All Questions
152,894
questions
7
votes
2
answers
18k
views
What is the structure of $SO(3)$ and its Lie Algebra? [closed]
First I want to give you some background how the question arised, before actually asking it.
Recently, in the context of quantum mechanics, I thought about the group $SO(3)$ and its Lie Algebra $so(3)...
1
vote
1
answer
272
views
Euclidean neighborhoods on Polyhedral surface
Let $(X, Vertex(X))$ be a Polyhedral surface (defined like in Polthier) , $x_0 \in X$ a vertex. Let $B_\epsilon(x_0)$ the euclidean ball centred at $x_0$ with radius $\epsilon$, $\epsilon > max ...
7
votes
2
answers
1k
views
Hecke algebra and $H^*(G/B)$
Given a complex reductive group, with Weyl group $W$, one can associate to it lots of "algebras of size $|W|$". For example $B$ equivariant functions on $G/B$ with convolution, grothendieck groups of ...
12
votes
4
answers
1k
views
Wanted: an example of a natural non-K\"ahler metric on a Kahler manifold
Let $X$ be a Kahler manifold. Associated to any hermitian metric $h$ on $X$ is a smooth real $(1,1)$-form $\omega = -\text{Im } h$, called the Kahler form of $h$. One of several equivalent conditions ...
19
votes
1
answer
2k
views
Motivation for Hall-Witt identity
I've wondered for a while about the (Hall-)Witt identity in group theory:
$[[a,b^{-1}],c]^b \cdot [[b,c^{-1}],a]^c \cdot [[c,a^{-1}]],b]^a = 1$.
(Here, $x^y$ means $y^{-1}xy$ and $[x,y]$ means $...
2
votes
1
answer
632
views
minimizing functions over simple matrix inequalities
I'm wondering if anything is known about minimizing convex, not necessarily linear functions subject to "simple" matrix equalities. To be precise, consider the following example:
$min \Sigma x_i ln ...
24
votes
9
answers
13k
views
Graduate ODE textbook
Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, ...
5
votes
0
answers
453
views
If $p=0$ and $df=0$, is $f$ a $p$th power?
This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...
3
votes
3
answers
2k
views
Conditional geometric distributions
If $p<1$ and $X$ is a random variable distributed according to the geometric distribution $P(X = k) = p (1-p)^{k-1}$ for all $k\in \mathbb{N}$, then it is easy to show that $E(X) = \frac 1p$, $\...
11
votes
0
answers
1k
views
Linear algebra of elliptic curves over p-adic fields
Unfortunately, the following question is somewhat ill-posed. However, I hope to make what I'm looking for sufficiently clear.
Given two elliptic curves (or Abelian varieties) over $\mathbb{C}$, one ...
5
votes
2
answers
4k
views
Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
I am especially interested in solving polynomial nonlinear matrix equations.
For instance, let $X$ be some matrix ...
18
votes
1
answer
946
views
Can there exist a `natural' finitely generated group with an undecidable word problem?
There are naturally occurring groups that have undecidable algorithmic problems. For instance, $F_2\times F_2$ has undecidable generalized word problem (membership problem for subgroups) and there is ...
28
votes
2
answers
4k
views
Galois theory for polynomials in several variables
I feel a bit ashamed to ask the following question here.
What is (actually, is there) Galois
theory for polynomials in
$n$-variables for $n\geq2$?
I am preparing a large audience talk on Lie ...
1
vote
0
answers
499
views
Applying the ideas of power series to certain convolutions - which identities transfer?
Let's suppose I'm working with some set of functions $f_k(n)$. $f_1(n)$ is essentially the root of my functions, and could be nearly anything, and then $f_k(n) = (f_1(n) * f_{k-1}(n))$ for some ...
11
votes
1
answer
916
views
Non standard algebraic geometry: shadows of varieties
In this question $\mathbb F$ is a field and $P({\mathbb F}^{n+1})$ is the projective space of dimension $n$ over $\mathbb F$. The term algebraic variety means a subset of $P({\mathbb F}^{n+1})$ ...
51
votes
1
answer
6k
views
Does $2^X=2^Y\Rightarrow |X|=|Y|$ imply the axiom of choice?
The Generalized Continuum Hypothesis can be stated as $2^{\aleph_\alpha}=\aleph_{\alpha+1}$. We know that GCH implies AC (Jech, The Axiom of Choice, Theorem 9.1 p.133).
In fact, a relatively weak ...
0
votes
0
answers
119
views
A continuous map from a T2 & compact space to a uniform space is uniformly continuous.
Can you recommend some literature that give a proof of this statement, and who allegedly prove it first?
BTW, is there any use of uniform spaces or topological spaces in mathematical (or theoretical) ...
8
votes
2
answers
267
views
Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?
I am attempting to solve the argument maximization problem
$$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$
where the functions $f_1$ and $f_2$ are concave but ...
5
votes
0
answers
781
views
In what sense does the Berezinian generalize the determinant?
One way of defining the determinant of a endomorphism of a vector space $\varphi:V \to V$ is by using the action of $End(V)$ on the underlying $\mathbb{Z}$-graded vector space of the exterior algebra $...
8
votes
3
answers
726
views
Natural statements independent from true $\Pi^0_2$ sentences
I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...
18
votes
1
answer
2k
views
can a common mortal understand why the affine line is not smooth in brave new algebraic geometry?
In the introduction to HAGII Toen and Vezzosi write that in brave new algebraic geometry (that is, algebraic geometry over the category of symmetric spectra) Z[T] is not smooth over Z.
I am told that ...
7
votes
1
answer
719
views
How is called a semigroup...
Does anyone know, how is called a semigroup in which every equation $ax=b$ has only a finite set (maybe empty) of solutions?
7
votes
0
answers
419
views
The Arnol'd family of circle maps - origins and density of hyperbolicity
$\newcommand{\R}{\mathbb{R}}\newcommand{\Z}{\mathbb{Z}}$
The Arnol'd family or standard family of circle maps is defined by
$$F_{\mu_1,\mu_2}:\R/\Z\to\R/\Z;\quad t\mapsto t + \mu_1 + \mu_2\sin(2\pi t);...
0
votes
1
answer
333
views
Hyperbolicity of a fundamental group
Let G be the fundamental group of a compact 3-manifold which supports on its interior a complete non positively curved Riemannian metric and is a cilinder near de metric. Is G hyperbolic?
9
votes
2
answers
864
views
Are there standard examples of stable theories that are undecidable?
What is known about decidability of various first order theories studied in stable model theory, geometric model theory, o-minimality? For example, is there a natural example of an undecidable first ...
7
votes
3
answers
2k
views
Algorithm for the smallest (algebraic) eigenvalues of a symmetric (sparse) matrix
Hi,
I'm looking for a way to get the negative eigenspace of a large (sparse) symmetric matrix. This matrix is basically a discretized version of the operator $-\Delta + V$, $V$ negative, on some ...
8
votes
2
answers
901
views
"Composition of Morita equivalences" or "Morita equivalence and the Nakayama functor"
This problem occured to me, when trying to find a Morita invariant for finite dimensional algebras.
Suppose $\Lambda$ and $\Gamma$ are two self-injective $k$-algebras ($k$ being a field) which are ...
1
vote
0
answers
276
views
When is the henselization ('the smallest etale neighbourhood') of the intersection of locally closed subvarieties $Z_1,Z_2\subset X$ isomorphic to the product of the henselizations of $Z_i$ over the one of $Z_1\cup Z_2$?
For a locally closed subscheme $Z\subset X$ (I am interested in the case when $X$ is a variety) one can consider its henselization in $X$ i.e. the 'smallest' pro-etale morphism $U\to X$ such that $Z$ ...
3
votes
1
answer
459
views
Software for calculating products and sums of Kronecker deltas
I am looking at a Kahler metric $g$ on a certain manifold $M$, which has the good taste to be invariant under a transitive group of isometries, and I want to say something about its holomorphic ...
1
vote
1
answer
210
views
fourier transform of cumulative function
Hi
I've encountered a test that uses the cumulative value of a finite time series to deterime the data set's stationarity.
I would like to know the characteristics of this test in frequency space,...
0
votes
1
answer
2k
views
How to find out the sum involving floor function [closed]
I came up with an equation while solving a question. The question is - suppose we have n numbers, from 1 up to n. How many groups of 3 numbers (repetition allowed) can be formed whose sum will be n.
...
3
votes
1
answer
633
views
On lattice points "far inside" convex lattice polygons
Let $\mathcal{P}$ be a convex lattice polygon with $n$ vertices and let $\mathcal{L}$ be the set of all lattice points inside $\mathcal{P}$. For every $n \geq 5$, does there exist a point in $\mathcal{...
2
votes
1
answer
224
views
Is there an invariant similar to the delta invariant that distinguishes an $A_2$ node form an $A_1$ node?
Consider the following question: If two nodes collide what do you get?
First of all it can not be a strict $A_2$ node, because the delta invariant
of that is $1$. So it has to be more singular than ...
2
votes
0
answers
325
views
Finite index normal subgroup in a normalised union of two groups. [duplicate]
Possible Duplicate:
Existence of simultaneously normal finite index subgroups
$G$ is a group, $A$ and $B$ are two subgroups of $G$. Let $H$ be a subgroup of $A\cap B$ which is of finite index in ...
2
votes
2
answers
627
views
Measure on $\omega_1$
Let $\mathcal{O}$ be the $\sigma$-algebra on $\omega_1$ generated by its countable subsets. Is there a ($\sigma$-additive) probability measure on $\mathcal{O}$ that is not concentrated on a countable ...
16
votes
6
answers
4k
views
Homology of loop space
I've been reading Galatius's Park City notes on the Madsen-Weiss theorem (available here).
On page 8, he states the following theorem. Let $X$ be a space such that $\pi_1(X)$ is abelian and acts ...
0
votes
2
answers
389
views
definition of group operation in elliptic curves
Hi,
Using the isomorphism between an elliptic curve $E$ and its $Pic_1(E)$ group, one can
easily give $E$ the structure of a group variety after choosing a point $O\in E$. The
operation that one gets ...
31
votes
2
answers
5k
views
A question about rejected journal submissions, similar results, and discrepancies between the order of submission and the order of publication [closed]
Today I just received the decision of my paper from a journal. The paper was submitted last December and my paper is kind of long (about 40 pages), so I think it's reasonable to take such a long time ...
5
votes
0
answers
631
views
Calculating Hilbert Symbol for over field extensions,
Let $F$ be an unramified extension of $Q_2$. How can I compute the Hilbert symbol (a,b) for
$a,b \in F^*$. Here, (a,b) is 1 if $ax^2+by^2=z^2$ has a nontrivial solution, and -1 otherwise.
In the ...
2
votes
2
answers
403
views
epsilon-Manifold with curvature at one point
I remember briefly hearing about this notion (stated in the title), of a manifold where there is a nonzero curvature at precisely one point (a delta-function distribution), and such that there is a ...
14
votes
0
answers
525
views
When are the fibers of a resolution of singularities reduced?
I apologize if this is too much of a fishing expedition, but I've had bad luck searching for any literature on this subject, and I was hoping someone could tell me if it's too easy to be worth ...
28
votes
4
answers
4k
views
When is it appropriate to entitle a paper "A note on..." or "On the ..."? [closed]
I rarely find modern research papers (on mathematics) that are less than 5 pages long. However, recently I came across a couple of mathematical research papers from the 1960/1970's that were very ...
32
votes
0
answers
2k
views
Is there software to compute the cohomology of an affine variety?
I have some affine varieties whose cohomology (topological, with $\mathbb{C}$ coefficients) I would like to know. They are very nice, they are all of the form $\mathbb{A}^n \setminus \{ f=0 \}$ for ...
7
votes
0
answers
311
views
What is known about the locus of zero-divisors in the group ring of a (non-abelian) finite group?
Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Left multiplication by $\alpha\in \mathbb{C}G$ is a linear map $\alpha:\mathbb{C}G \to \mathbb{C}G$, and so $\alpha$ has a left determinant ...
3
votes
0
answers
167
views
A trilinear forms question about representations of real linear group.
This might be a naive question. Suppose $\pi_i$ are irreducible generic unitary representation of $GL_{n}(\mathbb{R})$ with its associated Whittaker models $\mathcal{W}(\pi_i)$. Let $E_m$ be the space ...
4
votes
1
answer
406
views
Hyperbolicity of a semidirect product
Let F be a finitely generated free group and let $\gamma : F \rightarrow F$ be an automorphism. Is the semidirect product $F \rtimes \mathbb{Z}$ an hyperbolic group? where $\mathbb{Z}$ acts in F via $...
2
votes
0
answers
199
views
Non-regular (Non-coherent) subdivisions of a polygon.
There are many papers and books which study about the regular subdivision of a convex lattice polytope.
My question is about "Non"-regular subdivisions of a 2-dimensional convex lattice polygon.
I ...
2
votes
3
answers
336
views
Can one approximate "close" smooth functions?
(This question was originally asked on StackExchange: https://math.stackexchange.com/questions/82437/can-one-average-close-smooth-functions )
Suppose $M$ is a connected, smooth, second-countable ...
9
votes
1
answer
418
views
Left orderable linear groups
Are all torsion-free finitely generated linear groups over $\mathbb{C}$ left orderable? In particular, are torsion-free congruence subgroups of $SL_n(\mathbb{Z})$ left orderable?
1
vote
2
answers
262
views
Representation as an eigenspace of a finite set of differential operators
Let $G$ is a semisimple algebraic group over characteristic 0. I'm curious about the finite set of differential operators $D_i$ on algebra $\Bbbk [V]$ ($V$ is a representation of $G$), this ...