# All Questions

54 views

### Euler method(path s1s2=s2s1)

Given a differential equation $\frac{dy}{dx}=f(x,y(x)), y(x_0)=y_0$. What is the condition for function of f(x,y) such that the result of $y(x_0+S_1+S_2)$ by using Euler forward method, a step size ...
192 views

63 views

### Question about Corollary I.6.6 in Hartshorne

I am having trouble understanding something in Corollary I.6.6 of Hartshorne. Let $K$ be a function field of dimension one over $k$ (by which he means a finitely generated extension of transcendence ...
21 views

### Getting the average of values with errors.

I have five data values each with an associated error. I want to find the mean of these values but also take the errors into account. How do I do this? Lets say the data values and errors are: ...
175 views

### Calculate limit of ratio of these definite integrals

How do I evaluate the following limit? $\lim_{n\to\infty}\frac{\int_{0}^1\left(x^2-x-2\right)^n dx}{\int_{0}^1\left(4x^2-2x-2\right)^n dx}$
26 views

### Prove $P(X<2<4X)=P(\frac{1}{2}<X<2)$

Is $P(X<2<4X)=P(\frac{1}{2}<X<2)$ ?? I came across a question that requires me to find the probability that 2 lies between X and 4X. which is $P(X<2<4X)$ , so this was impossilbe ...
22 views

### My question is in which condition $M_m=P_m$?

Suppose I have a collection of operators $\{P_m\}$ on a finite dimensional Hilbert Space which satisfies $\sum_{m} P_m= I$ where $m$ is eigen value. and I have a collection of positive definite ...
175 views

### Proving the Maximum Principle and the Continuous Dependence on Initial Condition and Boundary Conditions.

This is a two part problem that uses the Maximum/Minimum Principle and the Continuous Dependence. I already got the answer for the Maximum/Minimum Principle, but now I have to apply the Continuous ...
31 views

### GMM estimation of linear regression with intercept restriction

Say I have a time series regression as follows: $$y_t = a_i + \beta_i x_t + \varepsilon_t^i \ \ ; \ \ t = 1, 2, \cdots, T \ \ \text{for each } i$$ Now say I impose the following restriction on the ...
230 views

### Evaluating $\int\limits_0^{\pi} \frac{dx}{1+2\sin^2x}$

After making u substitution two times, I am getting indefinite integral as $$\int\limits\dfrac{dx}{1+2\sin^2x} = \dfrac{\arctan(\sqrt{3}\tan(x))}{\sqrt{3}}+ C$$ I am stuck at working the bounds ...
20 views

### Is the set of polynomials an $x^n+ a_{n-1}x^{n-1}+\ldots+a_1x +a_0$ such that $2^k+1$ divides $a_k$ an ideal in $\Bbb Z[x]$?

Is the set of polynomials $a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$ such that $2^k+1$ divides $a_k$ an ideal in $\Bbb Z[x]$? I think it is true for $2^k+1$ and it will be true for all the divisors as ...
97 views

### Difference between type and similarity type

In usual terminology, is there a difference between the type and similarity type? Is there a general consensus for the definition of the two terms? Please suggest to me books where I can study these ...
40 views

### Normal modes of a drum and Kac's question: Can one hear the shape of a drum?

I consider a vibrating membrane $D\subset {\mathbb{R}}^2$, fixed on $\partial D$. The vertical displacement $f=f(x,y,t)$ of the membrane satisfies the wave equation. I search solutions of the form ...
98 views

### Combination Transversion

Suppose we have a lottery, consisting of 5 balls. The range of balls is 1-39. In any given pick, there will be no duplicate values, and the order need not matter. The upper limit of combinatorial ...
329 views

### If $f$ is continuous on $\mathbb R$ then $\exists c\in\mathbb R: f(x)=c$ has only one solution

I have to prove that there is no continuous function $f: \mathbb{R} \to \mathbb{R}$ such that, for each $c \in \mathbb{R}$ the equation $f(x)=c$ has exactly two solutions. My attempt: We have that ...
42 views

### Kolmogorov Exponential Bounds (Upper)

This is one version of Kolmogorov exponential bound from Allan Gut's Probability: A Graduate Course (2005, p385-386). Let $Y_k$ be an independent sequence of random variables with zero mean and ...
203 views

### Minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$.

I want to minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$, and I want to find the values of $a, b,$ and $\lambda$. This is what I've ...
40 views

33 views

### Mathematical expression for map from $[0,1]$ to $S^2$

A topological space is called arcwise connected if, for any points $x,y\in X$, there exists a continuous map $f: [0,1]\rightarrow X$ such that $f(0)=x$ and $f(1)=y$. Although it is intuitively ...
115 views

### What is the meaning of $dy=dx^2$?

When I read the mathematical analysis ,I think if the differential is $dy=Adx^2$ $A$ is a function about x, what will happen? Maybe, it is not proper defined ,but I think the "function" meet ...
48 views

72 views

### Connectedness of $O(3)$ group manifold
A topological space is said to be connected if it cannot be written as $X=X_1\cup X_2$, where $X_1,X_2$ are both open and $X_1\cap X_2=\emptyset$. Otherwise, X is called disconnected. Is it wrong to ...