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The critical length of a space of functions can be described as the supremum of the length of the intervals where Hermite interpolation problems are unisolvent for any choice of nodes. We analyze the critical length for spaces containing products of algebraic polynomials and trigonometric functions. We show the relation of these spaces with spherical Bessel functions and bound above their critical length by the first positive zero of a Bessel function of the first kind.

Disk polynomials form a basis of orthogonal polynomials on the disk corresponding to the radial weight ${\alpha +1 \over \pi }(1-r^{2})^{\alpha }$ . In this paper, the stability properties of disk polynomials are analyzed. A conditioning associated with the representation of the least squares approximation with respect to this basis is introduced and bounded. Among all disk polynomials, the least bounds are obtained for Zernike polynomials corresponding to α = 0.

In this paper, an accurate method to construct the bidiagonal factorization of Gram (mass) matrices of Bernstein bases of positive and negative degree is obtained and used to compute with high relative accuracy their eigenvalues, singular values and inverses. Numerical examples are included.

We construct a six-dimensional space of polynomials containing approximations of trigonometric functions. In this way, an approximation of a circle can be represented as a parametric curve in this space. We show that the space has shape preserving representations and we construct the basis with optimal shape preserving properties.

In this paper an accurate method to construct the bidiagonal factorization of collocation and Wronskian matrices of Jacobi polynomials is obtained and used to compute with high relative accuracy their eigenvalues, singular values and inverses. The particular cases of collocation and Wronskian matrices of Legendre polynomials, Gegenbauer polynomials, Chebyshev polynomials of the first and second kind and rational Jacobi polynomials are considered. Numerical examples are included. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

In this paper we provide algorithms for computing the bidiagonal decomposition of the Wronskian matrices of the monomial basis of polynomials and of the basis of exponential polynomials. It is also shown that these algorithms can be used to perform accurately some algebraic computations with these Wronskian matrices, such as the calculation of their inverses, their eigenvalues or their singular values and the solutions of some linear systems. Numerical experiments illustrate the results.

Cycloidal spaces are generated by the trigonometric polynomials of degree one and algebraic polynomials. The critical length of a cycloidal space is the supremum of the lengths of the intervals on which the Hermite interpolation problems are unisolvent. The critical length is related with the critical length for design purposes in computer-aided geometric design. This paper shows an unexpected connection of critical lengths with the zeros of Bessel functions. We prove that the half of the critical length of a cycloidal space is the first positive zero of a Bessel function of the first kind.

This paper proposes a method for learning the process of curve fitting through a general class of totally positive rational bases. The approximation is achieved by finding suitable weights and control points to fit the given set of data points using a neural network and a training algorithm, called AdaMax algorithm, which is a first-order gradient-based stochastic optimization. The neural network presented in this paper is novel and based on a recent generalization of rational curves which inherit geometric properties and algorithms of the traditional rational B&eacute, zier curves. The neural network has been applied to different kinds of datasets and it has been compared with the traditional least-squares method to test its performance. The obtained results show that our method can generate a satisfactory approximation.

For a given totally positive space of continuous functions, we analyze the construction of totally positive bases of the space of antiderivatives. If the functions of the totally positive space have continuous derivatives, normalization properties can be used to describe totally positive bases of the space of derivatives and relate them with properties of the Greville abscissae. Construction of totally positive (TP) bases of the space of antiderivatives.Relationship between normalized TP basis and that of the space of derivatives.Characterization in terms of bases with increasing Greville abscissae.

Weighted φ-transformed systems include many systems of functions useful in C.A.G.D. It is proved that these systems inherit some geometric properties of a given initial system, such as shape preservation or optimal shape preservation. A general class of important rational bases can be obtained as a particular example of weighted φ-transformed systems. For these bases, evaluation and subdivision algorithms are presented. Some relevant applications are pointed out.

Exponential polynomials as solutions of differential equations with constant coefficients are widely used for approximation purposes. Recently, mixed spaces containing algebraic, trigonometric and exponential functions have been extensively considered for design purposes. The analysis of these spaces leads to constructions that can be reduced to Hermite interpolation problems. In this paper, we focus on spaces generated by algebraic polynomials, hyperbolic sine and hyperbolic cosine. We present classical interpolation formulae, such as Newton and Aitken-Neville formulae and a suggestion of implementation. We explore another technique, expressing the Hermite interpolant in terms of polynomial interpolants and derive practical error bounds for the hyperbolic interpolant.

Hermite interpolation problems on cycloidal spaces are analyzed. Newton and Aitken-Neville formulae are obtained. Numerical examples are included.

The rational de Casteljau algorithm for the evaluation of rational Bezier curves is extended to very general rational spaces. Many examples are included.

The cycloidal spaces C n , generated by the trigonometric polynomials of degree 1 and algebraic polynomials of degree n−2, have important applications in computer-aided geometric design. The critical length of C n is the supremum of the lengths of the intervals on which C n is an extended Chebyshev space. We provide a constructive procedure to obtain the critical lengths of the spaces C n .

Objectives We present the short- and long-term results of a series of patients with symptomatic intracranial arterial stenoses treated with angioplasty and stenting. Materials and methods We reviewed patients with symptomatic intracranial stenoses greater than 50% who were treated with angioplasty, stenting, or both. We recorded demographic data and risk factors (hypertension, diabetes, dyslipemia, ischemic heart disease). We classified all lesions treated according to their location, degree of stenosis, and length. The degree of stenosis was classified as moderate (50–70%) or severe (>70%). In the follow-up, we assessed cerebrovascular accidents, episodes of ischemic heart disease, and deaths in the first 30 days and in later follow-up. Results Between 2006 and 2010, we treated 26 patients (21 men and 5 women; age range, 44–79 years; mean age, 63 years) with 29 intracranial lesions. The endovascular procedure (angioplasty + stenting) was successfully performed in 23 cases (92.0%). In the first 30 days after the procedure, 3 (11.5%) patients had adverse effects of vascular origin: 1 stroke, 1 hemorrhage, and 1 death due to thrombosis of the stent. Long-term follow-up (5–46 months) in the 25 patients who survived more than 30 days detected no recurrence of symptoms. Conclusion Endovascular treatment of intracranial stenosis is technically feasible. Short-term complications are highly prevalent. No recurrence of symptoms was detected during long-term follow-up.

Resumen Objetivos Presentamos los resultados a corto y largo plazo de una serie de pacientes con estenosis arteriales intracraneales (EAIC) sintomaticas tratados en nuestro centro mediante angioplastia y endoprotesis. Material y metodos Revision de pacientes con estenosis intracraneales > 50% sintomaticas tratados en nuestro centro mediante angioplastia y/o colocacion de endoprotesis. Se recogieron los datos demograficos y los factores de riesgo vascular (hipertension, diabetes, dislipemia, cardiopatia isquemica). Todas las lesiones tratadas se caracterizaron por su localizacion, grado y longitud de la estenosis. El grado de estenosis se califico como moderado (50-70%) o grave (> 70%). En el seguimiento se evaluaron los accidentes cerebrovasculares, episodios de cardiopatia isquemica o fallecimientos ocurridos durante los 30 primeros dias y en el seguimiento posterior. Resultados Entre 2006 y 2010 se trataron 26 pacientes (21 varones y 5 mujeres) con edades comprendidas entre 44 y 79 anos (media: 63 anos), con 29 lesiones intracraneales. El procedimiento endovascular se realizo con exito (angioplastia + endoprotesis) en 23 casos (92,0%). En los 30 primeros dias postintervencion se produjeron tres (11,5%) efectos adversos de origen vascular: un ictus, una hemorragia y un fallecimiento por trombosis de la endoprotesis. El seguimiento a largo plazo se ha realizado sobre 25 pacientes (5-46 meses). No se han detectado recurrencias de la sintomatologia. Conclusion El tratamiento endovascular de las EAIC es tecnicamente factible. Las complicaciones del procedimiento a corto plazo tienen una prevalencia alta. No se ha detectado recurrencia de la sintomatologia a largo plazo.

We prove that, in contrast to the case for rational surfaces, some tensor product representations through spaces containing algebraic, trigonometric and hyperbolic polynomials are monotonicity preserving. The surface representations provided in this paper are the only known monotonicity preserving surfaces in addition to the tensor product Bezier and tensor product B-spline surfaces.

Background and purpose: Comprehensive indications for treatment of symptomatic vertebral stenosis remain unavailable. Even less is known about endovascular treatment of asymptomatic cases. We treated symptomatic and asymptomatic vertebral ostium stenosis with angioplasty and stenting and investigated the long term outcome. Methods: Consecutive patients with two different indications were included. Group 1 (G1) had symptomatic >50% stenosis. Group 2 (G2) had asymptomatic >50% stenosis and severe lesions of anterior circulation and were expected to benefit from additional cerebral blood supply. Results: Twenty nine vertebral origin stenoses in 28 patients (75% men, mean age 64 ± 9 years) were treated. There were 16 G1 and 13 G2 cases. Technical success rate was 100%. Immediate neurological complications rate was 3.4% (one G1 patient with vertebral TIA due to release of emboli). Two further strokes were seen during follow up (32 ± 24 months): vertebrobasilar stroke in a G2 patient with permeable stent in V1 segment, new ipsilateral V3 occlusion and high-risk cardioembolic source, and carotid stroke in a G1 patient who had had ipsilateral carotid stenting. There were no deaths of any cause. Asymptomatic restenosis was observed in one out of 19 patients from both groups who underwent a follow up angiography. Conclusions: Angioplasty and stenting appears to be technically feasible and safe in asymptomatic and symptomatic vertebral stenosis. More studies are needed in order to clarify its role in primary and secondary prevention of vertebrobasilar stroke. High risk anterior circulation lesions should be taken into account as a possible indication in patients with asymptomatic vertebral stenosis.

This paper presents a unified approach to deal with spaces containing simultaneously algebraic and trigonometric or hyperbolic polynomials. Bases with optimal shape preserving and stability properties are constructed. Evaluation and subdivision algorithms for them are provided. Bases for the corresponding and mixed spline spaces are also constructed. Some nice properties of these bases and the generated curves are shown.

We analyze the critical length for design purposes of six-dimensional spaces invariant under translations and reflections containing the functions 1, cos t and sin t. These spaces also contain the first degree polynomials as well as trigonometric and/or hyperbolic functions. We identify the spaces whose critical length for design purposes is greater than 2π and find its maximum 4π. By a change of variables, two biparametric families of spaces arise. We call shape preservation region to the set of admissible parameters in order that the space has shape preserving representations for curves. We describe the shape preserving regions for both families.

The evaluation of multivariate polynomials of n variables in Bernstein–Bézier form is considered. A forward error analysis for the corresponding de Casteljau algorithm and the VS algorithm is performed. We also include algorithms that simultaneously evaluate the polynomial and provide “a posteriori” error bounds, without increasing significantly the computational cost. The sharpness of our running error bounds is shown in the case of trivariate polynomials.

Running error analysis for the bivariate de Casteljau algorithm and the VS algorithm is performed. Theoretical results joint with numerical experiments show the better stability properties of the de Casteljau algorithm for the evaluation of bivariate polynomials defined on a triangle in spite of the lower complexity of the VS algorithm. The sharpness of our running error bounds is shown.

Interesting curves can be represented in the space Q=span{1,t,t2,cos t,sin t}, t∈[0,α] (0

We analyze the connection between two ideas of apparently different nature. On one hand, the existence of an extended Chebyshev basis, which means that the Hermite interpolation problem has always a unique solution. On the other hand, the existence of a normalized totally positive basis, which means that the space is suitable for design purposes. We prove that the intervals where the existence of a normalized totally positive basis is guaranteed are those intervals where the existence of an extended Chebyshev basis of the space of derivatives can be ensured. We apply our results to the spaces Cn generated by 1,t, …, tn-2, cos t, sin t. In particular, C5 is a space suitable for design which permits the exact reproduction of remarkable parametric curves, including lines and circles with a single control polygon. We prove that this space has the minimal dimension for this purpose.

We show that five is the minimal dimension of a space required to draw a complete circle with a unique control polygon. We identify all five-dimensional spaces invariant under translations and reflections where we can find shape preserving representations of a circle parameterized by its arc length.

It is proved that the C-B-spline segments introduced by Zhang are generated by a normalized totally positive basis. It is also constructed a normalized B-basis of C-B-splines, which presents optimal shape preserving and stability properties.

We discus several alternatives to the rational Bezier model, based on using curves generated by mixing polynomial and trigonometric functions, and expressing them in bases with optimal shape preserving properties (normalized B-bases). For this purpose we develop new tools for finding B-bases in general spaces. We also revisit the C-Bezier curves presented by Zhang (1996), which coincide with the helix spline segments developed by Pottmann and Wagner (1994), and are nothing else than curves expressed in the normalized B-basis of the space P 1 = span {1,t, cos t, sin t} . Such curves provide a valuable alternative to the rational Bezier model, because they can deal with both free form curves and remarkable analytical shapes, including the circle, cycloid and helix. Finally, we explore extensions of the space P 1 , by mixing algebraic and trigonometric polynomials. In particular, we show that the spaces P 2 = span {1,t, cos t, sin t, cos 2t, sin 2t} , Q= span {1,t,t 2 , cos t, sin t} and I = span {1,t, cos t, sin t,t cos t,t sin t} are also suitable for shape preserving design, and we find their normalized B-basis.

It is analyzed when a given space of functions admits a knot insertion algorithm. Techniques related to total positivity will provide the construction of such knot insertion algorithms in a general space admitting shape preserving representations. Many examples are included.

Given a space of functions which admits shape preserving representations using control polygons, we construct a corner cutting algorithm which will be called B-algorithm. It is an evaluation algorithm satisfying important properties such as subdivision property and convergence to the curve of the resulting control polygons. Many examples are given.

Corner cutting algorithms are used in different fields and, in particular, play a relevant role in Computer Aided Geometric Design. Evaluation algorithms such as the de Casteljau algorithm for polynomials and the de Boor–Cox algorithm for B‐splines are examples of corner cutting algorithms. Here backward and forward error analysis of corner cutting algorithms are performed. The running error is also analyzed and as a consequence the general algorithm is modified to include the computation of an error bound.

Resumen Se presentan los resultados obtenidos mediante tratamiento endovascular con balones desprendibles en cuatro casos de fistula carotido-cavernosa directa (FCC directa) tratados durante los dos ultimos anos. Se consiguio ocluir la fistula con un balon y conservar permeable la arteria carotida interna en todos los casos. La remision del cuadro clinico fue completa en la totalidad de los pacientes dos semanas despues del tratamiento. Como complicacion existio unicamente un episodio transitorio de afasia y monoparesia que cedio a las pocas horas.

We present the short- and long-term results of a series of patients with symptomatic intracranial arterial stenoses treated with angioplasty and stenting.We reviewed patients with symptomatic intracranial stenoses greater than 50% who were treated with angioplasty, stenting, or both. We recorded demographic data and risk factors (hypertension, diabetes, dyslipemia, ischemic heart disease). We classified all lesions treated according to their location, degree of stenosis, and length. The degree of stenosis was classified as moderate (50%-70%) or severe (70%). In the follow-up, we assessed cerebrovascular accidents, episodes of ischemic heart disease, and deaths in the first 30 days and in later follow-up.Between 2006 and 2010, we treated 26 patients (21 men and 5 women; age range, 44-79 years; mean age, 63 years) with 29 intracranial lesions. The endovascular procedure (angioplasty+stenting) was successfully performed in 23 cases (92.0%). In the first 30 days after the procedure, 3 (11.5%) patients had adverse effects of vascular origin: 1 stroke, 1 hemorrhage, and 1 death due to thrombosis of the stent. Long-term follow-up (5-46 months) in the 25 patients who survived more than 30 days detected no recurrence of symptoms.Endovascular treatment of intracranial stenosis is technically feasible. Short-term complications are highly prevalent. No recurrence of symptoms was detected during long-term follow-up.

Comprehensive indications for treatment of symptomatic vertebral stenosis remain unavailable. Even less is known about endovascular treatment of asymptomatic cases. We treated symptomatic and asymptomatic vertebral ostium stenosis with angioplasty and stenting and investigated the long term outcome.Consecutive patients with two different indications were included. Group 1 (G1) had symptomatic50% stenosis. Group 2 (G2) had asymptomatic50% stenosis and severe lesions of anterior circulation and were expected to benefit from additional cerebral blood supply.Twenty nine vertebral origin stenoses in 28 patients (75% men, mean age 64 +/- 9 years) were treated. There were 16 G1 and 13 G2 cases. Technical success rate was 100%. Immediate neurological complications rate was 3.4% (one G1 patient with vertebral TIA due to release of emboli). Two further strokes were seen during follow up (32 +/- 24 months): vertebrobasilar stroke in a G2 patient with permeable stent in V1 segment, new ipsilateral V3 occlusion and high-risk cardioembolic source, and carotid stroke in a G1 patient who had had ipsilateral carotid stenting. There were no deaths of any cause. Asymptomatic restenosis was observed in one out of 19 patients from both groups who underwent a follow up angiography.Angioplasty and stenting appears to be technically feasible and safe in asymptomatic and symptomatic vertebral stenosis. More studies are needed in order to clarify its role in primary and secondary prevention of vertebrobasilar stroke. High risk anterior circulation lesions should be taken into account as a possible indication in patients with asymptomatic vertebral stenosis.

We analyze some properties of bivariate tensor product bases. In particular, we obtain conditions so that a bivariate tensor product basis is optimally stable for evaluation. Finally, we apply our results to prove the optimal stability of tensor product bases of spaces containing algebraic, trigonometric and hyperbolic polynomials with a lot of interest in the representation of curves and surfaces in CAGD.

Carotid cavernous fistulas are abnormal communications between the cavernous sinus and the internal and/or external carotid artery. They may be spontaneous or acquired. Two basic types are distinguished as a function of the aetiology, clinical features and treatment. The direct type is usually due to trauma while the indirect is of varied aetiology.We present a series of seven cases of direct carotid-cavernous fistulas, four post-traumatic and three spontaneous, with initial symptoms of ocular involvement and ophthalmoparesia predominantly. All were treated, in our hospital, by embolization with a detachable balloon in the venous variant of the fistula in six cases and by carotid occlusion in one case. Only one severe complication due to distal embolization occurred.It is necessary to consider the diagnosis of carotid-cavernous fistulas in patients complaining of acute alterations of ocular motility, headache and/or other ocular symptoms, especially when there is a clinical history of head injury. There should be minimal delay between diagnosis and treatment since there is a high risk of hemorrhage and irreversible sequelae caused by alteration of the venous drainage when they are long-standing. The treatment of choice for direct fistulas is by detachable balloon, which has been shown to be superior to other techniques.

Objetivo. Aun hoy, a diferencia de la endarterectomia, la evolucion a largo plazo en la angioplastia carotidea no esta bien definida. El objetivo es describir la evolucion a corto y largo plazo y la tasa de reestenosis de una serie consecutiva de pacientes con estenosis de carotida sometidos a tratamiento endovascular en el Hospital Universitario La Fe, diferenciando entre pacientes con y sin sintomas. Pacientes y metodos. Desde 1999 hasta 2010 se incluyeron 426 pacientes a los que se les realizo angioplastia carotidea (el 25,5% eran pacientes asintomaticos). Se llevo a cabo seguimiento clinico a 374 pacientes. Fueron visitados anualmente, y cada ano se les realizo estudio neurosonologico. Se recogieron las complicaciones durante los 30 primeros dias y las complicaciones a largo plazo; se recogio la tasa de reestenosis en el seguimiento y se valoraron las posibles diferencias entre pacientes con y sin sintomas. Resultados. La morbimortalidad en el primer mes fue del 4,2% (4,4% en pacientes sintomaticos y 3,8% en pacientes asintomaticos). El seguimiento medio fue de 55 meses: un 8% de pacientes presento ictus, un 11% sufrio un infarto de miocardio y un 24,3% fallecio, sin diferencias significativas entre los pacientes con y sin sintomas. La tasa de reestenosis fue del 17,3% a los cinco anos, un 7,5% igual o superior al 50%, y de ellas, en el 1,1% fueron sintomaticas. Cinco pacientes con reestenosis significativa fueron tratados con exito con tecnicas endovasculares. Conclusiones. La angioplastia carotidea es una tecnica efectiva y segura en manos experimentadas; nuestros resultados son buenos, tanto a corto como a largo plazo, con una baja tasa de reestenosis. En caso de producirse esta ultima, el intervencionismo vascular es seguro y eficaz.

We consider spaces of the form span〈1,t,…,tn−4,u1(t),u2(t),u3(t),u4(t)〉, where the functions ui (i=1,…,4) are algebraic polynomials, or trigonometric or hyperbolic functions. We find intervals [0,α] where we can guarantee that the spaces possess normalized totally positive bases (and so, shape preserving representations). We construct their normalized B-bases (with optimal shape preserving properties). We also present a unified approach to deal with the associated spline spaces. A recursive procedure to obtain bases with optimal shape preserving and stability properties is presented.

It has been recently pointed out that several orthogonal polynomials of the Askey table admit asymptotic expansions in terms of Hermite and Laguerre polynomials [López and Temme, Meth. Appl. Anal. 6 (1999) 131–146; J. Comp. Appl. Math. 133 (2001) 623–633]. From those expansions, several known and new limits between polynomials of the Askey table were obtained in [López and Temme, Meth. Appl. Anal. 6 (1999) 131–146; J. Comp. Appl. Math. 133 (2001) 623–633]. In this paper, we make an exhaustive analysis of the three lower levels of the Askey scheme which completes the asymptotic analysis performed in [López and Temme, Meth. Appl. Anal. 6 (1999) 131–146; J. Comp. Appl. Math. 133 (2001) 623–633]: (i) We obtain asymptotic expansions of Charlier, Meixner–Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Hermite polynomials. (ii) We obtain asymptotic expansions of Meixner–Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Charlier polynomials. (iii) We give new proofs for the known limits between polynomials of these three levels and derive new limits.