All Questions
153,330
questions
3
votes
1
answer
198
views
Ultraproduct of non-commuative $L^p$-spaces
Let $1<p<\infty.$ Let $I$ be a non-empty set and $\mathcal{U}$ be an ultrafilter over $I.$ Let $M_i$ be von Neumann algebras equipped with normal faithful semifinite traces $\tau_i,$ $i\in I.$ ...
- 1,765
1
vote
1
answer
361
views
How to obtain a product-to-sum identity for the sinc function?
We know that
$$\text{sinc}(x)=\prod_{n=1}^\infty\cos\left(\frac{x}{2^n}\right)$$
and for some truncated $k$ we can write the following product-to-sum identity:
$$\prod _{n=1}^k \cos \left(\frac{x}{2^n}...
- 35
5
votes
0
answers
173
views
Large finite subsets of Euclidean space with no isosceles (or approximately isosceles) triangles
Here's a question in combinatorial geometry which feels very much like other questions I'm familiar with but which I can't see how to get a hold of. I'll actually propose two different questions on ...
- 19.1k
2
votes
0
answers
139
views
Noether’s “set theoretic foundations” of algebra. Reference
In [C Mclarty] we read
[Noether] project was to get abstract algebra away from thinking about operations on elements, such as addition or multiplication of elements in groups or rings. Her algebra ...
- 21
1
vote
1
answer
60
views
Performing Statistical Analysis on a Data Set With a lot of Null Responses
I am currently trying to perform some statistical analysis on some data to see if there is any meaningful conclusion for a research project I am working on; however, I have come across a problem. ...
3
votes
1
answer
339
views
Why control a continuous approximation of stochastic gradient descent instead of just the SGD?
In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in
$$x_{k+1} = x_k - \eta u_k \nabla f_{\...
- 637
0
votes
1
answer
228
views
Existence and smoothness for viscous Burgers equation?
What do we currently know about (references please!) the existence and smoothness of solutions to the viscous Burgers equation, in 1D, 2D, and 3D?
- 51
123
votes
14
answers
36k
views
What are some noteworthy "mic-drop" moments in math?
Oftentimes in math the manner in which a solution to a problem is announced becomes a significant chapter/part of the lore associated with the problem, almost being remembered more than the manner in ...
1
vote
0
answers
146
views
Clarification about the ϵ -net argument
I have been reading the paper Do GANs learn the distribution? Some theory and empirics.
In Corollary D.1, they reference the paper Generalization and Equilibrium in Generative Adversarial Nets which ...
- 19
8
votes
1
answer
681
views
Is there an abstract theory of club sets and stationary sets?
In set theory, there are several distinct notions of club sets, stationary sets, diagonal intersection, regressive function, normal filters, normal ultrafilters, etc. I am wondering if there is an ...
- 27.8k
5
votes
0
answers
158
views
Continuous open self maps on Cantor space
A continuous self map on the Cantor space $C = \{0,1\}^\mathbb{N}$ is a mapping $f = (f_i)_{i\in \mathbb{N}}$ such that each $f_i$ is a map from $C$ to $\{0,1\}$ that depends only on a finite number ...
- 2,582
2
votes
1
answer
128
views
Uniformly Converging Metrization of Uniform Structure
This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...
- 10.3k
1
vote
0
answers
61
views
Uniform convergence of holomorphic automorphisms
Let $X$ be a complete Kobayashi hyperbolic complex manifold. It is well-known that the automorphism group of $X$ is a real Lie group where the topology on the automorphim group is the compact-open ...
- 1,149
2
votes
2
answers
429
views
On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function
Define $\theta(x)=\sum_{p\leq x} \log p $, where $p>1$ denotes a prime.
Nicolas proved that if the Riemann zeta function $\zeta(s)$ vanishes for some $s$ with $\Re(s)\leq 1/2 + b$, where $b\in(0, 1/...
user136836
8
votes
2
answers
408
views
For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?
How explicit are the model structures for various categories of spectra?
Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit ...
- 61.5k
2
votes
0
answers
106
views
Deforming Modular Symbols
This is probably a silly question, as I don't really know a whole lot about modular symbols over arbitrary rings.
How do modular symbols over a finite field square with Katz modular forms? If they ...
user130124
2
votes
1
answer
102
views
Maximum number of $0$-$1$ vectors with a given rank
Let $k\ge2$. The maximum number of $0$-$1$ (column) vectors of length $2k-1$ which make a rank $k$ matrix with no zero row nor two identical rows is $2^{k-1}+1$. (The rank is over the rationals.)
I ...
- 33
4
votes
1
answer
244
views
Minimal cardinality of a filter base of a non-principal uniform ultrafilters
Let $\kappa$ be an infinite cardinal. An ultrafilter ${\cal U}$ on $\kappa$ is said to be uniform if $|R|=\kappa$ for all $R\in{\cal U}$. If ${\cal U}$ is a non-principal ultrafilter on $\kappa$, ...
- 45.4k
1
vote
0
answers
222
views
Equivariant vector bundles whose quotient map preserves the stabilizer
Let $G$ be a compact Lie group which act on a manifold $M$. We fix this action throughout our question.
Assume that $E\to M$ is a vector bundle which has the potential of admitting ...
- 312
5
votes
1
answer
454
views
Solving Linear Matrix Recurrences
Question:
Are there standard techniques available for solving the following kind of linear matrix recurrence relations:
$$M_1,\cdots,M_k\ \in\ \mathbb{R}^{m\times n}$$
$$ A_1,\cdots,A_k\ \in\ \mathbb{...
- 12.7k
0
votes
0
answers
39
views
Asymptotic upper densities in infinite binary stochastic processes
Consider an infinite binary process $X=X_1,X_2,\ldots$ (with corresponding probability $P$). For some bits $1$ is less probable than $0$. I am interested in the following asymptotic upper density :
$$\...
1
vote
1
answer
185
views
Is a polytope with vertices on a sphere and all edges of same length already rigid?
Let's say $P\subset\Bbb R^d$ is some convex polytope with the following two properties:
all vertices are on a common sphere.
all edges are of the same length.
I suspect that such a polytope is ...
- 12.6k
0
votes
1
answer
119
views
A characterization of maps that are homotopic relative to $A$ over $S$
Let $$\begin{array}{ccccccccc}
A & \rightarrow & X \\
i\downarrow & & \downarrow p \\
B & \xrightarrow{v} & S
\end{array} $$ be a commutative diagram of simplicial sets, with ...
- 765
2
votes
1
answer
242
views
Étale fibration for $K[[X_1,...,X_n]]$
Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...
- 563
1
vote
0
answers
60
views
Growth rate of the critical points of the Fibonacci terms $t_{n}(x,y)$ vs $t_{n}(1,1)$ in the classical Laver tables
The classical Laver table $A_{n}$ is the unique algebra $(\{1,\dots,2^{n}\},*_{n})$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and $x*_{n}1=x+1\mod 2^{n}$ for all $x,y,z\in A_{n}$.
Define the ...
- 27.8k
3
votes
1
answer
219
views
"Oddity" of $q$-Catalan polynomials: Part II
This question extends my earlier MO post for which I'm grateful for answers and useful comments.
The Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$ satisfy:
$\text{$C_{1,n}$ is odd iff $n=2^j-1$ for ...
- 41.8k
3
votes
1
answer
517
views
Computing affine Springer fibers
$\DeclareMathOperator\diag{diag}\DeclareMathOperator\Gr{Gr}\DeclareMathOperator\SL{SL}$I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $...
- 2,969
2
votes
1
answer
105
views
Attraction in Laver tables
If $X$ is a self-distributive algebra, then define $x^{[n]}$ for all $n\geq 1$ by letting $x^{[1]}=x$ and $x^{[n+1]}=x*x^{[n]}$. The motivation for this question comes from the following fact about ...
- 27.8k
33
votes
4
answers
2k
views
How many random walk steps until the path self-intersects?
Take a random walk in the plane from the origin,
each step of unit length in a uniformly random direction.
Q. How many steps on average until the path self-intersects?
My simulations suggest ~$8....
- 149k
3
votes
0
answers
486
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
11
votes
2
answers
444
views
Actions of locally compact groups on the hyperfinite $II_1$ factor
Let $R$ be the hyperfinite $II_1$ factor, and let $G$ be a locally compact group.
(1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$?
(2) If so, how does one ...
- 42.5k
10
votes
1
answer
717
views
Axiom of choice and algebraic tensor product
The first part of the question was asked on Math-stackexchange.
Let $V$, and $W$ be vector spaces. By the universal property of the tensor product,
there is a canonical map from $V^*\otimes W^*$ ...
- 975
5
votes
1
answer
242
views
Amalgamation via elementary embeddings
Can there exist three transitive models of ZFC with the same ordinals, $M_0,M_1,N$, such that there are elementary embeddings $j_i : M_i \to N$ for $i<2$, but there is no elementary embedding from $...
- 18.1k
3
votes
1
answer
481
views
Semi-orthogonal decompositions for Calabi-Yau varieties
In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry there is an exercise (Exercise 1.7 and 1.8) to prove the statement that the derived category $D^b(X)$ of a Calabi-Yau variety $X$ has ...
- 51
2
votes
1
answer
139
views
Perturbation of the adiabatic limit
Let $(M,g_M)$ be a closed oriented Riemannian manifold that has a fibration structure
$$
Y \rightarrow M \overset{\pi}{\rightarrow} B
$$
where $(Y,g_Y)$ and $(B,g_B)$ are closed Riemannian manifolds ...
- 2,525
3
votes
2
answers
363
views
Is this a submanifold?
Let $(M,g)$ be a compact Riemannian manifold with an isometric action $\rho : G \to \mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $\rho$ to $TM$ given by:
$$\psi : G \...
- 1,774
1
vote
0
answers
32
views
Algorithms for Detecting the Completion of a Triangle in a Stream of Edges
I need to efficiently determine in a complete weighted graph $G$ the sequence of triangles according to descending order of circumferences.
My idea would be to incrementally construct a new graph $H$, ...
- 12.7k
3
votes
1
answer
482
views
Yet another question on sums of the reciprocals of the primes
I recall reading once that the sum $$\sum_{p \,\, \small{\mbox{is a known prime}}} \frac{1}{p}$$
is less than $4$.
Does anybody here know what the ultimate source of this claim is?
Please, let me ...
- 8,722
10
votes
1
answer
372
views
Smooth vector fields on a surface modulo diffeomorphisms
Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.)
Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$...
- 32.8k
3
votes
0
answers
346
views
Irreducible Smooth Proper one-dimensional Schemes isomorphic to $\mathbb{P}^1$
I have a curious question about an argument/hint given in following thread:
https://math.stackexchange.com/questions/3062986/irreducible-smooth-proper-one-dimensional-mathbbz-schemes
The OP asked if ...
- 6,000
8
votes
1
answer
302
views
Laplacian spectrum asymptotics in neck stretching
Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....
- 1,197
1
vote
0
answers
56
views
The significant role of dual frames in the progress of Frame theory
For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
- 11
1
vote
0
answers
92
views
When does there exist a faithful normal expectation onto von Neumann subalgebra (finite vNa)?
Let $R$ be a finite von Neumann algebra and $S$ be a von Neumann subalgebra of $R$. What conditions must $S$ satisfy so that a faithful normal conditional expectation $\Phi : R \to S$ exists?
For ...
- 159
2
votes
0
answers
130
views
Factoring Finite Morphisms through Open Immersions
This is a somewhat silly question, but I was wondering the following. Assume X, Y, and Z are schemes and consider the composition
$X \xrightarrow{f} Y \xrightarrow{g} Z $ and assume $g\circ f $ is a ...
- 121
3
votes
1
answer
118
views
A converse of Cartan's automatic continuity theorem
Let $G$ be a compact real Lie group. We say that $G$ has property $(*)$ if every abstract automorphism of $G$ is continuous. A theorem of Cartan says that if $G$ has perfect Lie algebra, it has ...
- 35
4
votes
0
answers
248
views
Second derivative at 1 of L function of elliptic curve
Let $E$ be an elliptic curve over $\mathbb Q$ of conductor $N$ and rank $0$. It follows from the functional equation that
$$L'(E,1)=(\log(2\pi/\sqrt{N})+\gamma)L(E,1)$$
where $\gamma$ is Euler's ...
- 11.7k
4
votes
1
answer
371
views
Nearly eventually almost periodic functions
Call a function $f: [0, \infty) \to \mathbb R$ nearly eventually almost periodic with period $p > 0$ if for a.e. $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges.
Suppose $f: ...
- 2,039
7
votes
2
answers
957
views
Maximize $L^p$ norm over sphere
For $p \in \mathbb{R}$, consider the function
$$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$
My goal is to maximize this function under the constraints that
$$ \lambda_1^2 +...
- 9,591
2
votes
1
answer
205
views
$L^{2}$ Betti number
Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...
- 61
1
vote
0
answers
152
views
Are all even regular undirected Cayley graphs of Class 1?
Are even order Cayley graphs of Class 1, that is, can they be edge-colored with exactly $m$ colors, where $m$ is the degree of each vertex?
I think yes, because of the symmetry the Cayley graphs ...
- 2,027