All Questions
153,425
questions
10
votes
2
answers
1k
views
understanding the definition of $\infty$-operad of module objects
I'm just trying to understand the following definition:
Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following:
Let $O^\otimes$ be ...
- 702
3
votes
0
answers
284
views
(Non trivial) coidempotents(Co-$K$-theory)
I was interested to know about coalgebraic version of "Idempotents".
So I seached the web and I found the following interesting post :
https://math.stackexchange.com/questions/689322/co-idempotents-...
- 312
5
votes
1
answer
489
views
Finite group acting on sphere
Let $G$ be a finite abelian group (of odd order if it's significant) acting on sphere $S^2\subset\mathbb{R}^3$. So my question: is it true that $G$ has a fixed point?
- 133
7
votes
0
answers
586
views
Unique maximal ideal in group C*-algebras
Let $G$ be a discrete group. Let $C^*(G)$ denote the full group C*-algebra of $G.$ Let $\pi:C^*(G)\rightarrow \mathbb{C}$ be the *-homomorphism associated with the trivial representation of $G.$
...
- 2,679
3
votes
1
answer
76
views
H S class operator and its equality
$A \in S(K)$ iff $A$ is a subalgebra of some member of $K$
$A \in H(K)$ iff $A$ is a homomorphic image of some member of $K$
It is trivial to see the containment $SH \leq HS$. Taking a simple ...
- 155
6
votes
2
answers
829
views
eisenstein part of the theta function
If $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the $\textbf{theta series}$ $\theta_Q(z):=\sum_{m\in\mathbb{...
- 305
3
votes
1
answer
174
views
A problem related to routing in a graph
I have come across a new problem - I want to know whether this problem is similar to some existing problem or not.
The new problem is this. There is a tourist who has a having the following ...
- 31
4
votes
1
answer
2k
views
Did Nash prove that every game or every symmetric game has a symmetric equilibrium?
Most references seem to state that Nash showed every symmetric game has a symmetric equilibrium point, but as far as I can tell from Nash's paper, he actually showed the much more general statement ...
user17474
15
votes
1
answer
805
views
Lift chain complex from $\mathbb{F}_2$ to $\mathbb{Z}$
We start with a finite dimensional chain complex over $\mathbb{F}_2$, equipped with a basis. That is, we have finitely many finite dimensional $\mathbb{F}_2$-vector spaces $C_0,\dots,C_k$ with bases $...
- 488
2
votes
1
answer
336
views
How to minimize $-\sum p_b \ln{p_b}$?
Consider multisets of the form $A = \{a_1,\dots,a_n\}$ of integers. Let $q = P(a_i = a_j)$ when $i$ and $j$ are chosen independently and uniformly from $\{1,\dots, n\}$. Let $B$ be the set of ...
- 3,195
2
votes
0
answers
98
views
On Flajolet's analytic urn model: a unified approach or just an interesting trick?
Recently I'm reading Flajolet's work on analytic urn models. In around 2006 He introduced a new analytical method that can give exact solutions to many classical urn models in a unified way. For a ...
- 21
2
votes
0
answers
181
views
Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?
Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...
- 2,005
22
votes
5
answers
1k
views
Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?
Let $S \subset \mathbb{R}^n$ be the boundary of a centrally symmetric convex body and provide $S$ with the geodesic metric given by its embedding in Euclidean space (i.e., the distance between two ...
- 1,293
9
votes
0
answers
775
views
Isoperimetric inequality, isodiametric inequality, hyperplane conjecture... what are the inequalities of this kind known or conjectured?
I duplicate here a question I asked on math.stackexchange.
Question: Which inequalities similar to the famous isoperimetric inequality is known?
conjectured?
I recently learned about some ...
27
votes
4
answers
3k
views
Algebra and cancer research
Let me start by acknowledging the existence of this thread: Mathematics and cancer research
It is well-known that mathematical modeling and computational biology are effective tools in cancer research....
- 1,412
6
votes
4
answers
2k
views
Singular distributions: Applications and Instances
Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ...
- 722
3
votes
0
answers
148
views
Dead Flies Problem [duplicate]
If a set of points in the plane contains one point in each convex region of
area 1, then can it have finite density?
what is the density of the points? In my understanding, it means the average ...
user39815
2
votes
0
answers
60
views
abstract simplicity results for anisotropic quasi-simple algebraic groups defined over a non-archimedean local field
I was wondering if anyone has some references I can look at that deal with results that identify some abstractly quasi-simple normal subgroup of the group of rational points of an anistropic ...
- 2,005
4
votes
2
answers
295
views
Simple groups and words
Let S be a finite simple nonabelian group, w a word in a finite number of variables which is not a power of another word. Must there be a substitution of elements of S in w such that the resulting ...
- 11.2k
4
votes
2
answers
833
views
Schreier's index formula
A finitely generated group G is said to satisfy Schreier's index formula if for every subgroup H of index k in G we have: d(H) - 1 = k(d(G) - 1). For example, a finitely generated free group satisfies ...
- 11.2k
1
vote
0
answers
150
views
number of times Brownian motion hits boundaries
Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...
- 19
4
votes
2
answers
566
views
Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$
This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...
- 3,938
12
votes
0
answers
725
views
$2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients
Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...
- 3,151
3
votes
2
answers
831
views
What are the modular transformation properties of q-Pochhammer symbols?
Do q-Pochhammer symbols, defined as
$(a;q) = \prod_{k=0}^{\infty} (1- a q^k)$
have known modular transformation properties? That is, if we write $q = q[z] = e^{2\pi i z}$, is there any reasonably ...
- 31
21
votes
3
answers
2k
views
What is Chern-Simons theory expected to assign to a point?
Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a $3$-...
- 115k
3
votes
1
answer
4k
views
Non-hyperbolic fixed points in multidimensional systems
Consider first a one-dimensional dynamical system given by $dx/dt = f(x)$. Suppose that the origin is a fixed point, i.e. $f(0)=0$. Suppose that we're interested in whether trajectories that start ...
- 1,316
11
votes
2
answers
3k
views
Limit of distance between two random points in a unit $n$-cube
What is the limit, as $n \to \infty$, of the expected distance between two
points chosen uniformly at random within a unit edge-length hypercube
in $\mathbb{R}^n$?
For $n=1$, the average distance ...
- 149k
3
votes
2
answers
287
views
Perimeter of a 'trapped' convex set
Consider the following setup: three bounded, 'nice' convex sets $A \subseteq B \subsetneq C \subset \mathbb{R}^2$, and three points $x,y,z\in \partial A\cap \partial B\cap \partial C$ (see edit below)....
- 115
3
votes
1
answer
741
views
References for von Neumann Algebras
I have some -possibly- simple but broad questions: Where to begin the study of von Neumann Algebras? Which are the important questions in the field that guide current research? I'm interested in ...
- 193
8
votes
2
answers
815
views
Projective modules over noncommutative tori?
It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class ...
- 493
4
votes
1
answer
291
views
In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?
Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In
Meyer, Werner
Die Signatur von ...
- 41
24
votes
0
answers
909
views
The topologies for which a presheaf is a sheaf?
Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.
Suppose that $Q$ is a presheaf on $...
- 8,559
52
votes
1
answer
6k
views
Are the primes normally distributed? Or is this the Riemann hypothesis?
Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.
Let $\mathrm{Li}(x)$ be the offset logarithmic ...
- 8,423
1
vote
0
answers
46
views
Identification of model involving convex polynomials
I want to solve a nonlinear least squares problem on the following form
\begin{equation}
\begin{array}{l}
\min_{\theta,\phi} J(\theta,\phi) &=& \min_{\theta,\phi} \sum_{i=1}^k ([p_0(a_i^T\phi),...
- 11
1
vote
1
answer
198
views
infinitary logic and partial fixed point logic
Is there a property definable in finite-variable infinitary logic $L^{\omega}_{\omega\infty}$ but not definable in partial fixed point logic FO(PFP) ?
- 41
2
votes
1
answer
735
views
reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$
I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
- 5,163
1
vote
0
answers
143
views
Bounding Rayleigh quotient for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
- 121
11
votes
1
answer
471
views
Cardinals below the critical point of a generic embedding
This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension?
To focus on the ...
- 18.1k
2
votes
3
answers
80
views
a special filtration satisfying $0$-$1$ law
Let $\xi$ be a uniformly random variable on $[0,1]$ defined on some probability space $(\Omega,\mathcal{F})$. Define the process $\xi_t:=\min(\xi,t)$ for $0\le t\le 1$. And let $\mathcal{F}_t=\sigma(\...
- 1,837
4
votes
1
answer
706
views
Why tangent vector of statistical manifold is a function?
In differential geometry, tangent vectors are considered operators.
At point p, the local tangent space is defined as
$$
T_p(M)=\{X^i\partial_i|X\in R^n\}
$$
This is quite easy to understand for me.
...
- 59
23
votes
1
answer
3k
views
What is the status of the Friedlander-Milnor conjecture today?
For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...
- 231
2
votes
2
answers
239
views
If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...
- 9,636
6
votes
2
answers
1k
views
Geometric explanation of Hutton's formula?
$$\frac{\pi}{4} = 2 \tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{7} \;.$$
Is there some geometric construction that explains this beautiful equation
(known as Hutton's formula)?
Perhaps a "proof without ...
- 149k
2
votes
1
answer
530
views
On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski
In response to a comment posted under
Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd ...
- 2,158
3
votes
1
answer
242
views
Lyapunov Exponents for independent-nonidentically distributed matrices?
My question is highlighted in bold at the end.
$\mathrm{\underline{Background}}$
Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$
(with $\mathbb{E}\log\left\Vert A_{i}\right\Vert <...
- 155
3
votes
0
answers
339
views
Conjecture relating differential equation and sum of a function over partitions
The following is an addition to A function from partitions to natural numbers - is it familiar?; the function $f(\lambda)$ as defined there, when summed over all partitions of n, gives an unexpected ...
- 136
8
votes
1
answer
794
views
The sum over zeros in the explicit formula for $\zeta(s)$
The explicit formula for $\zeta(s)$ is:
$$
\psi(x)=x-\sum_{|\operatorname{Im}\rho|<T}\frac{x^\rho}{\rho}-\log(2\pi)-\log\left(1-\frac{1}{x^2}\right)+O\left(\frac{x\log^2T}{T}\right),
$$
where $\psi(...
- 139
2
votes
1
answer
434
views
Minimize a strictly convex quadratic function subject to linearly equality and nonnegativity constraints in finite time?
I am wondering if we can minimize a strictly convex quadratic function in finite time, subject to linearly equality and nonnegativity constraints.
Thanks!
- 65
1
vote
1
answer
131
views
Is there a name for functions f(x,y) that are only convex in x (and continuous in y). [closed]
This is just a notation question.
If I cannot find a preexisting name, I would try one-sided convex, or something of the sort.
- 247
11
votes
2
answers
873
views
Eilenberg-MacLane Spaces of "large" groups
It is well-known that if $G$ is a discrete group, then $BG=K(G,1)$. I'm interested in comparing classifying spaces of topological groups with the classifying spaces of the same groups but equipped ...