemes which are asymptotically equivalent to converging stationary or non stationary schemes

a new non stationary binary 6 point subdivision scheme in this paper the hyperbolic form of binary 6 point interpolating non stationary subdivision scheme has been constructed using the hyperbolic function some of the important properties of the proposed scheme has been discussed the comparison of the proposed scheme with the trigonometric form of binary 6 point interpolatory non stationary scheme is depicted through examples which indicate that the proposed scheme not only accommodates and is more consistent with the control polygon but also generates pleasing curves corresponding to the larger parametric values in and outside the parametric interval as compared to the trigonometric form of binary 6 point scheme developed by daniel et al comparison with some other interpolating non stationary subdivision schemes has also been demonstrated

an interpolating 6 point c2 non stationary subdivision scheme in this paper we consider a 6 point interpolating c 2 binary non stationary subdivision scheme and present some of its important properties we highlight some advantages of the scheme and demonstrate its performance by some examples

limits of level and parameter dependent subdivision schemes in this paper we present a new matrix approach for the analysis of subdivision schemes whose non stationarity is due to linear dependency on parameters whose values vary in a compact set indeed we show how to check the convergence in c r s and determine the holder regularity of such level and parameter dependent schemes efficiently via the joint spectral radius approach the efficiency of this method and the important role of the parameter dependency are demonstrated on several examples of subdivision schemes whose properties improve the properties of the corresponding stationary schemes

invariant subspace problem for classical spaces of functions abstract we construct continuous linear operators without non trivial invariant subspaces on several classical non banach spaces of analysis the schwartz space of rapidly decreasing functions the space of smooth functions on a compact smooth manifold the space of holomorphic functions on the unit disc or on an arbitrary polydisc the space of entire functions on c d for d 2 as these are reflexive spaces the result gives an analogous statement for the dual spaces e g the space of tempered distributions the construction works with kothe sequence spaces and is based on methods developed by read to construct his famous example on the space l 1

analysis of non stationary hermite subdivision schemes reproducing exponential polynomials abstract the aim of this paper is to study the convergence and smoothness of non stationary hermite subdivision schemes of order 2 in conti et al 2017 provided sufficient conditions for the convergence of a non stationary hermite subdivision scheme that reproduces a set of functions including exponential polynomials the analysis has been focused on the non stationary hermite scheme with the order 3 but the case of 2 which is practically most useful is yet to be investigated in this regard the first goal of this paper is to fill the gap we analyze the convergence of non stationary hermite subdivision schemes of order 2 next we provide a tool which allows us to estimate the smoothness of a non stationary hermite scheme by developing a novel factorization framework of non stationary vector subdivision operators using the proposed non stationary factorization framework we estimate the smoothness of the non stationary hermite subdivision schemes the non stationary interpolatory hermite scheme proposed by conti et al 2015 and a new class of non stationary dual hermite subdivision schemes of de rham type

a ternary 4 point approximating subdivision scheme in the implementation of subdivision scheme three of the most important issues are smoothness size of support and approximation order our objective is to introduce an improved ternary 4 point approximating subdivision scheme derived from cubic polynomial interpolation which has smaller support and higher smoothness comparing to binary 4 point and 6 point schemes ternary 3 point and 4 point schemes see table 2 the method is easily generalized to ternary 2n 2 point approximating subdivision schemes we choose a ternary scheme because a way to get smaller support is to raise arity and we use polynomial reproduction to get higher approximation order easily 2007 elsevier inc all rights reserved

smooth curve interpolation with generalized conics abstract efficient algorithms for shape preserving approximation to curves and surfaces are very important in shape design and modelling in cad cam systems in this paper a local algorithm using piecewise generalized conic segments is proposed for shape preserving curve interpolation it is proved that there exists a smooth piecewise generalized conic curve which not only interpolates the data points but also preserves the convexity of the data furthermore if the data is strictly convex then the interpolant could be a locally adjustable gc 2 curve provided the curvatures at the data points are well determined it is also shown that the best approximation order is o h 6 an efficient algorithm for the simultaneous computation of points on the curve is derived so that the curve can be easily computed and displayed the numerical complexity of the algorithm for computing n points on the curve is about 2 n multiplications and n additions finally some numerical examples with graphs are provided and comparisons with both quadratic and cubic spline interpolants are also given

regularity of non stationary subdivision a matrix approach in this paper we study scalar multivariate non stationary subdivision schemes with integer dilation matrix m and present a unifying general approach for checking their convergence and for determining their holder regularity latter in the case m mi m ge 2 m mi m 2 the combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius as an application we prove a recent conjecture by dyn et al on the holder regularity of the generalized daubechies wavelets we illustrate our results with several examples

nonuniform cascade algorithms abstract for a sequence of bounded linear operators hk k 0 1 on a banach space s the algorithm k n hk k 1 n 1 generates a family of sequences k n n 0 k 0 1 from an initial family of vectors k 0 s k 0 1 we study the convergence of k n as n and give an application on the convergence of cascade algorithms for nonuniform splines when s is the space of all sequences i i z with norm sup i z i and i i z belong to the banach space x l 2 r

fractal generation using ternary 5 point interpolatory subdivision scheme this paper is devoted to explore the generation of fractal curves and surfaces using ternary 5 point interpolatory subdivision scheme the fractal behavior of the limiting curves and surfaces is analyzed with the help of tension parameter it may be noted that the proposed fractal scheme determines two intervals of range of shape control parameter the behavior of the shape control parameter for the generation of fractals has been visualized through five examples which provides faster rate of generation of fractals

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

operator on the space of rapidly decreasing functions with all non zero vectors hypercyclic abstract we construct a continuous linear operator t on the schwartz space s r d of rapidly decreasing functions such that each non zero orbit of t is dense the construction is inspired by the work of c read on similar operators on the space l 1 the construction due to the structure of a frechet space can be made significantly simpler than the original construction of read