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2 answers
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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)...
Niki's user avatar
  • 335
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 ...
acmath's user avatar
  • 43
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 ...
Jan Weidner's user avatar
  • 12.8k
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 ...
Gunnar Þór Magnússon's user avatar
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 $...
Selim's user avatar
  • 479
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 ...
Fumiyo Eda's user avatar
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 ...
Jared Weinstein's user avatar
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 $E(X) = \frac 1p$, $\...
Vaughn Climenhaga's user avatar
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 ...
Moosbrugger's user avatar
  • 1,517
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 ...
gondolf's user avatar
  • 1,487
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 ...
Benjamin Steinberg's user avatar
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 ...
DamienC's user avatar
  • 8,103
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 ...
Nathan McKenzie's user avatar
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 $P({\mathbb F}^{n+1})$ ...
Daryl Cooper's user avatar
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 ...
Asaf Karagila's user avatar
  • 38.1k
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) ...
Alan's user avatar
  • 1,514
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 ...
JNM's user avatar
  • 101
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 $...
David Carchedi's user avatar
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 $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...
Kaveh's user avatar
  • 5,362
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 ...
Yosemite Sam's user avatar
  • 1,869
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?
Boris Novikov's user avatar
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);...
Lasse Rempe's user avatar
  • 6,455
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?
Luis Jorge's user avatar
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 ...
mmm 's user avatar
  • 1,299
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 ...
Antoine Levitt's user avatar
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 ...
Julian Kuelshammer's user avatar
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 $Z$ ...
Mikhail Bondarko's user avatar
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 ...
Gunnar Þór Magnússon's user avatar
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,...
user19330's user avatar
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. ...
Soumyajit De's user avatar
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{...
Cosmin Pohoata's user avatar
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 ...
Ritwik's user avatar
  • 3,235
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 ...
Drike's user avatar
  • 1,555
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 ...
Nate Ackerman's user avatar
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 ...
Carlos Garcia's user avatar
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 ...
unknown's user avatar
  • 647
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 ...
Soroosh's user avatar
  • 818
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 ...
Chris Gerig's user avatar
  • 17.1k
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 ...
Ben Webster's user avatar
  • 43.9k
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 ...
David E Speyer's user avatar
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 ...
Jonathan Fine's user avatar
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 ...
user1832's user avatar
  • 2,679
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 $...
Luis Jorge's user avatar
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 ...
Jessie's user avatar
  • 21
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 ...
Piotr Pstrągowski's user avatar
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?
user avatar
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 ...
zroslav's user avatar
  • 1,412

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