## What is the fastest algorithm to establish whether a linear system in $\mathbb{R}$ has a solution?

I know the best algorithm to solve a linear system in $$\mathbb{R}$$ with $$n$$ variables is Coppersmith-Winograd’s algorithm, which has a complexity of $$O\left(n^{2.376}\right).$$ How much easier is it to simply determine whether the same system has any solution?

More precisely, given a system of $$m$$ equations and $$n$$ unknowns, what is the complexity of establishing whether it has any solution?

## Example of a polynomial that fails to be non-zero over a higher field than $\mathbb{C}$ or $\mathbb{R}$?

It’s well known over the field $$\mathbb{F}_2$$ that the polynomial $$p(x) = x^2 – x$$ is equivalent to the $$0$$ polynomial over the field. Naturally however if we consider say $$\mathbb{R}$$ then it’s easy to see that $$x^2 -x$$ is not $$0$$ everywhere.

This leads to an interesting question: are there examples of (possibly multivariate) polynomial expressions $$P(x_0,x_1…x_k)$$ such that $$P$$ is $$0$$ everywhere over $$\mathbb{C}$$ or $$\mathbb{R}$$ but such that there is a field $$K$$ of which $$\mathbb{C}$$ or $$\mathbb{R}$$ is a quotient field (or sub-field), and the specific polynomial $$P$$ is not identically $$0$$ over the entirety of $$K$$.

## Proof verification that the sequence $x_n = \frac{1}{n}$ converges to every point of $\mathbb{R}$ on the cofinite topology

Let $$a \in \mathbb{R}$$. Then any open set $$U$$ in the cofinite topology containing $$a$$ is of the form $$U = \mathbb{R} – \{\alpha_1, \alpha_2, \cdots, \alpha_p\}$$ for some $$p \in \mathbb{N}$$ (and of course $$\alpha_i \neq a \ \forall 1 \leq i \leq p$$). Now, take $$N = c \left(\displaystyle{\frac{1}{\min \alpha_i}}\right) + 1$$, where $$c(x)$$ stands for the ceiling of $$x$$ (i.e, $$c(2.1) = 3$$). Then it’s clear that $$x_n \in U \ \forall n \geq N$$ and we’re done.

Is this alright? I actually think I could do away with all of this and just make the argument that the tail of $$x_n$$ is eventually contained in $$U$$ since $$\mathbb{R} – U$$ has only finitely many elements and $$\{x_n\}_{n \in \mathbb{N}}$$ is infinite, but I’m not sure if that’s fine too.

## If $f: \mathbb{R} \to \mathbb{R}$ is a continuous surjection, must it be open?

If $$f: \mathbb{R} \to \mathbb{R}$$ is a continuous surjection, must it be open?

I think not. I proved if $$f: \mathbb{R} \to \mathbb{R}$$ is an open continuous surjection, then $$f$$ is a homeomorphism. So, if the question is true, every continuous surjection must be a homeomorphism. But, I didn’t find a counterexample. Can someone help me?

## Show $\lim_{a \to 0} a \cdot \mu (\{ x \in \mathbb{R} : |f(x)| > a \}) = 0$ for $f \in L^1 (\mathbb{R})$, $a>0$

Let $$f \in L^1 (\mathbb{R})$$, and $$a > 0$$. Show

$$\lim_{a \to 0} a \cdot \mu (E_a) = 0$$

where $$E_a = \{ x \in \mathbb{R} : |f(x)| > a \}$$.

Try

Since

$$\int_\mathbb{R} |f| d\mu = \int_{E_a} |f| d\mu + \int_{\mathbb{R}\setminus E_a } |f| d\mu < \infty$$

we have $$\int_{E_a} |f| d\mu < \infty$$, $$\forall a$$.

But I’m stuck at how I can relate this to find $$\mu(E_a)$$.

Any help will be appreciated.

## If $A \times B$ is Lebesgue measurable in $\mathbb{R}^2$ and $B$ is Lebesgue measurable in $\mathbb{R}$ then $A$ is so?

Though it seems simple but I am struggling to find a proof for it being a starter to measure theoty. I think this statement is true and I am trying to prove it. Here is my trial: Write $$A= A \times \{0\}= \cup_i(A_{i} \times B_{i})$$,

$$m_1(A)=m_2(A \times \{0\})=\cup_i m_1(A_i)m_1(B_i)$$

I don’t know what to do next! Is my approach correct?

where $$m_1,m_2$$ are 1 and 2 dimensional measures in $$\mathbb{R},\mathbb{R^2}$$ respectively.

## Proving $\inf_{a \in \mathbb{R}} \mathbb{E}|\xi -a| = \mathbb{E}|\xi – m\xi|$

Let $$\xi$$ be a random variable form $$(\Omega, \mathcal{F}, \mathbb{P})$$. Than let $$m\xi$$ be a median of random variable $$\xi$$. I need to prove $$\inf_{a \in \mathbb{R}} \mathbb{E}|\xi -a| = \mathbb{E}|\xi – m\xi|.$$

So I know that $$\mathbb{P}(\xi.

I also know that $$\inf_{a\in \mathbb{R}} \mathbb{E}(\xi-a)^2=\mathbb{D}\xi$$ when $$\mathbb{D}\xi=\mathbb{E}(\xi-\mathbb{E}\xi)^2.$$

Can I somehow use it here or how should I prove it?

## Show that $f_n \rightarrow 0$ in $C([0, 1], \mathbb{R})$

I was given the following problem and was wondering if I was on the right track.

Let $$f_n(x) = \frac{1}{n} \frac{nx}{1 + nx}, \: 0 \le x \le 1$$

Show that $$f_n \rightarrow 0$$ in $$C([0, 1], \mathbb{R})$$.

I have this theorem that I figured I could use:

$$f_k \rightarrow f$$ uniformly on A $$\iff$$ $$f_k \rightarrow f$$ in $$C_b$$.

In this case, $$C_b$$ is the collection of all continuous functions on $$[0,1]$$. So if I can prove the function is uniformly continuous, this would prove that $$f_n \rightarrow 0$$. Can I apply this theorem like this to prove what I want? Also, if I can, would using the Weierstrass M test be best to prove uniform convergence here?

Thanks