2
votes
0answers
47 views
+50

Criterion for orientability: Derivative of transition map

The definition I have been given for a smooth abstract surface, $S$, to be orientable is that given a continuous family of maps $f_t: D \to S$ that embed the closed unit disk into $S$ with $f_0(D) = ...
3
votes
4answers
289 views
+50

Evaluate $\int\sqrt[5]{\frac{x+5}{x-5}}\,\mathrm dx$

How to compute this integral? I stuck at a point where I get $\displaystyle\int\frac{1}{t^5-1}+ \cdots $ $$\int\sqrt[5]{\frac{x+5}{x-5}}\,\mathrm dx$$ using $\displaystyle t=\sqrt[5]{\frac{x+5}{x-5}}$ ...
8
votes
0answers
153 views
+50

Determining information in minimum trials (combinatorics problem)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the ...
3
votes
1answer
69 views
+50

Need some facts about Newton-Schulz iterative method and its application to sparse matrices

I am studying Newton-Schulz iterative method for obtaining an approximate inverse , which is given by $V_{k+1}=V_{k}(2I-AV_{k})$, wherein $I$ is the identity matrix and it converges, when the ...
20
votes
2answers
405 views
+50

How to solve $\int_0^{\infty}\frac{\cos{ax}}{x^3+1}dx$?

QUESTION. It is looked for a closed solution for following real integrals $\displaystyle\int_0^{\infty}\displaystyle\frac{\cos{ax}}{x^3+1}dx$ and ...
7
votes
0answers
160 views
+50

Probability of a zero product given one previous zero product

Consider two random vectors $v=(v_1,\dots, v_n)$ and $w=(w_1,\dots, w_{n+1})$. Each element of $v$ is independently $\pm1$ with prob $1/2$. Each element of $w$ is independently $1$ with probability ...
12
votes
2answers
244 views
+150

Integer part of a sum (floor)

Let $\left(\, x_{n}\,\right)_{\,n\ \geq\ 1}$ be a sequence defined as follows: $$ x_{1}={1 \over 2014}\quad\mbox{and}\quad x_{n + 1}=x_{n} + x_{n}^{2}\,, \qquad\forall\ n\ \geq\ 1 $$ Compute the ...
1
vote
1answer
136 views
+150

Flexes of cubic curve

Which are the flexes of the cubic curve of Fermat $$x^3+y^3+z^3=0$$ at $\mathbb{P}^2(\mathbb{C})$ ? Could you give me a hint how we could find the flexes? Do we have to use maybe the following ...
1
vote
2answers
89 views
+50

Understanding proof of Peano's existence theorem

I'm studying the proof of Peano's existence theorem on this paper. At page 5 it is said that the problem $$\begin{cases} y(t) = y_0 & \forall t ∈ [t_0, t_0 + c/k] \\ y'(t) = f(t − c/k, y(t − ...
4
votes
0answers
64 views
+50

One more question about mapping quaternionic matrices into real matrices

Real matrices that lie in the image of the inclusion homomorphism $\rho_n: M_n(\mathbb C) \to M_{2n}(\mathbb R)$ are called complex linear. It is easy to see that a real matrix is complex linear if ...
3
votes
0answers
218 views
+100

Subcategories of $\mathbf{Top}$ which are closed under arbitrary products and coproducts

Is there a classification of those (full) subcategories of the category $\mathbf{Top}$ of topological spaces that are closed under arbitrary products and arbitrary coproducts in $\mathbf{Top}$? EDIT: ...
0
votes
1answer
50 views
+150

Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This has been cross-posted to MO.) Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$). ...
0
votes
1answer
52 views
+50

Words of the Normal Form of the Presentation of a Finite Monoid

Massive Edit: After consulting with a few mathematicians at my university, I got a better understanding of what I was actually looking for. $$ \langle\ s,\ t\ \vert\ s^2 = 1,\ t^n = 1\ \rangle $$ ...
0
votes
2answers
58 views
+50

How do you take the inner product of a vector whose components have different units?

For example, what is the inner product of <1m, 1s> and <2m, 3s>?
7
votes
4answers
267 views
+500

Construct quadrangle with given angles and perpendicular diagonals

The following came up when I worked on the answer for a different question (though it was ultimately not used in this form): Proposition. Given positive angles $\alpha,\beta,\gamma,\delta$ with ...
9
votes
0answers
145 views
+100

Arithmetic Progressions in an unusual sequence

An infinite sequence ($a_0$, $a_1$, ...) is such that the absolute value of the difference between any 2 consecutive terms is equal to $1$. Is there a length-8 subsequence such that the terms are ...
3
votes
1answer
60 views
+50

Branch Cuts of $f(z) + g(z) = \sqrt{p(z)} + \sqrt{q(z)}$

How does one find the branches of $$f(z) + g(z) = \sqrt{p(z)} + \sqrt{q(z)}$$ where $p$ & $q$ are second degree polynomials? It would be very nice to see this general method applied to, say, ...
3
votes
0answers
131 views
+50

Approximating an integral with another integral with finite limits

I came across the following integral in my work $$\int_{-\infty}^{\infty} \frac{\frac{1}{(1- \ \ 2 \pi j s \theta)^{m}}-1}{2\pi j s }\ e^{-2\pi j s\sigma^2}\ ds $$ Assuming $\theta,m,\sigma^2$ are ...
7
votes
0answers
108 views
+50

Why the Steinberg idempotent is idempotent?

Consider the group $GL_n(\mathbb{F}_p)$. We have the following subgroups : -$\Sigma_n$ the symmetric group (permutation matrices) -$B_n$ the Borel subgroup (upper triangular matrices) -$U_n$ the ...
5
votes
2answers
225 views
+50

A question on the unit tangent bundle of the sphere and $SO(3)$

Let the unit tangent bundle be defined as follows: $$T^1S^2=\{(p,v)\in \mathbb R^3 \times \mathbb R^3 | |p|=|v|=1 \text{ and } p \bot v \}$$ Let $SO(3)$ be the group of rotations of $\mathbb R^3$. ...
54
votes
2answers
1k views
+100

Some users are mind bogglingly skilled at integration. How did they get there?

Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this. Especially in a ...
24
votes
9answers
2k views
+450

Old oxford scholarship question: $a^ab^b \ge a^bb^a$

Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.
1
vote
3answers
89 views
+300

To find a measurable subset with arbitraray measure

Suppose $E$ is measurable subset of $\Bbb R$ s.t. $m(E)=1$ . Is exists $A$ that is measurable subset of $E$ and $m(A)=\frac 1 2$? $A\subset E$ so $m(A) \le m(E)=1$ . since $0 \le m(A)\le 1$, ...
2
votes
0answers
83 views
+50

Use of Poincare Lemma in solving $\nabla \times \textbf{A}(\textbf{r})=\frac{\textbf{r}}{r^3}$

Let $U = \mathbb{R}^3 \setminus \{(0,0,z) \}$ (ie $\mathbb{R}^3$ with the $z$-axis removed ) and consider $\beta$ on $U$ given by $\displaystyle \beta = \frac{x dy \wedge dz + y dz ...
103
votes
0answers
4k views
+100

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
0
votes
0answers
46 views
+50

Identity of the pushforward of a vector field using a Jacobi bracket.

Let $Z(u,v)$ be the vector field $Z(u,v)=(u^2+u,v^2+v)$, let $\Gamma_t$ denote its flow. I have shown that $[X,Z]=Z-X$. Show that $(\Gamma_t)_*X=e^{-t}X-(e^{-t}-1)Z$. Could someone please show me ...
7
votes
0answers
46 views
+50

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
10
votes
0answers
99 views
+50

Some four clubs have exactly $1$ student in common

There are $100$ students in a school, and they form $450$ clubs. Any two clubs have at least $3$ students in common, and any five clubs have no more than $1$ student in common. Must it be that some ...
3
votes
1answer
52 views
+100

How can I recursively approximate a moving average and standard deviation?

Consider a sequence of measurements $(x_1, x_2, ...)$. Let $\mu_n$ be the $p$-period moving average defined by $$\mu_n = \frac{1}{p}\sum_{i=n-p+1}^nx_i$$ and $\sigma_n$ be the $p$-period moving ...
4
votes
1answer
75 views
+50

Good source on current 'views and thoughts' on mathematics

Recently I've become interested in the history of the way people think about mathematics. What I mean by this is for example how Godels proof basically put an end to the whole school of thought ...
3
votes
1answer
90 views
+50

Describing non-vanishing $1$-forms on two dimensional manifolds.

Let $h_1 \mathrm{d}x_1 + h_2 \mathrm{d}x_2$ be a non-vanishing $1$-form on a $2$-dimensional manifold. Why do locally exist smooth functions $f,g$ with $f\mathrm{d}g= h_1 \mathrm{d}x_1 + h_2 ...
4
votes
1answer
80 views
+100

Show that there is a continuous $g$ with compact support

If $f$ is a measurable complex function (that means that it doesn't take the values $\pm \infty$) with compact support, then for each $\epsilon >0$ there is a continuous $g$ with compact support so ...
2
votes
0answers
72 views
+200

Prove that in a group iterated commutators with repeated generators is trivial implies that each generator commutes with all its conjugates

Let $G$ be a finitely generated group with generating set $S=\{x_1,\cdots,x_n\}$. Let $[x,y]=x^{-1}y^{-1}xy$ be the commutator of $x$ and $y$. Suppose that every iterated commutator with repeated ...
5
votes
1answer
111 views
+50

If $G/N$ and $N$ are finitely presented, then $G$ is finitely presented.

Let $G$ be a group and $N\triangleleft G$. Show that if $G/N$ and $N$ are finitely presented, then $G$ is finitely presented. Worked on this problem for about 2 hours before we all threw in the ...
3
votes
1answer
101 views
+50

Colimit in the category of (all) simply transitive group actions

Let $\mathcal{C}$ be the category of all group actions, i.e. : the objects are the pairs $(G,F)$ where $G$ is a group and $F$ is a functor $F\colon G\to\mathbf{Sets}$ a morphism between $(G_1,F_1)$ ...
0
votes
0answers
88 views
+50

Condition for a process to be a supermartingale

I am struggling in this question: Let $W$ denote a Brownian motion. Given that $ X_t = e^{- \lambda t} X_0 + \int_0^t \sigma e^{- \lambda (t-s)} \,dW_s$ solves the SDE \begin{equation} dX_t = - ...
2
votes
0answers
80 views
+50

From the viewpoint of modern geometry, is there a “best” definition of the term “triangle”?

I can think of at least six different possible definitions of the term "triangle" in Euclidean geometry. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three or fewer points. ...
4
votes
0answers
123 views
+100

Factorizations in the symmetric group

Notation notes: The cycle $(i,i+1,\dots,j)\in S_n$ is the permutation $(1)(2)...(i,i+1,\dots,j)\dots(n)$ in cycle notation. Motivation Given a factorization of a permutation into certain cycles ...
0
votes
1answer
50 views
+50

Problem integrating in problem using the Poincaré Lemma

a) It is easy to show that $d\beta=0$. b) $\begin{align}\hat{\mathbb{X}}_t &= (\frac{\partial}{\partial t}\hat{\Phi}_t) \hat{\Phi}_t^{-1}) \\ &= (\frac{\partial}{\partial t}\hat{\Phi}_t) ...
1
vote
1answer
69 views
+50

The intersection points are collinear

It is given a hexagon inscribed in a conic section. I want to prove that the pairs of opposite site intersect at three points of the projective plane that are collinear. How could we do this? ...
5
votes
1answer
64 views
+50

Finding taylor expansion for $\tanh(x)$

I am a high school student and am trying to find the taylor expansion of $\tanh(x)$ in terms of a summation form. I have gotten this far, and am aware it might get complicated very quickly. If someone ...
2
votes
0answers
40 views
+50

Induced Markov chain - verify Markov property and another property

First, here is how we defined induced Markov chains: Suppose that $(X,E,P)$ is an irreducible Markov chain, where $X=(X_i)_{i\in\mathbb{N}_0}$, $E$ is the state space and $P=(p_{i,j})_{i,j\in E}$ is ...
-2
votes
2answers
66 views
+100

Calculate weak derivative

I am supposed to calculate the weak partial derivatives of the function $f: B(0,\frac{1}{2}) \rightarrow \mathbb{R}, x \mapsto |\log(\|x\|_2)|^\alpha$ for all $\alpha \in \mathbb{R}$, where ...
6
votes
0answers
66 views
+50

History of the term “anodyne” in homotopy theory

There is a notion of an anodyne morphism (usually of simplicial sets). This type of morphisms is used, for instance, to establish basic properties of quasicategories. But the name is quite mysterious. ...
6
votes
2answers
110 views
+50

A finite sum over $\pm 1$ vectors

What is a nice way to show $$\sum_{u\in\{-1,+1\}^N}|\sum_{i=1}^Nu_i|=N\cdot\binom{N}{\frac{N}{2}}$$ when $N$ is even? Could there be a short inductive proof?
4
votes
1answer
60 views
+50

Reference Request: Thom Spectrum of a virtual vector bundle

Given a vector bundle $\xi$, one can construct a Thom spectrum $\mathbb{T}h(-\xi)$ associated to its additive inverse. I unsuccessfully browsed through literature to find a reference for an explicit ...
11
votes
1answer
227 views
+100

Integral: $\int_{-\infty}^{\infty} \frac{dx}{(e^x+x+1)^2+\pi^2}$

I am looking for real analytic methods to prove the following: $$\int_{-\infty}^{\infty} \frac{dx}{(e^x+x+1)^2+\pi^2}=\frac{2}{3}$$ I have seen a similar problem on the website but if I remember ...
4
votes
2answers
142 views
+100

What do Sylow 2-subgroups of finite simple groups look like?

What do Sylow 2-subgroups of finite simple groups look like? It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision ...
1
vote
0answers
17 views
+100

Commutator $[A_{p,q},A_{s,t}]$ in the pure braid group?

Let $B_n$ be the braid group; that is, a group generated by $\sigma_1,\cdots,\sigma_{n-1}$ with relations $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ for $i=1,\cdots,n-2$; ...
4
votes
0answers
98 views
+50

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...

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