7
votes
0answers
186 views
+500

Proving that there exists a saturated set with given highest weight

This is an question about an exercise in Humphreys book on Lie algebras. First of all a bunch of definitions and notation, see ยง13 in Humphreys for details. Let $\Phi$ be a root system, $\Delta$ a ...
10
votes
0answers
206 views
+150

Which harmonic numbers have prime numerators?

Let $$1+\frac12+\frac13+\cdots+\frac1n=\frac{q}{p}$$ with $(p,q)=1$ and $q$ is a prime number. (I) Prove or disprove that the quantity of $n$ is limited (II) Find all of the $n$ I use the matlab ...
1
vote
1answer
82 views
+50

When branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$.

I'm working on the following problem for several days without finding any solution: Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb ...
3
votes
1answer
132 views
+50

How to prove $\left\lfloor\frac{\sigma{(n-1)}+1}{2}\right\rfloor\le f(n)<\left\lfloor\frac{\sigma{(n-1)}+1}{2}\right\rfloor+n$

Question: let $x_{1},x_{2},\cdots,x_{n}$ be such that $$n\ge 3,x_{1}\le x_{2}\le\cdots \le x_{n},$$ $$x_{1}x_{2}x_{3}\cdots x_{n}=x_{1}+x_{2}+\cdots +x_{n}.$$ Let the number of ordered ...
37
votes
2answers
2k views
+500

A zero sum subset of a sum-full set

I had seen this problem a long time back and wasn't able to solve it. For some reason I was reminded of it and thought it might be interesting to the visitors here. Apparently, this problem is from a ...
12
votes
7answers
378 views
+50

The probability that x birthdays lie within n days of each other

This is a question that has bugged me for quite some time: what is the chance that x people happen to have their birthdays within n days of each other? A bit more specific, since this is how a ...
2
votes
0answers
87 views
+250

Convergence of empirical distribution from another observations

Let $$(X_1,Y_1),\dots,(X_n,Y_n)$$ are i.i.d. observation with continuous distribution function $F=(G,H)$ such that $\Phi^{-1}(G)=a\Phi^{-1}(H)$ for some unknown value of $a$ where ...
3
votes
1answer
283 views
+50

Why this recursively defined sequence of real numbers converges to -Pi?

Remy J. Cano in his private email described the sequence of real numbers, recursively defined as $$a(n) = a(n-1)+\frac{2 \cdot \cos(\frac{a(n-1)}{2})}{2 \cdot \sin(\frac{a(n-1)}{2})-1},a(0)=0$$ This ...
9
votes
1answer
336 views
+200

Properties of 4 by 4 Matrices

Define $ A=\begin{pmatrix} x_1 & x_2 & 0 & 0\\ 0 &1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}, B=\begin{pmatrix} 1 & 0 & 0 & 0\\ x_3 ...
3
votes
0answers
157 views
+50

A problem with sign of coefficients of a polynomial expression

Let $f$ be a real coefficient homogeneous polynomial in $n$ undeterminates, such that $f(x_1,\cdots,x_n)>0$ whenever $x_1,...,x_n$ are non-negative real numbers, not all $0$. Then how to show ...
4
votes
1answer
111 views
+50

For every $a>0$, show that $\langle f_a, \psi\rangle= \int_{|x|>a} \frac{\psi(x)}{|x|}dx+\int_{|x|>a} \frac{\psi(x)- \psi(0)}{|x|}dx$

To prove that the inner product is a distribution it must satisfy the following property" $$|T(\phi)|=|\langle T,\psi\rangle| \leq C_N \sum_{|\alpha| \leq N} \|\partial^\alpha \psi\|_\infty$$ Part ...
3
votes
2answers
186 views
+150

what is the limit of $\theta$ if $r\to 0$ when we have these conditions?

the problem is based on this picture. at beginning or we say $t=0$, $P$ is a circle of which the center is at the point $(0,r)$, $r_0=1$ is the initial radius of this circle. $AB$ is a vector which ...
4
votes
3answers
67 views
+50

Hyperplanes and Support Vector Machines

I have the following question regarding support vector machines: So we are given a set of training points $\{x_i\}$ and a set of binary labels $\{y_i\}$. Now usually the hyperplane classifying the ...
4
votes
1answer
132 views
+50

Formula for adjusting font height

INTRODUCTION AND RELEVANT INFORMATION: I am a software developer that needs to implement printing in my application. In my application user can choose different paper sizes ( A3, A4, A5 ...) which ...
4
votes
2answers
136 views
+50

About amicable numbers

My question might seem wrong but I will explain it now: We will define $\phi(a)$ as the sum of divisors of $a$ except $a$. A pair $(a,b)$ is an amicable pair $\iff ((\phi(a) = b) \land (\phi(b) = ...
15
votes
0answers
187 views
+250

Zombie outbreak on a $k$-regular graph

Suppose we have a zombie outbreak on a connected $k$-regular graph of order $n$. There are $n_0$ initially infected zombie nodes, and each turn, each zombie infects its neighbors with probability ...
8
votes
1answer
95 views
+100

Generalizing Newton's identities: Trace formula for Schur functors

We work over $\mathbb C$. A general linear group ${\rm GL}(V)$ acts diagonally on the tensor power $V^{\otimes n}$ as $$(A^{\otimes n})(v_1\otimes\cdots\otimes v_n):=(Av_1)\otimes\cdots\otimes ...
8
votes
1answer
272 views
+50

History of Algebraic Geometry: Motivation behind definition of schemes

I am trying to read an article by Jean Dieudonne which talks about development of Algebraic Geometry. The article was being published in the journal "Advances in Mathematics" Volume 3, Issue 3, Pages ...
16
votes
0answers
385 views
+50

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
4
votes
3answers
93 views
+200

Tangent and angle bisectors

The tangent to the incircle of a triangle ABC is reflected about the external angle bisectors. Show that the triangle formed by the resulting 3 lines is congruent to ABC .
4
votes
2answers
190 views
+100

Optimization problem using Reproducing Kernel Hilbert Spaces

I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernce to this paper [Olivier Chapelle, ...
5
votes
3answers
141 views
+100

Book series like AMS' Student Mathematical Library?

I had the joy of discovering AMS' Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect ...
0
votes
0answers
91 views
+100

Newton's Second Law

I don't follow the part of the solution, which I have underlined in green. Which equation would I get this from (if any)?
3
votes
4answers
56 views
+100

Pair of PDEs to be solved together

I have the following pair of equations to be solved together to find the functions $H_{x}$ and $H_{y}$, which are the components of a vector $\bar{H}(x,y)=H_{x}(x,y)\hat{x}+H_{y}(x,y)\hat{y}$ in ...
3
votes
0answers
85 views
+50

How many Groups there are on a finite countable set?

Let say cardinality of set S is $n=|S|$. We know that there are $n^{n^2}$ all binary operations on that set. To find out how many groups can be created by this set and by those operations, we need not ...
0
votes
1answer
57 views
+50

Conformal map between unit disk and parabolic region

The following is an old qualifying exam problem, Construct a conformal map from the unit disk to the parabolic region $ \{ z \in \mathbb{C}: \Im z > (\Re z)^2 \} $.
7
votes
1answer
93 views
+100

The largest value of $k$ for $\Bbb{Z}^{k}$ to be embedded in $\mathcal{GL}(n,\Bbb{Z})$.

Reading my course on group theory, I asked my self the following question : Suppose that $\Bbb{Z}^{k}$ can be embedded in $\mathcal{GL}(n,\Bbb{Z})$. What is the largest value of $k$?
0
votes
0answers
29 views
+50

Delayed System Help

It is well-known that a small delay may or may not cause stable equilibrium to become unstable. Can anyone help that if for $\tau=0$ the equilibrium solution is unstable and if $\tau>0$ is there a ...
1
vote
2answers
186 views
+50

Newly Developed With Details - Describing orthographic projection using simple 2D transformations

Thanks to Pedro for helping me further develop my question into something tangible. His (most recent) answer below clearly and formally outlines what I am asking. This is similar to this question, ...
12
votes
1answer
102 views
+250

How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
6
votes
1answer
57 views
+50

Is it true that the commutators of the gamma matrices form a representation of the Lie algebra of the Lorentz group?

Wikipedia claims (http://en.wikipedia.org/wiki/Gamma_matrices): The elements $\sigma^{\mu \nu} = \gamma^\mu \gamma^\nu - \gamma^\nu \gamma^\mu$ form a representation of the Lie algebra of the ...
1
vote
1answer
26 views
+50

Proof completion: Determine a simple expression for $\tau(G)$ in terms of the vertex degrees of $G$. (details inside)

I need some help completing a proof I wrote (or seeing a simpler solution to the same problem). For a list of paths $P_1, \ldots , P_m$, let $L(P_1, \ldots, P_m)$ be the number of paths in $P_1, ...
3
votes
1answer
40 views
+50

Interchanging pointwise limit and derivative of a sequence of C1 functions

Assume the following: $f_n$ is a sequence of C1 functions on $[0,1]$ $f_n(x) \rightarrow 0$ for pointwise. $f'_n(x) \rightarrow g(x)$ pointwise. Is it true that $g(x) = 0$ almost everywhere? I think ...
1
vote
0answers
37 views
+50

Lenstra's Elliptic Curve Algorithm

I am currently trying to understand Lenstra's Elliptic Curve Algorithm. As a source I use "Rational Points on Elliptic Curves" by Joseph H. Silverman and John Tate. They describe the algorithm as ...
0
votes
0answers
28 views
+50

Solving system of two linear odes

I am trying to solve \begin{align} y_1'+B_{12}y_1=\beta_{12}y_2\\ Ay_2'+B_{21}y_2=\beta_{21}y_1, \end{align}with $y_1(0)=y_2(0)=y_0$. I find the eigenvalues to be ...
3
votes
0answers
93 views
+50

Proof of external product theorem using K-theory

I am using Hatcher's K-Theory book to work through the proof of the external product theorem: $\mu:K(X) \otimes \mathbb{Z}[H]/(H-1)^2 \to K(X) \otimes K(S^2) \to K(X \times S^2)$ is an isomorphism ...
1
vote
2answers
72 views
+50

Verifying Stokes' Theorem

Could someone help me set up this problem? I'm not sure where to start... I know I have to prove that the line integral equals the surface integral of the curl, but how do you figure out what line to ...
5
votes
0answers
110 views
+50

Seeking proof for the formula relating Pi with its convergents

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n) ? $$(-1)^n\cdot(\pi ...