I must solve $$\int_{-\infty}^{\infty}\frac{x \text{sin}x}{x^2+4} dx$$ I simplified this to $$\int_{-\infty}^{\infty}\frac{x \text{sin}x}{x^2+4} dx = \frac{1}{i}\int_{-\infty}^{\infty}\frac{x e^{ix}}{x^2+4} dx$$ Now consider only $$\int_{-\infty}^{\infty}\frac{x e^{ix}}{x^2+4} dx $$ I considered ...

I want to find the following: $$ \int_{0}^{2}\int_{y/2}^{1}y{\rm e}^{-x^3} \,{\rm d}x\,{\rm d}y $$ using change of variables. I've solved $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} {\rm e}^{-x^2}\,{\rm d}x\,{\rm d}y $$ by changing to polar coordinates, but it doesn't look like that will be ...

A line with equation $L = (1, 0, -2) + s(2, -1, 2)$ intersects with the plane $x + 2y + z = 2 $ at an angle of $\theta$ degrees. I am having some difficulty understanding why the dot product is necessary and why I cannot simply obtain $\theta$ by using the right-angled triangle formed between the...

Let $X$ be a compact Riemannian 4-manifold, $P$ a principal $G$-bundle over $X$, and $\mathfrak{g}$ be its adjoint bundle. Let $\omega$ be a self-dual conneciton on $P$ (i.e. its curvature $\Omega \in \Omega^2(\mathfrak{g})$ is self-dual). It induces a covariant derivative $d^\omega:\Omega^0(\mat...

In R.W. Sharpe's Differential Geometry, a flat fibre bundle is defined as a bundle whose transition functions are constant. I don't understand the difference between this and a trivial bundle. Because, surely if the transition functions are constant you can just perform a rescaling so that they a...

Let $G$ be an abelian group and $M$ a $G$-module. The basic definitions: Let $H < G$ be a subgroup of finite index. We have a map $tr: H^0(H, M) \rightarrow H^0(G, M)$ on group cohomology defined by $m \mapsto \sum_{g \in G/H} gm$. This can be extended to a map $H^*(H, M) \rightarrow H^*(G, M)$ o...

I am asked to find the isolated singularities of the function and determine their typing, order, and finding the principal part at each pole. So for the first function, which is $\frac{e^z-e}{z^2-1}$, I know that it has poles of order $1$ at $\pm1$. However, when trying to find the principal part...

This is a strengthening of a question another user asked, here: Are irrational numbers order-isomorphic to real transcendental numbers?. In the answer to that question, it was stated that the irrationals are order-isomorphic to the transcendental reals. My question is this. Suppose $S$ is a conti...

Two robots, Aaron and Erin, have made it to this year’s final! Initially they are situated at the center of a unit circle. A flag is placed somewhere inside the circle, at a location chosen uniformly at random. Once the flag is placed, Aaron is able to deduce its distance to the flag, and Erin is...

I am somewhat confused about the different flavours in which the statement "the peripheral (group) system is a complete knot invariant" usually comes, and I believe not all of them have precisely the same strength, meaning that some might determine the knot up to orientation and/or mirror image. ...

I am reading Atiyah, MacDonald "Commutative Algebra" and there it is said that ring $A$ is Noetherian if three equivalent conditions are satisfied. (1) Every nonempty set of ideals in $A$ has maximal element. (2)Every ascending chain is stationary (3) Every ideal is finitely-generated. I see why ...

This is more a reference request question than a technique one. I hope someone who has read Zariski, Samuel's Commutative Algebra (volume I) could answer it. (It seems few people use it nowadays). In Zariski, Samuel's Commutative Algebra (volume I), the definition of the length of an ideal is as ...

Is there a simple description of those colimits that exist in the simplex category $\mathbf{\Delta}$ (of finite linear orders and non decreasing maps) ? It is easy to find examples of diagrams for which a colimit exists or not, but I would be interested in a concise and exhaustive description of ...

Differential equations can be used to describe functions. Some famous ones include $$\frac{\partial f}{\partial x} - f(x) = 0$$ $$\frac{\partial^2 f}{\partial x^2} - f(x) = 0$$ The iterated running xor working on a sequence of numbers $\{x_0,\cdots,x_N\}$ on $\mathbb Z_2$ : $$\mathcal{X}\{x\}_{k}...

I took my examination in differential equations earlier; i would've gotten a perfect score, but i tripped in this problem: $$(1+5y\sin x)dy + y^4\cos xdx = 0$$ I used substitution suggested by the equation: $w = \sin x$, $dw = \cos x dx$. I transformed it into a linear D.E., however, i was stuck ...

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$, where $n \in \mathbb N$ is an arbitrary fixed integer that stands for the dimension of the euclidian space $\mathbb R^n$. In everything that follows, I am dealing with the usual Lebesgue integral on $\mathbb R^n$. Question. Let $\psi \in C_c^\in...

So, according to Humphreys:$\mathfrak{gl}_n$ is all $n \times n$ matrices, $\mathfrak{sl}_n$ is $n\times n$ matrices with sum of diagonal elements (trace) equal to zero and $\mathfrak{s}_n$ is set of all $\lambda I$ where $\lambda \in F$ (underlying field) and $I$ is the identity matrix (diagonal...