Trending questions
159,066 questions
3
votes
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answer
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views
Descriptive set theoretic complexity of computable maps with respect to the Turing jump of the input
For natural numbers $e$, $n$ and elements of Cantor space $X$ let $\{e\}^X(n)$ be the result of running the $e$th Turing machine with oracle $X$ on input $n$. Let $X'$ be the Turing jump of X.
Suppose ...
- 2,221
0
votes
2
answers
223
views
What is the definition of Tr in the context of Hilbert modular forms?
I am currently reading Garrett's book "Holomorphic Hilbert Modular Forms". But I meet trouble at the starting line. Let $F = \mathbb{Q}(\sqrt D)$ be a real quadratic field, $u= a + b\sqrt{D}\...
- 185
5
votes
1
answer
539
views
Under what circumstances Is a symmetric matrix representable as a Coulomb matrix?
Question:
I am exploring a neural network architecture inspired by physical interactions, where each neuron has associated "mass" and "position" vectors. The weight matrix between ...
- 3,074
6
votes
1
answer
347
views
Is a ball the hardest body to approximate by polytopes (in the Banach–Mazur metric)?
$\DeclareMathOperator\conv{conv}\DeclareMathOperator\Vol{Vol}$In the paper "An extremal property of the hypersphere" by Macbeath, the following functionals were introduced (here $n$ is fixed,...
- 93
5
votes
1
answer
379
views
Why is this Brieskorn manifold a rational homology sphere?
In Némethi's book "Normal surface singularities", Example 5.1.17, there is a formula to find the Seifert invariants of a Brieskorn complete intersection $\Sigma(a_1,...,a_n)$. I am ...
- 197
1
vote
0
answers
29
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Change in active constraints when perturbing the objective of a QP
Suppose I have a quadratic program (with positive semidefinite cost matrix) with affine (polytopic) constraints. It is known that the solution to this is piecewise affine, with the ``pieces'' defined ...
- 11
0
votes
2
answers
116
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Upper bounds on quotients of binomial coefficients
Let $\gamma>1$ be a real number and let $n\in \mathbb{N}$.
Define $f\colon\mathbb{N}\to[0,1]$
$$
f(n_0) = \frac{\binom{n-n_0}{m}}{\binom{n}{m}},
$$
where
$$
m = \Big\lfloor{\frac{n}{\lceil\gamma ...
6
votes
1
answer
856
views
Is integration semi-algebraic?
I am learning a bit of semi-algebraic geometry and I have looked into a bunch of examples of functions which are semi-algebraic. In particular I have tried to understand whether the function
$$ F: (0,\...
- 1,045
3
votes
0
answers
96
views
Commutator of $A\otimes I$ and $I \otimes B$ vanishes?
Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
- 31
2
votes
1
answer
103
views
LCH spaces $X$ such that if $Y$ is a perfect image of $X$, then $Y$ is zero-dimensional
I am looking for locally compact Hausdorff spaces $X$ with the following property:
If $f:X\to Y$ is a perfect map onto locally compact Hausdorff space $Y$, then $Y$ is zero-dimensional.
One can see ...
- 1,211
7
votes
1
answer
297
views
Unoriented cobordism of oriented manifold
We can regard an oriented manifold as an unoriented manifold by forgetting the orientation. This gives a homomorphism from the oriented cobordism group to the unoriented cobordism group. What is the ...
- 73
2
votes
0
answers
83
views
The support of the functions in the closed span of the Rademacher functions in $L_1(0,1)$
Given a measurable function $f:(0,1)\to \mathbb{R}$, we denote by $M(f)$ the measure of the set $\{t\in (0,1) : f(t)\neq 0\}$.
It is not difficult to prove that if $(f_n)$ is a normalized sequence in $...
- 4,461
0
votes
0
answers
98
views
Does the smooth locus of any toric variety built from a fan always contain a rational point?
Let $k$ be an arbitrary field and $X$ be a toric variety built from a fan, defined over $k$.
Does the smooth locus of $X$ always contain a $k$-rational point? Why?
- 639
5
votes
1
answer
249
views
Integral closure in characteristic 0
Let A be a Noetherian domain of characteristic 0, K be its field of fractions. Is the integral closure of A in K always finitely generated as A-module?
- 51
3
votes
1
answer
117
views
Sobolev inequalities and Wiener algebra
It follows from the Gagliardo-Nirenberg inequality that for a locally integrable function $f$ defined on $\mathbb R^d$ (we assume $d\ge 3$)
such that $\nabla f$ belongs to $L^2(\mathbb R^d)$ and
$$
\...
- 16.2k
1
vote
0
answers
58
views
Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 305
6
votes
1
answer
859
views
How many Fourier coefficients vanish?
Let $G$ be a compact abelian connected metric group with Haar measure $\mu$ and let $f\colon G\to S^1$(=unit circle in $\mathbb{C}$) be a continuous function (not necessarily a group homomorphism) ...
- 3,031
3
votes
0
answers
91
views
Are the reductions of the cuspidal characters of GL2(Fq) distinct?
Let $p$ be an odd prime and $q=p^n$ for some $n \geq 1$. If $\mathbb{F}_q$ is the unique, up to isomorphism, finite field with $q$ elements then the cuspidal representations of the group $\rm{GL}_2(\...
- 117
5
votes
0
answers
234
views
Avoiding Cartan subalgebra in a Lie algebra
Let $G$ be a simple complex algebraic group acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation.
What is the largest integer $d$ such that every subspace $U \subseteq \mathfrak{g}$ ...
- 269
2
votes
3
answers
184
views
Existence and sharpness of Bernstein-type bounds on the moment-generating function
Let $X$ be a centred random variable with variance $\sigma^2$, and whose moment-generating function exists in an open neighbourhood of the origin.
Say that $X$ satisfies a 'Bernstein-type' MGF bound ...
- 801
7
votes
1
answer
339
views
Is $C(X, \{0,1\})$ locally compact?
Let $X$ be a locally compact Hausdorff space. Let $C(X, \{0,1\})$ be the space of continuous functions $X \to \{0,1\}$ with the compact-open topology, that is, the topology generated by the following ...
- 1,485
0
votes
1
answer
96
views
A question on finite Fourier series
Let $\mathcal F(N)$ denote the space of finite Fourier series up to frequency $N > 0$, i.e. $f\in \mathcal F(N)$ if and only if it can be written as
$$f(x) = \sum_{k=0}^N a_k\cos(kx+\theta_k)$$
for ...
- 71
3
votes
1
answer
112
views
Generalization of a result of Kostant related to Gauss decomposition and Toda lattices
I found myself needing a generalization of a result of Kostant in his famous paper
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math, Volume 34, 1979, ...
- 173
0
votes
0
answers
190
views
About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
4
votes
4
answers
2k
views
I want a smooth orthogonalization process
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 155
1
vote
0
answers
153
views
Does it make sense to take a formal power series over a non-ring? [closed]
Suppose we have the set of nonnegative integers. This is obviously not a ring, and not even a group. But are we able to identify ring-like properties for it if we take a formal power series over it?
...
- 29
5
votes
0
answers
139
views
Cone avoidance and $\Pi^0_1$-classes
Suppose $X \subseteq 2^{\omega}$ is nonempty and $\Pi^0_1$ relative to $a$. Assume $c_0 \nleq_T b_0 \oplus a$ and $c_1 \nleq_T b_1 \oplus a$. Must there exist some $y \in X$ such that $c_i \nleq_T ...
- 51
0
votes
1
answer
51
views
Reconstruction of law of diffusion process from call option values
Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the
$$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$
Then, ...
- 5,405
3
votes
1
answer
420
views
Riemannian manifold with two geodesics
If any two dinstict points in a complete Riemannian manfiold can only be joined by two different geodesics, is the Riemannian manifold isometric to round sphere?
- 51
3
votes
0
answers
161
views
First time reviewing an applied mathematics paper: how to evaluate it?
I am in the pure math camp but was invited to referee an applied/interdisciplinary paper because I'm a specialist in the underlying mathematical tool. I want to ask for general guidance about ...
- 565
12
votes
1
answer
816
views
Do linear groups over a commutative ring satisfy the Tits alternative?
A group $G$ is said to satisfy the Tits alternative if any finitely generated subgroup of $G$ is either virtually solvable or contains a nonabelian free subgroup. Tits proved this for linear groups ...
- 863
34
votes
7
answers
3k
views
A hat puzzle question—how to prove the standard solution is optimal?
I am currently writing an essay on hat puzzles, and for the warm-up section I introduce some of the standard finite hat puzzles. One of these proceeds as follows:
You and two friends are each given a ...
- 236k
6
votes
1
answer
189
views
Fully faithful embeddings of derived category of projective space into derived category of a higher dimensional projective space
Let $N>n$. Are there are any known cases of a fully faithful embedding $D^b(\mathbb{P}^n) \hookrightarrow D^b(\mathbb{P}^N)$?
- 123
3
votes
0
answers
196
views
What is the trace of the category of divisibility posets?
$\newcommand{\Tr}{\mathrm{Tr}}$Lately I've been trying to gather more examples of centres and traces, hoping to write a comprehensive treatment on those on Clowder.
One of the examples I've been ...
- 11.8k
10
votes
5
answers
999
views
Integral of $\log|e^{it}-1|$
Does there exist an elegant proof of
$$
\int_0^{2\pi}\log|e^{it}-1|\,dt=0 \ ? \label{1}\tag1
$$
Of course, one can do some $\varepsilon$-$\delta$ stuff and get it, but I look for a nice proof. In the ...
- 1,294
0
votes
1
answer
72
views
The asymptotic at infinity of ODE
It may be a simple problem in ODE.
Let $u$ be the positiv solution to
$$u''(t)-f(u,t)u(t)=0, u(0)=1, \lim_{t\to+\infty}=0,$$
with $f>0$ and $\lim_{t\to+\infty}f(u(t),t)=1$.
Can we prove that there ...
- 705
236
votes
36
answers
35k
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Conway's lesser-known results
John Horton Conway is known for many achievements:
Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger $15$-...
8
votes
2
answers
382
views
Suppose that $G$ is a finite subgroup of ${\rm SO}(3)$. Is there a *smooth* self-map of ${\bf R}^3$ whose fibers are precisely the orbits of $G$?
$\newcommand\R{\mathbf R}\DeclareMathOperator\SO{SO}\newcommand\C{\mathbf C}$The question is motivated by the theory of orbifolds. If $\mathcal O$ is an orientable $3$-orbifold (without boundary), an ...
- 83
6
votes
1
answer
407
views
Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 2,907
8
votes
1
answer
367
views
Why do we have fewer distinct Gauss sums over a field of characteristic $2$?
Let $p$ be a prime number and $q$ be a power of $p$. Fix a non-trivial additive character $\psi\colon \mathbb{F}_q\to \mathbb{C}^\times$ and, for each non-trivial multiplicative character $\chi\colon \...
- 773
3
votes
1
answer
257
views
Reflections on affine quadric hypersurfaces
Let $f\colon\mathbb{Z}^n\otimes\mathbb{Z}^n\to\mathbb{Z}$ be a non-degenerate symmetric bilinear form and consider the affine quadric hypersurface $$
X:=\{f(x,x)+2=0\}\subseteq\mathbb{Z}^n.
$$
For ...
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10
votes
5
answers
742
views
Dissection proof of Heron's formula?
In his recent book, Love Triangle, Matt Parker playfully complains that Heron's formula is an "opaque formula, and I feel like you just chuck in the side-lengths, turn a series of arbitrary ...
- 82.7k
0
votes
0
answers
88
views
Is there a name for "applying linear operations to vector sequences from the right"?
Let $v_1,...,v_n\in\Bbb R^d$ be a sequence of vectors. When we say that we "linearly transform" this sequence, we mean that we apply a linear transformation $T\in\Bbb R^{d\times d}$ to each ...
- 13.6k
3
votes
0
answers
126
views
Parametrization of indecomposable modules via quiver varieties
Let $k$ be an algebraically closed field, $Q$ a quiver without oriented cycles and $m^\alpha (Q)$ the variety of quiver representations with dimension vector $\alpha$. There is a canonical algebraic ...
- 696
1
vote
1
answer
73
views
"Gray code" for $[\omega]^{<\omega}$
Let $\newcommand{\oo}{[\omega]^{<\omega}}\oo$ denote the collection of finite subsets of the set of non-negative integers $\newcommand{\o}{\omega}\o$.
If $A,B$ are any sets, let $A \,\triangle \, B ...
- 48.9k
1
vote
1
answer
135
views
A distributive identity for products of partition functions
An $r$-composition of a non-negative integer $s$ is an expression $s=s_1+s_2+\cdots+s_r$ where the $s_i$ are also non-negative integers. Define $k(r,s):=\sum \pi(s_1)\pi(s_2) \cdots \pi(s_r)$ where ...
- 161
0
votes
1
answer
99
views
Is every subgroup closed in this complete, nondiscrete topological group?
Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
- 361
0
votes
1
answer
99
views
Only special permutations result in a constant expression when permuting coefficients in a sum involving binomials?
Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$.
Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that
$$
\sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}.
$$
Must it be ...
- 15.8k
4
votes
2
answers
381
views
Gibbs measure as stationary distribution of SDEs
I have been trying to understand how one can mathematically explain some of the results from statistical mechanics, especially regarding certain distributions like the Gibbs distribution. It would be ...
- 807
0
votes
1
answer
53
views
Exponentially weighted norms are not equivalent
Let $\|u\|^2_{L^2_\eta}$ be the exponentially weighted norm of the space of functions $u(x)$ for which $u(x)\mathrm{e}^{\eta\cdot x}$ with $\eta\in \mathbb{R}$ is in $L^2(\mathbb{R})$. How can I show ...
