Let $D= \text{diag}\{d_1,d_2,...,d_n\} \in \mathbb R^{n\times n}$ be the diagonal matrix, $v \in \mathbb R^{n}$ be the column vector. Then the eigenvalues of the rank one update matrix $D+\alpha vv^T$ can be found as roots of the secular equation:$$f(\lambda) = 1+\alpha \sum_{i=1}^n \frac{v_i^2}{...

On pg 210 and 211(section 3.2, example 3.11), Hatcher attempts to prove that $H^{k}(T^{n},R)$ has basis the cup products $\alpha_{i_{1}} \smile \cdots \smile \alpha_{i_{k}}$ where $i_{1}< \cdots < i_{k}$ and $T^{n} $ is the n-Torus. He first shows that if $\alpha \in H^{1}(I,\partial I;R)$ is a g...

Let $C(\mathbb{R}^2,\mathbb{R})$ be the space$^1$ of all continuous functions $\mathbb{R}^2\rightarrow \mathbb{R}$. I'm interested in analyzing subspaces of $C(\mathbb{R}^2,\mathbb{R})$ determined by first-order theories, especially finite equational theories. Specifically, for $T$ a set of first...

Call a sentence $\varphi$ in the language of arithmetic $Q$-like iff $\mathbb{N}\models\varphi$ and $\{\varphi\}$ is essentially incomplete (= no computably axiomatizable theory containing $\varphi$ is complete and consistent). The standard example is the conjunction of the finitely many axioms o...

Here $k$ and $n$ are fixed positive integers. I need to find the maximum of $f(x) = E[5\min(m, k) +\frac{kx}2]$ for $x$ between 0 and 10. Here we have $m \sim Bin(n, 1-\frac x{10})$. The problem here is that $x$ lies in the probability of the binomial random variable. We can actually write $f...

Let $P(x)$ be a polynomials of degree $N$ with real coefficients such that $P(0),P(2),P(3),\cdots,P(N+1)$ are integers. Find all integer $z$ for which there exist a polynomial $P$ that $P(z)$ is not an integer. Note that we don't refer to $P(1)$. my question: Is there any rules for $z$? ex...

Is it true for any $N\geq 1$, there is a value $f(N)$ such that for any integer $x\geq f(N)$ the integers $x+1,x+2,x+3,\ldots ,x+N$ are always multiplicatively independent (i.e. the relation $(x+1)^{e_1}(x+2)^{e_2}\ldots (x+N)^{e_N}$=1 with the $e_k\in{\mathbb Z}$ is possible only when all those ...

Clearly, if a map between Stone spaces is surjective on points it is an epimorphism. In the category of topological spaces, surjections coincide with epimorphisms. In the category of Hausdorff spaces, epimorphisms are precisely the continuous functions with dense image: in one direction, dense ma...

The integer-valued sequence $u_n=\binom{2n}{n}$ satisfies the recurrence relation $(n+1)u_{n+1}=2(2n+1)u_n$. My question : for which values $a,b,c,d\in{\mathbb Z}$ is there a sequence $(u_n)$ (not eventually constant) satisfying the recurrence relation $(an+b)u_{n+1}=(cn+d)u_n$ ? My thoughts : ...

I just don't get it, like at all. $U_{n}$ is an iteration defined on $\mathbb{N}$, btw. The question was: $$\begin{align} U_{n+1} &= \frac{8U_n - 8}{U_{n} + 2} = 8 - \frac{24}{U_{n} + 2}\\ U_0 &= 3 \end{align} $$ "Prove that $3 \leqslant U_n \leqslant 4$ by using Mathematical Induction" ...

I was wondering why the following Lie group action is Hamiltonian. Equip $\mathbb{C}^{k\times n}\cong\mathbb{R}^{2kn}$ with the canonical symplectic form $\omega_0$ on $\mathbb{R}^{2kn}$. We have an action by the Lie group $G=U(k)$ on $\mathbb{C}^{k\times n}$ by matrix multiplication, which is ...

I want to show that there exists a constant $C>0$ such that for all functions $f\in S(\mathbb{R})$, $$\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|f(x+h)+f(x-h)-2f(x)|^2}{|h|^3}dxdh=C\int_{\mathbb{R}} |f'(x)|^2dx.$$ My idea was to use Plancherel's theorem to obtain the equivalent equality (taking Fou...

Here are two related problems I have encountered several times before: $i)$ Show that there can only be countably many pairwise disjoint crosses in the plane. $ii)$ Show that there can only be countably many pairwise disjoint figure eights in the plane. A cross is defined as the union of...

Let $P(x)$ denote the sixth-order Taylor polynomial of $$e^{-2x}-3x\cos x+5\sin x$$ at $x=0$. Let $a_1,a_2,a_3,a_4,a_5,a_6$ denote the six roots (complex roots are allowed) of the equation $P(x)=0$. If $a_1+a_2+a_3+a_4+a_5+a_6=\frac mn$ where $m$ and $n$ are two positive integers with no c...

Suppose there are infinite coaches with infinite members in each coach. They stay at the hotel for infinite days. I know that guests can be accommodated using various methods like the prime powers method, but there's a slight variation in the question which is that the guests have to change their...

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, as of December 2018, there are $51$ known examples of even perfect numbers -- on the other hand, we still do not know whether there are any odd perfect numbers...

I've been trying to work on this problem and I can't seem to solve it. Could you help? Prove by induction that: $$ f(x) = e^x \sin(x) $$ $$ f^{(n)}(x)=2^{\frac{n}{2}}e^x\sin(x+\frac{n\pi}{4}) $$ I have proven it for n = 0 and then assumed it be true n = k. Then I am trying to prove it for n = ...

I want to find the integer solutions to this Diophantine equation: $5x^3=y^2+1$ I have seen a lot of problems with monic variables, but not with a constant on the $x^3$ such as this. I know I can factorise the right hand side and get: $5x^3=(y-i)(y+i)$ and work in $\mathbb Z[i]$ But I am uns...

I'm working on some data which use length and height and return one value. Something like : And I would like an equation that use the width and the height as parameters to have an approximate number of the value in the table. For example Width: $50$ and Height: $60$ would return something c...

I have the following convex optimisation problem that I really do know how to prove is unbounded below. minimise $\Sigma_{i=1}^{n} \alpha_i - \Sigma_{i=1}^{n} \beta_i$ subject to $y_i[\Sigma_{j=1}^{n}\lambda_j y_j x_i^t x_j - b] = \beta_i - \alpha_i$ $0 \le \lambda_j \le 1, \alpha_i...

I am currently studying the following proof, which may be found on this article (page 12): However, I'm facing difficulties to proper understand some of the steps. Here are my questions: Why is it possible to say that "there exists an integer $N$ and $\alpha > 0$ such that $\Sigma_n(x) \leq ...

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. (c) Prove Theorem 22.37. And here is the t...

Problem: Consider a wide sense stationary stochastic process $X(t)$, with zero mean and auto correlation function $R_X(\tau)$. Consider its transformation by a linear time invariant derivative filter, which is the first derivative of $X(t)$, that is $Y(t)=X'(t)=\frac{dX(t)}{dt}$. Verify that...

Assume that $X:(\mathbb R^+ \times \Omega) \to \mathbb{R}^n$ is a solution to an SDE of the form $dX = \mu(X,t) dt + \sigma(X,t)dW$ where $\mu, \sigma$ are continuous, Lipschitz continuous in the first variable (with a Lip constant idependent of $t$) and satisfy $||\mu(x,t)|| \le c||x|| +1 $ and...