Trending questions
159,032 questions
3
votes
0
answers
78
views
Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$
In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
4
votes
1
answer
283
views
Do the two orientations on an orientable manifold $M$ uniquely witness lifts of $\tau_M: M \to B\text{O($n$)}$ to $B\text{SO($n$)}$?
For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort:
There are two orientations on $M$. Is it ...
- 113
3
votes
0
answers
51
views
Harmonic map in the homotopy class of the identity map
Eells and Sampson's existence Theorem states that if $(N, h)$ is nonpositively curved, then a given map $f : (M, h') \to (N, h)$ can be deformed into a harmonic map in its homotopy class. Here smooth ...
- 1,799
9
votes
1
answer
159
views
Eigenfunctions of the Laplace–Beltrami operator on the coadjoint orbit of $\mathfrak{su}(n)$
$\DeclareMathOperator\SU{SU}$For $\mathfrak{su}(2,\mathbb{C})$, the generic coadjoint orbit is $\mathbb{S}^2$, and the Laplace–Beltrami operator on it is given by
$$
\Delta \equiv \frac{1}{\sin\theta} ...
- 195
13
votes
0
answers
141
views
Chromatic number of unit distance graph of R^n
Consider the graph whose vertices are the elements of $\mathbb{R}^n$ with an edge between $x$ and $y$ iff $d(x,y)=1$. There is an extensive literature on the chromatic numbers $\chi(\mathbb{R}^n)$ (...
- 199
5
votes
1
answer
433
views
Number of roots of a quadratic form over GF(2)
If $Q(x) = x^T A x$ with $x \in GF(2)^n$ and $A \in GF(2)^{n \times n}$, is there a way to find how many roots $Q(x)$ has based on any properties of $A$ (e.g., rank, etc.)?
- 63
1
vote
1
answer
83
views
Solution to a quadratically constrained quadratic program with unit ball constraint
I am working on a quadratically constrained quadratic program (QCQP) of the form:$$ \min_{x} \quad \frac{1}{2} x^T P x + q^T x + r$$
$$ \text{subject to} \qquad x^{T}x \leq 1 $$
where $P \in S^{++}_{...
- 13
-3
votes
0
answers
33
views
Bayesian Inference for Parameters Estimation in ARMA Model [closed]
In the usual sense, Maximum Likelihood Estimation is the common method for Parameter estimation in ARMA(p,q) model.
If I am looking to estimate parameters for ARMA(p,q) with Bayesian Inference, how ...
- 1
9
votes
1
answer
298
views
Connected open sets in the topology generated by the collection of connected open sets
Let $(X,\mathcal{T})$ be a connected topological space. Let $\mathcal{T}'$ be the topology on $X$ that is generated by the collection of connected open sets in $(X,\mathcal{T})$. That is, the ...
0
votes
0
answers
94
views
The ratio $\Omega(n)/\omega(n)$ for a special set of integers
It is known that every positive integer can be expressed as $n=s\cdot q $, where $s$ is a powerful number and $q$ a squarefree number, with
$(s,q)=1$. It is also known that the set of integers such ...
- 471
1
vote
0
answers
62
views
Bipartite Representation of a Directed Graph
I am working on a combinatorial optimization problem and have constructed a bipartite graph as a representation of a directed graph.
The construction is as follows:
Given an initial directed graph $G$ ...
2
votes
1
answer
157
views
Does the support of a smooth compactly supported function have a smooth boundary?
Let $U$ be a bounded domain in $\mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ an infinitely differentiable function with compact support. My question is whether $\operatorname{supp}(f)$ has a smooth ...
11
votes
0
answers
327
views
+50
Sobolev's PDE Scottish Book Problem (Problem 188)
In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution.
In 2015, when the second edition of the Scottish Book with updates and commentary on ...
- 13k
3
votes
1
answer
102
views
Which compact Lie groups have an upper bound on the dimension of irreducible continuous representations?
To fix notation, if $G$ is a compact Lie group, $Rep(G)$ denotes the set of continuous irreducible unitary representations of G, and $\widehat{G}$ denotes the quotient $Rep(G)/\sim$, which identifies ...
3
votes
1
answer
144
views
Jordan plane curve such that $\frac{d(g(x),g(y))}{d(x,y)}\to0$?
Write $g$ as the inverse of $f$.
Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that
$$
\displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0
$$ as $r\to0$?
If you like,...
- 211
1
vote
1
answer
86
views
Sequence of MMP with scaling cannot be isomorphism
Let $(X,B)$ be a projective klt pair, H is an $\mathbb{R}$-divisor s.t. $K_X+B+H$ is nef. Suppose that we can run $(K_X+B)$-MMP with scaling $H$(that is, the flip exists) and denote $\alpha_i:X \...
- 11
5
votes
0
answers
85
views
When the set inert primes of a number field described by congruence relations?
We know that the following theorem:
Theorem: If the set of splitting primes of a number field $F$ can be described by congruence relations. Then $F$ is an abelian extension of $\mathbb{Q}$.
I want to ...
- 125
4
votes
2
answers
165
views
Connectedness of degeneracy loci
Let $E, F$ be vector bundles of ranks $e, f$ on a smooth variety $X$ of dimension $n$.
Let $\varphi : E \to F$ be a morphism and let $k$ be such that $n > (e-k)(f-k)$.
Fulton-Lazarsfeld's theorem ...
- 331
4
votes
2
answers
508
views
Rational independence of higher golden ratios
The higher Golden Ratios $(\phi_n)_{n=0,1,2,3,\dots}$ are the real numbers defined recursively by setting
$\phi_0=0$ and for $n\ge1$,
$\phi_n$ is the positive (real) root of $X^2-\phi_{n-1}X-1$. We ...
- 43
9
votes
1
answer
459
views
A conjecture related to Frankl's conjecture
Let $\mathcal{F}\subseteq2^{[n]},\emptyset\in\mathcal{F}$ be an union-closed family of sets. For $S\in\mathcal{F}$, let $w(S)$ be the number of subsets of $S$ in $\mathcal{F}$. Does there always exist ...
- 1,462
6
votes
1
answer
230
views
Can one define second-order equinumerosity in MSO via first-order cardinality quantifiers?
Let $L$ be MSO (Monadic Second Order Logic) extended with an $n$-ary second-order cardinality predicate (i.e., a first-order cardinality quantifier) $C_R$ for every $n$-ary relation $R$ on the ...
- 2,555
0
votes
0
answers
71
views
Reducible quartic space curve that is set-theoretic complete intersection
$\newcommand\P{\mathbb P} \newcommand\C{\mathbb C}$For a degree 4 curve, the Macaulay quartic curve is the only irreducible space curve (up to isomorphism) for which it remains an open problem to find ...
- 2,275
4
votes
0
answers
73
views
Integration of volume forms over manifolds with corners
Suppose that $M$ is a (compact, oriented, smooth) manifold with corners.
Let $M_{\leq 1}$ the manifold with boundary of all points of index $\leq 1$ (following the notations here). This manifold is an ...
- 1,035
-4
votes
0
answers
144
views
Divisors of n and n + 1
Suppose $a$ is a proper divisor of $n$ (where $n$ is a positive integer), and $b$ a proper divisor of $n + 1$.
Is there a general criterion (or general property of $n$) which enables one to conclude ...
- 4,555
2
votes
1
answer
163
views
Existence of Kähler metric of bounded geometry on the Hermitian vector bundle on projective spaces
$\DeclareMathOperator\Tot{Tot}$A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity ...
- 21
5
votes
1
answer
132
views
Relation between Tor amplitude and $p$-complete Tor amplitude for a ring of characteristic $p$
Fix a prime number $p$. Let $A$ be a commutative ring, and consider an $A$-algbera $B$ of characteristic $p$. So we have a sequence of ring homomorphisms
$$
A \to A/pA \to B.
$$
Assume that we want to ...
- 125
4
votes
0
answers
170
views
Intuition on geometry of sections
Premise: I have asked the same question on math.stackexchange about a week ago, without receiving answer. Therefore I've decided to ask it also here. If this violates the rules, I apologize and I'll ...
3
votes
1
answer
227
views
Compute generators for group of totally positive units of a number field?
Given a number field $K$, I would like to compute (in Sage) generators for the group of totally positive units of $K$.
Update: I've tried some code (details below), which I've received some help on in ...
- 111
13
votes
2
answers
1k
views
What's the deal with De Morgan algebras and Kleene algebras?
The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is ...
- 32.5k
7
votes
1
answer
177
views
Ergodicity of action of finite index subgroups in the boundary
Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}...
- 1,101
-4
votes
2
answers
173
views
Why exactly is Simpson's rule better than the Trapezoidal rule? [closed]
I am reading up on numerical integration and have trouble to really understand why or rather in what sense Simpson's rule is better than the Trapezoidal rule in general. There is a lot of stuff ...
3
votes
0
answers
91
views
References on smoothness of minimal surfaces in Riemannian manifolds
It's well known that $C^1$ minimal surfaces (surfaces that are locally area minimzing) in $\mathbb{R}^n$ are automatically smooth, and one can prove this result by solving the Dirichlet problem of the ...
- 438
17
votes
4
answers
2k
views
What intuitive notion is formalized by condensed mathematics?
Preface: I ask this question from the position of a curious layperson who is excited about new conceptual advances in mathematics.
It is often said that the notion of "topology" formalises ...
- 1,433
5
votes
0
answers
239
views
Classification of principal $\mathrm{SO}(3)$-bundles on a 4-manifold via characteristic classes
I am interested in a reference with a detailed (as simple and topological as possible) proof of the following fact:
Theorem. A principal $\mathrm{SO}(3)$-bundle on a compact oriented 4-manifold are ...
- 6,962
8
votes
0
answers
202
views
Logarithm of a $p$-group in $\mathrm{GL}_n(p)$
$\def\GL{\operatorname{GL}}\def\ZZ{\mathbb{Z}}\def\FF{\mathbb{F}}\def\Id{\mathrm{Id}}\def\fu{\mathfrak{u}}$Let $p$ be prime, let $n<p$, let $U_n(\FF_p)$ be the group of $n \times n$ upper ...
- 156k
-2
votes
0
answers
82
views
Every well-ordered set is isomorphic to an unique ordinal? [closed]
Every well-ordered set $W$ is isomorphic to a unique ordinal
Proof: We start with some well-ordered set $W$. We attempt to construct a bijective map from $W$. Let us consider this class
$$\{(x, \...
- 1
6
votes
0
answers
127
views
Dependence on Urysohn's Lemma in Cartan's Construction of Haar Measure
This question was posted by someone else on stackexchange three months ago, but no one has answered as of yet:
Cartan's 1940 paper, Sur la mesure de Haar, claims to provide a proof of the existence ...
- 85
1
vote
1
answer
154
views
Is smoothness preserved under an isometric isomorphism?
Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
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3
votes
1
answer
160
views
Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
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4
votes
1
answer
218
views
Reference request: Algebras over monoid objects in a monoidal category [duplicate]
Looking for a reference for the following easy-to-prove fact:
Say $T$ and $S$ are monads on $\text{Set}$ admitting a monoid homomorphism $\phi : S \to T$ (i.e., a morphism in $\text{Mon}([\text{Set},\...
4
votes
1
answer
550
views
Question on Lorentzian geometry
I apologize in advance if this is a too basic question.
Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
- 1,171
6
votes
3
answers
752
views
Clarification and Proof of Inequality (8.11) in Analytic Number Theory by Iwaniec and Kowalski
I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski. It is found on top of page 200. In bottom of page 199, the authors prove that
$$
|S_f(N)|^2 \leq N + \frac{2N^2}{q} ...
- 111
1072
votes
296
answers
351k
views
Examples of common false beliefs in mathematics
The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
1
vote
0
answers
70
views
About ratio $\frac{\Omega (n)}{\omega (n)}$
What is the asymptotic estimate of $\sharp\left\{n\leq x,\frac{\Omega (n)}{\omega (n)}> 1+\varepsilon \right\}$, with fixed $\varepsilon > 0$, where $ n=s\cdot q$ ; $s$-a powerfull part of $n$ ...
- 471
2
votes
0
answers
76
views
When is a first-order delay differential equation equivalent to a higher-order ordinary differential equation?
The proportional delay differential equation
$$
xf'(x)+2xf'(x/2)+C+4f(x/2)-5f(x)=0
$$
with initial condition $f(0)=C$ expresses that Simpson's rule exactly integrates $f$ over any interval $[0,x]$ and ...
- 3,065
5
votes
1
answer
378
views
Convergence of random functions
Suppose I have a sequence of random continuous functions, $f^{n} : [0, t] \to \mathbb{R}$. Suppose there also exists a random continuous function, $f: [0, t] \to \mathbb{R}$, defined on the same ...
- 85
1
vote
0
answers
267
views
Are there connections between Calabi-Yau manifolds and number theory?
I am interested in understanding whether there are any significant connections between Calabi-Yau manifolds and number theory. Calabi-Yau manifolds are central objects in algebraic geometry and string ...
9
votes
2
answers
431
views
Hermite–Fourier expansion for the median
Let $n$ be an odd positive integer. Let $M : \mathbb{R}^n \to \mathbb{R}$ be the median function: $M(x_1,\dots,x_n)$ is the median of $x_1,\dots,x_n$. What can be said about the Hermite–Fourier ...
- 24.7k
12
votes
0
answers
342
views
Does every finite group have a small projective representation (over some ring)?
Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
...
- 602
1
vote
0
answers
88
views
Density of a subset of Schwartz space in the fractional Sobolev space
It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$, (where $0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}...