So, I was playing this video game where you could use a jewel to add value to an item. Item values goes from +0 to +6. Each time you use the jewel you have a 50% chance of success, now, the Expected value of jewels I should use to get it to +6 would be easily calculated, it's 12. (6/0.5). But he...

I meet an exercise in my homework: Let $f\in \mathcal{D}'(\mathbb{R}^n)$ and $g\in \mathcal{D}(\mathbb{R}^n)$, show that the biliner map $(f,g)\mapsto f*g $ is continuous in $f$ and $g$, respectively. I have proved that for any fixed $f\in \mathcal{D}'(\mathbb{R}^n)$, if $g_{i}\in \mathcal{D}(\m...

If $\Omega$ is a subset of a matric space $X$ with metric, $d$, then the signed distance function $f$, is defined by $f(x) = \begin{cases} d(x, \partial \Omega) & x \in \Omega\\ -d(x, \partial \Omega) & x \in \Omega^c \\ \end{cases}$ where $\partial \Omega$ denotes the boundary o...

This is an extended question of the classical rolling dice and give face value question. You roll a dice, and you'll be paid by face value. If you're not satisfied, you can roll again. You are allowed $k$ rolls. In the old question, if you are allowed two rolls, then the expected payoff i...

let $a_{i},b_{i}>0$, show that $$\sum_{i=1}^{n}a_{i}b_{i}\ge \dfrac{2}{n+\sqrt{\sum_{i=1}^{n}\dfrac{b_{i}}{a_{i}}\sum_{i=1}^{n}\dfrac{a_{i}}{b_{i}}}}\sum_{i=1}^{n}a_{i}\sum_{i=1}^{n}b_{i}\tag{1}$$ I try:since $$\sum_{i=1}^{n}\dfrac{b_{i}}{a_{i}}\sum_{i=1}^{n}\dfrac{a_{i}}{b_{i}}\ge n^2$$ We jus...

Here is the problem: Suppose $R$ is a field. (a) Show that $h^{n}(?) = Hom_{R}(H_{n}(?; R), R)$ is a cohomology theory defined on (at least) the category of finite CW complexes. (b) Show that $u$ is the natural transformation of cohomology theories. Where $u$ is defined in the following parag...

Let's consider the set $X := \{(x,\,0,\,0)\in \mathbb{R^{3}}: 0 < x < 1\}$. Under the usual topology of $\mathbb{R^3}$, is this set open? My guess it is not, if we sketch it, but how can one analytically prove this, in terms of open balls and? Thanks in advance!

I see here and their that there is a closed form expression of the least-square fit of a sphere of radius $r$ and center $\mathbf{c}$ to $N$ data points $\lbrace \mathbf{x_i}\rbrace_{i\in(1...N)}$. How this expression is obtained? I considered this energy to be minimized : $$ E(r,\mathbf{c})...

I am trying to understand the accepted answer by @Fabian to the post Asymptotic expansion of the complete elliptic integral of the first kind. After a substituion the following integral is obtained $$ K(k) = \int_{1-k^2}^1\!dy\,\frac{1}{2 \sqrt{y(1-y)(y-1 +k^2)}}$$ and the integral is expand...

I found an example problem in Set Theory: The Structure of Arithmetic by Norman T. Hamilton. The example comes from a brief section about logo, and it goes as follows: Consider the sentences Three is a purple cow If three is a purple cow, then 17 is a number. The second sen...

Let ${X_t}$ be an ergodic Markov chain such that $E(X_{t+1}-X_t)=-\epsilon$ for $X_t\in [2,n-1]$, $E(X_{t+1}-X_t)<-\epsilon$ for $X_t=n$, and $E(X_{t+1}-X_t)=\beta$ for $0\leq X_t\leq 1$, where $\epsilon$ is a small positive number, $\beta>\epsilon$, and $n$ is a big positive constant. Intuit...

I have functions $f(x)$ and $g(t,x)$, where $x \in \mathbb{R}^d; t \in \mathbb{R}$. I further assume that these functions fall off fast enough for $|x| \rightarrow \infty$. I want to show the following equality: $$\int_{\mathbb{R}^d} (f(x) * \partial_t g(t,x) ) g(t,x) dx = \int_{\mathbb{R}^d} (f(...

For a positive integer $n$, define a function $ f_n (x) $ at an interval $ [ 0, n+1 ] $ as $$ f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 .$$ Let $ a_n $ be the minimum value of $f_n (x) $. Find the value of $$\sum_{n=1}^{11} (-1)^{n+1} a_n . $$ It is said the answer is...

I would like to prove that (almost surely) $$f_n(B_{\tau_1 } , \dots, B_{\tau_{n-1 }}, -1) < B_{\tau_{n-1 }} < f_n(B_{\tau_1 } , \dots, B_{\tau_{n-1 }}, 1)$$ Where the context is as follows: we have a martingale $(X_n )$ such that for each $n \ge 1$ there exists a Borel measurable func...

In the book "Fundamentals of the Theory of Operator Algebras" by Kadison & Ringrose on page 204 I don't quite understand one detail of the proof for the spectral radius formula $r(A) = \lim \|A^n\|^{1/n}$ in Banach algebras. We know, that the map $z \mapsto (I-zA)^{-1}$ has a convergent power se...

First of all, if you think this problem belongs on a different stack exchange, I am happy to move it. The exercise is as follows: ACI has decided to put an order for golf shoes twice every year and expects to receive one shipment of $960$ pallets of shoes by the beginning of January and a...

I’m working on the following problem. I have a few ideas, however if anyone has any inspiration it would be greatly appreciated. Not asking for outright answers, just suggestions. Forgive me if it’s blatantly obvious, I’m going on little sleep 😂. Prove: No combination of $(2x + 5)$ and $(3)...

Let $\mathbb{K}$ be a field and $n$ a positive integer. The notation $\text{Mat}_{n\times n}(\mathbb{K})$ represents the set of all $n$-by-$n$ matrices with entries in $\mathbb{K}$. The subset $\text{GL}_n(\mathbb{K})$ of $\text{Mat}_{n\times n}(\mathbb{K})$ is composed by the invertible matric...

I've been finding it hard to verify that given maps are open or closed in practice. For example, in his discussion of the Mobius bundle, Lee's Introduction to Smooth Manifolds says without justification that if $E$ is the quotient of $\mathbb{R}^2$ by the equivalence relation $(x,y) \sim (x',y')$...

Let $$\sum_{n=1}^{\infty} g_n(x)$$ converge uniformly to a function $g(x)$ on $A \subseteq \mathbb{R}$. Can we say that the series defined by $$\sum_{n=1}^{\infty} f_n(x) = \sum_{n=1}^{\infty} \frac{1}{n}g_n(x)$$ converges uniformly to a function $f(x)$ on $A \subseteq \mathbb{R}$? If so, prove i...

As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but onl...

Let $A, B$ be square matrices over $\Bbb C$. Prove that matrices $AB$ and $BA$ have the same characteristic polynomial. I know it's a famous problem and found various answers. However, I am at my first year of math degree and my knowledge is very limited. I have never seen matrix which...

We haven't learned L'Hopital's Rule in class so I can't use it and I have tried substitution, factoring, and multiplying by conjugate but nothing seems to work. Is there a way this problem could be solved without the L'Hopital's Rule? Also, I do not know how to input math equations into this so I...