All Questions
153,436
questions
2
votes
0
answers
171
views
Why does the following construction describe the Serre functor?
In the book "Spectral Agebraic Geometry" that Jacob Lurie is currently writing, he gives a construction (11.1.5.1), which describes the Serre functor: it has already been shown that any proper $A$-...
- 21
18
votes
3
answers
2k
views
Why should we study derivations of algebras?
Some authors have written that derivation of an algebra is an important tools for studying its structure. Could you give me a specific example of how a derivation gives insight into an algebra's ...
- 211
2
votes
1
answer
786
views
Sheaf / de Rham cohomology of a stack with values in a complex of abelian sheaves
I am reading Differentiable Stacks and Gerbes to understand about (hyper) cohomology groups of a stack $\mathcal{X}$ with values in a complex $\mathcal{M}$ of abelian sheaves over $\mathcal{X}$.
...
- 5,941
1
vote
1
answer
276
views
reference required: slope stability is an open condition
It is hard to find any reference which contains a proof of the following statement: slope stability is an open condition in a flat family.
There is one I have found, '65 paper of Narasimhan and ...
- 307
15
votes
1
answer
825
views
Homotopy pullback of a homotopy pushout is a homotopy pushout
Let's assume that we have a cube of spaces such that everything commutes up to homotopy.
The following holds:
- The right square is a homotopy pushout and
- all the squares in the middle are ...
- 181
2
votes
1
answer
98
views
Is the site for cubical sets with connections equivalent to a full subcategory of posets?
Cubical sets with connection form a presheaf category on some category $C$. Is $C$ just the full subcategory of the category of posets whose objects are products of the interval $\Delta[1]$?
- 61.6k
7
votes
1
answer
198
views
Distributing $N$ points on the sphere so that the sum of their mutual distances is maximized?
Generaliation the result in our paper for sum and similarly my previous question for product. I have a question:
My question: Distributing $N$ points on the sphere so that the sum of their mutual ...
- 1,086
2
votes
0
answers
46
views
generalisations of module maps
For an algebra $A$ over a field $\mathbb{K}$, consider left modules $E,F$. The two obvious collections of maps to take from $E$ to $F$ are the left module maps ${}_A\mathrm{Hom}(E,F)$ and the linear ...
- 1,213
19
votes
0
answers
699
views
Eckmann-Hilton argument / Grothendieck
In terse terms, the “Eckmann-Hilton argument” says that a monoid in the category of monoids is commutative. It is essentially used, for example, to prove that homotopy groups of topological groups are ...
- 12.8k
3
votes
0
answers
163
views
Non-special $\aleph_2$-Aronszajn trees in the Laver-Shelah model for $\aleph_2$-Souslin hypothesis
Assume $V=L$ and let $\kappa$ be a weakly compact cardinal. Let $G$ be $Col(\aleph_1, < \kappa)$ generic over $V$. Working in $V[G]$ force with a countable support iteration of forcing notions of ...
1
vote
1
answer
608
views
Eigenvalues of n matrices
For $i \in \{1,...,n\}$, let $A_i=[a^i_{jk}]$ be a symmetric matrix.
For $i \in \{1,...,n\}$, we assume that $a^i_{jj}=0$ for all $j$ and rank $(A_i)=2$.
Then it has one positive and one negative ...
12
votes
1
answer
278
views
Example of group cohomology not annihilated by exponent of $G$?
Is there an example of a finite group $G$ and an action on $M=\mathbb{Z}^n$ such that $H^2(G,M)$ has exponent greater than the exponent of $G$?
(Especially, can we have $G=\mathbb{Z}/2\mathbb{Z}\...
user39380
3
votes
1
answer
509
views
A generalization of Newton-Girard Identities
Let $x_1, ..., x_n$ be formal variables. One variant of the Newton-Girard identities expresses
$$\sum_{\pi \in S_n} x_{\pi(1)} x_{\pi(2)} \cdots x_{\pi(k)}$$
as a polynomial in the power sums of the $...
- 13.4k
7
votes
1
answer
203
views
Examples of probability measures with `fake' decay
To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ ...
- 323
16
votes
1
answer
521
views
Chromatic numbers of infinite abelian Cayley graphs
The recent striking progress on the chromatic number of the plane by de Grey arises from the interesting fact that certain Cayley graphs have large chromatic number; namely, the graph whose vertices ...
- 19.1k
5
votes
2
answers
318
views
Reference Request: Finite dimensional submanifolds of the space of smooth mappings
I apologize for my ignorance, but hope that someone would provide some pointers to what I am sure is a reasonably well-developed body of theory. Consider $C^\infty(U,V)$ where $U \subset R^k$ and $V \...
2
votes
0
answers
144
views
Multiset or Bag monad on Finite-Dimensional Hilbert Spaces
Edit: I will be happy if someone can get me the Bag monad on a 2-category of groupoids, regardless of any reference to Hilbert Spaces. (It's a fire sale!!)
I am trying to create the quantum ...
- 1,319
1
vote
0
answers
201
views
Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
- 11
4
votes
0
answers
211
views
Conjugacy class representatives for the automorphism group of a finite abelian group
Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...
- 286
5
votes
1
answer
192
views
Holomorphic functions with equal inverse images of unit circle
Let $f,g:\mathbb{C} \to \mathbb{C}$ be holomorphic and have the property $f^{-1}(S)=g^{-1}(S)$ where S is the unit circle centered at 0. What can be said about $f$ and $g$.
- 751
5
votes
2
answers
253
views
The number of co-circular four tuples
Let $A,B ⊂ \mathbb{R}$ such that $|A| = |B| = n$. What is the best-known upper bound on the number of four-tuples in $A \times B$ where the four points are co-circular, they lie on the same circle?
- 389
3
votes
3
answers
629
views
Asymptotic forms of Legendre functions for large degree
Does anyone know where to find (or how to obtain) expressions for the Legendre functions for large degree, to second order? For example, to first order the expressions are
$$
P_n(\cosh(x)) ~
\...
- 573
1
vote
1
answer
100
views
Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$
Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...
- 133
4
votes
1
answer
832
views
Twisted group rings and cohomology
Let $R$ be a commutative ring with unity and let $G$ be a finite group. Let $\gamma \in Z^{2}(G,R^{\times})$ be a $2$-cocycle. The twisted group ring (of $G$ over $R$ with respect to $\gamma$) $R *_{\...
- 1,651
10
votes
1
answer
439
views
Spacing of fractions with prime denominator
Let $X\ge 2$ be large. Let $$A = \left\{\frac{a}{q}: 1\le a < q\le X,\ (a, q) = 1\right\},$$ and let $$B = \left\{\frac{a}{p}: 1\le a < p\le X,\ (a, p) = 1,\text{ and $p$ is prime}\right\}.$$ ...
- 1,831
1
vote
0
answers
50
views
Calculating the density of data points around a specified point in a k-dimensional space [closed]
I am looking for a way of calculating how close data points are to a specified point in a k-dimensional space. My current method involves pythag to calulate the distance between the specified point ...
- 11
2
votes
3
answers
598
views
Multivariable vs single variable Alexander polynomial for links?
If we take a $n$-component link $L$, we have the multivariable Alexander polynomial $\Delta(L)(t_1,\ldots,t_n)$. Is there a standard single-variable Alexander polynomial? If yes, is it just euqal to $\...
- 1,425
5
votes
0
answers
163
views
Theoretical justification of time-series forecasting using Takens' embedding
This is a cross-posting
where I couldn't get an answer. In the meantime I have tried to improve the original logic:
As in Takens original paper about his embedding theorem, consider a compact $m$-...
- 51
4
votes
0
answers
87
views
Good range and fair range
Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
- 831
9
votes
0
answers
189
views
Analogue of strong stationary reflection from MM
Foreman-Magidor-Shelah proved that if Martin’s Maximum holds, then for every regular $\kappa>\omega_1$, every stationary $S \subseteq S^\kappa_\omega$ contains a club of ordertype $\omega_1$. This ...
- 18.1k
1
vote
0
answers
648
views
structure of flat modules over a DVR
I. Kaplansky has proved that if a torsion free module (over a complete DVR) is countably generated and does not contains infinitely divisible elements then it is free.
Is there any analog of this ...
- 21
0
votes
1
answer
773
views
Exponential Sequence of Sheaves
Let $(X, \mathcal{O}_X)$ be a complex analytic space in the sense of Grauert, i.e., a $\mathbb{C}$-analytic ringed space which is locally isomorphic to a local model. We may assume that $X$ is a ...
- 1,349
1
vote
1
answer
141
views
Gluing locally defined continous functions over complex domain
This is a cross-post to the question I asked at MSE over almost a month ago.
Suppose $n, l, m \in \mathbb N$ and $n \ge l > m$. Let $T: \mathbb C \to \mathcal M(n \times l; \mathbb C)$ be ...
- 195
6
votes
2
answers
382
views
An inequality related to area and sidelengths of a polygon $Area(A_1A_2....A_n) \le \frac{1}{n}cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$
$\DeclareMathOperator\Area{Area}\DeclareMathOperator\cotg{cotg}$I am looking for a proof (or a reference) of an inequality related to area and the sidelengths of a polygon as follows:
Let $A_1A_2\...
- 1,086
12
votes
1
answer
646
views
Relations between coefficients of expansions of a rational function at 0 and infinity
This question goes in the bucket of "this must be well known, but I don't see it and am not sure where to look it up."
Given two Laurent power series $A(t)=\sum_{k>N}a_kt^k$ and $B(t)=\sum_{k>M}...
- 44k
1
vote
1
answer
109
views
Completeness of Lowner order in separable Hilbert space
Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the interesection of positive operator and ...
- 1,493
4
votes
1
answer
154
views
Simplicial Pseudomanifolds with Boundary - Bounding number of maximal faces in terms of number of vertices and dimension
A simplicial pseudo-manifold of dimension $d$ with boundary is a simplicial complex satisfying the following conditions.
Every maximal face has dimension $d$
Each face of dimension $d-1$ is a face ...
- 43
3
votes
1
answer
308
views
Completely Positive Maps and their dual in Separable Hilbert Space
Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the set of positive operator with trace less ...
- 1,493
3
votes
1
answer
214
views
Are Holder Condition and signal to noise ratio (SNR) related?
This question was posted in https://math.stackexchange.com but I got hardly any view. If posting here is an objection please let me know I would delete it immediately.
This question has evolved from ...
- 435
6
votes
1
answer
643
views
Generalized projective spaces, spheres, and exotic spheres [closed]
I like to explore and ask for proper references for the relations between generalized projective spaces, spheres, and exotic spheres:
The real projective space
$\mathbb{RP}^1 \simeq S^1,$
is ...
- 10.3k
12
votes
3
answers
610
views
Is there a discrete lattice analogue of conformal transformations?
There is a simple discrete combinatorial analogue of manifolds and homeomorphisms: Replace manifolds by simplicial complexes and homeomorphisms by Pachner moves. Equivalence classes of manifolds under ...
- 2,901
4
votes
0
answers
117
views
A quantity that distinguishes finer than Hausdorff dimension
Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have ...
- 1,640
2
votes
0
answers
37
views
Reference request: semimarkov processes
What are some good modern introductions to the theory of semimarkov processes? To be clear, by a semimarkov processes, I mean a Markov chain, together with "waiting times" between transitions, the ...
- 495
9
votes
3
answers
389
views
Are there invariants of cell complexes similar to the Euler characteristic?
The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely
\begin{equation}
\chi=\...
- 2,901
2
votes
0
answers
115
views
Probability bound involving random, convex sets
Let $X,Y$ and $Z$ be random vectors in $\mathbb{R}^d$ defined on $(\Omega,\mathcal{H},\mathsf{P})$ s.t. $Z$ is $\mathcal{F}$-measurable for some $\mathcal{F}\subset\mathcal{H}$. Define a family of ...
- 185
5
votes
0
answers
224
views
Vanishing cycles for elliptic fibration on K3 surface?
Let $X$ be an elliptic K3 surface (over $\mathbb{C}$). Assume we have an elliptic fibration on $X$ that only has $I_1$ singular fibers.
If we fix a smooth fiber $F$ of such a fibration and a ...
user74900
3
votes
0
answers
147
views
Parallel transport for variety over finite field
I was wondering: Given a variety over a finite field, say the projective plane or sphere over $\mathbb{F}_q$. Then I can try to define parallel transport along (geodesic) curves. In particular, I can ...
- 1,746
2
votes
1
answer
180
views
Support of functions in Fourier domain
Let $\mathcal F$ be the Fourier transform. I would like to understand whether being in a Sobolev space implies that the Fourier transform of a function is necessarily supported on a compact ball up to ...
- 23
4
votes
1
answer
1k
views
Derivative of Lipschitz continuous functions
Given a real analytic family of Lipschitz continuous functions $f_t:\overline{U}\rightarrow\mathbb{R}^n$, $t\in\mathbb{R}$, with $U\subset \mathbb{R}^n$ some open and bounded domain. For each $t_0\in \...
- 237
9
votes
2
answers
895
views
Kuznetsov trace formula, orthogonality of Bessel functions
Sorry if this is a vague question. I remember from my younger days that
before proving his trace formula, Kuznetsov had a pretty result on
orthogonality of Bessel functions. The formulas that I am ...
- 11.8k