Recent content by phonic

Dear All, I am simulating a random walk on a sphere with unit radius. I want to move from current location p_t to the new location p_{t+1} along the big circle, whose arc has an angle omega relative to p_t's latitude. I tried using the law of cosine. But at the poles, the law of cosine...

Hi All, I try to solve second order PDE: \frac{\partial^2 f(x,y)}{\partial x^2}=-a^2f(x,y) \frac{\partial^2 f(x,y)}{\partial y^2}=-a^2f(x,y) where a >2, f(x,y) is a periodic function in x, but has fixed boundaries in y. Is there a way to solve it? What does the solution look like? Any...

Dear All, I have two functions that are defined on a table, i.e. f1(x,y), f2(x,y), where x and y are bin indices, and 1\leq f1(x,y) \leq 1 , 1\leq f2(x,y) \leq 1 . I would like to perform some test to show whether f1(x,y) and f2(x,y) are significantly different. Is there some way to do...

Using your idea, define u=(e^{ik(x-x_0)+il(y-y0)}+e^{-ik(x-x_0)-il(y-y0)})/2 and v=(e^{-ik(x-x_0)-il(y-y0)}+e^{-ik(x-x_0)+il(y-y0)})/2, then the intergration region A looks simpler: A: u+v>b , but then \cos(k(x+y))\cos(ly) is not possible to write as a function of u and v.Is there other...

Dear All, I want to calculate some convolution like integrations: g_1(k,l)=\int\int_A \cos(k(x+y))\cos(ly) f(x,y)dx dy g_2(k,l)=\int\int_A \cos(k(x+y))\sin(ly) f(x,y)dx dy g_3(k,l)=\int\int_A \sin(k(x+y))\cos(ly) f(x,y)dx dy g_4(k,l)=\int\int_A \sin(k(x+y))\sin(ly) f(x,y)dx dy f(x,y)...

Dear all, I have a function in the form of f(y) = \int_{-h(y)}^{h(y)} g(x,y)dx . Then how to calculate its derivative w.r.t. y? i.e. \frac{d f(y)}{dy} Thank you very much for your input!

Dear All, I got some trobule in solving the following simple-looking PDE's. Can anyone give a hint about how to solve it? thanks a lot! I guess the solution is of the form y(u,v)=A[\cos(k(u-f(v))-B]\cosh(v)-C. But I don't know a formal way to solve. \frac{\partial^4y(u,v)}{\partial u^2 \partial...

Dear All, I wonder how to calculate the sum of the following Cosines: \sum_{n=1}^\infty \cos(nx) Can anyone give a hint? Thanks a lot!

Xevarion and EnumaElish, Thanks for helping. Actually, I am working on 2). It is about error analysis. Each of the exponential function in the summation is an error term. I want to bound the total error by a function f(x), which is preferably an exponential function. In my case, x_is are...

upper bound on exponential function? Dear All, I am searching for an upper bound of exponential function (or sum of experiential functions): 1) \exp(x)\leq f(x) or: 2) \sum_{i=1}^n \exp(x_i) \leq f(x_1,\cdots,x_n, n) . Since exponential function is convex, it is not possible to use...

Dear All, I am searching for an upper bound of exponential function (or sum of experiential functions): 1) \exp(x)\leq f(x) or: 2) \sum_{i=1}^n \exp(x_i) \leq f(x_1,\cdots,x_n, n) . Since exponential function is convex, it is not possible to use Jenssen's inequality to get an upper bound...

Tanks for your reply. Then the problem is to bound the tail probability of this normal variable. I know one inequality is (R. D. Gordon, The Annals of Mathematical Statistics, 1941(12), pp 364-366) P(z \geq x) = \int_x^\infty \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}z^2} dz \leq \frac{1}{x}...

Dear all, I wonder wheather there exsits a probability inequality for the sum of independent normal random variables (X_i are i.i.d. normal random varianble with mean \mu and variance \sigma^2): P\left(\frac{1}{n}\sum_{i=1}^n X_i - \mu> \epsilon\right)\leq f(\epsilon, \sigma^2,n) \right). We...

Dear All, Is it possible to have an analytical inverse function of y=x e^{-\frac{1}{x^2}}. Since y is monotonously increasing, its inverse function exists. But is it possible to get a close form? Thanks a lot! Phonic

s can be considered as a constant number. Since the Markov inequality holds for any s. Is there some bounds on the tail probability of a normal distribution?