Question: Find $$\int_1^\infty \frac{\{x\}}{x(x+1)}dx,$$ where $\{x\}$ means $x - \lfloor x \rfloor$. I have attempted to split this into two integrals, namely $$\int_1^\infty \frac{x}{x(x+1)} - \int_1^\infty \frac{\lfloor x \rfloor}{x(x+1)},$$ however did not get anywhere significant. I have als...

Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study in some paper the author stated that $$\int_{\mathbb{T}^2} J^s (\partial^3_x w) (J^s w)dxdy + \int_{\mathbb{T}^2} J^s (\partial_x \partial_y^2 w)J^s wdxdy=0.$$ I could not r...

Following is Question 9 of Chapter 10 of Abstract Algebra by Dan Saracino: Suppose $G$ is a group and $H$ and $K$ are subgroups of $G$ such that $|H|=39$ and $|K|=65$. Prove that $H \cap K$ is cyclic. Up to this chapter, Sylow Theorem has not been introduced, and we should solve this by Lagrange ...

I'm trying to understand the proof of this theorem: which is: I don't inderstand the highlighted lines, can somenone explain why this occur? for the first I think it must be the direct sum for the second, I don't see how he identified the couple of morphisme to one morphisme for the third It...

I have the following PDE, $$(xz-y)p+(yz-x)q=xy-z$$where $p=z_x,\quad q=z_y$ Now having a hard time to get two solution from, $$\frac{dx}{xz-y}=\frac{dy}{yz-x}=\frac{dz}{xy-z}$$ I can't think of any multipliers trick which can help me to get one. In fact, $dx-dy$ or $dx+dy$ also couldn't help here...

I was reading about multiple angle formulas to expand $\sin{(nx)}$ or $\cos{(nx)}$ to be in terms of $\sin{x}$ and $\cos{x}$ on Wolfram MathWorld, and came across these formulas: These formulas look so much like the Taylor series expansions of the respective functions that I feel there must be ...

John wants to build a set of numbers, from the range of 1 to 100. The only rule is that in that set no number can be the average of any other two. For example, if the set contains the numbers 1 and 3, then 2 cannot be present. What’s the size of the biggest set that John can build? More precisely...

Prove or disprove. Suppose that $\left(M_{n}\right)_{n}$ is a martingale with $M_{n} \geqslant-10 \quad \forall n$, a.s. Is it true that $$ \sum_{i=1}^{\infty}\left(M_{i-2} M_{i}-M_{i-1} M_{i-2}\right)<\infty, \text { a.s? } $$ This is what I have done so far I thought of applying the lemma of di...

Let $C_1, C_2$ be two projective smooth curves over $\mathbb{C}$. Is it possible to say when $C_1 \times C_2$ a complete intersection in some projective space? For three curves the answer is "it is never a complete intersection" since a product of three curves has too big Picard group and it woul...

Consider I have boundary operators $\partial_1$, $\partial_2$: $\partial_1 \circ \partial_2 = 0$. Then if interested in $\text{ker}\,\partial_1 / \text{im}\,\partial_2$ one can study $\text{ker}\,(\partial_1^T\partial_1 + \partial_2\partial_2^T)$. However this is only true if I can define non-de...

Let $\phi:X\to Y$ be a morphism between irreducible quasiprojective varieties. If $\phi$ has a dense image in $Y$ can we conclude that its image has an interior? It really feels like it, but I couldn't show it. Maybe there is a counterexample? I guess for specific type of quasi-projective varieti...

First let me say that I have read tons of questions and answers on AC but none helps me get past this issue, so I am posting my own question. Let's say we have an infinite family of disjoint non-empty sets $\{ X_{\lambda} \}_{\lambda \in I}$. What's wrong with the following reasoning: $\forall \...

By long "s", I mean this $\int$ What I am referring to is when we use separation of variables in solving differential equations; Eg: $\frac{dy}{dx} = \frac{y}{x-3}$ $\frac{1}{y} dy = \frac{1}{x-3} dx$ Since this is my first time learning about the separation of variables, I got stuck here. In a...

If I have a set A, comprising of numbers from 1 to 10: $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Let's say I want to make another set by "including" all even numbers: $\{2, 4, 6, 8, 10\}$ Or I wanted to make a different set by "excluding" all odd numbers: $\{2, 4, 6, 8, 10\}$ These sets are of cou...

Wikepedia KKM says that Brouwer fixed-point theorem, Sperner's lemma, and Knaster–Kuratowski–Mazurkiewicz lemma are equivalent. In "A course of topological combinatorics", I find a proof that Sperner implies Brouwer. I am wondering is there a proof that Sperner implies KKM?

$\newcommand{\Exp}[1]{\mathbb{E}\left[#1\right]}$ I am interested in understanding wether the following approach holds when calculating the expectation of two correlated random variables. Suppose $X\sim F_X$ and $Y\sim F_Y$ have positive support, and assume the joint density may be implied from t...

The question states: Let $A$ be the intersection of the balls $x^2+y^2+z^2\leq 9$ and $x^2+y^2+(z-8)^2\leq 49$ I am asked to just set up the iterated triple integral that represents the volume of $A$ in cartesian coordinates. What I am able to determine so far is that for the equation $x^2+y^2+z^...

I'm struggling with producing a proof of the following result: Let $X = \overline{\mathbb{C}}$ be the Riemann sphere, and consider $M(X)$ the space of finite Borel measures on $X$ with norm given by the total variation. Let $\mathcal{H}$ be a basis of open sets for the standard euclidean topology...

This MO question https://mathoverflow.net/questions/249541/formal-power-series-is-taylor-expansion-of-rational-function-iff-hankel-determin states that if $k$ is a field and $k[[T]]$ is the power series ring over $k$ then $u(T)\in k[[T]]$ is the power series of some rational function if and only ...