I want to learn how to compute examples of dualizing complexes and it seems like the first step in this direction is learning how to construct an injective resolution for $\omega_{\mathbb{P}^n}$. This is because we can define the dualizing complex as $$ \omega_X^\bullet =\textbf{RHom}_{\mathbb{P}...

The Corollary 3.39 on page 250 says: If $M$ is a closed connected orientable $n$-manifold, then an element $\alpha \in H^k(M;\mathbb{Z})$ generates an infinite cyclic summand of $H^k(M;\mathbb{Z})$ iff there exists an element $\beta \in H^{n-k}(M;\mathbb{Z})$ such that $\alpha \smile \beta$ is a...

I have 2D point clouds which are 4-way symmetrical (invariant by 90° rotation). The points are usually arranged on the nodes of a square grid, densely populated, but some cases can be more complicated. I do know the point pattern in advance. I need to find the symmetry axis, which are arbitraril...

The usual ( but simple version) Lucas-Lehmer primality test, as done on Mersenne numbers ( of form $2^n-1$) is as follows: $$s_0=4\\s_n=(s_{n-1})^2-2 \pmod {2^n-1}\\if\;s_{p-2}\equiv0\pmod{2^n-1}\\2^n-1\;is\;prime$$ Are there other ways to doing this tests ?

Is Mersenne number Trial factoring, similar to the Lucas-Lehmer test ?

I know FFT is used in signal processing ( at last check), the Lucas-Lehmer Test and probably many other things. But what is the Fast Fourier Transform and what area's of math will help me understand transforms like it ( and yes I know of the area Fourier analysis, just not if anything about it) ...

Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$$ Easy to show that $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq\frac{a+b+c}{a+b+c-\sqrt[3]{abc}}$$ is true ...

For the non-negative real numbers $a, b, c$ prove that $$(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$$ What I did is applying Holder's inequality in LHS:$$(a^2+(\sqrt{2})^2)(b^2+(\sqrt{2})^2)(c^2+(\sqrt{2})^2) \geq (abc + 2\sqrt{2})^2$$ Then it suffices to prove that $$(abc+2\sqrt2)^2 \geq 3(a+b+...

$a$,$b$,$c$,$d>0$ $abc+bcd+cda+dab=a+b+c+d$ Prove the following inequality $\sqrt{a^{2}+1}+\sqrt{b^{2}+1}+\sqrt{c^{2}+1}+\sqrt{d^{2}+1}\leq\sqrt{2}(a+b+c+d)$

$a, b, c$ are positive real numbers such that $ab+bc+ca=3abc$ Prove∶ $$\sqrt{\frac{a+b}{c(a^2+b^2 )}}+\sqrt{\frac{b+c}{a(b^2+c^2)}}+\sqrt{\frac{c+a}{b(c^2+a^2 )}}\;\;\leq\; 3$$

Let $x,y,z,w>0$ show that $$\sqrt{\dfrac{x}{x+2y+z}}+\sqrt{\dfrac{y}{y+2z+w}}+\sqrt{\dfrac{z}{z+2w+x}}+\sqrt{\dfrac{w}{w+2x+y}}\le 2$$ I tried C-S, but without success.

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds. I spent a lot of time trying to solve this and, having consulted some books, I came to this: $$2x^2+2y^2+2z^2 \ge 2xy + 2xz + 2yz$$ $$2xy+2yz+2xz = 1-(x^2+y^2+z^2) $$ ...

$a, b,c $ are positive real numbers such that $a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$ Any ideas ?

Let $m=(abc)^{\frac{1}{3}}$, where $a,b,c \in \mathbb{R^{+}}$. Then prove that $\frac{b}{ab+b+1} + \frac{c}{bc+c+1} + \frac{a}{ac+a+1} \ge \frac{3m}{m^2+m+1}$ In this inequality I first applied Titu's lemma ; then Rhs will come 9/(some terms) ; now to maximise the rhs I tried to minimise the de...

Given $a,b,c \in \mathbb{R^+}$ such that $a+b+c=12$ Find Minimum value of $$S=\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2$$ My Try: By Cauchy Schwarz Inequality we have $$\left(a+\frac{1}{b}\right)+\left(b+\frac{1}{c}\right)+\left(c+\frac{1}{a}\right...

Let $a,b,c>0$, show that $$\sqrt{\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}}+\sqrt{3}\ge\sqrt{\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}}+\sqrt{\dfrac{b}{a}+\dfrac{c}{b}+\dfrac{a}{c}}$$ I tried C-S, AM-GM, Holder and more, but without success. following maybe is idea:$$\dfrac{a}{b}+\...

For $a,b,c>0$ satisfy $ab+bc+ca\ge \frac{4}{3}$. Prove that $$\sqrt{a^2+\frac{1}{\left(b+1\right)^2}}+\sqrt{b^2+\frac{1}{\left(c+1\right)^2}}+\sqrt{c^2+\frac{1}{\left(a+1\right)^2}}\ge \frac{\sqrt{181}}{5}$$ My try: By Minkowski: $LHS\ge \sqrt{\left(a+b+c\right)^2+\left(\frac{1}{a+1}+\frac{1...

In $\Delta ABC$ show that $$\cos{\frac{A}{2}}+\cos\frac{B}{2}+\cos\frac{C}{2}\ge \frac{\sqrt{3}}{2} \left(\cos\frac{B-C}{2}+\cos\frac{C-A}{2}+\cos\frac{A-B}{2}\right)$$ since $$\frac{\sqrt{3}}{2}\left(\cos\frac{B-C}{2}+\cos\frac{C-A}{2}+\cos\dfrac{A-B}{2}\right)=\frac{\sqrt{3}}{2}\sum\cos\...

Let $x,y,z\ge 0$,and such $x+y+z=2$,show that $$\sum\sqrt{\dfrac{x}{y^2+z^2}}\ge \dfrac{2}{15}\sum\sqrt{\dfrac{2+47x}{2-x}}$$ I tried C-S,Holder but without success.$$\left(\sum_{cyc}\sqrt{\dfrac{x}{y^2+z^2}}\right)^2(\sum x(y^2+z^2))\ge (x+y+z)^3$$

Suppose $\vec h : [a, b] \to R^n$ is continuous. Then show that; $$\bigg|\bigg| \int_a^b \vec h(t) dt \bigg|\bigg| \leq \int_a^b \| \vec h(t) \| dt$$ Note: This is used as Lemma in one of the lecture videos of Theodore Shifrin and he says this follows simply from Cauchy-Schwarz, ie $|\vec f...