-4
votes
1answer
119 views

Outline approach to Collatz 3n+1 conjecture / Criticism needed

// Instead of trying to show there are no loops and no sequences that increase without bounds, consider how any "deviant set of sequences" must partition the natural numbers into two infinitely large ...
2
votes
1answer
63 views

Exactness and Naturality

I'm trying to read this blog post about exact functors, and I see mentions of naturality which I have not stumbled upon elsewhere. In particular, in the proof of the Theorem, the author says By ...
39
votes
4answers
2k views

Do “Parabolic Trigonometric Functions” exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) ...
1
vote
2answers
32 views

When does homology commutes with arbitrary direct sums

Is it necessary to have the criteria that the direct sum of a collection of monics is a monic, to show that homology commutes with arbitrary direct sums? Because when I tried to prove the result, I ...
0
votes
0answers
23 views

finite index subgroups of profinite completions

Let $G$ be a finitely generated, residually finite group, and let $\widehat{G}$ denote its profinite completion. Is there a 1-1 correspondence between finite index subgroups of $G$ and open subgroups ...
0
votes
0answers
23 views

Use the chain rule to convert the Laplace equation in (x,y) coordinates into an equivilent differental equation in (r,theta) coordinates.

use the equations $r=\sqrt{x^2 +y^2}$ and $\theta=\arctan(\frac{y}{x})$. I was able to get the partial derivative of of $r$ with respect to $x$ and $y$ and the partial derivative of $\theta$ with ...
0
votes
4answers
57 views

Monty Hall problem (shifting probabilities)

I was explaining the Monty Hall problem to someone, and I explained it in this way: You have three doors, and you pick one, giving you a $1/3$ chance of being right. The presenter opens one of the ...
1
vote
1answer
33 views

A non-UFD where prime=irreducible [duplicate]

It is easy to see that in an atomic domain (where every element factors into irreducibles), we have that all irreducibles are prime iff the domain in question is an UFD. I think it is not true for a ...
3
votes
2answers
79 views

difficult complex integral $\int_\gamma \frac{1}{z^2+i}dz$

We are asked to calculate $\int_\gamma \frac{1}{z^2+i}dz$ where $\gamma$ is the straight line from $i$ to $-i$ in that direction. My parametrization is simple, I chose $z(t)=i-2it$. Notice that ...
-1
votes
2answers
75 views

C*-Algebras: Contractive Morphism

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with $\mathbb{1}_\mathcal{A}\in\mathcal{A}$. Consider an algebraic morphism $\pi:\mathcal{D}\subseteq\mathcal{A}\to\mathcal{B}$ with ...
2
votes
4answers
307 views

Evaluate $ \lim\limits_{x\to 1} \frac{(\ln x)^2}{1-x} $

I have trouble finding the limit of the following : $$ \lim\limits_{x\to 1} \frac{(\ln x)^2}{1-x} $$ using the rule from L´Hopital. Since both quotients converge to $0$, I should be able to use ...
2
votes
1answer
15 views

Rates of convergence of an OLS estimator

I have a linear regression model $$ y_t=x_t\beta+e_t,\quad t=1,\ldots,N. $$ Here $x_t$ is non-random and given by $(1,\delta_t t)$ where $\delta_t$ is 1 for odd $t$ and $0$ otherwise. Moreover, ...
7
votes
1answer
494 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
2
votes
2answers
26 views

Family bounded in $\mathcal{L}^1$ has limit a.e.

Let $(X, \mathcal{F} , \mu )$ be a measure space. Suppose $\lbrace X_n \rbrace$ is a family of functions in $\mathcal{L}^1$, bounded in $\mathcal{L}^1$ i.e. there exist $K \geq 0 $ such that ...
7
votes
0answers
65 views

Group theoretic solution to an IMO problem

Is there a (strictly) group theoretic interpretation (and possibly a solution) to this problem (taken from the 27th IMO)? "To each vertex of a regular pentagon an integer is assigned in such a way ...
3
votes
2answers
170 views
+100

3-dimensional light up cube, # of rows/cols/diags in/on a 4 × 4 × 4 cube

Imagine a 3-dimensional cube (much like a 4 × 4 × 4 Rubik's cube) except the planes of the cube cannot be twisted individually and instead of faces with different colors, it is clear (see ...
3
votes
2answers
60 views

Group of order 30 can't be simple

I have this following question from my class note on Sylow Theorem: Show that a group of order 30 can not be simple. For that I know the followings: (1) A simple group is one that does not have ...
1
vote
4answers
505 views

Find the values of $x$ which makes $\det (A)=0$ without expending determinant

Find the values of $x$ which makes $\det(A)=0$ without expending determinant: Let $A$ : $$\begin{bmatrix}1 & -1 & x \\2 & 1 & x^2\\ 4 & -1 & x^3 \end{bmatrix} $$ How can I ...
2
votes
1answer
30 views

Associated Legendre polynomials

The associated Legendre ODE is given by $$ \left( (1-x^2) f'(x) \right)' - \frac{m^2}{1-x^2} f(x) = \lambda f(x)$$ The eigenfunctions have certain properties that I would like to understand by ...
0
votes
1answer
39 views

Determining the equation of this 3D object

Does anyone know how I can determine the equation of the 3D object below? (Maybe there's a program that can do it?) I am looking for a formula to define this 3D object, but am having trouble finding ...
0
votes
0answers
20 views

Convergence and limit of Muller sequence

The Muller sequence is given by the recursive definition: $U(n+1)=111-\frac{1130}{U(n)}+\frac{3000}{U(n)U(n-1)}$ with $U(0)=5.5$ and $U(1)=\frac{61}{11}$. This sequence is interesting in ...
2
votes
1answer
34 views

$\dim V = \dim \phi(V)+\dim \ker \phi$

I want to show that $\dim V = \dim \phi(V)+\dim \ker \phi$. I know this proof can be found in any linear algebra textbook. However, my question is not exactly about the proof, but on a statement I ...
0
votes
2answers
107 views

Show the sum is equal to a product of six primes

On a set of math challenges, one of them is to prove that $$145678+456781+567814+678145+781456+814567$$ is the product of six different primes. This sounds like number theory to me, but I have no ...
4
votes
2answers
98 views

Inverse of a set, possible?

Just like ordinary algebraic operations have inverses, could we imagine the inverse of a set? Like $x\in\{x\}$ then maybe the inverse denoted $[|x|]$ would mean $$\{\ [|x|]\ \}=x$$ Would this idea ...
2
votes
0answers
18 views

Canonical algebra isomorphism $k[D(f)]\cong k[S_0,\dots,S_n]_{(f)}$?

Here's a common set up. Suppose you have $f\in k[S_0,S_1,\dots,S_n]$ is a homogeneous polynomial with $\deg(f)=d$, over some closed field $k$. Let $D(f)$ be the principal open set of $f$ in projective ...
3
votes
0answers
46 views

A question about the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$

I wanted to clarify some confusion I was having on the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$, which I call $Aut(\mathbb{Z}_{2} \times \mathbb{Z}_{4})$. I considered the ...
3
votes
3answers
1k views

Area Between Three Circles of Differing Radii

From the link in wikipedia http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii OPEN QUESTION: What is the equation, in three variables, relating the radii of ...
3
votes
3answers
46 views

3D coordinates of circle center given three point on the circle.

Given the three coordinates $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$ defining a circle in 3D space, how to find the coordinates of the center of the circle $(x_0, y_0, z_0)$?
1
vote
1answer
14 views

Showing that the Brownian Bridge is Gaussian

Take $X_t = (1-t)B_{t/(1-t)}$ for $t\in[0, 1)$ where $B_t$ is a $1$-dimensional Brownian motion. I want to show that $X_t$ is Gaussian. I have actually never been able to find a precise definition ...
0
votes
2answers
31 views

Random variables and Linearity

I have an equation $Y = 5 + 3\times X$ and I assume that $X$ is a random variable taking values from a uniform distribution. Can I consider that also $Y$ is a random variable which takes values from a ...
0
votes
1answer
23 views

2 question about supremum of subset and a sequence that converge to it.

Let $A$ be a bounded subset of $\mathbb{R}$. 1. Show that there exists a sequence $a_n$ of elements of $A$ such that $\lim _{ }\left(a_n\right)\:=\:sup\left(A\right)$ 2. Show that we can build a ...
1
vote
1answer
17 views

Prime numbers $p$ and $q$ and possession of normal subgroup of order $p$

I have this following question from my class note on Sylow Theorem: Let $p$ and $q$ be prime numbers such that $p \nmid (q-1).$ Show that each of order $pq$ possesses a normal subgroup of oder ...
8
votes
1answer
47 views

Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$?

It is known that given a solution to, $$a^4+b^4+c^4 = d^4\tag1$$ then either $-c+d,\;c+d$ is always divisible by $2^{10}$. For example, $$95800^4+414560^4+217519^4=422481^4$$ then ...
3
votes
3answers
48 views

Is every countable ordinal homeomorphic to a subspace of $\mathbb R$?

I know that every countable ordinal is isomorphic to some subset of $\mathbb R$ as ordered sets. Is it also the case that every countable ordinal (with the order topology) is homeomorphic to some ...
12
votes
3answers
582 views

Evaluate $ \displaystyle \lim_{x\to 0}\Bigg( \frac {(\cos(x))^{\sin(x)} - \sqrt{1 - x^3}}{x^6}\Bigg) $

Evaluate $$ \displaystyle \lim_{x\to 0}\Bigg( \frac {(\cos(x))^{\sin(x)} - \sqrt{1 - x^3}}{x^6}\Bigg) $$ I tried to use L'Hospital rule but it got very messy. Moreover I also tried to analyze ...
1
vote
3answers
90 views

Education problem - what to do ? Very intriguing problem [on hold]

I really love maths, I see some people complaining of being afraid of maths, or finding it difficult to have motivation to do maths and I can't understand them. Since I'm blinded by my love of maths; ...
2
votes
2answers
71 views

Knapsack problem NP-complete

Show that the knapsack problem (Given a sequence of integers $S=i_1, i_2, \dots , i_n$ and an integer $k$, is there a subsequence of $S$ that sums to exactly $k$?) is NP-complete. Hint:Use the exact ...
-5
votes
0answers
84 views

How meaningful is “$\infty=-\frac12$”? [duplicate]

edit I'd like to dispute the dupe-closing: The other question ($1+1+1+\cdots=−\frac12$) asks for a proof of that formula, while this questions asks whether its implication is valid in a more general ...
2
votes
2answers
36 views

How to tell if two spherical coordinates lie on the same plane

I have the rho, theta, and phi values of two points, how can one tell that two vectors are normal to the same plane by looking at their spherical coordinates?
1
vote
0answers
35 views

Probability of Posting a Quad and Trip on 4chan

Important Pre-Requisite Knowledge On the image board 4chan, every time you post your post gets a 9 digit post ID. An example of this post ID would be $586794945$. A Quad is a post ID which ends with ...
0
votes
2answers
37 views

Alternate element disjoint exhaustive Subsets with same cardinality

Suppose $U$ is an ordered set, I want to construct subsets $A$ and $B$ such that: (1) (Disjoint) $A \cap B = \phi $ (2) (Exhaustive) $A \cup B = U$ (3) (Alternate elements) $\forall x,y \in A, ...
0
votes
2answers
398 views

Question on compound interest

If you deposit $100 at the end of every month into an account that pays 3% interest per year compounded monthly,the amount of interest accumulated after months is given by the sequence. I tried the ...
6
votes
3answers
72 views

Evaluating sums using residues $(-1)^n/n^2$ [duplicate]

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
2
votes
1answer
63 views

Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?

When learning mathematics we are told that infinity is undefined. (*) Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this ...
2
votes
2answers
56 views

Stone-Weierstrass: Examples

By Stone-Weierstrass one has: $f\in\mathcal{C}(K):\quad p_n\to f$ Now, for analytic functions this is just Taylor: $$f\in\mathcal{C}^\omega([a,b]):\quad f(x)=\sum_{k=0}^\infty a_kx^k$$ But, how does ...
0
votes
0answers
30 views

Proving least upper bound property implies greatest lower bound property

In Rudin 1.11 Theorem Proof he claims the following Suppose $S$ is an ordered set with the least upper bound property $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of ...
1
vote
0answers
11 views

Is there a Knot Theory software to analyze general curves in 3D?

So I happen to like proteins quite a lot and one thing that is very similar to a protein, when represented as the bare minimum, is a 1D curve embedded in the 3D space. They form beautiful and unique ...
1
vote
2answers
79 views

What is some pure math news website by a publisher? [on hold]

Why aren't there be any pure math website by a publisher? I google a lot and resulting only applied math news or math journal that is difficult and inaccessible even to advanced reader I am looking ...
2
votes
3answers
60 views

a limit property at infinity

Let $k\in(0,1)$ is fixed and $L$ is a finite value. Is it possible to say if $\lim_{x\to\infty}f(x)=L$ then $\lim_{x\to\infty}f(kx)=L.$
4
votes
1answer
71 views

If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous?

Question: If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous? Background: I thought of the question when proving that "If a function is periodic and continuous, ...

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