I am have been solving this problem since a month. I solved a more difficult ones but I do not know why I stuck at this one. There is a clue I can not understand. I solved the first point and stuck with the second one. The problem statement mentions **Theorem 1.2 ** in Chapter 3, Section 3.1 in t...

Consider $$\tag{1} y'(x)+y(x)=\frac{1}{x} $$ For reference, the exact solution is $$\tag{2} y(x)=e^{-x}(C+\operatorname{Ei}(x)) $$ Where $\operatorname{Ei}$ is the exponential integral and $C$ is the integration constant. I want to study the/a particular solution of (1) as $x \to 0^+$. Using domi...

From Michigan State University's Herzog contest, 1981: Three points are taken at random on a unit sphere. What is the probability that the area of the spherical triangle exceeds the area of a great circle? I assume we always take the unique proper spherical triangle, with sides and angles not g...

Theorem: Let $\phi$ be a homomorphism of group $G$ onto group $\bar{G}$ with kernel $K$. Then $G/ K \cong \bar {G}$ I have visited a lot of pages in our forum in order to find out the importance and essence of above theorem but unfortunately I found something but they were not useful and inform...

I was thinking about linear tramsformations and i came up with this example: $$f:\mathbb{R}^n \to \mathbb{C}^n\\ f(x)=ix$$ for this example, domain and co-domain are not defined over the same field and all linear transformations that i encountered by now had domain and co-domain defined over the...

Let $\prec$ be a binary relation. Two logically-equivalent ways to express transitivity are \begin{align} & a \prec b \to \forall x (x \prec a \to x \prec b) \\ & a \prec b \to \forall x (b \prec x \to a \prec x) \end{align} They have two distinct converses: [1] \begin{align} & \forall x (x \prec...

$$\lim_{x\to 0}⁡\frac{\ln^2 (1+\tan⁡(2x))}{\cos⁡(6x)-\cos⁡(2x)}$$ i tried a lot of things with trigonometric formulae but it doesn't help. help me please.

I've already seen a similar question here: Is an Anti-Symmetric Relation also Reflexive? But my question is rather, if you know that a relation is reflexive, then, can this relation also be antisymetric? As far as I know, by definition, a relation is antisymetric if for two elements in R, xRy and...

Wolfram.mathworld.com defines a unique prime in the following way: "Following Yates (1980), a prime $p$ such that $\frac{1}{p}$ is a repeating decimal with decimal period shared with no other prime is called a unique prime. For example, $3$, $11$, $37$, and $101$ are unique primes." OEIS has uniq...

While messing around with factorials, I noticed this: $$3! - 2! + 1! = 6 - 2 + 1 = 5$$ $$4! - 3! + 2! - 1!= 24 - 6 + 2 - 1=19$$ $$5! - 4! + 3! - 2! + 1! = 5! - 19 = 101$$ $$6! - 5! + 4! - 3! + 2! - 1! = 6! - 101 = 619$$ $$7! - 6! + 5! - 4! + 3! - 2! + 1! = 7! - 619 = 4421$$ Notice that all of the...

The real Schur decomposition theorem states that for any matrix $A\in\mathbb R^{n\times n}$, there exists an orthogonal matrix $Q$ and a "quasitriangular" matrix $T$ such that $A=QTQ^T$. Here, "quasitriangular" means that $T$ has the form $$T=\begin{pmatrix}B_1&&&*\\&B_2&&\\&&\ddots&\\0&&&B_n\end...

I'm wondering on how one can go about proving that $$\frac{1}{2\pi \imath} \int_{\left(\mathcal{C}\right)} \frac{\zeta^2(1-n)\,z^{-n}}{2\cos\left(n\pi /2\right)}\,\mathrm{d}n = -\gamma -\frac12 \log z - \frac{1}{4\pi z}+\frac{z}{\pi}\sum_{n= 1}^{+\infty} \frac{\tau(n)}{z^2+n^2}$$ where $\tau(n)$ ...

In the university course I'm taking, a predicate is defined as a mathematical statement whose truth value depends on the variables involved in the statement. This definition makes me wonder whether a predicate can be a contradiction. For instance, if we use $P(x)$ to denote the statement "$x$ is ...

Is it possible to divide the equilateral triangle into 4 pieces to build a square with those four pieces, provided that one or two pieces are flipped over to the other side? If possible, I wish to find a solution, and if not, I wish to find proof. Let me remind original solution: Now imagine tha...

To keep things relatively simple I'm presenting a somewhat-butchered version of the IMH; for more details, see S.-D. Friedman, Internal consistency and the inner model hypothesis. Throughout, "ctm" means "countable transitive model of $\mathsf{ZFC}$," an inner model of a ctm $\mathcal{A}$ is a ct...

Consider an elliptic curve $E: y^2 = x^3 + ax + b$ over some finite field $F_{q^k}$ and a point $P$ on $E$ of order $n$. Miller's algorithm tells us how to efficiently construct a rational function $f_{n,P}(x,y)$ on $E$ with divisor $div f = n [P] - n [\mathcal{O}]$. I have worked through exampl...

As sine functions are nested more and more in manner shown below, the shape of the function approaches that of a square wave. \begin{align} f^1(x)&=\sin(x)\\ f^2(x)&=\sin(\,\sin(x)\,)\\ f^3(x)&=\sin(\,\sin(\,\sin(x)\,)\,)\\ \end{align} Is it possible to define $f^n(x)$ where $n$ can be any ration...

In Hirsch's book, there is a wonderful theorem 2.11: where a structure functor is simply a presheaf, and continuous means it is a sheaf (it has the gluing property). Nontrivial means there is at least one local section. Locally extendable means, in his words: As an example, if $X$ is a $C^r$ ma...

Let $V$ be a real inner product space with inner product $\langle,\rangle$. For $u,v,w\in V$, how to show the following inequality $$\langle u,v \rangle \langle u,w\rangle \leq \frac{1}{2}(\langle v,w\rangle +\|v\| \|w\|)\|u\|^2?$$ I tried with Cauchy-Schwartz inequality but failed to prove above.

I'm looking at old past papers and found this question: "Solve the quadratic equation $3x^2 + 4x - 5$ giving your answer in the form $\frac{a}{b\pm\sqrt{19}}$, where $a$ and $b$ are integers." I've never seen a quadratic solution in this form, with the surd root on the bottom. Does anybody have...