All Questions
23,892 questions
0
votes
1
answer
101
views
Limit sequence of regular function in $L_1$‘s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
- 349
0
votes
0
answers
118
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Find the maximum of an expression under the logconcave assumption
Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 105k
49
votes
3
answers
3k
views
What happens if you strip everything but the “between” relation in metric spaces
Given a metric space $(X,d)$ and three points $x,y,z$ in $X$, say that $y$ is between $x$ and $z$ if $d(x,z) = d(x,y) + d(y,z)$, and write $[x,z]$ for the set of points between $x$ and $z$.
Obviously,...
- 32.1k
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
9
votes
1
answer
223
views
$p$-adic analytic pro-$p$ group satisfies a pro-$p$ identity?
Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity
$w$ if for every homomorphism $ f:...
2
votes
1
answer
670
views
Integral on level sets
Let $g_\epsilon : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the ...
CommunityBot
- 1
0
votes
0
answers
121
views
How to find the inverse of this linear integral operator?
Let $f(x): \mathbb{R}^d \rightarrow \mathbb{R}$ be a function that decays ``fastly enough'' at infinity.
We can define the following linear operator
$$L[f](x):= \int_{\mathbb{R}^d} d^d y \, \frac{f(y)}...
- 1
5
votes
0
answers
160
views
Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
14
votes
1
answer
678
views
History of well founded relations
I have changed the title in the hope of attracting the attention of someone who knows about the history of set theory as well as intuitionistic logic:
Who was the first to state the definition of well-...
- 8,481
2
votes
0
answers
37
views
Theta series of well-rounded lattices
I've started looking into well-rounded Euclidean lattices and I was interested in learning whether their theta series have any interesting properties, but haven't found much in terms of bibliography ...
- 223
55
votes
2
answers
3k
views
Is it known? A sum over lattice parallelograms of area one is equal to $\pi$
I recently discovered a formula, my proof is really a high school proof in three lines.
$$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\...
- 5,063
4
votes
1
answer
800
views
Bound the operator norm of the Fréchet derivative of a Lipschitz function in this setting
I want to find a bound for the operator norm of the Fréchet derivative of a Lipschitz continuous function in the following setting:
Let
$E$ be a $\mathbb R$-Banach space;
$v:E\to[1,\infty)$ be ...
CommunityBot
- 1
3
votes
1
answer
290
views
Derived Koszul complex
Let $X$ be a projective variety over $\mathbb{C}$ and $V$ be a vector bundle over $X$. Let $\pi: V\to X$ be the natural projection.
Let $i: X\to V$ be the zero section map. Let $V^\vee$ be the dual ...
- 2,806
3
votes
1
answer
231
views
Are principal parabolic group scheme bundles Zariski locally trivial?
Let $P$ denote a parabolic subgroup scheme of $\operatorname{Sp}(2n;F)$, where $F$ is a field (I am interested in $K=\mathbb{Q}_p$ so possibly okay to assume local with characteristic $0$ if it makes ...
- 777
16
votes
2
answers
590
views
Can you perturb an inscribed polytope so all its edges grow?
Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point.
My question is the following:
Let $P, P'$ be two non-...
- 13.6k
2
votes
1
answer
152
views
Co-locating slowly increasing smooth functions in two different ways
This question is subsequent from my previous one.
I will write everything in detail for the sake of completeness.
Let $g_1$ and $g_2$ be smooth functions on $\mathbb{R}$, whose derivatives of all ...
- 128k
3
votes
0
answers
287
views
What did Mirimanoff say about Intuitionism?
Dmitry Mirimanoff, "L'intuitionisme", Alma Mater n° 6, Geneva, 1945.
Most of Mirimanoff's work was in number theory, but he wrote three papers about set theory that were way ahead of their ...
- 8,481
7
votes
0
answers
160
views
What happens if we add an initial object to a Lawvere theory?
Motivation
There is a theory of smooth spaces (for example, diffeological spaces) as certain sheaves on the site $\mathrm{Cart}$, which is the "category of cartesian spaces" whose objects ...
- 355
3
votes
1
answer
490
views
Space derivative of flow of ODE with monotone source
Consider the ODE
$$
\begin{cases}
\partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\
\Phi(0,x) = x, & x \in \mathbb R
\end{cases}
$$
where $f$ is function which is a non-...
CommunityBot
- 1
2
votes
0
answers
102
views
Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential
Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
- 12.9k
0
votes
1
answer
71
views
Asymptotic expansion inverse discrete Fourier transform
Let $\ell^1(\mathbb{Z})$ be the space of biinfinite sequences $f = (f(n))_{n \in \mathbb{Z}} \subset \mathbb{C}$ such that it is absolutely summable. The discrete Fourier transform or Fourier series ...
- 151
3
votes
1
answer
182
views
Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product
Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing).
For any pair of ...
5
votes
2
answers
256
views
On the closed convex hull of a weakly compact set
Let $H$ be an infinite-dimensional real Hilbert space and let $B$ be the closed unit ball of $H$. Let $K\subset B$ be a weakly compact set whose closed convex hull agrees with $B$. Question: does $K$ ...
- 54.2k
4
votes
1
answer
204
views
Stationary phase formula for a complex valued phase
I'd be interested in computing an asymptotic expansion when $h \rightarrow 0$, of an integral of the form
$$
I_h = \int_{\mathbb{R}}{e^{\frac{i}{h}\varphi(x)}dx}
$$
where $\varphi : \mathbb{R} \...
- 6,367
6
votes
1
answer
168
views
Laplacian is surjective from $\mathcal{C}^{\infty}(B)$ to $\mathcal{C}^{\infty}(B)$
Let $B$ denote the open unit ball in $\mathbb{R}^n$. Let $\mathcal{C}^{\infty}(B)$ represent the space of smooth functions on $B$.
Is the Laplacian operator $\Delta$ surjective as a map from $\mathcal{...
- 16.4k
7
votes
2
answers
284
views
Bounded geometric morphisms, origin and motivation for the terminology
Bounded geometric morphisms serve as a generalization of Grothendieck topoi; $T$ being a Grothendieck topos iff global section $T \rightarrow Set$ is bounded. I managed to only track this down to the ...
- 35.5k
3
votes
0
answers
148
views
Solutions of a quadratic Diophantine equation over algebraic integers
The problem is not very exactly Diophantine in the classical way. I am trying to find some algebraic integer solutions in a number ring to a quadratic equation over the same ring.
Precisely, let $\...
- 287
5
votes
1
answer
250
views
About generalized Springer theory for spin groups
I am interested in the detailed computation of the generalized Springer theory for spin groups (type B or D). In the last sentence in Section 14 of Lusztig's Intersection cohomology complex on a ...
- 12.9k
0
votes
1
answer
119
views
Reference request: hyperfinite cross product
Given a countable essentially free ergodic non-singular group action $G \curvearrowright (X, \mu)$ on a standard measure space, suppose $\mu$ is a non-atomic probability measure and $\alpha: G \...
- 17k
1
vote
1
answer
119
views
Reference request for the isomorphism $H^1(G_{K_v},E)[n]\cong (E(K_v)/nE(K_v))^*$ in the context of Tate-duality
Let $E/K$ be an elliptic curve over a number field $K$. Let $M_K$ be the set of all places of $K$. Let $K_v$ be a completion of $K$ at $v$.
I'm searching for a reference for the statement of the ...
- 8,945
8
votes
2
answers
897
views
Can you do geometry with persistent homology?
Setup
In practice, persistent homology of data $X$ is often used to infer the homology of the underlying (Riemannian) manifold $M$ that the data is sampled from.
However most filtrations (Vietoris, ...
- 363
8
votes
1
answer
390
views
Order bounded version of monotone complete $C^*$-algebras
Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
- 12.6k
7
votes
1
answer
394
views
Inverse limit in the category of $C^{\ast}$-algebras or operator spaces
Does the inverse limits (projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces?
I tried to search but could not find a proper reference. Any reference or comments about ...
CommunityBot
- 1
1
vote
1
answer
73
views
N-soliton, The Lax operator and the transmission coefficient
I'm interested in the soliton stability result given in HERBERT KOCH and DANIEL TATARU's paper
"MULTISOLITONS FOR THE CUBIC NLS IN 1-D AND THEIR
STABILITY", published in IHES. However, I ...
- 1,826
2
votes
0
answers
30
views
An algorithm to decompose a directly indecomposable permutation group into a wreath product
I am considering the following two binary operations on permutation groups:
the direct product, and
the wreath product.
It turns out that there is an efficient algorithm to factor a given ...
- 5,822
4
votes
0
answers
70
views
Permutation matrix in terms of an $\mathfrak{su}(r)$-basis (generalised Gell-Mann matrices)
Let $V \cong \mathbb{C}^r$ be the defining representation of $\mathfrak{su}(r)$. Then the permutation on $V \otimes V$ can be expressed as
$$ P = \frac{1}{r} \, 1 \otimes 1 + \frac{1}{2} \sum_{a=1}^{r^...
- 1,996
1
vote
1
answer
175
views
Original text about Riesel numbers?
Riesel numbers appeared in the answer to my question
Solving $2^{x+1} m -1=p^y$ for prime $p$ and natural $x,y$.
The original reference
Riesel, Hans (1956). "Några stora primtal". Elementa. ...
- 105k
8
votes
4
answers
1k
views
Counting with trees
Let $\mathcal{U}_n$ denote the set of unrooted unlabelled trees with $n$ edges. For $T\in\mathcal{U}_n$, let $1^{u_1}2^{u_2}\cdots n^{u_n}$ be its degree distribution, that is, $u_i=\#$ of vertices ...
- 43.2k
0
votes
0
answers
121
views
Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?
Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
- 7,127
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 ...
- 63.9k
9
votes
4
answers
3k
views
Textbook suggestions for rigorous fluid dynamics
I am interested in studying fluid dynamics and am searching for a good introductory textbook. I know just the very basics of fluids on the physics side. For mathematical prerequisites, I have ...
- 35.5k
16
votes
2
answers
887
views
Mumford–Tate 1962 "Algebraic geometry seminar" citation
In FGA 3.V, there is a citation for
Mumford D. and Tate J., Séminaire de géométrie algébrique, Harvard University, Spring term 1962 (à paraître).
This seems to be the same seminar mentioned by ...
- 101
14
votes
2
answers
852
views
Where is Grothendieck's “Résidus et Dualité: Prénotes pour un Séminaire Hartshorne”?
About the book Residues and Duality, from R. Hartshorne, Giacomo Aiello wrote in a footnote to Grothendieck's bibliography by topic:
This is the famous seminar held by Robin Hartshorne at Harvard in ...
- 101
2
votes
1
answer
215
views
Number of binary matroids of rank $r$ on a ground set with $n$ elements
How many simple binary matroids are there, up to isomorphism, of rank $r$ on an $n$-element ground set, where $r \le n < 2^r$? Write this number as $a_r(n)$. Is there somewhere where I can get this ...
- 331
5
votes
1
answer
183
views
What is a natural interpretation of the commutator of the conditional expectation operator?
Notation: We denote by $\mathbb E_{\mathcal F} X$ the conditional expectation of the random variable $X$ with respect to the $\sigma$-algebra $\mathcal F$.
Given two $\sigma$-algebras $\mathcal G, \...
- 12.9k
1
vote
0
answers
99
views
Shortest loop through vertices of a convex polytope
Let $P$ be a convex polytope in Euclidean space $\mathbf{R}^3$ and $\Gamma$ be a closed curve which passes through all vertices of $P$. How small can the length $L$ of $\Gamma$ be? More specifically, ...
- 7,194
-1
votes
1
answer
168
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 6,215
4
votes
0
answers
125
views
Minimal model for $A_\infty$-categories
Is there a reference for existence and construction of the minimal model of an $A_\infty$-category? Most references I found ultimately refer to Lefèvre-Hasegawa's thesis but there doesn't seem to be a ...
- 489
4
votes
0
answers
80
views
Interpolation-extrapolation scales of H. Amann
I am currently reading H. Amann's notes titled "Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems", and I have a question regarding the abstract ...
- 63.9k