Претендент на третью интегральную теорему о среднем

Подобно тому, как формула Лагранжа является частным случаем формулы Коши о среднем в дифференциальном исчислении, также можно показать, что первая теорема о среднем является частным случаем интегральной теоремы Коши. В работе рассмотрены две формы интегральной теоремы Коши. Первая из них следует непосредственно из дифференциальной формулы Коши о среднем, а вторая является ее обобщением, подобно тому, как существует первая и вторая интегральные теоремы о среднем и их обобщенные варианты.У статті розглянуто дві теореми про середнє в інтегральному численні. Перша з них є інтегральним аналогом теореми Коші у диференціальному численні. Друга теорема є узагальненням першої теореми Коші і проводитися з використанням властивостей інтегральної міри. Теореми розширюють поняття про середнє функції. Згідно нашої теорії середнє у загальному сенсі є середнім функції відносно іншої функції. У частковому випадку відносне середнє перетворюється у звичайне середнє. Третя теорема про середнє може бути використана для оцінки визначних інтегралів.Two mean value theorems in the integral calculus have been considered in the article. The first theorem is an integral analogue of the Cauchy’s theorem from differential calculus. The second one is a generalization of the Cauchy’s first theorem. These theorems expend our imagination about the function mean value. The mean value concept is a function mean value with respect to the other function. The third integral mean value theorem can be used for estimation of some definite integrals

Characterization of Monoped Equilibrium Gaits

We characterize equilibrium gaits of a small knee monoped in terms of manifest parameters by recourse to approximate closed form expressions. We first eliminate gravity during stance and choose a very special model of potential energy storage in the knee. Next, we introduce simple closed form approximations, motivated by the mean value theorem, to the elliptic integrals arising in the more general case. In so doing, we derive a conjectured generalization applicable to small knee monopeds with an arbitrary knee potential. Finally, we introduce a new closed form perturbation intended to adjust the approximate coordinate transformations to the presence of gravity. Simulation data is offered as evidence for the efficacy (to within roughly 5-10% accuracy) of both the proposed generalization across knee potentials and the proposed perturbation for the presence of gravity during stance

Análise Matemática I

A. Cálculo diferencial em R: Revisão de alguns conceitos e resultados. Teorema dos Acréscimos Finitos (Lagrange). Diferenciais de Funções de uma variável - Definição. Regras de cálculo e aplicações. Aproximação Polinominal - Polinómios de Taylor e fórmula de Taylor com resto; aplicações. Série de Taylor como limite dos polinómios de Taylor. Séries numéricas: propriedades das séries, critérios de convergência, séries alternadas. Breve referência às séries de funções. Conceito de intervalo de convergência. B. Integral de Riemann em R: Integração de funções reais de variável real - Integral de Riemann, sua definição e propriedades. Teoremas do valor médio para integrais. Teoremas Fundamentais do Cálculo. O Conceito de Primitiva - Regras de Primitivação por substituição e por partes. Aplicações do integral ao cálculo de áreas em coordenadas cartesianas e polares e ao cálculo de volumes. Primitivação de fracções racionais algébricas. Primitivação de expressões racionais trigonométricas. Primitivação de expressões irracionais por substituição trigonométrica. C. Tópicos adicionais: Funções hiperbólicas. Integrais impróprios. Equações diferenciais de primeira ordem.A. Differential Calculus in R:Review of fundamentals of differentiation.Increments, differentials and linear approximations. The mean-value theorem for derivatives.Polynomial approximations to functions: The Taylor polynomials generated by a function.Taylors formula with remainder. Estimates for the error in Taylors formula. The Taylor series as a limit of Taylor polynomials. Numerical series: properties, convergence criteria, alternating series.Reference of functional series. Concept of convergence interval.B. Integral Calculus in R:Riemann sums and the integral. Integrability of bounded monotonic functions. The integrability theorem for continues functions. Properties of the integral. Mean-value theorem for integrals.The derivative of an indefinite integral. The first fundamental theorem of calculus.Primitive functions and the second fundamental theorem of calculus.Integration by substitution. Integration by parts. Areas of plane regions. Polar coordinates. Area calculation in polar coordinates. Volume calculations by the method of cross sections.Integration by rational partial fractions. Rational trigonometric integrals. Integrals containing quadratic polynomials.C. Additional topics:Hyperbolic functions. Improper integral.First order differential equation

Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series

We consider some versions and generalizations of the approach to expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series. The expansions of iterated stochastic integrals with respect to martingale Poisson measures and with respect to martingales were obtained. For the iterated stochastic integrals with respect to martingales we have proved two theorems. The first theorem is the generalization of expansion for iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series. The second one is the modification of the first theorem for the case of complete orthonormal with weight $r(t_1)\ldots r(t_k)\ge 0$ systems of functions in the space $L_2([t, T]^k)$ (in the first theorem $r(t_1)\ldots r(t_k)\equiv 1$). Mean-square convergence of the considered expansions is proved. The example of expansion of iterated (double) stochastic integrals with respect to martingales with using the system of Bessel functions is considered.Comment: 37 pages. Minor changes. arXiv admin note: text overlap with arXiv:1712.09746, arXiv:1801.05654, arXiv:1801.01564, arXiv:1712.08991, arXiv:1801.00231, arXiv:1801.03195, arXiv:1712.0951

The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof

In this review article we collected more than ten theorems on expansions of iterated Ito and Stratonovich stochastic integrals, which have been formulated and proved by the author. These theorems open a new direction for study of iterated Ito and Stratonovich stochastic integrals. The expansions based on multiple and iterated Fourier-Legendre series as well as on multiple and iterated trigonomectic Fourier series converging in the mean and pointwise are presented in the article. Some of these theorems are connected with the iterated stochastic integrals of multiplicities 1 to 5. Also we consider two theorems on expansions of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k)$ as well as two theorems on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized iterated Fourier series converging pointwise. On the base of the presented theorems we formulate 3 hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k).$ The mentioned iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion. Moreover, the considered expansions from these 3 hypotheses contain only one operation of the limit transition and substantially simpler than their analogues for iterated Ito stochastic integrals. Therefore, the results of the article can be useful for the numerical integration of Ito stochastic differential equations. Also, the results of the article were reformulated in the form of theorems of the Wong-Zakai type for iterated Stratonovich stochastic integrals.Comment: 35 pages. Section 12 was added. arXiv admin note: text overlap with arXiv:1712.09516, arXiv:1712.08991, arXiv:1802.04844, arXiv:1801.00231, arXiv:1712.09746, arXiv:1801.0078

Pathwise stability of likelihood estimators for diffusions via rough paths

We consider the classical estimation problem of an unknown drift parameter within classes of nondegenerate diffusion processes. Using rough path theory (in the sense of T. Lyons), we analyze the Maximum Likelihood Estimator (MLE) with regard to its pathwise stability properties as well as robustness toward misspecification in volatility and even the very nature of the noise. Two numerical examples demonstrate the practical relevance of our results.Comment: Published at http://dx.doi.org/10.1214/15-AAP1143 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org