# All Questions

63 views

### Model a function with specific shape

I need a function with a specific shape: Quadratic/gaussian concave shape ($-x^2$ like) Centered in $\frac{1}{2}$ where it reaches the max value 1 On 0 and ...
194 views

### Series of iterates of a power series

Let $f = \sum_{n \ge 2} a_{n} x^{n}$ be a formal power series of order higher/equal two. By $$f^{\circ n} := \underbrace{f \circ \dots \circ f}_{\text{ n times}}$$ we denote the $n^{th}$ iterate of ...
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### Determine the limit of $\frac{1}{n}\sum_{i =2}^n \frac{1}{\ln i}$ as $n \to \infty$

For the sequence $a_n =\frac{1}{n}\sum_{i =2}^n \frac{1}{\ln i}$ (with $n \ge 2$), I would like to determine if the limit exists, and, if so, find its value. Some observations I have made so far: ...
201 views

### A problem on continuous i.i.d random variable

Let $X_1, X_2, X_3, X_4$ be i.i.d continuous random variables with a common distribution function $F$. How to prove that "all the 4! possible orderings of $X_1, \dots, X_4$ are equally likely" without ...
92 views

### van der pol and liapunov

i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated. a) show that the second-order differential equation for ...
135 views

### What points of affine space can be mapped to zero by an étale morphism?

Let $K$ be a field and $n$ a positive integer. For what points $x\in\mathbb{A}^n_K$ can I find an étale morphism $f_x:\mathbb{A}^n_K\to \mathbb{A}^n_K$ mapping $x$ to zero and how does such a ...
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### Is $\sum_{n=2}^\infty\log\left(1+\frac{(-1)^n}{\sqrt{n}}\right)$ convergent?

The question is motivated by the following exercise: Find an example of a sequence of complex numbers $\{a_n\}$ such that $\sum a_n$ converges but $\prod(1+a_n)$ diverges. A necessary condition ...
355 views

### Center of Mass double integral

A lamina occupies the region which is the intersection of $x^2+y^2-2y \leq 0$ and the first quadrant of the $xy$-plane. Find the center of mass if the density at a point of the lamina is twice the ...
748 views

### Given one primitive root, how do you find all the others?

For example: if $5$ is a primitive root of $p = 23$. Since $p$ is a prime there are $\phi(p - 1)$ primitive roots. Is this correct? If so, $\phi(p - 1) = \phi(22) = \phi(2) \phi(11) = 10$. So ...
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### number of elements in vector space

Given that $k$ is a finite field with $q$ elements and $V$ is a $n$-dimensional $k$-vector space, then by basis representation, we know that for $v \in V, v=a_1v_1+a_2v_2+\cdots+a_nv_n$ uniquely. ...
117 views

### What's the lowest real $x$ such that $\zeta(x)$ converges?

It's easy to prove that$$\zeta(1)=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...$$ diverges, and $$\zeta(2)=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...$$ converges to $\frac{\pi^2}{6}$. Intuiting the ...
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### Determine all subgroups of $\mathbb{R}^*$ (nonzero reals under multiplication) of index $2$.

Determine all subgroups of $\mathbb{R}^*$ (nonzero reals under multiplication) of index $2$. how can I able to solve this problem
Let $F:I\times J\to\mathbb{R}$ be a $C^k$ (or analytic) function, with $I,J$ real open intervals. Set $f_\lambda(x):=F(\lambda,x)$ and consider the parametric equation $$f_\lambda(x)=0\,.$$ Assume its ...