All Questions
152,871
questions
0
votes
0
answers
205
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How is this pairing $\langle\,,\rangle$ defined of cocharacter and character of an algebraic group?
Let $G$ be a semisimple linear algebraic group. Let $X^*$ be the group of characters and $X_*$ be the group of cocharacters. Then I know that there exists a pairing $\langle\,,\rangle : X^*(G) \times ...
63
votes
6
answers
12k
views
Why isn't integral defined as the area under the graph of function?
In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...
3
votes
0
answers
190
views
Trivial action in the Hochschild-Serre spectral sequence
I probably don't understand something very basic about Hochschild-Serre spectral sequence.
Let $G$ be a group with normal subgroup $N$ and $M$ a $G$-module with trivial action. Then as far as I ...
13
votes
1
answer
495
views
Extending monads along dense functors
Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$.
$\require{...
4
votes
0
answers
224
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Looking for U.K. problem column (?) from 1980s
While digging through some dusty corners of my file cabinet, I found a photocopied sheet of eight (handwritten) problems from 1985 that I recall receiving from my secondary school mathematics teacher ...
2
votes
0
answers
137
views
$\omega$-categorical algebra
Let us consider a 1-category $C$. For any commutative and unital ring $k$ the the free $k$-module generated by the morphisms of $C$ can be equipped with an algebra structure by setting $fg$ to be ...
1
vote
0
answers
58
views
Martingales limit theorems (reference)
I have a sequence of processes $\{X^N(t)\}_{t\in [0,T]}$, $N\in\mathbb N$ such that
$X^N(t)=x+M^N(t)$,
where $M^N(t)$ is a martingale with expectation $0$ and with quadratic variation $<M^N>(t)$ ...
1
vote
1
answer
1k
views
Countable intersections in topological space
If a T1 topological space is closed under countable intersections, does this necessarily make the topology discrete? It is easy to construct a counterexample if the topological space is not assumed to ...
1
vote
0
answers
479
views
Connection between Fourier analysis and Galois theory
Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e.
$$x \equiv x\ \%\ m\pmod{m}$$
Let $\mu^n_m(x)$ denote multiplication by $n$ modulo $m$, i.e.
$$\mu^n_m(x) = nx\ \%\ m$$
Consider the Fourier ...
2
votes
0
answers
65
views
self-dual integral transform
Is it possible to describe all integral transforms where the inverse transform is implemented by the same formula (with maybe a sign flipped somewhere). Fourier is such an example obviously. I am ...
0
votes
1
answer
263
views
Decomposing functions to Taylor-Fourier series
[Cross posted from Math.SE due to lack of attention]
A great many functions can be expressed as a series of the form
$$ U_0(x) + U_1(x) x + U_2(x) \frac{1}{2!}x(x-1) + ... $$
Where $U_r(x)$ are ...
2
votes
0
answers
107
views
Is the action of free self-distributive algebras on racks computable in polynomial time?
Let $B_{\infty}$ denote the infinite strand braid group. Let
$\mathrm{sh}:B_{\infty}\rightarrow B_{\infty}$ be the mapping where
$\mathrm{sh}(\sigma_{i})=\sigma_{i+1}$ whenever $i\geq 1$. Then
$B_{\...
10
votes
3
answers
1k
views
Is $\mathrm{Graph}$ cartesian-closed?
Let $\mathrm{Graph}$ be the category of simple, undirected graphs with graph homomorphisms. For any graphs $G, H$ we denote by $\text{Hom}(G, H)$ the set of graph homomorphisms $f:G\to H$. (Note that $...
2
votes
0
answers
102
views
Does shifted conjugacy still give you free self-distributive algebras on one generator for quotient groups of the braid groups?
Let $B_{\infty}$ denote the infinite strand braid group. Let $\mathrm{sh}:B_{\infty}\rightarrow B_{\infty}$ be the group homomorphism where $\mathrm{sh}(\sigma_{i})=\sigma_{i+1}$ for all $i>0$.
...
8
votes
2
answers
4k
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Metric measure spaces: in what sense is analysis on these spaces "non-smooth"
I understand the basic definition of a metric measure space to be the following:
A metric measure space is a triple of a space $X$, metric $d$, and measure $m$: $(X,d,m)$ in the sense that the ...
4
votes
1
answer
312
views
Cyclotomic fields and splitting of central simple algebras
Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $\mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $\mathbb{Q}$-algebra embedding $K \...
2
votes
1
answer
259
views
Is this condition sufficient for a variety to be reversible?
A variety $V$ is said to be reversible, if for each $n>0$ and fundamental operation $f$ there are $m\geq n$ and $r$ along with terms $T_{2},\dots,T_{r}$ and $S_{1},\dots,S_{m}$ such that if $G,H$ ...
2
votes
0
answers
133
views
Computing Hochschild Invariants of Positselski's Coderived Categories
Positselski's work allows one to frame Koszul duality very elegantly in terms of so called coderived categories of modules over coalgebras, these are somewhat exotic dg categories of comodules over $C$...
2
votes
0
answers
152
views
Baker map-like problem
Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...
2
votes
0
answers
51
views
Inductive limits of unitary groups and quantum mechanics
I'm curious if someone has seen concrete applications of $U(\infty)$ in quantum mechanics. Is it possible, for example, in some particular cases to write down the propagator as a limit of a sequence ...
2
votes
0
answers
79
views
discrete parabolic Harnack inequality
I am currently looking for a discrete version of the parabolic Harnack inequality in the following "$L^1$ to $L^\infty$" form:
If $u(t,x)\geq 0$ is a (say, smooth) subsolution of
\begin{equation}
...
1
vote
0
answers
229
views
Short question on functions of bounded variation
For a function $f: \mathbb R \to \mathbb R$ of locally bounded variation, when is $$\liminf_{e \to 0} V(f)[x, x+e]/e $$finite everywhere? Here $V(f)[a, b]$ denotes the total variation of the function ...
2
votes
0
answers
62
views
Differences among various index theories in critical point theory
Index theories help characterize critical points of functionals having certain symmetries. What are the differences (regarding problems they can be applied to) between for example these ones?
the ...
1
vote
1
answer
112
views
Graphs formed of vertices of distance $2$
Let $G=(V,E)$ be a finite, simple, undirected graph. Let $D_2(G)$ be the graph with vertex set $V$, and two vertices form an edge if and only if they have distance $2$ in the original graph $G$.
...
1
vote
0
answers
118
views
Strongly isovariant (aka fixed point reflecting / stabiliser preserving) morphisms
I have some questions about what feels like basic topics in quotients of schemes by group actions. Consequently I suspect there are well-known references; I couldn't find them, though.
First ...
0
votes
0
answers
266
views
Local "boundary comparison principle" for harmonic functions
Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...
7
votes
1
answer
495
views
Smallest Mazur's good prime
Let $p$ and $\ell$ be primes $\geq 5$ such that $\ell$ divides $p-1$. Following Mazur, we say that a prime $q$ is a $\textit{good prime}$ if $\ell$ does not divide $q-1$ and $q$ is not a $\ell$th ...
1
vote
0
answers
46
views
Show that $\Phi_{P(X)}=\hat{X}$
Let $X$ be a compact subset of $\mathbb {C^n}$. The polynomial convex hall of $X$ is the set
$\hat{X}=\{z\in \mathbb {C^n}: \left|P(z) \right|\leq \left||P |\right|_\infty , \text{for all polynomial ...
5
votes
0
answers
102
views
Tensor square of duals over a domain
The title is motivated by my needs ($M=N$ in the sequel).
Linked to the question here and there (in the case of products) is the following.
Let $M,N$ $k$-modules ($k$ a commutative ring), then we ...
3
votes
0
answers
96
views
What are the composition series for these series of groups?
A rack is an algebra $(X,*,*^{-1})$ that satisfies the identities
$x*(y*z)=(x*y)*(x*z)$ and
$x*(x*^{-1}y)=x*^{-1}(x*y)=y$.
If $X$ is a rack then define a homomorphism $\phi_{n,X}:B_{n}\rightarrow \...
5
votes
1
answer
337
views
Classification of quasi-lisse vertex algebras
Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in Quasi-lisse vertex algebras and modular linear differential equations . They satisfy the property that the normalized character ...
3
votes
0
answers
198
views
Longest known polynomial progression of distinct primes
Is Euler’s quadratic progression of forty distinct primes (the values of $n^2-n+41$ for $n$ between 1 and 40) still the longest known sequence of this kind?
I’d also be curious to know the longest ...
1
vote
1
answer
142
views
Lemma for proof of Jordan-Hölder Theorem [closed]
Lemma
Let $G$ be a group, $K \triangleleft G$ a normal subgroup and $H_j \triangleleft H_i$ two subgroups of $G$ with $H_j$ a normal subgroup of $H_i$.
Then there is an isomorphism
$$(H_i K)/(H_j K)...
2
votes
0
answers
152
views
Most important results for Shalika germs
This is more of a general question, but what do you think are the most important results for Shalika germs if you were giving a presentation? You can assume the target audience to be 2nd-3rd year ...
3
votes
0
answers
87
views
Handlesliding a two component, linking number 1 link
Let $L = K_1 \cup K_2 \subset S^3$ be a two component framed link with $lk(K_1,K_2) = 1$. Let $\hat{L}$ denote the set of all links obtained by handlesliding $L$ around an an arbitrary number of ...
2
votes
0
answers
30
views
Graph vertex label dynamics, statistical model reference request
I am modeling some type of social interaction, and came up with the following natural question.
Let $K_n$ be the complete graph on $n$ vertices, with some initial edge labeling in some alphabet $A$.
...
21
votes
2
answers
2k
views
Applications of derived categories to "Traditional Algebraic Geometry"
I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
2
votes
0
answers
562
views
Intersection of a reduced projective variety with a general hyperplane is reduced
Let $X\subset \mathbb{P}^n$ be a reduced closed subscheme. For a general hyperplane $H$, $X\cap H$ is again reduced (and of dimension one less). Is there an easy proof of this result?
Algebraically, ...
6
votes
1
answer
327
views
Valuation Rings and Ultrafilters
I notice there is a certain similarity between the definition of a valuation ring and the definition of an ultrafilter.
To begin, take a field $K$ and let $\mathcal{A}$ be the set of subrings of $K$. ...
10
votes
4
answers
1k
views
An interesting sum over lattice points in a large disk centered at the origin
Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits_{(m,n) \in D_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^2 + n^2$, where $D_r$ denotes the closed ...
0
votes
1
answer
104
views
Size of edge set of infinite hypergraphs with $\chi(H) = |V(G)|$
Let $H=(V,E)$ be a hypergraph such that for every $e\in E$ we have $|e|\geq 2$. A map $c:V\to \kappa$, where $\kappa$ is a cardinal, is said to be a (hypergraph) coloring if for all $e\in E$ the ...
2
votes
0
answers
217
views
Does the Burau representation of braids distinguish between distinct elements of the free self-distributive algebras on one generator?
A well-known but now mostly solved problem in group theory is the question of whether the Burau representation of the braid groups is faithful. It turns out that this representation is not faithful ...
-1
votes
1
answer
379
views
Is there any intrinsic ( without any reference to embedding ) and coordinate free , basis free definition of a differentiable manifold? [duplicate]
Is there any way to uniquely characterise manifolds ( their geometrical and topological properties) without refering to charts or to a particular hyperspace containing the manifold. For example could ...
1
vote
0
answers
127
views
Word length norm in the symmetric group $\mathfrak{S}_r$
Consider on the symmetric group $\mathfrak{S}_r$ the generating system $\{\tau_i;\,1\le i\le r-1\}$ with $\tau_i = \langle i,i+1\rangle$ and the corresponding word length norm $N$. Now let $\tau\in\...
1
vote
1
answer
473
views
Sparse, left-looking LU factorization
I'm trying to understand the left-looking LU factorization algorithm for sparse matrices, by reading T.A. Davis' book, and have trouble in one step (sorry for the specific question) about returning ...
3
votes
2
answers
139
views
Rank of order-3 tensor with all slices being rank-1
If some tensor $T=(t_{ijk})$ has that all of its (2 dimensional) slices (along all 3 axes) are of rank-1, does it follow that the tensor is also rank-1? That is, can be written as
$$ t_{ijk}=a_i b_j ...
1
vote
1
answer
302
views
Existence of a Lyapunov function for a log-concave measure
Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\...
1
vote
0
answers
87
views
On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ over $\mathbb N$
Let $a,b,c,d,e,f$ be integers with $a\ge c\ge e>0$, $b>-a$ and $a\equiv b\pmod2$, $d>-c$ and $c\equiv d\pmod 2$, $f>-e$ and $e\equiv f\pmod2$. If each $n\in\mathbb N=\{0,1,2,\ldots\}$ can ...
4
votes
2
answers
156
views
$\sum_{k=1}^dA_k^*A_k$ and $\sum_{k=1}^dA_kA_k^*$ have the same norms if $A_k$ are commuting
Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$.
Let $A_1,\cdots,A_d$ be pairwise commuting operators on $E$. Is the equality
$$\left\|\displaystyle\...
0
votes
0
answers
16
views
Linear quadratic regulator equivalence of formulations
I don't see why the following three forms of the LQR optimal control problem are equivalent:
For $\begin{cases} x'=Ax+Bu \\ x(t_0)=x_0\end{cases}$ find
$$\min_{u\in L^2(t_0,T; \mathbb{R}^m)} J(u)=\...