# asymptotic

1. ### Looking for an expression to control concavity of a curve

Hi, Apologies if this doesn't really belong in the Algebra sub-forum - I thought this would be the most appropriate place, but if not then the admin staff should move this thread. I'm looking for a relatively simple algebraic expression which contains within it a parameter which controls...
2. ### Asymptotic expansion?

I need to solve the equation: 2ABB'' - A B'^2 + A'BB' = 0 for functions A(x) and B(x) for a project that I'm working on. As it happens I have a degree of freedom here: I can essentially choose either A or B to be just about any function I want, so long as they are continuous. In fact I've...
3. ### Asymptotic approximation of a series

I have never really studied asymptotic approximation, so I am mostly looking for where to begin my research. I have the following recurrence relation for a family of sequences that I hope to approximate: a_n = qa_{n-1} + p^{\lfloor (n-1)\log_p q \rfloor }, a_0 = 0 where p,q are two relatively...
4. ### Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows $$\epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber$$ $$\frac{ds}{dt}= \lambda_b P_C \ \epsilon \ c (1-s)- \lambda_r (1-q) \ s, \nonumber$$...
5. ### Deriving the sampling (or asymptotic) distribution

Assume we have the following function: $$f(p) = \frac{1}{(1-p)d}\ln\left(\frac{1}{T}\sum_{t=1}^{T}\left[\frac{1+X_t}{1+Y_t} \right]^{1-p} \right)$$ where $d$ is a constant $T$ is a constant $X_t$ for $t = 1, 2, \cdots, T$ are random variables $Y_t$ for $t = 1, 2, \cdots, T$ are random...
6. ### Asymptotic error formula for the trapezoidal rule

I need to use the asymptotic error formula for the trapezoidal rule to estimate the number n of subdivisions to evaluate $\int_{0}^{2}e^{-x^2}dx$ to the accuracy $\epsilon=10^{-10}$. I also need to find the approximate integral in this case. I would like to know if my attempt is correct. Thanks...
7. ### Asymptotic analysis to the WKB approx

Hi i have the WKB approx of: u_{+} = \sqrt{1-\frac{bm}{f}}e^{i\int f_{k} dt} + \sqrt{1+\frac{bm}{f}}e^{-i\int f_{k} dt} to the differential equation: \frac{d^{2}u_{+}}{dt^2} + [f_{k}^{2} + i(\frac{d(bm)}{dt})]u_{+} =0 This equation can be written as: \frac{d^{2}u_{+}}{dN^2} + [p^2 - i +...
8. ### Asymptotic curves

Consider the helicoid S given by the parametrization x(u,v)=(vcosu, vsinu,u). Find the asymptotic curves on S.
9. ### Asymptotic Analysis of Recursive Powering

I am currently going through MITs course on Algorithms online for my own knowledge. One particular problem I'm having trouble with is doing asymptotic analysis on the recursive powering algorithm provided on the following page: MIT's Introduction to Algorithms, Lecture 3: Divide and Conquer -...
10. ### searching an asymptotic solution of an nonlinear ODE

I am finding an asymptotic solution for the following ODE: (eq1) dy/dt = yp [1 + log-a(2 + y2)] with p > 1, a > 0, y(0) = y0 > 0 . We already know the solution of the following equation (eq2) dy/dt = yp with p > 1, y(0) = y0 > 0 can solved by taking v = y1-p ==> y(t)...
11. ### Asymptotic theory

Let (Xi)i=1....n de iid according to a uniform distribution U (g-t,g+t). Assuming g is known, the null hypothesis Ho: t=1 is rejected in favor of H1:t>1 at the alpha level of significance when: max 1<=i<=n |Xi-g|>1+ (1/n)log (1-alpha) i) show that the test with rejection region (1) has...
12. ### Problem of Asymptotic theory

Let (Xi)i=1....n de iid according to a uniform distribution U (g-t,g+t). Assuming g is known, the null hypothesis Ho: t=1 is rejected in favor of H1:t>1 at the alpha level of significance when: max 1<=i<=n |Xi-g|>1+ (1/n)log (1-alpha) i) show that the test with rejection region (1) has...
13. ### Asymptotic Variance

What is the asymptotic variance of an estimator? There seems to be several definitions on the web but I know the one I'm after includes the expectation of the second derivative of the log-likelihood.
14. ### Asymptotic Variance

What is the asymptotic variance of an estimator? There seems to be several definitions on the web but I know the one I'm after includes the expectation of the second derivative of the log-likelihood.
15. ### Asymptotic Variance of estimator

Very simple question. What is the definition?
16. ### asymptotic curve on a complete surface.

Hello, We have a 2-dimensional Riemannian Manifold immersed in \mathbb{R}^3, which is complete and has constant negative curvature. Then the statement is, that for every point p\in M, there is an asymptotic curve c:\mathbb{R}\rightarrow M, parametrized by arclength, s.t. c(0)=p. I don't...
17. ### Find asymptotic expansion for complex function

If Ei(x) = \int^\infty_x \frac{e^{-u}}{u} du, using integration by parts we can show that Ei(x) = e^{-x} \left( \frac{1}{x} - \frac{1}{x^2} + \frac{2!}{x^3} - \cdots + \frac{(-1)^n (n-1)!}{x^n} \right) + (-1)^n n! \int^\infty_x \frac{e^{-t}}{t^{n+1}} dt. Hence obtain an asymptotic expansion of...
18. ### Finding asymptotic expansion using Laplace's method

Consider the integral I defined by \displaystyle I = \int^1_0 e^{-xt} t \cos t dt. Use Laplace's method to obtain the first term of the asymptotic expansion of the integral as x \to \infty. How can I start? Can someone point me to somewhere with a similar example? Thank you.
19. ### Asymptotic equivalent of a sequence

Hi everyone, I've a little trouble with a small exercice. So, given the equation x^n+x=1, it can ben easily shown that \displaystyle \exists ! x_n \in \mathbb{R}^+ / x_n^n + x_n = 1. I also prooved that the sequence (x_n)_{n > 0} is convergent and its limit is l=1. Now, the question is : Find...
20. ### Volterra equation, asymptotic behaviour

Dear all, I want to solve the Volterra integral equation (of 2nd kind). But I only need to solve it analytically for large times "tau", i.e. I only need the asymptotic behaviour as "tau -> infinity". By simple algebra, I obtain an approximative analytical expression in this limit. However...