All Questions
153,439
questions
2
votes
0
answers
52
views
About direct products of groups [duplicate]
Let $G$ be a group. Suppose that $G\simeq G\times G\times G$ (here $\simeq$ is an isomorphism of groups). Is it true that in this case $G\simeq G\times G$? Of course, this question is slightly ...
- 21
4
votes
1
answer
367
views
Classifying/characterising groups acting on CAT(0)-cube complexes
What are some potential approaches to classifying/characterising (finitely presented?) groups that act (without fixed points and potentially more conditions) on CAT(0)-cube complexes? As so many ...
user114152
1
vote
1
answer
108
views
Shedding vertex
Let $G$ be a finite simple graph on the vertex set $\{x_1, \ldots,
x_n\}$ and $I(G) := (\{x_ix_j \mid \{i,j\} \in E(G)\}) \subset R=K[x_1,
\ldots, x_n]$ be the edge ideal corresponding to the graph $G$...
- 393
4
votes
1
answer
292
views
Inclusion-Exclusion for Coherent Sheaves
Let $X$ be a (reduced) affine variety with two irreducible components, $X_1$ and $X_2$, and let $\mathcal{E}$ be a torsion-free coherent sheaf on $X$. Denote the pullbacks of $\mathcal{E}$ to the ...
8
votes
1
answer
260
views
Two questions about basic sequences
Suppose $(x_n)$ and $(y_n)$ are two basic sequences in a separable Banach space $X$ such that $\overline{span}\{(x_n), (y_n)\}=X$. Can we always pass to subsequences $(x_{n_k})$ and $(y_{n_k})$ such ...
- 355
23
votes
3
answers
2k
views
What are the applications of the Atiyah-Bott Yang Mills paper?
I recently finished a seminar going through Atiyah and Bott's paper ''The Yang-Mills Equations over Riemann surfaces''. The ideas going into the proof were surprising and very beautiful to me.
...
- 5,565
2
votes
0
answers
110
views
How to compute explicit equations for the Jacobian of a variety over a field [duplicate]
Suppose we start with a projective curve $X$ over a field $K$, given as a closed subvariety of $\mathbb P^n_K$ by some explicit list of equations. I would like to find an explicit representation of ...
- 374
2
votes
2
answers
474
views
matrix-valued differential forms on complex manifolds
I'm pretty clear in my understanding of scalar-valued differential $(p, q)$-forms (resp. holomorphic $(p,0)$-forms) on a complex manifold $M$ and the related Hodge theory. What I'm not sure about is ...
- 329
3
votes
1
answer
163
views
Reference request: Spectral properties of real operators
Let $A:D(A)\subseteq E \to E$ be a closed operator on a complex Banach lattice $E.$ Then $A$ is said to be real if $x+iy \in D(A) \implies x,y \in D(A)$ for all $x,y \in E_{\mathbb R}$ and $A(D(A) \...
- 343
11
votes
3
answers
753
views
Non embedding of the Heisenberg group
It is well known that Heisenberg groups cannot be bi-Lipschitz embedded into Euclidean spaces. A standard proof uses the fact that a Lipschitz mapping from a Heisenberg group into a Euclidean space is ...
- 27.3k
1
vote
1
answer
57
views
Probabilistic meaning of maximal rectangle under probability distribution function graph
Let $\xi$ be a random variable with $p(x)$ density function, which is like a normal distribution (i.e. $p(x)$ is increasing on $(-\infty,0]$ and decreasing on $[0,\infty)$). Denote by $s$ the maximum ...
- 111
3
votes
0
answers
122
views
PDE background for curve shortening flow
Soon I will be learning curve shortening flow and I am under the impression that a knowledge of PDEs is essential. Specifically, what from PDEs is required?
For example, I plan on brushing up using ...
- 143
14
votes
3
answers
522
views
Proving convergence of sum over $\mathbb{Z}^n$
In my research, I am trying to use the following construction by Benson Farb and John Franks, which proves that for all $n$, the group of $n\times n$ matrices with 1's on the diagonal, 0's above the ...
- 143
1
vote
1
answer
97
views
Polynomial Eigenvalue Problem with few non-zero coefficients
Let us define a diagonal matrix $\mathbf{D}(\lambda) = diag(\lambda^{m_1}, \dots, \lambda^{m_n})$ with $\lambda\in\mathbb{C}$ and positive integers $m_1, \dots, m_n$. The generalized characteristic ...
- 909
0
votes
1
answer
266
views
Hyperbolic structures on infinite type surfaces
Let S be a surface whose fundamental group is NOT finitely generated. Does there exist a complete hyperbolic metric on S for which the area is finite? I suspect the answer in general is NO, but I do ...
- 627
0
votes
1
answer
304
views
Have you seen this prime distribution before?
The basic question is : has this system been considered before, and how do I find it? References to the literature would be most welcome, but I am asking for reasonable search terms. I will try the ...
- 13k
11
votes
3
answers
814
views
Are there any books/articles that apply abstract coordinate free differential geometry to basic thermodynamics?
The mathematical structure of thermodynamics by Peter Salamon (pdf) would be an example, but i would like a more abstract natural formulation of application of differential geometry or even geometric ...
- 309
5
votes
1
answer
243
views
Looking for a tractable algorithm or formula for the determinant of a tensor
It is possible to define the determinant of a tensor.
We think of a tensor as a collection of numbers but this collection easily extends to a proper multilinear map.
If $T:\{1,....,n\}^m\to \mathbb C$ ...
- 183
3
votes
2
answers
2k
views
Prime counting. Meissel, Lehmer: is there a general formula?
I am looking for a general forumla to count prime numbers on which the Meissel and Lehmer formula are based:
$$\pi(x)=\phi(x,a)+a-1-\sum\limits_{k=2}^{\lfloor log_{p_{(a+1)}}(x) \rfloor}{P_k(x,a)}$$
...
- 81
3
votes
0
answers
265
views
Prove A Skipping Prime Conjecture For Rio?
I am writing a paper to accompany a Short Communication I plan to give in Rio this August. The paper regards work on jumping primes, a project on which Jose Brox has been working with me. I was going ...
- 13k
5
votes
1
answer
1k
views
One point compactification of $(\mathbb{C}^{\ast})^n$
I would like to know if there is a closed form formula for the homotopy type of $\widehat{(\mathbb{C^{\ast}})^n}$? For example, it is not difficult to see that $\widehat{\mathbb{C^{\ast}}}$ has the ...
- 1,017
1
vote
1
answer
284
views
Translations between S4 and S5 modal logics
$\textbf{Question}$: Is there a translation from $\textbf{S5}$ modal logic to $\textbf{S4}$ such that
$$\text{If} \hspace{0.3cm} \textbf{S5} \vdash F \hspace{0.3cm} \text{then } \hspace{0.3cm} \...
- 629
14
votes
1
answer
407
views
Definability in the field of reals with a predicate for some powers of two
In "The field of reals with a predicate for the powers of two", Van den Dries has proved that the set of integers is not definable in $(\mathbb{R}, +,\cdot, \leq, 0, 1, 2^{\mathbb{Z}})$, where
$2^{\...
5
votes
1
answer
299
views
The division problem for tempered functions
It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...
- 193
2
votes
0
answers
110
views
Rate of convergence of generalized polynomial chaos
Let $\eta=g(\xi_1,\ldots,\xi_M)$ be a random variable expressed as a function of the random vector $\xi=(\xi_1,\ldots,\xi_M)$. Assume that $\xi_1,\ldots,\xi_M$ are absolutely continuous and ...
- 31
3
votes
0
answers
161
views
Nekrasov Partition function and the leading term of Prepotential
I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY
AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.
In (4.25) the author expressed the partition function ...
- 415
6
votes
1
answer
236
views
Does the existence of a unique chromatic (possibly transfinite) number for every (possibly non-finite) simple graph imply the axiom of choice?
Assuming the axiom of choice I can write for any cardinal number $\kappa$ and any simple graph $G$ that a function $f$ is a $\kappa\text{-coloring}$ of $G$ if and only if the cardinality of the image ...
- 2,506
31
votes
3
answers
1k
views
Non embedding of $Y\times Y$ into $\mathbb{R}^3$
I know that this is a well known result, but where can I find a proof? I am also interested to see more general non-embedding results of this type.
Theorem. Let $Y$ be the union of two segments ...
- 27.3k
5
votes
0
answers
125
views
What is the minimum number of steps for two elements of a Lie algebra to generate the whole Lie group?
Consider a compact, connected and simply connected Lie Group $G$, and two elements in the corresponding Lie algebra $X$ and $Y$. By successive action of exponential map you can get the following ...
- 101
8
votes
1
answer
506
views
Intersection of two generic extensions
It is well known that the intersection of two models of ZFC does not have to be a model of ZFC (or even ZF). Now what if we restrict ourselves to models $M[G]$, $M[H]$ which are generic over $M$ for ...
5
votes
0
answers
123
views
Complex algebraic submersions
Let $X$, $Y$ and $Z$ be smooth complex algebraic varieties and let $f:X\to Y$ and $g:X\to Z$ be two morphisms. Suppose that $f$ is surjective, that $df_x:T_xX\to T_{f(x)}Y$ is surjective for all $x\in ...
- 1,373
3
votes
0
answers
180
views
Identity for the product of two different associated Legendre polynomials
In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated:
$$
\small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}...
- 16.4k
1
vote
0
answers
125
views
The space of Riemannian structures as an orbifold.
Consider a smooth closed manifold $M$. The space of Riemannian metrics is an open cone in the space of sections of some vector bundle. On this space the group of diffeomorphisms of $M$ acts by ...
- 7,373
9
votes
2
answers
729
views
Do two dimensional representations with the same adjoint representation differ by a character?
Let $K$ be a field of characteristic not equal to $2$. Let $\text{ad} : \text{GL}_2(K) \to \text{GL}_3(K)$ be the adjoint representation, obtained by $\text{GL}_2(K)$ acting on $2 \times 2$ matrices ...
- 93
0
votes
0
answers
976
views
Weak convergence can imply strong convergence [duplicate]
In $\ell^1(\mathbb N)$, weak convergence implies strong convergence. Is there a classification of infinite-dimensional Banach spaces for which such a property holds true ?
- 15.2k
1
vote
0
answers
123
views
Rational points of torsors over a separable closure
I already asked this question on Math Stack few days ago ( torsors over a separable closure ), but did not receive any answer, so I post it here.
Let $G$ be a smooth linear algebraic group defined ...
- 1,661
6
votes
2
answers
501
views
Torsors over complete local fields
Let $G$ be a linear algebraic group scheme, and let $R$ be a complete discrete valuation ring, with quotient field $K$ and residue field $k$.
If $T$ is an $R$-torsor, it yields by base change a $k$-...
- 1,661
1
vote
0
answers
74
views
If $f$ takes values in $L(H,L(H,\Bbb R))$ and $μ$ is a $H\hat ⊗_πH$-valued measure, how are $\int f\:dμ$ and $\int f⊗_π\text{id}_Hdμ$ related?
Let
$H$ be a separable $\mathbb R$-Hilbert space
$H\:\hat\otimes_\pi\:H$ denote the projective tensor product of $H$ and $H$
$(\Omega,\mathcal A)$ be a measurable space
$\mu$ be a $H\:\hat\otimes_\pi\...
- 161
24
votes
1
answer
2k
views
In what ways is ZF (without Choice) "somewhat constructive"
Let me summarize what I think I understand about constructivism:
"Constructive mathematics" is generally understood to mean a variety of theories formulated in intuitionist logic (i.e., not assuming ...
- 30.2k
4
votes
0
answers
276
views
Analytic class number formula for orders
In the article "The analytic class number formula for orders in products of number fields" (https://arxiv.org/pdf/1604.04564.pdf), it is shown that the analytic class number formula holds for ...
4
votes
0
answers
672
views
Euclides' sieve
This is probably a well-known problem. Given a set or multiset of natural numbers let us construct its "Euclides" closure: on each step we take all possible products $P_i$ of the elements in the set, ...
- 4,761
2
votes
1
answer
568
views
Duality of Bochner $L^{\infty}$ space
Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space
$$
E:=L^{\infty}([0,1],X),
$$
i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. ...
3
votes
2
answers
396
views
Central limit theorem for weak dependent bernoulli random variables
Suppose $\epsilon_1,\epsilon_2,...$ are i.i.d bounded random variables with compact support. Let $X_k=g_k(\epsilon_k,...,\epsilon_1)$ be Bernoulli random variables with the covariance between $X_i$ ...
- 339
1
vote
0
answers
55
views
geometry of intersection of 2 polytope in higher dimension [closed]
Suppose $P_{1}$ is a $\frac{n}{2}$-dimension polytope in $ R^{n}$ with barycenter $c$, and $P_{2}$ is a $(n-1)$-simplex in $R^{n}$ with the same barycenter as $P_{1}$, i.e $c$ . And also suppose they ...
- 111
8
votes
1
answer
253
views
Differential operators and quasi-finite morphisms
Let $ X, Y $ be smooth affine varieties over $ \mathbb C $. Let $ T : X \rightarrow Y $ be a dominant quasi-finite morphism and let $ T^\# : \mathbb C[Y] \rightarrow \mathbb C[X] $ be the resulting ...
- 4,467
6
votes
0
answers
372
views
closed substack of quotient stack
The question concerns quotient stacks. I am not very comfortable with stacks, so feel free to edit the question if I am saying nonsense. It is also the reason why I may be spelling in too much detail ...
- 371
10
votes
0
answers
180
views
k-th Pontryagin class of $\Lambda^{2k}_{\pm}$ on an oriented $4k$-manifold
If $M^{4k}$ is an oriented Riemannian $4k$-manifold, then the star-operator splits the bundle $\Lambda^{2k}$ into $\pm 1$-eigenspace bundles denoted $\Lambda^{2k}_{\pm}$.
I'm curious if anyone has ...
- 777
3
votes
2
answers
360
views
Extreme points of a convex set
Let $S$ denote the set of all complex non-negative definite matrices with all diagonal elements being less that or equal to one. Can we show that any matrix which belongs to the set of all non-zero ...
- 455
5
votes
1
answer
849
views
Moments of maximum of independent Gaussian random variables
Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for
$$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, ...
- 509
15
votes
4
answers
4k
views
What are good Morse Theory lecture notes and books?
Searching on the net I couldnt find any recent lecture/course notes on Morse Theory. I found an old set of notes (http://www.math.toronto.edu/mgualt/Morse%20Theory/mfp.pdf) by Mike Hutchings and these ...
- 2,146