The Markov Inequality states that $$P(X>\alpha) \le \frac{E[X]}{\alpha}$$. If I flip a fair coin 100 times, I was trying to calculate the probability that there were at least 90 tails in order to determine whether this series of events was an outlier or not. I was expecting that this would be an ...

How many rectangle(s) with two “#” is/are there in the figure below? I tried to count it and my answer is 36. My teacher shared the answer and ask us to find the reason why. He said it is just $2 * 3 * 2 * 3 $ I found out that my answer is correct. But, what do these factors mean?

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous bounded function, $(X_k)_k$ a sequence of i.i.d random variables such that $$\forall x \in \mathbb{R},f(x)=\int_{\mathbb{R}}f(x+y)dP_{X_1}(y).$$ Let $x \in \mathbb{R},Y_k=f\left(x+\sum_{q=1}^kX_q\right),\mathcal{F}_k=\sigma(X_1,...,X_k).$ Verify t...

I'm sure I'm just being silly, but I've run into a claim in a paper I'm reading which I don't understand. Suppose $\mathcal{F}$ is a filter on $\mathbb{N}$. There is a natural topology $\tau_\mathcal{F}$ on $\mathbb{N}$ associated to $\mathcal{F}$ - namely, just take as opens exactly the sets in...

Suppose I have two lines in $3$D space passing through origin. The smallest angle formed between them would be between $0$ and $\pi/2$. Minimizing the cosine of this angle we'll get $\cos {(\pi/2)}=0$. For $3$ lines there will be in total $3$ angles between them. Let's again suppose these angles ...

Let $\mathbb{T}\subseteq(\mathbb{R}^2)^3$ be the set of ordered noncollinear triples of points in the plane. Say that a pseudovertex is a function $\mu:\mathbb{T}\rightarrow\mathbb{R}^2$ such that: $\mu$ is order-invariant and is continuous with respect to the usual topologies on domain and cod...

What is the power series expansion $f(\alpha)$ as $\alpha\to 1$ and also in the limit $\alpha\ll 1$, where $$f(\alpha)=\int_0^\infty\frac{x^\nu J_\nu(x\alpha)}{e^x-1}{\rm d}x$$ At $\alpha=1$ the integrand is analytic so we should have a convergent power series. The power series expansion for the ...

My professor gave us the following polynomial: $f(x) = 3x^3-4x^2-x+2$ Given is that $x = 1$ is a root of this function. We are asked to find the other ones. He then told us that, given $x=1$ is a root, we now know that we can factorize this polynomial into $(x-1)$, and a second factor starting wi...

I'm looking for the "most general" case in which the following statement is true: Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be ordered fields and $f\colon \mathcal{F}_1\to\mathcal{F}_2$ an injective function, then $f$ is strictly monotonic. I'm well aware that "the most general" isn't well define...

I'm new here on the site, I'm a final year student in computer science. In a machine learning course, there was a question on a test that I could not understand. The question goes like this: Suppose that $ F\subseteq\{0,1\}^\Omega $ is some collection of Boolean functions over $\Omega $. Define $...

I know that if $T:V\to W$ is a linear transformation where $V,W$ are finite dimensional.Then we have $Ker(T^t)=Im(T)^0$.But how to geometrically interpret this thing.What does it mean and why this has to be true?Can someone give me a clue?

I understand the concept behind this and it is quite obvious to me as to why but I am having problems proving this rigorously. To make things more "simple," imagine that there is function $f: [-2,2] \rightarrow \mathbb{R}$ and I must prove that $\lim_{x\rightarrow0}{f(x)}=L$ is true if and only i...

I have a abelian group $(G,+)$, and element $a \in G$ and some positive integer $n$. I would like to add an element $b$ to $G$ such that $nb=a$. To be more precise, I want to construct an Abelian group $G'$ such that $G$ is a subgroup of $G'$ and $G'$ has an element $b$ such that $nb=a$. What I a...

In my algebraic geometry course these result appeared: If X is a normal variety, then the set of singular points $S$ have codimension $\geq 2$. (Here normal means that $\mathcal{O}_x$ is integrally closed for all $x\in X$) I've read many proofs of this fact in books and I understood them. However...

That's the definition we got for the symplectic form: Let $$\omega : \: \mathbb{C}^n \times \mathbb{C}^n \rightarrow \mathbb{C}$$ be a bilinear, anti-symmetric and non-degenerate ($\forall_{y \in \mathbb{C}^n} \: \omega(x,y)=0 \: \Rightarrow \: x=0$) map. Let $e_1, ..., e_n$ be the base of $\mat...

How many ways can $11$ apples and $9$ pears be divided between 4 children so that each child receives five fruits? (Apples are identical. just like pears). Solution: $f\left(x,y\right)=\left(x^5+x^4y+x^3y^2+x^2y^3+xy^4+y^5\right)^4$ $f\left(x,y\right)=\left(\frac{x^6-y^6}{x-y}\right)^4$ $f\left(x...

Let $A_i$ be the event that the student passes the ith exam. The probability of passing the 1st exam is 0.80. If he passes the 1st exam then the chance of passing the second is 0.81 but if fails the first then the chance he passes the second drops to 0.40. If he passes both then his chance of pas...

We draw $m$ uniform independent random values among $n$, with $m\ge n$. We consider the probability $p(m,n)$ that not all $n$ values have been drawn. We want an upper bound within a constant factor. The probability that a given value was not drawn is $q(m,n)=(1-1/n)^m$. If these probabilities wer...

I have been working on this problem for a long time and I would like some help. They ask me to find for each $ n $ a hypothesis class $ \mathcal {H} \subset \{\pm 1 \}^{\mathbb {N}} $ with $ n $ elements such that the growth function is $ \tau_{\mathcal{H}}(m) =\min\{ m+1,n \}$. And they ask me t...

In the wikipedia article on the Faà di Bruno formula, one considers the composition of two "exponential/Taylor-like" series $\displaystyle f(x)=\sum_{n=1}^\infty {\frac{a_n}{n!}} x^n\; $ ($a_0=0$) and $\displaystyle g(x)=\sum_{n=0}^\infty {\frac{b_n}{n!}} x^n$ $$g\left(f(x)\right)=\sum_{n=0}^\inf...

Given a set $S$ of complex numbers such that $z\in S \implies \bar{z}\in S$ can I find a real matrix $M$ (in a space of dimension $|S|$) whose eigenvalues are precisely those in $S$? More generally is there a way to know when a real matrix M does exist? I'm struggling to come up with any counter ...

On page 19 in Atiyah-Macdonald, if $M$ is an $A$-modules and $(M_i)_{i \in I}$ is a collection of submodules of $M$, then their sum $\sum M_i$ is defined to be the set of all finite sums $\sigma x_i$ where $x_i \in M_i$ for all values of $i$ and all but finitely many values of $i$ are zero. It g...

I’m interested in the foundations of mathematics. I just completed the book First Order Mathematical Logic by Angelo Margaris. Next planning to study the books The Foundations of Mathematics followed by Set Theory, both by Kenneth Kunen. While it is usually said that set theory has no prerequisit...

This is a follow up of my last question where I was not clear on my problem. What I want to ask is : Given a $n \times m$ matrix $A$ and a $m \times n$ matrix $B$ such that $$BA = I_m$$ why $AB$ is always different from $I_n$ if $ n \neq m$ ? And specifically why is the theorem A function $T : V...

Let $p,q$ be coprime integers with $1\leq q<p$. Let $\Gamma_{p,q}$ be the action of the cyclic group $\Bbb Z_{p^2}$ on $\Bbb C^2$ generated by $(z_1,z_2)\mapsto (z_1, \zeta^{pq-1} z_2)$ where $\zeta=e^{2\pi i / p^2}$. Simiarly let $G_{p,q}$ be the action of $\Bbb Z_p$ on $\Bbb C^3$ generated by $...

Let's suppose I've got a function $f(x)$ where I'd like to differentiate with respect to $t$, but $t$ depends on $x$: $t(x)$. Thus the whole linked derivative thing: $\dfrac{\mathrm{d}f(x)}{\mathrm{d}t(x)}$. Is this possible at all? Alternatively I had to find $t^{-1}(x)$: $x(t)$ and then calcula...