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6
votes
1answer
676 views

A Coincidence concerning 163 and the Monster?

There is a McKay-Thompson series for the Monster group, namely $T_{1A}$, responsible for, $e^{\pi\sqrt{163}} \approx 640320^3 + 744$ Another one ($T_{2A}$) for, $e^{\pi/2\sqrt{232}} \approx 396^4 -...
2
votes
3answers
332 views

Recursion equations: $T(n)=T(\frac{n}{4})+T(\frac{3}{4}n)+1$

What's the simplest way to prove that the solution for this recursion equation: $T(n)=T(\frac{n}{4})+T(\frac{3}{4}n)+1$ , is $T(n)=\theta (n)$? I think that it is $T(n)=\theta (n)$ because it is ...
11
votes
3answers
1k views

Ash's construction of the Lebesgue-Stieltjes Measure from a distribution function

I'm reading this book Probability & Measure Theory by Ash. I think I've come across a part that is a little hand-wavy. We are trying to build a Lebesgue-Stieltjes measure from a distribution ...
2
votes
2answers
4k views

NP verifier-based definition

I'm a computer science student and I'm having some problem understanding the verifier based definition of NP problems. The definition says that a problem is in NP if can be verified in polynomial ...
23
votes
3answers
32k views

Lagrange multipliers with inequality constraints: minimize $f$ on the region $0 \leq x,y \leq 1$

I do not have much experience with constrained optimization, but I am hoping that you can help. My current problem involves a more complex function, but the constraints are similar to the ones below. ...
1
vote
2answers
435 views

Ptolemy's theorem. How to prove?

For distinct points $A, B, C, D$ on a plane, we have $(AB).(CD) + (AD).(BC) \geq (AC).(BD)$. Equality happens if and only if $A,B,C,D$ are collinear or concyclic with $A,C$ separating $B,D$.
13
votes
3answers
364 views

For which $n$ is $ \int \limits_0^{2\pi} \prod \limits_{k=1}^n \cos(k x)\,dx $ non-zero?

I can verify easily that for $n=1$ and $2$ it's $0$, $3$ and $4$ nonzero, $4$ and $5$ $0$, etc. but it seems like there must be something deeper here (or at least a trick).
1
vote
2answers
1k views

What is the meaning of $\int_\omega f\, \partial \Omega$?

In a paper I am reading, I have come across the following formula: $$\int_\omega f\, \partial \Omega$$ in which $\Omega$ is a bounded region in $\mathbb{R}^n$, $f:\Omega \to \mathbb{R}$ a function, ...
6
votes
2answers
408 views

“Change-of-base” between enriched categories

I would like to prove that a monoidal functor $$\Phi\colon \mathbf{V}\to \mathbf{V'}$$ induces a functor $$\Phi^\#\colon \mathbf{V}\text{-Cat}\to \mathbf{V'}\text{-Cat}$$ and in particular I ...
2
votes
2answers
264 views

Given any polynomial p(x) over Z, can one construct a graph with characteristic polynomial p(x)?

Given any polynomial $p(x)$ over $\mathbb{Z}$, can one construct a graph with characteristic polynomial $p(x)$? [Edit: Title question added to post.} Further questions include: Are there classes of ...
8
votes
2answers
2k views

Class field theory for function fields and a curious statement

Let $X_0$ be a smooth curve over a finite field $\mathbb{F}_q$, and let $X$ be the base-change to the algebraic closure. I read that, according to class field theory in function fields, "the image ...
5
votes
2answers
2k views

Creating a matrix of rank r from r number of rank 1 matrices?

I am told that all matrices of Rank $r$ can be formed out of the combinations of $r$ number of Rank 1 matrices. So that's the original matrix can be broken down into $r$ number of rank 1 matrices. But ...
8
votes
1answer
750 views

Several questions around the exponential law

$\mathrm{DISCLAIMER~:~}$I am not interested in working with compactly generated spaces. This post is related to this one : Exponential Law for based spaces. I learned about the exponential law for ...
1
vote
1answer
115 views

Probability: Terminology Question for Convergence in Distribution

I'm currently probability from two different sources: the classic text by Billingsley and the course notes of an instructor at my university. I've run into a terminology conflict that I was hoping ...
2
votes
2answers
7k views

method of moments of an uniform distribution

Let $ X_1, ... X_n $ a sample of independent random variables with uniform distribution $(0,$$ \theta $$ ) $ Find a $ $$ \widehat\theta $$ $ estimator for theta using the method of moments ...
2
votes
2answers
2k views

Three boxes with a check inside

I recently saw a math video in the online edition of a Spanish divulgation magazine. In the video, they had three boxes, one yellow, one red and one blue. A 1,000,000€ check was put inside one of the ...
4
votes
1answer
183 views

A technique for deciding satisfiability in fragments of first-order logic

By Goedels completeness theorem satisfiability in first-order logic is $\Pi_1$. So to obtain decidability in some fragment, it is enough to show that satisfiability is $\Sigma_1$ in this fragment. I ...
5
votes
5answers
1k views

Questions about right ideals

The question is from the following problem: Let $R$ be a ring with a multiplicative identity. If $U$ is an additive subgroup of $R$ such that $ur\in U$ for all $u\in U$ and for all $r\in R$, then $...
7
votes
1answer
905 views

Sphere eversion and Smale-Hirsch theorem

For two manifolds $M^m$ and $N^n$ with $m<n$ the Smale-Hirsch theorem says that the differential map $d:\operatorname{Imm}(M,N)\to\operatorname{Mon}(TM,TN)$ is a weak homotopy equivalence, where $\...
2
votes
1answer
113 views

Trouble with notation $I:J^{\infty}$

I am not sure I understand this notation correctly. The definition says, for a ring $R$ with $I,J$ ideals of $R$, we define $I:J^{\infty}=\cup_{i=1}^{\infty} I:J^i$. Now, $I:J$ is the set of elements ...
10
votes
2answers
13k views

What is a “free parameter” in a computational model?

In many articles regarding computational models of some particular phenomenon, there seems to be a consensus: "the smaller the number of 'free parameters' in the model, the better". So, what is meant ...
2
votes
1answer
150 views

Nonhomogeneous Linear O.D.E

I found this question and I cannot seem to answer it correctly and its kinda bothering. I am not seeing what I am not getting right with this particular problem. I took the same route as the OP and ...
4
votes
1answer
303 views

Why weak formulations in numerical mathematics?

Regard the Poisson equation on the domain $\Omega = [-1,1]^n$ with $f \in H^{-1}$ $- \triangle u = f$ with homogenous Neumann boundary conditions. From standard regularity theory we know $u \in H^1$....
2
votes
0answers
115 views

Probability metric spaces for which $r \leq R$ implies $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$

I'm looking for examples of spaces $X$ such that: $X$ is a probability space. $X$ is a metric space. If $x \in X$ and $0 < r \leq R$ then $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$. I ...
15
votes
5answers
868 views

Natural isomorphisms and the axiom of choice

The definitions of "natural transformation", "natural isomorphism between functors", and "natural isomorphism between objects" captures - among other things - the ...
6
votes
1answer
448 views

How to correct a wrong proof about the Birman exact sequence?

I've given a proof of the exactness of the Birman exact sequence of groups: $$1\to\pi_1(S_{g,r}^s)\to MCG(S_{g,r}^{s+1})\overset{\lambda}{\to} MCG(S_{g,r}^s)\to 1$$ making use of classifying spaces ...
4
votes
3answers
320 views

Diverging random walk

I have a process $X_{n+1} = X_n\xi_n$ where $\xi_n\sim\mathcal N(1,1)$ and $\xi_n$ is independent of $X_n$. I need to prove that if $X_0\neq0$ then $$ \mathsf P\{|X_n|>1\text{ for some }n\geq0\} = ...
4
votes
5answers
378 views

finding remainder by dividing a sum

Suppose I am dividing 4^30-2^50 by 5. I do understand that 4^30 will get converted to ...
3
votes
1answer
110 views

Average result of game

I'm trying to find the average result for a game where you have 12 items (containing a score), you can pick up to five of them, one of them is a multiplier, that when selected multiplies your result ...
3
votes
4answers
476 views

Mathematics necessary for a Master's degree in CS

I'm contemplating doing a Master's degree in Computer Science at night school. What sort of mathematics am I likely to encounter?
1
vote
1answer
154 views

Convergence of Bayes Error to zero

Given, $$ 0<p_{i} ,q_{i} <1 $$ define ${E}{}_{k}$ as, $$\begin{array}{l} {E_{1} =min(p_{1} ,q_{1} )+min(1-p_{1} ,1-q_{1} )} \\ {E_{2} =min(p_{1} p_{2} ,q_{1} q_{2} )+min(p_{1} (1-p_{2} ),q_{1}...
6
votes
4answers
30k views

What is the highest number that can be got from 4383 by moving exactly 2 matches?

What is the highest number that can be got from 4383 by moving exactly 2 matches? Number 1 has got 2 matches, so I thought it will be 47831 as I remove two matches from second number (3), but it isn'...
5
votes
1answer
1k views

injectivity of a group action

The action of a group $G$ on $X$ is always "injective" in the following sense: if $x\not = y$ then $\forall g\in G$, $gx\not = gy$ indeed if $gx=gy$ then $g^{-1}(gx)=(g^{-1})gx=x=g^{-1}(gy)=(g^{-1}g)y=...
5
votes
1answer
1k views

what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?
1
vote
1answer
178 views

Degree of a retraction

I am wondering about how to compute the degree of $r(z) := \frac{z}{| z |}$. I know that $deg f_n = n$ where $f_n (z)= z^n$ i.e. the degree is how many times it goes around $0$. I also know that $...
9
votes
2answers
7k views

Identifying the product of two Fourier series with a third?

Given the product of two functions defined explicitly through their Fourier coefficients (of unknown undeveloped form): $\sum_k{c_k e^{i k t}} \cdot \sum_k{c'_k e^{i k t}}$ Is there any way to ...
4
votes
2answers
633 views

Local versus global implicit function

Suppose the equation $f\left(x,y\right)=0$, with $x\in I_{1}$ and $y\in I_{2}$, $I_{1}$ and $I_{2}$ being open intervals. Additionally, consider that the conditions required to apply the Implicit ...
39
votes
1answer
3k views

How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $?

Note the $ p < x $ in the sum stands for all primes less than $ x $. I know that for $ s=1 $, $$ \sum_{p<x} \frac{1}{p} \sim \ln \ln x , $$ and for $ \mathrm{Re}(s) > 1 $, the partial sums ...
3
votes
1answer
8k views

polar coordinates of Gaussian Distribution with non zero mean

I found that the polar coordinates of 2-dimensional Gaussian distribution with mean zero $$\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\big(-({x^2+y^2})/{2\sigma^2}\big) \,...
36
votes
3answers
6k views

When can two linear operators on a finite-dimensional space be simultaneously Jordanized?

IN a comment to Qiaochu's answer here it is mentioned that two commuting matrices can be simultaneously Jordanized (sorry that this sounds less appealing then "diagonalized" :P ), i.e. can be brought ...
4
votes
3answers
472 views

Confused about quadratic forms

My book states: if $q: \mathbb{R}^n \to \mathbb{R}$ is a positive definite quadratic form then there exists a basis $B=(v_1,...,v_n)$ such that $q(x)=x_1^2+...+x_n^2=||x||^2$ for every $x=x_1v_1+......
2
votes
2answers
122 views

Sparseness for a matrix

I would like to define a function $f$ whose range is $[0,1]$ such that it takes a matrix $C \in R_+$ of dimension $m \times n$. The entries in the matrices are also in the range $[0,1]$. In addition, ...
7
votes
1answer
622 views

Is $SL(n,K) \cap D(n,K)$ always a maximal torus in $SL(n,K)$?

Let $K$ be an algebraically closed field. Let $SL(n,K), GL(n,K)$ denote the special linear group and general group respectively, and $D(n,K)$ is the diagonal subgroup of $GL(n,K)$. Then $SL(n,K) \cap ...
17
votes
3answers
12k views

Proper way to read $\forall$ - “for all” or “for every”?

I was asked in class the other day by a professor for whom English is a second language why $\forall$ is sometimes read "for all" while other times read "for every." Is there a rule for this? I was ...
10
votes
4answers
3k views

Why is the direct product of a finite number of nilpotent groups nilpotent?

I read that a direct product of a finite number of nilpotent groups is nilpotent. Here the definition of a nilpotent group is one that has a central series. A comment in my book following this claim ...
12
votes
6answers
8k views

A tricky geometry problem

I already have my own solution for the following question. But I am still interested in other elegant solutions without trigonometry if possible. This is my own solution. I am lazy to upload the TeX ...
4
votes
2answers
910 views

Order type and its reverse

I am given the following definition: For an arbitrary order type $\Theta$, denote by $\Theta$* (the reverse of $\Theta$) the order type $type\Theta$*$=typeA(\succ)$, where $\langle A,\prec \rangle$ ...
3
votes
1answer
384 views

Automorphism Group of a graph

If $X$ is a locally finite graph, (i.e. each vertex has finite index), is it true that the automorphism group Aut($X$) of the graph X is locally compact? Here, Aut($X$) has compact open topology; and ...
0
votes
1answer
101 views

Idea for proof that this $\Sigma$-formula holds in every nonempty structure

Can someone give me a hint, how to prove that the $\Sigma$-formula $$ \neg (\psi_{x \rightarrow t} \ \& \ \exists x \psi)$$ where $\psi$ is an arbitrary $\Sigma$-formula, $t$ is a $\Sigma$-...
3
votes
1answer
1k views

Finding the point where the tangent is the x-axis

Consider a simple example of a cubic: $px^3+qx^2+rx+s$ with $p,q,r,s\in\mathbb{R}$. I want to find the point $a$ in the following figure, where the cubic has a double root, as a function of the ...

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