# All Questions

0answers
15 views

### Number of zeros of $f^n$

Let $f:\Bbb R\to \Bbb R$ be infinetly differentiable function that vanishes at $10$ distinct points in $\Bbb R$.suppose $f^{n}$ denote $n$-th derivate of $f$, for $n \ge 1$. Then which of following ...
1answer
31 views

### Complex square matrices. Difficult proof.

$det(I+A\cdot\bar{A}) \ge 0$ Is it possible to prove the inequality is true for all complex square matrices $A$ where $I$ is the identity matrix and $\bar{A}$ is the complex conjugated matrix.
0answers
8 views

### Area between two functions

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 94. Exercise 16. Let $f(x)=x-x^2$, $g(x)=ax$. Determine $a$ so that the region above ...
0answers
1 views

### Permutations and inverzion of permutations

We have a permutation $a_1,a_2,...,a_n$ of the set {1,2,...,n}. For a pair $(a_i,a_j)$ we say it is an inversional permutation if $i<j$ and $a_i>a_j$. Find the number of permutations in which ...
4answers
46 views

### Calculate $\lim_{n\to+\infty}\cos{\big{(}\pi \sqrt{n^2+n}\big{)}}$

I need to prove that $$\lim_{n\to+\infty}\cos{\big{(}\pi \sqrt{n^2+n}\big{)}}=1$$ But this is something that seems highly unlikely to have a limit, but I am probably wrong. Wolfram Alpha says "$0$ ...
3answers
25 views

### Image of (0,1] under continous function

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Which of the following sets cannot be image of $(0,1]$ under $f$. {$0$} $(0,1)$ $[0,1)$ $[0,1]$ My initial guess was using intermediate ...
0answers
28 views

### Number of adjacent permutations.

What is the number of permutations for any number of adjacent elements swapping places at the same time in an array of length $n$? My solution: I think that we only need to count the number of ...
1answer
24 views

### Difficulty in finding a counter example

Finding difficulty in finding a counter example that if $f : (0,\infty) \to(0,\infty)$ is uniformly continuous then implying that $\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1$.
2answers
20 views

### Limit and Fibonacci

How to prove $$\LARGE{\lim_{n \to \infty} \frac{F_{kn}}{{F_n}^k} = 5^{(k-1)/2}}$$ Non-induction method is prefered.
0answers
26 views

1answer
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### Proof that the finite product of nonempty sets is nonempty without axiom of choice from ZF [duplicate]

How do you prove that for $X_{i} \neq \emptyset$, $i \in \{1,...,n\}$ that $\prod_{i=1}^{n} X_{i} \neq \emptyset$ only using the ZF axioms but not the Axiom of Choice? I would like to see a rigorous ...
1answer
67 views

### Paul Erdős showed a simple estimate for $\pi(x) \ge \frac{1}{2}\log_2 x$; is it possible to tweak his argument to improve the estimate?

Paul Erdős gave a simple argument to show that $\pi(x) \ge \dfrac{1}{2}\log_2 x$. Is it possible to tweak the argument and get a better estimate? I am wondering how good an estimate for $\pi(x)$ can ...
1answer
9 views

3answers
31 views

### Will someone explain this polynomial regression equation?

I am in high school and I need to write a program that does polynomial regression to any degree on a set of data for a personal project. I think that this Wikipedia Article has the equation that I ...
0answers
78 views

### Can the cube of every perfect number be written as the sum of three cubes?

I found an amazing conjecture: the cube of every perfect number can be written as the sum of three positive cubes. The equation is $$x^3+y^3+z^3=\sigma^3$$ where $\sigma$ is a perfectnumber (well it ...
0answers
53 views
+50

### How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space \$(\theta_A, ...
1answer
21 views

### Plotting Step Functions - connecting the steps?

How does one choose whether or not to draw vertical lines connecting the steps of a step function? Let's take the cdf of some discrete random variable as an example. My intuition tells me to graph ...

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