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In this paper, we obtain a simple formula based on the computation of some derivatives for determining the branches and the asymptotes of curves that are defined by a parametrization. For this purpose, we use some previous results and notions presented in Blasco and Pérez-Díaz, 2014a, Blasco and Pérez-Díaz, 2014b, Blasco and Pérez-Díaz, 2015, Blasco and Pérez-Díaz, 2020. From these results, we show how the generalized asymptotes of the input curve can be easily computed and we present some applications related to the ramification index and degree of the asymptote, the infinity form and the multiplicity of the infinity points. Furthermore, we show how to construct all the families of parametric curves having some given asymptotes. We develop this method for the plane case but it can be trivially adapted for dealing with rational curves in n-dimensional space. In addition, the formulaes presented can be similarly obtained for curves defined by a parametrization not necessarily rational., Agencia Estatal de Investigación

The finite element method is a powerful and general numerical method used to simulate subsurface processes. In this paper, we take recent hydraulic fracturing propagation algorithms and assess their performance when used within the finite element framework. In particular, we evaluate aperture and energy-based methodologies that are capable of extracting the propagation velocity of a hydraulic fracture propagating throughout the toughness and viscous regime. Such algorithms have the benefit of a quicker convergence on the fracture front. The aperture-based methodology consists of the multi-scale aperture asymptote that is yet to be applied with finite elements. On the other-hand, the energy-based methodology consists of a recently developed procedure for predicting the propagation velocity from the energy release rate, which is calculated using a J-integral devised for hydraulic fracturing. A comparison of the accuracy and the number of iterations required to converge on the fracture length is undertaken, and found to produce similar results for both methods. Consequently, we conclude that the higher accuracy of energy-based methods in extracting stress intensity factors does not immediately translate to a higher accuracy in extracting propagation velocities, most notably in the toughness-dominated propagation regime. Given the similar performance of the methods, and the simplicity of the aperture-based approach, we then extend the evaluation of the multi-scale aperture asymptote to the case of buoyancy-driven propagation. As a result, the aperture asymptote is shown to be a simple and efficient method for the simulation of subsurface processes using a finite element framework., Engineering Fracture Mechanics, 274, ISSN:0013-7944, ISSN:1873-7315

In this paper we deal with the following problem: given an algebraic plane curve C, implicitly defined, we determine its “asymptotic family”, that is, the set of algebraic curves that have the same asymptotic behavior as C., Agencia Estatal de Investigación

In this paper, we summarize two algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. The approach is based on the notion of perfect curves introduced from the concepts and results presented in previous papers of the same authors. From these results, we derive a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic curve parametrically defined in n-dimensional space., Agencia Estatal de Investigación

Background Whether antibiotic treatment of asymptomatic bacteriuria (AB) can prevent acute graft pyelonephritis (AGP) in kidney transplant (KT) recipients has not been elucidated. Methods In this multicenter, open-label, nonblinded, prospective, noninferiority, randomized controlled trial, we compared antibiotic treatment with no treatment for AB in KT recipients in the first year after transplantation when urinary catheters had been removed. The primary endpoint was the occurrence of AGP. Secondary endpoints included bacteremic AGP, cystitis, susceptibility of urine isolates, graft rejection, graft function, graft loss, opportunistic infections, need for hospitalization, and mortality. Results We enrolled 205 KT recipients between 2013 and 2015. AB occurred in 41 (42.3%) and 46 (50.5%) patients in the treatment and no treatment groups, respectively. There were no differences in the primary endpoint in the intention-to-treat population (12.2% [5 of 41] in the treatment group vs 8.7% [4 of 46] in the no treatment group; risk ratio, 1.40; 95% confidence interval, 0.40–4.87) or the per-protocol population (13.8% [4 of 29] in the treatment group vs 6.7% [3 of 45] in the no treatment group; risk ratio, 2.07, 95% confidence interval, 0.50–8.58). No differences were found in secondary endpoints, except for antibiotic susceptibility. Fosfomycin (P = .030), amoxicillin-clavulanic (P < .001) resistance, and extended-spectrum β-lactamase production (P = .044) were more common in KT recipients receiving antibiotic treatment for AB. Conclusions Antibiotic treatment of AB was not useful to prevent AGP in KT recipients and may increase antibiotic resistance. However, our findings should be regarded with caution, due to the small sample size analyzed.

In this paper, we summarize two algorithms for computing all the generalizedasymptotes of a plane algebraic curve implicitly or parametrically de-fined. The approach is based on the notion of perfect curve introducedfrom the concepts and results presented in [Blasco and Pérez-Díaz(2014)],[Blasco and Pérez-Díazz(2014-b)] and [Blasco and Pérez-Díaz(2015)]. From these results, we derive a new method that allow to easily compute horizontal and vertical asymptotes., Agencia Estatal de Investigación

Modelling polynomial curves or arcs with Bezier curves can be seen as a basis conversion not so easy for the rational curves. The classical representation of Rational curves based on controlled points with non negative weights as in NURBS does not cover all rational curves. This can be fixed by using the rational Bezier representation by mass points that are weighted points with negative or null weights. The curve of any rational function includes arcs denoted as connex components. These curves and their asymptotic lines are here modelled by the use of mass control points. The asymptotic lines are described by a point that are one weighted point or a vector. An algorithm proposes to represent any arc of a rational function f = P Q where P, Q are polynomials. The algorihtm starts by the rational Bezier representation by mass points with the parameter running on [0, 1], isolates any connex component of the curve by obtaining the zeroes of the denominator. The real line isthus split in subintervals. The next step consists in a homographic parameter change that transforms the [0, 1] interval into any previous interval. The parameter change does not modify the degree curve and points out the mass control points Bézier representation including the asymptotic line., La modélisation de courbes ou d'arcs de courbe par des courbes de Bézier, qui peut être vu comme un simple changement de base pour les courbes polynomiales, est moins évidente pour les courbes rationnelles. En effet, les courbes rationnelles ne se sont pas complètement capturées par le modèle classique des courbes de Bézier rationnelles à points pondérés de contrôle (i.e. les poids sont strictement positifs comme dans les NURBS), il est nécessaire d'introduire les courbes de Bézier rationnelles à points massiques de contrôle (i.e. les poids peuvent être nuls ou négatifs). La courbe représentative C f d'une fraction rationnelle f est généralement constituée de plusieurs branches (composantes connexes de la courbe). Dans cet article, nous montrons comment chaque branche est modélisée par une courbe de Bézier rationnelle à points massiques de contrôle. Pour ces courbes, la difficulté est de pouvoir modéliser les asymptotes. Cette difficulté est résolue en utilisant un vecteur (i.e. un point massique de poids nul) qui donne la direction de cette asymptote et un point pondéré (i.e. un point massique de poids non nul) qui fixe la position de cette asymptote. Pour déterminer les courbes de Bézier rationnelles (à points massiques de contrôle) modélisant chaque branche de la courbe représentative C f d'une fraction rationnelle f = P Q où P et Q sont des polynômes dépendant de la variable x, la méthode proposée commence par déterminer la courbe de Bézier ration-nelle représentant C f sur [0; 1], puis détermine, pour chaque composante connexe, la courbe de Bézier rationnelle en découpant la droite réelle R en intervalles où les extrémités sont les racines de Q(x) auxquelles sont ajoutées −∞ et +∞. Les points massiques de contrôle sont alors déterminés par un changement de paramètre homographique qui envoie l'intervalle [0; 1] sur l'un des intervalles précités.

International audience; We review Busemann's work on Hilbert geometry.

Salivary gland choristoma (SGCh) is defined as the presence of normal salivary tissue in an abnormal location. It is a rare entity in oral and maxillofacial region and its ethiology is unknown. The typical presentation of salivary gland heterotopia is an asymptomatic mass that may or may not produce saliva. Some examples of ectopic salivary tissue in the pituitary gland, in the lymph nodes, in the middle ear, in the neck, in the jaw, in the thyroid gland, in the mediastinum and in the rectum have been documented in literature. We report the case of a 61-year-old male presented with a bilateral tumorlike mass in the cheek. The mass was painless, of fibrous consistency and had size change with time. The histological diagnosis was salivary gland choristoma in the buccinator muscle. In this article, we will revise the characteristic of salivary gland heterotopias and we present a report case that has not been described in literature: a bilateral choristoma of salivary gland in the buccinator muscle, which should be included in the differential diagnosis of head and neck masses. Key words:Choristoma, heterotopias, ectopic tissue, salivary gland, buccinator muscle.

In this paper, drawing on international trade theory, I derive the asymptotic relation that holds for oligopolists’ iso-profit curves within Cournot’s game and I discuss the properties of the Nash equilibrium. Thereafter, I provide an economic rationale for such a mathematical relation. The results of this exploration suggest that for each firm the asymptotes of the iso-profit curves convey the boundaries towards which output competitors become net purchasers of the produced good. - In questo lavoro, con riferimento alla teoria del commercio internazionale, deriviamo la relazione asintotica verificata dalle curve di iso-profitto di due ipotetiche imprese che in un mercato oligopolistico competono nelle quantità. In aggiunta, dopo aver discusso le proprietà del risultante equilibrio di Nash, forniamo una spiegazione economica di questa relazione matematica. I risultati di questa indagine teorica suggeriscono che per ogni impresa gli asintoti delle curve di iso-profitto rappresentano i limiti produttivi verso i quali i rispettivi contendenti sul mercato del prodotto diventano acquirenti netti nella merce presa in considerazione.

In this paper, we generalize the results presented in [5] for the case of real algebraic space curves. More precisely, given an algebraic space curve C (parametrically or implicitly defined), we show how to compute the generalized asymptotes. The approach is based on the notion of perfect curve introduced from the concepts and results presented in [4]., Comment: 30 pages, 9 figures. arXiv admin note: text overlap with arXiv:1307.6153

This paper presents a comprehensive material invariant analysis on asymptotes and resonances of absorbed power distribution. A closed-form solution for absorbed power in the presence of distributed microwave source has been derived from the first principle, which shows the existence of two distinct asymptotic regimes: thin and thick samples. Within thin samples, absorbed power is shown to be weak function of position, while it decays exponentially within thick sample. In between thick and thin samples, absorbed power is shown to exhibit resonances, where a detailed analysis has been performed to reveal various resonating characteristics of average power. The main purpose of the analysis is to predict the occurrence of resonances and establish correlations for a priori prediction of resonating sample dimensions in the presence of various distributed microwave incidences. The closed-form analysis has been shown to be useful in forecasting the heating characteristics: hot spot vs uniform heating. It has been shown that the heating characteristics can be tuned by suitable distribution of microwave source towards optimal material processing. � 2006 Elsevier Ltd. All rights reserved.

In this paper we solve for the length of the slip zone emanating from the edge of a 90° semi-infinite contact which is nominally adhered to an elastically similar half-plane but has a small region of slip close to the edge. The solution is found using the monolithic three-quarter plane as the basic solution, which incorporates the adhered asymptotes as an ‘outer’ solution, and using a distribution of dislocations to quantify the region of slip, and also to define an inner slipping asymptote. The solution is applied to a finite square block pressed onto an elastically similar half-plane, and compared with a numerically obtained solution.

This article offers a power theory of value analysis of Wal‐Mart’s contested expansion in the retail business. More specifically, it draws on, and develops, some aspects of the capital as power framework so as to provide the first clear quantitative explication of the company’s power trajectory to date. After rapid growth in the first four decades of its existence, the power of Wal‐Mart appears to be flat‐lining relative to dominant capital as a whole. The major problems for Wal‐Mart lie in the fact that its green‐field growth is running into barriers, while its cost cutting measures seem to be approaching a floor. The article contends that these problems are in part born out of resistance that Wal‐Mart is experiencing at multiple social scales. The case of Wal‐Mart may tell us about the wider limits of corporate power within contemporary capitalism; and the research methods outlined here may be of use to scholars seeking to conduct political‐economic research on the pecuniary trajectories of other major firms.

We develop a method for computing all the generalized asymptotes of a real plane algebraic curve C implicitly defined by an irreducible polynomial f ( x , y ) ∈ R [ x , y ] . The approach is based on the notion of perfect curve introduced from the concepts and results presented in Blasco and Perez-Diaz (2013) . In addition, we study some properties concerning perfect curves and in particular, we provide a necessary and sufficient condition for a plane curve to be perfect. Finally, we show that the equivalent class of generalized asymptotes for a branch of a plane curve can be described as an affine space R m for a certain m.

Resumen: El bajo rendimiento académico, en general, la apatía y el temor que se manifiesta hacia la matemática hace que los docentes desarrollen nuevas estrategias para acercar el conocimiento a ellos, conservando su desarrollo personal, y la autonomía escolar. Para los estudiantes de décimo grado, el estudio de las leyes físicas, paralelamente con la trigonometría, donde el capítulo de las cónicas se ha tomado de último por efectos del desarrollo del presente trabajo, despierta el interés porque, se puede ver en estas dos áreas, la transversalidad de los conceptos que se han visto en las charlas y videos relacionados con la matemática y la astronomía, en el proyecto de uso del tiempo libre. El presente trabajo pretende contribuir al acercamiento de las cónicas mediante el uso de las aplicaciones de ellas en la astronomía, además en la indagación preliminar se encuentra que muchos estudiantes de la Institución Educativa Josefina Muñoz, de los grados décimos, evidencian relevante interés por la astronomía, y temas afines con la matemática, gusto desarrollado en los talleres lúdicos de aprovechamiento del tiempo libre, perteneciente a los proyectos extracurriculares, que se desarrollan en las instituciones educativas, y en este caso se aprovechará para potenciar el aprendizaje de las cónicas. Abstract: His poor academic performance in general apathy and fear manifested toward math makes teachers develop new strategies to bring knowledge to them, preserving their personal development, and school autonomy. For sophomores, the study of the laws of physics, parallel with trigonometry, where the conical section of the latter is taken for the purposes of development of this work, it arouses interest, you can see in these two areas the mainstreaming of the concepts that have been in talks and videos related to mathematics and astronomy, in the proposed use of leisure time. This paper aims to contribute to the rapprochement of conics using these applications in astronomy, and in the preliminary investigation found that many students of the educational Institution Josefina Muñoz, tenth grade, show significant interest in astronomy and issues related to mathematics, developed in workshops like playful use of leisure time, part of extracurricular projects that are developed in educational institutions, and in this case will be used to enhance learning of conics.

In this note, I derive the asymptotic relation verified by oligopolists' iso-profit curves within Cournot's game. Thereafter, I provide an economic rationale for such a mathematical relation. The results of this exploration suggest that for each firm the asymptotes of the iso-profit curves convey the boundaries beyond which output competitors become net purchasers of the good supplied in the market.

In my dissertation, I present the results of numerical investigations of the spectrum of the damped hyperbolic equation involving damping terms, both in the equation and in the boundary conditions. This equation describes the vibrations of a realistic string having distributed (Kevin-Voight) damping and smart material inclusions (self-straining actuators) in the string. The action of the actuators is modeled through specific boundary conditions that involve two independent parameters reflecting the strengths of smart materials. The dissertation has two distinct parts: the first part is devoted to the physical background of vibrational motion, and the second part is devoted to the formulation of a specific problem, presentation, and discussion of the main findings of research. In my dissertation I consider a non standard Strum-Liouville problem. The problem is nonstandard due to two factors, the model incorporates two energy decay mechanisms, i.e., the energy of a string dissipates through the internal friction (Kevin- Voight damping) and through the end points of the string. Many researchers have studied the Strum-Liouville problem for the string equation in the absence of any damping. My problem involves two different types of damping, which makes the problem much more difficult. My particular goals were to investigate the distribution of the eigenfrequencies for the string (in mathematical language, the distribution of the eigenvalues of some linear operator), to analyze dependence on the damping coefficient, and finally to support a well known conjecture in the mathematical community about the multiple eigenfrequencies. In the course of my work, an unexpected discovery was made. I already mentioned, my main goal originally was to support a well known idea concerning the behavior and properties of purely imaginary eigenvalues for the aforementioned Strum-Liouville problem. Since I have a string with energy dissipation, the corresponding spectrum is a countable set of complex points. These points geometrically converge to some horizontal asymptote; these points form a set symmetric with respect to the imaginary axes. It has been assumed tin the mathematical community that with a change in the damping of small steps, the two eigenvalues closest to the imaginary axes would move toward each other, and finally they merge into one double eigenvalue. A minor change in the damping instantly breaks the double eigenvalues into two different simple purely imaginary eigenvalues. A goal of this thesis was to support that idea numerically. Unexpectedly, it was observed that the actual behavior of the two eigenvalues closest to the imaginary axes is totally different from the prediction. Namely, when I change the damping by a very small step (10"^^), for the first dozen steps, the eigenvalues behave as expected: they slowly move in such a way that the distance between them decreases. However, at some moment, those two eigenvalues totally change their behavior and begin moving apart in opposite directions. The movement continues along two branches of a hyperbola like curve until the moment when both eigenvalues reach the imaginary axes. When I continue changes in the damping, the pair of eigenvalues moves along the imaginary axes. So, my calculations show that contrary to the widespread opinion, the eigenvalues never merge to create a multiple one. This interesting and important behavior had not previously been observed. It should be emphasized that in my research I have changed the values of the coefficient that stands for the first order derivative in time of the unknown function {Ut). In the dissertation of R. Plant II [9], the dependence of multiple eigenvalues upon changing the density of the string, i.e., the coefficient standing before the highest derivative in time {Uu)- The fact that R. Plant obtained similar results, validates the importance of the discovery made in this thesis. The results here create more unsolved questions, and opens a rich and exciting area for future research in this area.

On approximating sums by maximums and vice versa We discuss the approximation max (x, y) ≈ x + y for x, y > 0, which is found to be useful in obtaining simple and transparent approximate solutions and interpretations for analytically complicated problems.

Network optimization problems with a "scalable" structure are examined in this report. Scalable networks are embedded in a normed space and must belong to a closed family under certain transformations of size (number of nodes) and scale (dimension of the norm). The transportation problem of linear programming (TLP) with randomly distributed points and random demands, the earthwork minimization problem of highway design, and the distribution of currents in an electric grid are examples of scalable network problems. Asymptotic formulas for the optimum cost are developed for the case where one holds the scale parameter constant while increasing the size parameter, N. As occurs in some applied probability problems such as the Ising model of statistical mechanics, and the first passage of time of a random walk, the nature of the solution of linear problems depends on the dimensionality of the space. In the linear case, we find that the cost per node is bounded from above in 3+ dimensions (3+-D), but not in 1- and 2-D. Curiously, zone shape has no effect (asymptotically) on the optimum cost per point in 2+-D, but it has an effect in 1-D. Therefore, the 2-D case can be viewed as a transition case that shares some of the properties of 1-D (unbounded cost) and some of the properties of 3-D (shape-independence). A simple formula for the 2-D, Euclidean TLP is given. Asymptotic results are also developed for a class of non-linear network problems with link costs that are a concave power function of flow. It is found that if these functions are strictly concave than the solution in 2+-D is bounded.

ÖZET SİMETRİK GRUPLARIN KARAKTERLERİNİN ASİMTOTİK TEORİSİ Elif Kurtaran Matematik Bölümü Yüksek Lisans Danışman: Prof.Dr. Alexander Klyachko Ağustos, 1996 Bu çalışmada Riemann yüzeyleri arasındaki n. dereceden % : X - » Y dallanmış örtüleri ile Sn simetrik grubu arasındaki bağıntıyı inceledik. Ayrıca, n sonsuza giderken Sn simetrik grubunun karakterlerinin asimtotikleri ile normalize edilmiş ^Ml karakterleri bazı kısıtlamalar altında ele aldık. y x(ı) Anahtar Kelimeler : Örtüler, Riemann yüzeyleri, üçgenleştirme, simetrik grup, karakterler. iv ABSTRACT ASYMPTOTIC THEORY OF CHARACTERS OF THE SYMMETRIC GROUPS Elif Kurtaran M.S. in Mathematics Advisor: Prof.Dr. Alexander Klyachko August, 1996 In this work, we studied the connection between ramified coverings of Rie- mann surfaces tt : X - > Y of degree n and characters of symmetric group Sn- We considered asymptotics of characters of Sn as n - » oo and normalized characters 4fj of Sn under some restrictions. Keywords : Coverings, Riemann surfaces, triangulations, symmetric group, characters. m 71

La théorie de couche limite du second ordre a été développée par Van Dyke en 1962 en appliquant la technique des développements asymptotiques raccordés aux équations de Navier-Stokes. Cette théorie n'est pas totalement satisfaisante lorsque l'écoulement externe présente des variations importantes selon la normale à la paroi. Ceci se produit notamment dans les écoulements hypersoniques sur des corps de rentrée, ou la couche limite interagit avec le gradient d'entropie créé par l'onde de choc courbe. Une théorie améliorée est proposée ici, basée sur l'emploi simultané des développements asymptotiques et de la formulation déficitaire. Les équations déficitaires de couche limite sont tout d'abord établies pour des écoulements incompressibles bidimensionnels sur une paroi plane et résolues pour plusieurs formes de cisaillement du fluide parfait. En deuxième partie, on considère un écoulement de gaz parfait sur un corps de révolution, avec application à un hyperboloïde hypersonique simulant l'intrados de la navette spatiale américaine. Les résultats sont comparés à ceux de la théorie de Van Dyke ainsi qu'à des solutions des équations de Navier-Stokes. Pour un coût équivalent, la formulation déficitaire fournit généralement de meilleures prévisions que la théorie classique, en assurant notamment un raccord satisfaisant entre la couche limite et le fluide parfait. Une nette amélioration est également sensible sur les flux thermiques et sur le frottement pariétal. sans objet

Pas disponible / Not available, Notre travail est une contribution à l'étude du comportement des milieux hétérogènes ayant une structure géométrique périodique. Les outils essentiels utilisés sont la méthode d'homogénéisation et la méthode des perturbations singulières. Le travail s'est fait en trois parties. La première partie est consacrée à l'étude du transfert simultané de chaleur et de masse dans un milieu poreux humide. On clarifie l'influence des phénomènes d'évaporation-condensation sur la conductivité thermique apparente, et on étudie la dépendance du comportement global du matériau par rapport à l'ordre de grandeur relatif de deux petits paramètres : la conductivité de la phase gazeuse et la taille de caractéristique des pores. Dans la deuxième partie, on s'intéresse à l'écoulement macroscopique de deux fluides non-miscibles à travers un milieu poreux. Les "lois de Darcy généralisée" classiques sont réexaminées d'un point de vue théorique en utilisant la méthode d'homogénéisation. il est apparu qu'elles devraient être couplées, de façon à prendre explicitement en compte les interactions aux interfaces fluide-fluide. Dans la troisième partie, on étudie l?écoulement asymptotique de deux fluides immiscibles à travers des interstices de faibles épaisseurs entre deux solides. Par une méthode de perturbation, on établit des équations limites qui généralisent la loi de Reynolds classique au cas diphasique, en lui superposant des termes de couplage traduisant les effets d'un fluide sur l'autre

The hazard rate function h{x) = f{x)/[1-F{x)], corresponding to a distribution function F with density function / , is one of the most important parameters in reliability and other fields, since h{x)dx can be interpreted as the probability that an object fails in the time interval [x,x + dx] given that the object has survived to time x. A problem of considerable interest, especially to reusability engineers, is the estimation of h from a sample of n independent and identically distributed nonnegative lifetimes X1 , X2,…,Xn , with or without censoring. By censoring we mean that the occurrence of the random event of interest (called a failure) is prevented by the previous occurrence of another event (called a censoring event). If the censoring event is also a random variable, we have the randomly censored model. If there are several randomly censoring events, we have the multiple competing risks model. This paper discusses three nonparametric estimators of the hazard rate function h: the Fourier integral estimator, the nearest neighbor estimator in the multiple competing risks model, and the discrete maximum penalized-likelihood estimator in the randomly censored model. The asymptotic behavior of these estimators will be studied and conditions for strong consistency will be given.

PhD (Natural Sciences), North-West University, Potchefstroom Campus In hierdie hoofstuk word daar eerstens 'n oorsig gegee van die stogastiese proses benadering tot die bepaling van die asimptotiese verdeling van sekere passingstoetsgroothede. (Sien bv. Anderson en Darling (1952); Kac, Kiefer en Wolfowitz (1955); Parzen (1964) Bl. 99). Daar word ook gedui op sekere probleme van hierdie benadering wat veroorsaak dat dit nie toegepas kan word in gevalle wat ons graag sou wou beskou nie. Vervolgens gee ons 'n toetsstatistiek wat gebasseer is op rangorde statistieke en wat nou verwant is aan die sogenaamde Cramér-Smirnov statistiek. Indien die stogastiese proses benadering op hierdie statistiek toegepas word, sal bogemelde probleem ook hier voorkom. As alternatiewe benadering word aangetoon hoedat die statistiek effektief geskryf kan word as 'n kwadratiese vorm in onafhanklik en iden= ties verdeelde stogastiese veranderllkes. Die probleem herlei dus na die bepaling van die asimptotiese. verdeling van 'n kwadratiese vorm en hierdeur word die pro= bleem hierbo gemeld, vermy. Doctoral