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We state and prove a theorem showing how iterates of the Volterra operator can be used to evaluate indeterminate forms of type0/0. This general result allows one to recover the Boltzmann-Gibbs entropy as a limit of a wide class of generalized entropies, as considered in an earlier work.

As was pointed out in Chap. 2, the central idea in analysis is that of limit, which was introduced and studied for sequences of real numbers, i.e., for functions \(x:\mathbb{N}\rightarrow \mathbb{R}.\) In particular, the behavior of the term x n ​: = x(n) was studied under the assumption that the element n in the domain of our sequence was approaching infinity.

Background Chagas disease patients with right bundle-branch block (RBBB) have diverse clinical presentation and prognosis, depending on left ventricular (LV) function. Autonomic disorder can be an early marker of heart involvement. The heart rate recovery (HRR) after exercise may identify autonomic dysfunction, with impact on therapeutic strategies. This study was designed to assess the HRR after symptom-limited exercise testing in asymptomatic Chagas disease patients with RBBB without ventricular dysfunction compared to patients with indeterminate form of Chagas disease and healthy controls. Methods One hundred and forty-nine subjects divided into 3 groups were included. A control group was comprised of healthy individuals; group 1 included patients in the indeterminate form of Chagas disease; and group 2 included patients with complete RBBB with or without left anterior hemiblock, and normal ventricular systolic function. A symptom-limited exercise test was performed and heart rate (HR) response to exercise was assessed. HRR was defined as the difference between HR at peak exercise and 1 min following test termination. Results There were no differences in heart-rate profile during exercise between healthy individuals and patients in indeterminate form, whereas patients with RBBB had more prevalence of chronotropic incompetence, lower exercise capacity and lower HRR compared with patients in indeterminate form and controls. A delayed decrease in the HR after exercise was found in 17 patients (15%), 9% in indeterminate form and 24% with RBBB, associated with older age, worse functional capacity, impaired chronotropic response, and ventricular arrhythmias during both exercise and recovery. By multivariable analysis, the independent predictors of a delayed decrease in the HRR were age (odds ratio [OR] 1.11; 95% confidence interval [CI] 1.03 to 1.21; p = 0.010) and presence of RBBB (OR 3.97; 95% CI 1.05 to 15.01; p = 0.042). Conclusions A small proportion (15%) of asymptomatic Chagas patients had attenuated HRR after exercise, being more prevalent in patients with RBBB compared with patients in indeterminate form and controls.

Doze pacientes com idades entre 7 a 12 anos, na forma indeterminada da doença de Chagas, com sorologia e xenodiagnóstico positivos, receberam tratamento específico. Dois pacientes tomaram 7mg/kg de nifurtimox durante 60 e 90 dias e 10 usaram 5-7mg/kg de benznidazol durante 60 dias. A evolução clínica foi verificada através de exame clínico, eletrocardiograma, exame radiológico contrastado do esôfago. Após o tratamento somente uma (8,3%) paciente apresentou todos os exames negativos. Oito deles foram avaliados após oito anos do tratamento e 4 acompanhados durante 20 anos. Sete (58,4%) permaneceram na forma indeterminada e 4 (33,3%) chagásicos progrediram clinicamente para cardiopatia grau II e/ou esofagopatia, apesar do tratamento precoce. São necessários estudos com maior número de crianças na fase indeterminada e acompanhamento a longo prazo para se estabelecer a influência do tratamento específico na evolução da doença de Chagas. Twelve chagasic patients between the ages of seven and twelve, in the indeterminate phase with serology and xenodiagnosis positive, received the specific treatment. Eight of these were evaluated after an eight-year treatment period and four were followed-up during 20 years. Two patients took 7mg/kg of nifurtimox during sixty and ninety days and ten of these used 5-7mg/kg of benznidazole during 60 days. The clinical outcome was verified through clinical examination, electrocardiogram and contrasted X-ray of the esophagus. After the treatment, only one patient presented negativity in all the examinations. Seven (58.4%) remained in the indeterminate form and despite the precocious treatment four chagasic patients (33.3%) progressed clinically to second degree cardiopathy and/or megaesophagus.

We obtain logarithmic behaviours of a four-point correlation function in the c=-2 conformal field theory by using the Feigin-Fuchs construction. It becomes an indeterminate form by a naive evaluation, but is obtained by introducing an appropriate regularization procedure., Comment: LaTeX, 7 pages

The objective was to study the disorders of upper esophageal sphincter in 37 patients with indeterminate clinical form of Chagas’ disease. Eighty (48.6%) patients with synchronic waves were found. The average maximum pressure value of the upper esophageal sphincter was significantly higher in the synchronic group. Subjects with indeterminate clinical form of Chagas’ disease may have functional disorders demonstrated by an increase in the pressure of the upper esophageal sphincter.

A forma indeterminada da doença de Chagas é definida pela presença de infecção pelo Trypanosoma cruzi na ausência de manifestações clínicas, radiológicas e eletrocardiográficas de acometimento cardíaco ou digestivo. Pacientes na forma indeterminada podem apresentar anormalidades cardiovasculares significativas à propedêutica mais avançada. Entretanto, a validade do conceito de forma indeterminada tem sido reafirmada, pela simplicidade diagnóstica e benignidade do prognóstico. Na prática clínica, dificuldades diagnósticas são freqüentes, relacionadas à subjetividade e ao significado incerto de achados clínicos, eletrocardiográficos e radiológicos. Adicionalmente, o prognóstico na forma indeterminada não é uniformemente bom: após cinco a 10 anos, postula-se que um terço dos pacientes evoluirão para a forma cardíaca. A morte súbita, uma complicação rara, pode ser a primeira manifestação da doença. É necessária uma reavaliação do conceito de forma indeterminada, com redefinição dos critérios diagnósticos e da conduta terapêutica. A estratificação do risco individual, através de métodos clínicos e não-invasivos, pode permitir o reconhecimento de grupos de risco aumentado, passíveis de intervenções terapêuticas. Como o tratamento etiológico pode prevenir o aparecimento da cardiopatia, seu papel no manejo da forma indeterminada deve ser reavaliado. The indeterminate form of Chagas' disease is defined by the absence of clinical, radiological and electrocardiographic manifestations of cardiac or digestive involvement in Trypanosoma cruzi chronic infected persons. When submitted to advanced cardiovascular tests, these patients may present significant abnormalities. However, the indeterminate form concept was reaffirmed as valid, since diagnostic criteria are simple and prognosis is benignant. In clinical practice, diagnostic difficulties are frequent, related to subjectivity and uncertain meaning of clinical, electrocardiographic and radiological findings. Moreover, indeterminate form prognosis is not equally good: after five to 10 years, a third of patients will have cardiopathy. Sudden death, a rare complication, may be the first manifestation of Chagas' disease. It is necessary to reappraise indeterminate form concept, redefining diagnostic criteria and therapeutic management. Clinical and noninvasive evaluation may allow individual risk stratification; therapeutic interventions may be beneficial in high risk groups. Since etiologic treatment may prevent cardiopathy, its role in indeterminate form management must be reassessed.

Purpose: Up to 20 to 30% of chronic Chagas' disease patients are expected to develop the cardiac form of the disease. The identification of left ventricular (LV) diastolic dysfunction or left atrial (LA) dysfunction may yield a strategy to recognize early cardiac involvement. Our aim was to analyze these parameters in Chagas' disease patients with normal LV systolic function using new echocardiographic techniques. Methods: Echocardiograms from 52 Chagas' disease patients with the indeterminate form, 29 patients at the stage A of the cardiac form (changes limited to the electrocardiogram) and 25 controls were consecutively acquired. LV diastolic function was analyzed by interrogation of the mitral inflow, pulmonary vein flow, color M-mode flow propagation velocity, LV untwist and tissue Doppler of the mitral annulus. LA function was analyzed by real-time 3 dimensional echocardiography and LA strain (ɛ) analysis including global peak positive LA ɛ (ɛ pos peak), peak negative LA ɛ and total LA ɛ. Results: All groups presented similar age, gender distribution, chamber diameters, LV mass and LV systolic function. Diastolic dysfunction was more prevalent in stage A patients. While all controls presented normal LV diastolic function, 11.5% of the patients with the indeterminate form presented delayed relaxation and 2% pseudo-normal LV diastolic function and 35.5% of the patients at the stage A of the cardiac form presented delayed relaxation and 13.5% pseudo-normal LV diastolic function (p=0.0001). Tissue Doppler was the best index to discriminate the presence of diastolic dysfunction. E/E' ratio was progressively higher (C: 5.8±1.6; indeterminate: 7.0±1.7; stage A: 8.0±2.9; p=0.0008) and E'/A' ratio was progressively lower (C: 1.5±0.3; indeterminate: 1.2±0.5; stage A: 1.0±0.3; p=0.0004) from controls towards stage A patients. LA volumes were higher in stage A patients than in controls, but LA emptying fractions did not differ among the groups. LA ɛ pos peak was lower in stage A patients (15.1±4.2%) than in patients with the indeterminate form (17.9±4.8%) and controls (17.8±4.3%; p=0.02), while the other LA ɛ parameters did not differ among the groups. Conclusions: Patients with Chagas' disease and normal LV systolic function present LV diastolic dysfunction, which is more prevalent in patients at the stage A of the cardiac form. Tissue Doppler was the best index to discriminate the presence of diastolic dysfunction and routine evaluation of E'/A' ratio may enable its early recognition. Depression of LA conduit function in stage A patients was observed only by LA strain analysis.

Universidade Estadual Paulista Julio de Mesquita Filho (UNESP), Departamento de Doencas Tropicais e Diagnostico Por Imagem, Botucatu, Rubiao Junior, CEP 18618-000, SP, Brasil

The tendency of one event F to cause a later event E was explicated by a simple formula in terms of probability by Good (1961/62), more specifically in terms of ‘weights of evidence’ (logarithms of Bayes factors). The argument was based on the concept of causal circuits. The same explication, and a generalization, are reached here without making use of that concept, and the mathematical level is elementary.

by stating something along the lines of "it measures the strength of the linear relationship between paired x and y values." Students generally easily understand that -1 < r < 1, with increasingly large values of Jrl corresponding to increasingly stronger correlation. However, the question "What is the value of r if all of the y-values are equal?" prompted some interesting discussions and surprising results in a recent course I taught. In this capsule, we will recount some simple, yet nonintuitive, phenomena about correlation that arose while we explored some answers to this question. It is known that the sample correlation coefficient r cannot be computed if all the y-values or all the x-values are equal [D. Sheskin, Handbook of Parametric and Nonparametric Statistical Procedures, CRC Press (1997) 549-550]. Indeed, if Yi = c for all i, then the formula above for r results in the indeterminate form 0. Attempts to use intuition to define r in this case are equally unsuccessful. If one focuses on the word linear in the first sentence above, then the answer to the question could be 1. Of course, there is no reason that the correlation should be positive, so one could argue for r = -1 just as well. On the other hand, if one focuses on the phrase strength of the relationship between paired x and y, then it is easy to argue that r = 0 since for any (x, y) pair, the x-value seems to have no bearing on the value of y. As any good calculus student knows, however, some discontinuities are removable. Perhaps this happens here. Certainly if one of the Yi values is altered ever so slightly from the others, the formula for r will be valid and we won't have changed the correlation much, right? We tried this as a class using the software Excel. For the six pairs of data {(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2.01)} the linear correlation was 0.65465367....

A atividade colinesterásica foi determinada no plasma e em eritrócitos de 10 indivíduos normais e em 26 pacientes chagásicos crônicos, sendo 6 com insuficiência cardíaca congestiva compensada, 10 com cardiopatia sem insuficiência cardíaca congestiva e 10 com a forma indeterminada. Os valores enzimáticos encontrados foram (média ±sd): 2196 ± 442 UI/L no plasma e 19,4 ± 3,3 Ul/g Hb em eritrócitos do grupo controla; 2291 ± 317 UI/L no plasma e 19,2 ±3,7 Ul/g Hb em eritrócitos dos chagásicos com insuficiência cardíaca congestiva compensada; 2445 ± 357 UI/L no plasma e 18,3 + 3,0 Ul/g Hb em eritrócitos dos cardiopatas chagásicos sem insuficiência cardíaca congestiva; 2006 ± 327 UI/L no plasma e 17,8 ± 3,1 Ul/g Hb em eritrócitos de chagásicos com a forma indeterminada. Nossos dados mostram que o comprometimento neuronal que ocorre nos cardiopatas chagásicos crônicos não é suficiente para levar a alterações significativas (p > 0,05) das atividades colinesterásicas no plasma e em eritrócitos.Cholinesterase activity was measured in the plasma and erythrocytes from 10 healthy individuals and 26 chronic chagasic patients. The chronic chagasic patients were distributed in 3 groups: 6 chagasic patients with compensated heart failure; 10 chagasic patients with cardiopathy but without heart failure; and 10 chagasic patients with the indeterminateform. The enzymatic levels achieved in plasma and erythrocytes were. respectively: 2,196 ± 442 UI/L and 19.4 ± 3.3 Ul/g Hb from Controls; 2,291 ±317 UI/L and 19.2 ±3.7 Ul/g Hb from chagasic patients with compensated heart failure; 2,445 ±357 UI/L and 18.3 ± 3.0 Ul/g Hb from chagasic patients with cardiopathy but without heart failure and 2,006 ± 327 UI/L and 17.8 ± 3.1 Ul/g Hb from chagasic patients with the indeterminate form. Our data show that neuronal injury in the chronic cardiopathy is not sufficient to promote significant alterations (p > 0,05) in the cholinesterase activity in the plasma and erythrocytes.

Autologous mixed lymphocyte reactions (AMLR) were studied in 18 individuals chronically infected with Trypanosoma cruzi. These individuals were further classified into three clinical groups: the asymptomatic indeterminate form (n = 5); the mega disease form (n = 5); and the cardiomyopathy form (n = 8). While patients with mega disease showed a normal proliferative response when compared with normal controls, the indeterminate group showed a lowered response in sharp contrast with the heart disease group, which presented a very high proliferative response to autologous non-T cells. These abnormal AMLR represent direct evidence for an immunoregulatory malfunction which may be involved in the inflammatory cardiac damage in chronic Chagas' disease.

A tangent stiffness matrix of a frame member has been formulated and used in previous papers for the nonlinear analysis of frameworks. However that formulation gives rise to indeterminate forms of the type 0/0 when the axial force in the member vanishes. The evaluation of the expressions in the limit and for vanishingly small values of the axial force is provided in the present paper.

A method for the automatic evaluation of algebraic limits is described. It combines many of the techniques previously employed, including top-down recursive evaluation, power series expansion, and L'Hôpital's rule. It introduces the concept of a special algebraic form for limits. The method has been implemented in MODE-REDUCE.

The availability of sophisticated computer graphics provides an extremely powerful tool for investigating the geometrical aspects of many concepts and methods in mathematics. This is particularly applicable to calculus where much of the underlying material is geometric in nature. When teaching certain topics in calculus, most notably limits of indeterminate forms and logarithmic differentiation, we routinely deal with rather complex functions which are almost impossible to graph accurately by hand. In the present paper, a survey of some of the more interesting and surprising graphs which actually arise are displayed when the corresponding functions are viewed through the eye of the computer.

In a clinical, radiological and electrocardiographical, follow-up study of the "case control" type performed in Virgem da Lapa, Minas Gerais State, Brazil, 124 chagasic patients were followed during six years. The results of the patients, the majority in the indeterminate form, did not register any change, in 32.2% there was a progress in the disease and in 5.6% the electrocardiogram returned to normal. These results when compared to that achieved by the control group, composed of pairs of non chagasic persons with the same age and sex, was shown to be 27.4% higher than among patients with positive serology. This factor represents the excess risk or exclusively chagasic component in the development of the disease. No differences were observed by sex related to the development of the disease. It was more premature and seven times more frequent however when related to the cardiopathy than to the megaesophagus. Both conditions occurring mainly in slight or moderate degree. In 192 chagasic patients and 188 non chagasic persons observed in that area in the same period, the mortality was 3.6 times higher among the chagasic patients with a letality due to cardiopathy of 8.9% without difference between sexes but more premature among the males. Sudden death was more frequent than that one caused by cardiac insufficiency. The prognostic was good for the patients with indeterminate and digestive forms and reserved for patients with the highest degree of cardiopathy.

Data on the epidemiology and the natural history of the indeterminate form of human chronic Chagas' disease (IFCCD) are discussed, revealing its great importance in endemic areas of Brazil. The work shows that IFCCD presents a gradual and very slow course, causing a benign picture in the studied patients. Evolution patterns, prognostic and anatomopathological features are also discussed. For practical purposes, the classical concept of IFCCD proved to be simple, operational and consistent, It is defined by the absence of symptoms and clinical findings in chronic infected patients with positive serology and/or parasitological examinations for Trypanosoma cruzi coupled with normal electrocardiographic and radiological exams (heart, oesophagus and colon X-Rays). If a patient is submitted to more rigorous and sophisticated tests, these can reveal some alterations, generally small ones and unable to interfere with the prognosis of the infection. It is suggested that research lines specially related to the evolution ary factors and immunological involvement during this phase be adopted.Analisam-se epidemiologia e a história natural da forma indeterminada da doença de Chagas (FCI), confirmando-se sua grande importância epidemiológica em áreas endêmicas da tripanosomose, no Brasil. Os dados mostram que a evolução da FCI é geralmente lenta, com bom prognóstico, discutindo-se alguns fatores evolutivos, o prognóstico e o substrato anatomopatológico desta entidade. Sob o prisma prático, verifica-se que o conceito clássico da FCI se mostra simples, consistente e operacional: define-se no paciente crônico e assintomático, portador de provas sorológicas e/ou parasitológicaspositivas para T. cruzi, com exames clínico, eletrocardiográfico e radiológico (área cardíaca, esófago e cólon) normais. Exames outros mais rigorosos e sofisticados podem demonstrar algumas alterações e anormalidades nestes pacientes, geralmente discretas e pouco interferentes no prognóstico. Fazem-se algumas sugestões de pesquisa sobre a FCI, principalmente com relação ao envolvimento imunológico e a fatores de evolução.

This study was designed to investigate free-recall learning of trigrams as a function of intralist similarity and form of structure. Similarity refers to the extent to which stimuli on the list share the same letters; form of structure refers to the contingencies between the letters which occur in the subset of trigrams. With similarity and form used to generate items, three different lists emerged: high similarity, good form (G); high similarity, poor form (P); and low similarity, indeterminate form (L). The results indicate that high-similarity items are learned more readily than low-similarity items (L) only if the high-similarity items also have good form (G); when high-similarity items have poor form (P), the L stimuli are more readily acquired. Theoretically, Underwood's stage-analysis position is supported when performance on G trigrams is compared to that on L items. Garner's prediction, that G items will be learned more readily than P items, is also supported.

The object of this work is to find the limit of an expression ƒ(x) as the variable x approaches a value x0, finite or not, and when replacing x by x0 does not allow us to compute this limit. This is a case-study in the fields of theorem-proving and of formula manipulation. Using canonical forms, the program is able to operate on polynomials inside the expression, to use finite MacLaurin's series and to compare powers with exponentials or logarithms. Unfortunately, this is not always sufficient because some expressions have to be transformed one or several times before these methods can succeed. Then the program will use heuristics to find the best transformation to apply to the given expression. The expression will be transformed as many times as necessary until the methods can give the solution.

(1978). Indeterminate Forms of Exponential Type. The American Mathematical Monthly: Vol. 85, No. 6, pp. 484-486.

The subject of indeterminate forms is usually treated in algebra or in calculus; and hence it seldom occurs to the student that these forms may occur in trigonometry. It is sometimes desirable to examine critically some of the operations in proving identities, so that a false conclusion may not be drawn. Frequently the process of dividing both numerator and denominator by a factor holds only for certain values of the variable, or variables, involved. The occasional presence of indeterminates in solving spherical triangles teaches us that a spherical triangle is not always determined when three parts are given, as is sometimes supposed.

This chapter discusses indeterminate forms and improper integrals. If f and g are two functions having the property that lim x → x 0 f ( x ) = 0 and lim x → x 0 g ( x ) = 0, the function f/g has the indeterminate form 0/0 at x 0 . The chapter presents a rule that is called L'Hopital's rule for finding lim x → x 0 [ f ( x )/ g ( x )] in certain cases when f / g has the indeterminate form 0/0 at x 0 . It also presents the proof of this result and considers other indeterminate forms. There are other situations in which lim x → x 0 [ f ( x )/ g ( x )] cannot be evaluated directly. The chapter defines one such important case. The definitions states, “Let f and g be two functions having the property that lim x → x 0 f ( x ) = ± ∞ and lim x → x 0 g ( x ) = ± ∞, where x 0 is real, +∞ or − ∞. Then the function f / g has the indeterminate form ∞/∞ at x 0 .” There are other indeterminate forms that can be dealt with by applying L'Hopital's rule. The chapter explains these forms presenting some examples.

Infinity has many faces. The layman often perceives it as a kind of “number” larger than all numbers. For some primitive tribes infinity begins at three, for anything larger is “many” and therefore uncountable. The photographer’s infinity begins at thirty feet from the lens of his camera, while for the astronomer—or should I say the cosmologist—the entire universe may not be large enough to encompass infinity, for it is not at present known whether our universe is “open” or “closed,” bounded or unbounded. The artist has his own image of the infinite, sometimes conceiving it, as van Gogh did, as a vast, unending plane on which his imagination is given free rein, at other times as the endless repetition of a single basic motif, as in the abstract designs of the Moors. And then there is the philosopher, whose infinity is eternity, divinity, or the Almighty Himself. But above all, infinity is the mathematician’s realm, fir it is in mathematics that the concept has its deepest roots, where it has been shaped and reshaped innumerable times, and where it finally celebrated its greatest triumph.

The indeterminate form of Chagas' disease is characterized by positive serology for the disease in the absence of clinical findings and in the presence of both normal esophagogram and electrocardiogram. When more sensitive methods were used, abnormalities have been described either in the esophagus or in the heart. The authors have studied simultaneously the esophagus and the heart in the same subjects. In thirteen adults with diagnosis of indeterminate form and nine adult controls, the esophageal manometry both in basal conditions and after stimulus (bethanecol) and vectorcardiogram were performed. In the control group none of the subjects presented concomitant esophageal and cardiac alterations while in the chagasic group 92,3% of the patients presented results simultaneously altered. It is concluded that the studied patients showed indications of parasympathetic denervation manifested by simultaneously esophageal and heart alterations. A forma indeterminada da Doença de Chagas é caracterizada por sorologia positiva com ausência de manifestações clínicas, na presença de resultados normais aos exames radiológico do tubo digestivo e eletrocardiográfico. No presente trabalho, os autores estudam simultaneamente o esôfago e o coração, nos mesmos indivíduos. Treze adultos com diagnóstico de forma indeterminada da Doença de Chagas e nove adultos controles foram submetidos ao exame vetorcardiográfico e à manometria esofágica em condições basais e sob estímulo com cloridrato de betanecol (0,08 mg/kg p.c). No grupo controle nenhum dos indivíduos apresentou concomitância de alterações esofágicas e cardíacas, enquanto no grupo chagásico 92,3% dos pacientes apresentaram exames concomitantemente alterados. Concluem que os pacientes estudados apresentam evidências de desnervação parassimpática manifestada por alterações simultâneas esofágicas e cardíacas.

In this chapter we shall introduce a few basic concepts involving units, Newton’s second law of motion, and some “common sense” statements of basic transport relationships. After a brief calculus review, there will be presented statements of continuity, differentiability, and differentials, L’Hopital’s rule, and the Leibnitz rule for differentiating integrals. We conclude with some vector and tensor operations, and a discussion of matrix algebra.

This chapter discusses continued fractions and indeterminate equations. Any positive rational number can be represented in the form of a so-called continued fraction. The process of finding the greatest common divisor of two numbers by means of consecutive divisions is called Euclid's algorithm. The properties of terminating fractions hold infinite continued fractions and permit the finding of rational numbers as close as desired to sufficient partial denominators. An indeterminate equation is an equation for which there is more than one solution. For example, 2x = y is a simple indeterminate equation, as are ax + by = c and x 2 = 1. Indeterminate equations cannot be solved uniquely.

Publisher Summary This chapter discusses the longitudinal dynamic motions of the helicopter. The linearized, small-perturbation equations of motion are taken into consideration. There are three equations in three unknowns. Two equations are for forces resolved along body axes, and one is for pitching moments. It is assumed that the flight is at approximately constant altitude. The X axis is approximately horizontal, while the Z is approximately vertical. Within these restrictions, the equations are meaningful and directly usable over a complete range of speed from hovering to high-speed forward flight. The use of velocity components Δu and Δw as variables in place of V and α avoids difficulty over definition of α at low speeds, and associated indeterminate forms. The equations in this form are appropriate for tracing the small perturbation dynamics over the whole speed range, without a gap between hovering and forward flight. They are even useful for some simple kinds of simulation, in which the derivatives are programmed as functions of velocity.

Publisher Summary One of the most important properties of a continuous function is the intermediate value property. Even though derivatives are not always continuous, they do always possess the intermediate value property. The derivatives are sometimes described as being “Darboux continuous,” not because they are always continuous, but because they always possess the intermediate value property characteristic of continuous functions. It is not always possible to obtain higher order derivatives. Derivatives are used to calculate the value of indeterminate forms. The method used for this is known as L’Hopital's rule.

A reparameterisation procedure is investigated for embedded model problems. The procedure is given by solving differential equations determined by indeterminate forms of limit. Some properties are provided for the existence of an embedded model. Note that an embedded model may include another embedded model. We introduce the concept of embedded model of kth generation and discuss the use of one-by-one elimination procedure to construct graphs of embedded models. As examples, we derive embedded models for some distributions, to which existing method cannot be applied. Our method includes the method given by Cheng et al. [1] as a special case.

Recent data from our laboratory have shown that patients with the indeterminate form of Chagas' disease can have impairment of left ventricular contractility, as evaluated by the slope of the left ventricle end-systolic pressure-dimension relationship. We also showed that Chagas' disease patients with minimal baseline wall motion abnormalities detected by two-dimensional echocardiography have more intense contractility impairment when compared to patients with the indeterminate form of the disease without this abnormality. The prognostic implications of these findings have not been established. We evaluated 59 patients (37-76 years, mean = 55 years) with different clinical forms of Chagas' disease, who had normal left ventricular global systolic function at baseline (57.6 ± 6.9%) and who had at least one additional echo during clinical follow-up (0.4-17.6; mean 4.6 years). Group 1 consisted of 14 patients with minor baseline left ventricle wall motion abnormalities and group 2 consisted of 45 patients without these abnormalities. During follow-up, global left ventricle systolic function deterioration was observed in 10 group 1 patients (71.4%) and in only 10 group 2 patients (22.2%; P < 0.005). Age and duration of follow-up were not independent determinants of left ventricular function deterioration in these patients. The present data indicate that mild segmental left ventricular wall motion abnormalities are associated with worsening of systolic function in Chagas' disease patients who have normal baseline global systolic performance.

The teaching of traditional units in this area of calculus may be facilitated by the use of numerical demonstrations.

It is preferable to show students that proper solutions can be derived directly from the integrated form of a rate equation. This can best be accomplished by the use of L'Hospital's rule as will be shown in this article.

In what follows, we shall try to obtain an expression for lim v { [c(n)]I't}, where 4(n)?O and 0(oo)= o. In case r=1, use is made of the series Zn-O(n)Zn and its radius of convergence p. For on the one hand i/p= limn1n-c{0(n)"/} and on the other i/p=limn-.00{0(n)/0(n-1)1 when this limit exists. When both limits exist, they are equal and limn--. {1(n)lln } limn1n{.(n)/0(n-1) }. For r> 1, we proceed as follows: Let limn_00A(?) exist and equal A. Define A(') by