# All Questions

**0**

votes

**1**answer

39 views

### Lightning strike fractal formula

I need to generate random gold ore channels for a game, I was thinking they would look kinda like lightning strikes. Anyone know any good fractals (recursive functions) that looks like it? Or ...

**-2**

votes

**0**answers

38 views

### How can I (iteratively) solve these equations? [on hold]

I am by no means a mathematician at all (programmer) so I need some pointers on how to solve the following equations - if someone could point me to a method that would work, that would be very ...

**-1**

votes

**1**answer

11 views

### generate analytically bivariate correlated data

How does one generate correlated binomial data when one is given marginal probabilities of each and also the correlation coefficient.
The following code in SAS for example works best when we want ...

**1**

vote

**0**answers

15 views

### Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?

This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations"
In Part (2) of Theorem 19.8.4 of that book it says
If ...

**-2**

votes

**0**answers

23 views

### How can we find the presentation of a group? [on hold]

Is it possible that to find the presentation of a group $G$ such that it is extension of $\mathbb{Z}_2\times \mathbb{Z}_2$ by $\mathbb{Z}_2$?

**0**

votes

**0**answers

17 views

### Isotropic subspaces in a symplectic vectorspace over $GF(q)$

Let $V$ be a symplectic vectorspace of dimension $2n$, and $r\mid n$. Is this statement true?"There is an isotropic spread of $r$ dimensional subspaces in $V$". By an isotropis subspace I mean a ...

**2**

votes

**0**answers

56 views

### Prove that the Log-Euclidean distance is negative-definite

Let $\Bbb{S}_{++}^n$ be the $\frac{n(n+1)}{2}$-dimensional Riemannian manifold of the symmetric positive definite (SPD) $n\times n$ real matrices.
The Log-Euclidean distance between two points of ...

**1**

vote

**0**answers

28 views

### Does null geodesic flow live on a natural compact bundle?

Let $(M,g)$ be a compact pseudo-Riemannian manifold (closed or with boundary).
A geodesic $\gamma:(a,b)\to M$ is called null if $g_{ij}\dot\gamma^i\dot\gamma^j=0$.
The geodesic flow can be seen as a ...

**-3**

votes

**0**answers

59 views

### What is the symmetry of SU(3) - when seen as a manifold? [on hold]

Simply asked: is it more correct to state that the symmetry of the SU(3) manifold is $Z_3$ or $S_3$? Or neither of the two?
SU(3) has a kind of threefold symmetry; but which one exactly? When ...

**2**

votes

**1**answer

73 views

### Jacobson-Morozov theorem

Jacobson-Morozov theorem for a semisimple algebraic group $G$ (presumably I am working over algebraically closed field) states that: given a unipotent u, there exists a homomorphism $\phi$ from $SL_2$ ...

**6**

votes

**5**answers

583 views

### Algorithms for calculating R(5,5) and R(6,6)

Calculating the Ramsey numbers R(5,5) and R(6,6) is a notoriously difficult problem. Indeed Erdős once said:
Suppose aliens invade the earth and threaten to obliterate it in a year's time unless ...

**0**

votes

**1**answer

44 views

### Comparison of Lp norm of matrix and its entry wise norm. [on hold]

I need to know the relation between operator norm of a matrix i.e. $ \Vert A\Vert_p$ for case of p=1 and 2 and its entry wise Frobenius norm $ \Vert A\Vert_F$.

**3**

votes

**2**answers

258 views

### Idea of using etale site

I have just read an article which mentions that, when Grothendieck considered using etale morphism, he did borrow the idea from Riemann that multivalued function on an open subset of complex plane ...

**9**

votes

**3**answers

519 views

### Determining if a matrix is orthogonal

Let g be an element of $GL_n(\mathbb C)$. We know that there are orthogonal groups $O(\beta)=\{X\in GL_n(\mathbb C) \mid X^t\beta X=\beta\}$ for any $\beta$, invertible symmetric matrix. Though these ...

**3**

votes

**0**answers

44 views

### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...

**2**

votes

**0**answers

43 views

### Relation between linear independence of lattice vectors and the toric variety defined by that lattice

I have been reading some basic and elementary work on toric varieties, but even though people assured me that toric varieties are very well understood, several questions remained.
Setup. Let ...

**3**

votes

**0**answers

85 views

### What is Jantzen's formula for the determinant of the Shapovalov form in the case of generalized Verma modules?

The best reference I found is
[Kac, Kazhdan '79]
which extends the results of Shapovalov and Jantzen to the case of infinite dimensional Lie algebras.
Theorem 1 of this paper gives the Shapovalov ...

**1**

vote

**1**answer

114 views

### Subgroup of Projective general linear group on complete discrete valuation ring

Let $R$ be a complete dvr and $k$ its residue field of positive characteristic.
Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$.
Is there some ...

**5**

votes

**1**answer

168 views

### Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...

**11**

votes

**1**answer

472 views

+50

### Subsequence and integers as a sum of $\frac{1}{n}$

For all $M \in \mathbb{Z}$, is there a finite sequence of positive integers (not necessarily distinct) $(n_i)_{i \in I}$, s.t. $\sum_{i \in I} \frac{1}{n_i} = M$, and there is no subsequence $(n_i)_{i ...

**3**

votes

**2**answers

223 views

### Representations of complex semi-simple algebraic group “defined over $\mathbf{Z}$”?

If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over ...

**0**

votes

**0**answers

86 views

### Invariant mesures for expanding maps of the circle

Is there any characterization for the support of T-invariant measures? where T is a C¹ expanding map of the circle i.e. T'(x)>Lambda>1 for all x in the circle.
I know there are periodic and total ...

**2**

votes

**1**answer

251 views

### Structure of symplectic group over finite fields

We are working over the finite field $\mathbb{F}_{q}$ of odd prime characteristic $p$ and of cardinality $q$ some power of $p$. We recall the symplectic group $Sp(4,\mathbb{F}_{q})$ as the group of ...

**42**

votes

**34**answers

6k views

### Are there any books that take a 'theorems as problems' approach?

Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Theorem: Each group has exactly one identity" and then show a proof ...

**105**

votes

**7**answers

10k views

### How to memorise (understand) Nakayama's lemma and its corollaries?

Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...

**37**

votes

**15**answers

5k views

### Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...

**18**

votes

**12**answers

6k views

### How seriously should a graduate student take teaching evaluations?

Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...

**66**

votes

**18**answers

15k views

### Depressed graduate student. [closed]

How does a depressed graduate student go about recovering his enthusiasm for the subject and the question at hand?
Edit: I am not that grad student; it is a very talented friend of mine.
Moderator's ...

**54**

votes

**20**answers

8k views

### What should we teach to liberal arts students who will take only one math course?

Even professors in academic departments other than mathematics---never mind other educated people---do not know that such a field as mathematics exists. Once a professor of medicine asked me whether ...

**17**

votes

**3**answers

2k views

### Famous vacuously true statements

I am interested to know other examples vacuously true statements that are non-trivial. My starting example is Turan's result in regards to the Riemann hypothesis, which states
Suppose that for each ...

**13**

votes

**7**answers

2k views

### Bijection between irreducible representations and conjugacy classes of finite groups

Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?

**34**

votes

**13**answers

2k views

### Equality vs. isomorphism vs. specific isomorphism

This question prompted a reformulation:
What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit?
I believe this to be a serious question because ...

**38**

votes

**12**answers

5k views

### Cures for mathematician's block (as in writer's block) [closed]

What kind of things do you find that help you get the "creative juices flowing," to use a tired cliche, when you're stuck or burnt out on a problem? I've read about some studies that suggest listening ...

**25**

votes

**4**answers

6k views

### Is the Jaccard distance a distance?

Wikipedia defines the Jaccard distance between sets A and B as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a book claiming that this is a metric. However, I couldn't find any ...

**10**

votes

**1**answer

479 views

### Can there be a global linear ordering of the universe without a global well-ordering of the universe?

This question arose in the answers to Asaf Karagila's
question Does ZFC prove that the universe is linearly orderable?. The answer there was that one can have a ZFC model with no global linear ...

**0**

votes

**1**answer

87 views

### Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?

I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise.
The other variables in the linear ...

**22**

votes

**4**answers

2k views

### In what sense is the étale topology equivalent to the Euclidean topology?

I have heard it said more than once—on Wikipedia, for example—that the étale topology on the category of, say, smooth varieties over $\mathbb{C}$, is equivalent to the Euclidean topology. I have not ...

**7**

votes

**1**answer

337 views

### Reference request for Plancherel measure

I need a good reference for the basic definitions of the dual of locally compact group (not necessarily abelian), its natural topology, $\sigma$-algebra, and the Plancherel measure on it (when they ...

**7**

votes

**4**answers

1k views

### Geometric interpretation of Universal enveloping algebras

Given a complex Lie algebra $\mathfrak g$, we can form its universal enveloping algebra and interpret it as a noncommutative space.
Is this perspective useful? What does this space "look like"?
How ...

**5**

votes

**3**answers

768 views

### Explicit isomorphism between distributions and universal enveloping algebra

The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is isomorphic to the algebra of distributions on the Lie group $G$ with support at the identity. The proof I have of this fact uses the ...

**1**

vote

**1**answer

231 views

### Turning a measurable function to a bijection

Let $f:(0,1)\rightarrow (0,1)$ be a borel measurable function such that for every $y$ in $(0,1)$ , $f^{-1}(y)$ is a borel set and $\mu(f^{-1}(y))=0$ and also $\mu (f((0,1)))=1$ where $\mu$ is the ...

**5**

votes

**1**answer

504 views

### Infinite direct products and derived subgroups

Suppose $G_1, G_2, \dots, G_n, \dots$ are groups (I use countable sequences, though the question is also applicable for uncountable collections of groups). Suppose G is the unrestricted external ...

**0**

votes

**0**answers

1 view

### Convergence of the Lyndon - Hochschild - Serre spectral sequence

I have some trouble understanding the notion of convergence of a spectral sequence conceptually (in general). More specifically, I'm trying to understand the convergence of the Lyndon - Hochschild - ...

**0**

votes

**0**answers

36 views

### Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...

**0**

votes

**0**answers

52 views

### Integration by parts? [on hold]

Let $f:{\mathbb R}\rightarrow {\mathbb R}_+$ be a density function with finite expectation. This is,
$$\int_{\mathbb R}x f(x)dx<\infty.$$
Suppose that we want to integrate $I(a)=\int_a^{\infty} x ...

**1**

vote

**0**answers

47 views

### Twisting by a multiplicative Character in Katz, Perversity and Exponential sums

Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients.
In his Proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided ...

**-1**

votes

**0**answers

28 views

### Euler equation formula [on hold]

when I am using Euler equation for Fourier transform integrals of type $
\int_{-\infty}^{\infty} dx f(x) exp[ikx] $ I am getting following integrals:
$\int_{-\infty}^{\infty} dx f(x) cos(kx)$ (for ...

**6**

votes

**0**answers

56 views

### $F[[T]] \times F[[1/T]]$ fundamental domain, show compactness

Let $p$ be a prime number. What is the easiest way to see that $(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$ is compact? Here $\mathbb{F}_p[T, 1/T]$ is embedded in ...

**17**

votes

**2**answers

720 views

### History of Geometric Analogies in Number Theory

My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens?
For example, was it before Grothendieck introduced schemes to ...

**2**

votes

**2**answers

93 views

### Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve

Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem.
My bounded open sets can be assumed to be pretty well-behaved, but I wonder if the above conditions are ...