Newest Questions
159,035 questions
7
votes
1
answer
178
views
Comonoid homomorphisms in the bicategory of profunctors
Cartesian bicategories axiomatize the intuitively evident but mathematically elusive "cartesian" product on bicategories such as Rel, Span, and Prof. An important concept for cartesian ...
- 171
3
votes
2
answers
387
views
Definition of weak conditional convergence of random variables
I am looking for a definition of conditional convergence. Suppose that $X_1, X_2, \dots, X_n$ are $\mathbb R$-valued random variables with finite second moments, and $W_1, W_2, \dots, W_n$ are iid $\...
- 103
-1
votes
1
answer
466
views
Is there a single function describing this piecewise function?
I am examining the piecewise function given by the following. $f(x) = \pi - \arctan(\frac{2x}{1-x^2})$ when $0 \leq x < 1$, $f(x) = \frac{\pi}{2}$ when $x=1$, and $f(x) = \arctan(\frac{2x}{x^2-1})$ ...
- 1,109
2
votes
0
answers
89
views
On dense subspaces of $L^p$-spaces of finitely additive measures
Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for ...
- 557
12
votes
1
answer
525
views
Is the Tukey order well-founded
Consider the Tukey order restricted to directed orders of the form $(U,\supseteq)$, where $U$ is an ultrafilter on $\omega$. It is defined as follows:
For two ultrafilters $U,W$ on $\omega$, we say ...
- 123
5
votes
1
answer
262
views
Radicands of square roots of the 2020s, written in simplest radical form
As of the time of writing, the current decade is the 2020s. An interesting property of this decade is that there are 3 years that satisfy the property that the square-free part (https://oeis.org/...
- 425
7
votes
1
answer
356
views
High dimensional Fekete's subadditive lemma: does $|\vec x_{n+m}|\leq |\vec x_n+\vec x_m|$ imply the convergence of $\{\vec x_n/n\}$?
Let $d\geq 1$ be a positive integer. If $\{\vec x_n\}_{n=1}^\infty$ is a sequence of $d$-dimensional vectors satisfying $$\lvert\vec x_{n+m}\rvert\leq \lvert\vec x_n+\vec x_m\rvert\qquad \text{for all ...
- 517
8
votes
1
answer
413
views
Precipitous ideal and inner model
Assume $\kappa$ is measurable, $U$ is its unique normal measure, $V=L[U]$. We levy collapse $\kappa$ to make it become $\omega_1$.
If we don't have the inner model condition, then we only know that $\...
- 1,490
3
votes
0
answers
227
views
Classifying spaces beyond CW complexes
We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...
- 803
1
vote
1
answer
75
views
Number of intersection points of a system of quartic polynomials induced by the projected matrix commutator
Let $A, B \in \mathbb{R}^{d \times d}$ denote two symmetric positive definite matrices. I am interested in solutions $V_r \in \mathbb{R}^{d \times r}, 1 < r < d$ to the system of quartic ...
- 11
3
votes
0
answers
69
views
Example of a factorisation of functors $F = HK$ for which the Kan extension of $F$ along $K$ is not $H$ [closed]
I was reading Emily Riehl's book: Categorical Homotopy theory, and I encountered exercise 1.1.3:
Exercise 1.1.3: Construct a toy example to illustrate that if $F$ factors through $K$ along some ...
2
votes
0
answers
77
views
G-modules vs. $\Delta(NG)$-modules
Let X be a simplicial set. Its category of simplices, denoted by $\Delta(X)$, is the category whose objects are the pairs $(x,[n])$, with $x\in X_n$, and morphisms $\bar{c}:(y,[m])\to (x,[n])$, where $...
- 245
0
votes
0
answers
131
views
Can a non-separable C$^*$ algebra have separable GNS Hilbert space
Suppose we have a $C^*$ algebra $\mathfrak{U}$ that is non-separable. Consider a state $ω$ of $\mathfrak{U}$ and the GNS representation $(H_ω,π_ω,Ω)$. Is it possible for $H_ω$ to be separable, and if ...
- 151
1
vote
1
answer
611
views
The Krull dimension of the tensor product of rings
The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...
- 35
6
votes
0
answers
145
views
A function whose derivatives belong to $BMO(\mathbb{R}^n)$
I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert
s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp.
539-558. In this paper he defines ...
- 411
8
votes
1
answer
881
views
Examples of ZBMath reviews that motivated you to read the paper
This is community wiki question.
I will be writing my first review for ZBMath. I would like to take some suggestion through examples.
In general, abstract is too small and introduction is too lengthy ...
2
votes
1
answer
85
views
Jacobi fields in singular metric on quotient space
Consider the square $\Omega = (0,\pi) \times (0,\pi/2) \ni (r,\theta)$ endowed with the Riemannian metric
\begin{equation}
f^2 \big(\mathrm{d} r^2 + \sin^2(r) \, \mathrm{d} \theta^2 \big),
\end{...
- 5,048
4
votes
0
answers
394
views
Are there integers $x,y,z$ such that $(x+1)y^2-xz^2=x^3+2x+2$?
Is equation
$$
(x+1)y^2-xz^2=x^3+2x+2
$$
solvable in integers?
Motivation: For a polynomial $P$ consisting of $k$ monomials of degrees $d_1,\dots,d_k$ and integer coefficients $a_1,\dots,a_k$, define ...
- 7,182
1
vote
0
answers
113
views
MMP for surfaces over a curve where the geometric generic fiber is a rational curve
I am looking for an explaination or an reference for the following fact:
Let $\pi:X\rightarrow Z$ be a contraction from a smooth surface $X$ to a curve $Z$. Assume that the geometric generic fiber of ...
- 141
25
votes
2
answers
3k
views
What is the minimal density of a set A such that A+A = N?
Thinking about the four square theorem and related questions, I found myself wondering: What is the minimal density of a set $A \subset \{0, 1, 2, ... \}$ such that $A + A = \mathbb{N}$?
What I know:
...
- 1,643
5
votes
0
answers
136
views
When does an iteration not add functions $\eta\to V$ at the final stage?
I am interested in better understanding the following property:
Let us say that an iteration of forcings $\langle\mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma,\beta<\gamma\rangle$ is ...
- 2,206
2
votes
1
answer
423
views
Simple proof for convexity of a real valued matrix function
I am looking for a simple and short proof showing that $X \to \|X X^\top\|_F^2$ is a convex function where $\|\cdot\|_F$ is the Frobenius norm. I have one proof by showing that the derivative is ...
- 407
4
votes
1
answer
427
views
Inequality of inclusion-exclusion term
This question was initially posted on math.stackexchange.com but did not receive any answers for half a week.
While analyzing the properties of an algorithm I am working on (I'm a computer scientist), ...
- 45
5
votes
1
answer
351
views
Modularity of the Drinfeld center of the category of G-graded vector spaces
Background: Let $G$ be a finite group, and $\mathrm{Vect}_G$ be the category of finite dimensional $G$-graded vector spaces over some algebraically closed field $k$ of char 0. It is well-known that $\...
- 173
1
vote
1
answer
92
views
Existence of a strongly regular vertex ordering on cubic graphs
Definition: Let $G=(V,E)$ be a cubic (i.e. $3$-regular) graph, and $<$ a total order on $V$. For $v\in V$ let $v^\downarrow$ denote the set of nodes $w\in V$ such that $w<v$, and let $\alpha(v) =...
- 191
3
votes
0
answers
61
views
Are two homotopic principal bundles isomorphic?
Let $E_1 \to B$ and $E_2 \to B$ be two principal $G$-bundles, where $E_1$ and $E_2$ are two simply-connected manifolds and $G$ is a compact Lie group.
Suppose there exists a $G$-equivariant continuous ...
- 891
0
votes
1
answer
457
views
How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets
How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind ...
- 19
6
votes
1
answer
255
views
Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?
It's well-known that complex cobordism $MU^\ast$ is universal among complex-oriented associative, graded-commutative cohomology theories $E$. This means that if $E$ is a multiplicative cohomology ...
- 64k
2
votes
0
answers
95
views
Self adjoint operators from energy functionals
It is known that the equation
$$
\Delta f = 0
$$
on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet ...
- 115
0
votes
0
answers
66
views
Meromorphic functions converging in measure
Let $f_1, f_2, \ldots$, and $g$ be measurable complex-valued functions on the open unit disk. We say that the sequence $f_1, f_2, \ldots$ converges in measure to $g$ if, for all $\epsilon, \mu >0$, ...
- 974
3
votes
1
answer
146
views
Does every graph admit an embedding such that identically-colored edges do not cross?
Given a graph, is it always possible to color the edges of the graph using two colors such that there exists an embedding of the graph in the plane where only opposite-colored edges cross?
Simple ...
- 133
4
votes
2
answers
222
views
Does $\mathsf{ZF}$ prove $\operatorname{Col}(\lambda,\kappa)$ preserves cardinals below $\lambda$?
Let $\lambda<\kappa$ be cardinals and consider the forcing $\operatorname{Col}(\lambda,\kappa)$ adding a generic surjection $\lambda\to\kappa$. More formally, $\operatorname{Col}(\lambda,\kappa)$ ...
- 3,042
15
votes
1
answer
1k
views
Recognizing free groups
While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...
- 1,174
0
votes
1
answer
96
views
An example of module which is square-free, CS, NOT C3, and NOT nonsingular
Let $M$ be a right $R$-module ($R$ has unity). Recall that $M$ is called square-free if $M$ does not contain two nonzero isomorphic submodules with zero intersection. $M$ is called CS if every ...
- 324
2
votes
1
answer
238
views
Expected norm of a product of Gaussian matrices
Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof?
$$E[\|C_n\|_F^2]=d^{n+1}$$
This ...
- 2,252
4
votes
0
answers
174
views
Do the nearby cycle and Beilinson's vanishing cycle functors commute?
Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...
- 1,071
2
votes
1
answer
225
views
Is this theorem true in the case of a general measure space?
I'd would like to confirm if the following proposition is indeed true in the case of an arbitrary measure space.
Theorem: Let $(X,\Sigma,\mu)$ be a measure space and $\{f_n\}_{n\in\mathbb{N}}\...
- 649
2
votes
0
answers
154
views
Existence of a hyper plane
I am very new to algebraic geometry, and self-studying varieties. I have the following question.
Suppose $Y$ is a variety of dimension $r$ and degree $d>1$ in $\mathbb{P}^n$. Let $P$ be a ...
- 613
8
votes
2
answers
1k
views
A conceptual proof that bounded index subgroups of a bounded torsion abelian group contain bounded index complemented subgroups
Call an abelian group $G = (G,+)$ $m$-torsion for some natural number $m$ if one has $m \cdot x = 0$ for all $x \in G$. A subgroup $H$ of $G$ is said to be complemented if one can write $G = H \oplus ...
- 114k
14
votes
1
answer
572
views
How exactly are realizability and the Curry-Howard correspondence related?
Consider, on the one hand:
the Curry-Howard correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$-calculus, and on the other, propositions and ...
- 32.5k
2
votes
1
answer
237
views
On spectral calculus and commutation of operators
Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
- 1,171
1
vote
0
answers
227
views
Deformations over $A_{\inf}$
Setup:
Let $K$ be a perfectoid field of characteristic $0$ with tilt $K^{\flat}$.
Let $A_{\inf}=W(\mathcal{O}_{K^{\flat}})$ be the infinitesimal period ring.
Let $\mathcal{X}$ be a flat, projective $\...
- 282
2
votes
0
answers
69
views
Can the Weyl algebra be free over its invariant subalgebra?
Let $k$ be an algebraically closed field of zero characteristic, let $P_n$ denote the polynomial algebra in $n$ indeterminates, and let $G$ be a finite group of linear automorphisms. Then, by ...
- 3,318
15
votes
1
answer
738
views
Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$
Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
- 317
-2
votes
1
answer
112
views
How to prove that increasing the number of constant symbols of a first-order logic by the number of formulas keeps the number of formulas the same [closed]
Let $S$ be a set of theory symbols for a first-order logic, and let $C$ be a set of constant symbols in $S$ such that $|C| = |L(S)|$, where $L(S)$ is the set of all formulas generated by $S$ in the ...
- 1
1
vote
0
answers
208
views
Are there integers $x,y,z$ such that $1 + x - x^3 + x^2 y^2 + z + z^2 = 0$?
In my previous question Can you solve the listed smallest open Diophantine equations? I discuss the smallest equations (in some well-defined sense) for which it is not known whether they have any ...
- 7,182
3
votes
1
answer
255
views
Is this constraint convex?
I have an optimization problem where the following constraint causes DCP Rule Error.
$$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\...
- 31
4
votes
0
answers
78
views
Pfaffian elements and anomalies
If $X$ is a compact even dimensional spin manifold, then we have a family of chiral Dirac operators parametrized by $Met(X)$, the (infinite dimensional) manifold of Riemannian metrics on $X$. This is ...
- 6,777
2
votes
0
answers
102
views
Orthogonal representation of free products of two groups
Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...
- 301
0
votes
0
answers
209
views
Research directions related to the Hilbert-Smith conjecture
The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...
- 139