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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 ...
Johnny T.'s user avatar
  • 3,547
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 ...
user57888's user avatar
  • 1,229
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 ...
David Levit-Gurevich's user avatar
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{...
Ivan Di Liberti's user avatar
4 votes
0 answers
224 views

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 ...
Timothy Chow's user avatar
  • 78.1k
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 ...
User371's user avatar
  • 537
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 $<M^N>(t)$ ...
user268193's user avatar
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 ...
Daniel Elessar's user avatar
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 ...
Hans-Peter Stricker's user avatar
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 ...
Kphysics's user avatar
  • 121
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 ...
Sidharth Ghoshal's user avatar
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_{\...
Joseph Van Name's user avatar
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 $...
Dominic van der Zypen's user avatar
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 $i>0$. ...
Joseph Van Name's user avatar
8 votes
2 answers
4k views

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 ...
yoshi's user avatar
  • 397
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 \...
Sun Ra's user avatar
  • 173
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 $G,H$ ...
Joseph Van Name's user avatar
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 $C$...
user avatar
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 \...
James Baxter's user avatar
  • 2,029
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 ...
Ivan's user avatar
  • 445
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} ...
leo monsaingeon's user avatar
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 ...
James Baxter's user avatar
  • 2,029
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 ...
Riku's user avatar
  • 819
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 $G$. ...
Dominic van der Zypen's user avatar
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 ...
Tom Bachmann's user avatar
  • 1,951
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 = ...
user avatar
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 ...
Emmanuel Lecouturier's user avatar
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 ...
user62498's user avatar
  • 813
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 ...
Duchamp Gérard H. E.'s user avatar
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 \...
Joseph Van Name's user avatar
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 ...
user avatar
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 ...
James Propp's user avatar
  • 19.4k
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)...
Jürgen Böhm's user avatar
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 ...
Ioannis Zolas's user avatar
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 ...
user101010's user avatar
  • 5,319
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 $A$. ...
Per Alexandersson's user avatar
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, ...
user2718's user avatar
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 $K$. ...
Ronald J. Zallman's user avatar
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 ...
Wahome's user avatar
  • 737
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 ...
Dominic van der Zypen's user avatar
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 ...
Joseph Van Name's user avatar
-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 ...
Sheldon's user avatar
  • 23
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\...
FKranhold's user avatar
  • 1,623
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 ...
grok's user avatar
  • 2,489
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 ...
Student88's user avatar
  • 503
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 $\...
0xbadf00d's user avatar
  • 161
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 ...
Zhi-Wei Sun's user avatar
  • 14.4k
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\...
Student's user avatar
  • 1,154
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)=\...
Bogdan's user avatar
  • 1,330

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