Newest Questions
159,064 questions
1
vote
0
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107
views
Spectral sequence for a truncated semi cosimplicial space
Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...
- 177
8
votes
1
answer
443
views
When can $L$ sets of the form $\{a,b,a+b\}$ partition $\{1,2,\dots, 3L\}$?
Firstly, this question has been posted to Math StackExchange with no complete answer so far.
Consider a set of the form $\{a,b,a+b\}$ where $a$ and $b$ are positive integers with $b > a$. I will ...
0
votes
1
answer
142
views
Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul
In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...
- 43
2
votes
1
answer
329
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Can set theory be interpreted in infinite arithmetic?
Is the following system of infinite arithmetic consistent?
If so, can it interpret $\sf ZFC$?
Language: first order logic
Primitives: $\operatorname{Card}, <, + , \times,\text{^}$
where $\...
- 11.3k
1
vote
0
answers
64
views
Physical measure of a dynamical system in terms of its density
Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
In ergodic theory, the occupation measure is
$$\mu_{x, T}(...
- 247
2
votes
0
answers
123
views
An alternative proof that Buchsbaum rings are generalized Cohen-Macaulay
Let $(R,\mathfrak{m})$ be a Noetherian local ring. $R$ is said to be Buchsbaum if, for each ideal $\mathfrak{q}$ generated by a full system of parameters, the number $\lambda_R(R/\mathfrak{q})-e_{\...
- 253
1
vote
1
answer
264
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Question about the paper "Infinitely many monotone Lagrangian tori in del Pezzo surfaces"
In the paper "Infinitely many monotone Lagrangian tori in del Pezzo surfaces" by Renato Vianna, the author constructs an infinite amount of non-symplectomorphic monotone Lagrangian tori in ...
- 791
17
votes
3
answers
2k
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Are some congruence subgroups better than others?
When I first started studying modular forms, I was told that we can consider any congruence subgroup $\Gamma\subset\operatorname{SL}_2(\mathbb{Z})$ as a level, but very soon the book/lecturer begins ...
- 821
4
votes
1
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131
views
Does the Alexandrov angle define convex functions along geodesics in CAT(0) spaces?
Let $X$ be a CAT(0) space and suppose $a,b,c,d\in X$ satisfy
$$
\max\{\angle_a(b,c),\angle_a(b,d)\}<\frac{\pi}{2}.
$$
Let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)...
- 743
1
vote
0
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131
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Can we construct a general counterexample to support the weak whitney embedding theorm?
The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$.
...
- 109
0
votes
1
answer
122
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Permutation of the natural numbers from operation related to binary expansion of $n$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
Let $T(n,k)$ be a $(k+1)$-th bit from the right side in the binary expansion of $n$. Here
$$
T(n, k) = \left\lfloor\frac{n}{2^k}\right\rfloor \...
- 4,948
1
vote
1
answer
115
views
About the power of numbers primes distribution
Let $r>0$, $p\neq q$ two primes numbers and $A=\{(m,n)\in\mathbb N^2; |p^m-q^n|\leq r\}$.
Is it true that $A$ is a finite set?
- 4,074
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70
views
A differential operator of differential operators
Consider a differential operator of the form $x^2-2\frac{\partial}{\partial x}$, where $x$ itself is the laplacian.
Does such an operator make sense? The motivation for this is that such an object is ...
- 9
5
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0
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293
views
On the deformation theory of associative algebras
Let us start by recalling the notion of a formal deformation:
Let $K$ be a field of characteristic zero and $A$ be an associative $K$-algebra. Consider a commutative augmented $K$-algebra $R$, with ...
- 541
1
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0
answers
69
views
A kind of weak convergence for Sobolev spaces with zero on boundary
Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. Is it true that if $(\varphi_n)_{n\geq 1}\subset C^{\infty}_c(\Omega)$ with $\varphi_n\to \...
- 1,759
9
votes
3
answers
850
views
Epimorphisms of relations
Let $\bf Rel$ be the category whose objects are sets and whose morphisms are relations.
What is an epimorphism in this category?
I have a sufficient condition, which is: $R$ is epic if the associated ...
- 1,083
5
votes
1
answer
713
views
Is there an infinite chain of endofunctors of finite sets?
We consider the category of endofunctors of finite sets with natural transformations.
Is there an infinite chain $F_1, F_2, \ldots$ of endofunctors such that there is a natural transformation from $...
- 185
1
vote
0
answers
245
views
A sort of dual to nondegenerate random variables
I was motivated by this classical puzzle/1992 Putnam problem.
Suppose 4 points are independently and uniformly distributed on a sphere in 3d. What is the probability the tetrahedron they form contains ...
- 646
0
votes
0
answers
73
views
Can we construct general counterexample to support the Weak Whitney theorem? [duplicate]
Can we construct an example for the weak Whitney theorem to illustrate the existence of a continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold that cannot be smoothly ...
- 109
1
vote
0
answers
92
views
Lagrangian Floer theory for negative monotone symplectic manifolds and Lagrangians
In the paper "Floer cohomology of Lagrangian intersections and pseudo-holomorphic Disks I", Oh shows that for a compact monotone Lagrangian $L$ in a closed monotone symplectic manifold $M$ ...
- 791
2
votes
0
answers
165
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A direct proof that every projectivity between parallel lines is affine
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
- 42k
2
votes
0
answers
147
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
1
vote
1
answer
132
views
Completion of $\mathbb F_q(T)$
It is easy to prove that for a an irreducible polynomial $P$ of degree $d$ of $\mathbb F_q[T]$, one can embed $\mathbb F_{q^d}$ in $\mathbb F_q(T)_P$ (the completion of $\mathbb F_q(T)$ at $P$) and ...
- 3,998
12
votes
4
answers
1k
views
Understanding the condition $\frac{1}{p} + \frac{1}{q} = 1$ in the estimate $xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$
I just read a proof of Holder's inequality in measure theory, which boils down to the following inequality:
$$xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$$
where $x,y\ge 0$ and $\frac{1}{p} + \frac{1}{q} = ...
- 7,547
5
votes
0
answers
155
views
A non-trivial (not a concatenation of de Bruijn sequences) infinite binary sequence whose initial $2^{n+1}$ bits contain all $n$-bit words for any $n$
Does there exist an infinite binary sequence $B$ that satisfies all of the following three properties?
It is possible to prove that for any integer $n$ the initial $2^{n+1}$ bits of $B$ contain all $...
- 959
1
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0
answers
126
views
Looking for information on "Pouzet's lemma"
I am looking for information on "Pouzet's lemma," which I learned about in slides of Chris Hartman on Sperner's lemma. I have copied the statement of the result from that site. I would love ...
- 111
1
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0
answers
94
views
Is there a way to linearize matrix quadratic forms?
Say $x$ is a random vector in $\mathbb{R}^n$. Then, given a (deterministic) symmetric real positive definite matrix $A$, if we want to calculate the expectation of the quadratic form, we can use the ...
- 380
1
vote
0
answers
79
views
A general theory of pairings
Bilinear forms and bilinear maps for vector spaces over a field are standard material for an introductory course in linear algebra.
There are also text books for bilinear forms and related quadratic ...
- 231
5
votes
1
answer
210
views
Relation between row space and column space resp. null space and left null space over general rings
Let $R$ be a ring and $M\in\text{Mat}(R,m\times n)$ a matrix for $m,n\in\mathbb{N}$. What results are known about the relation between column space (cs, image) and row space (rs), resp. null space (...
- 231
2
votes
0
answers
110
views
relative entropy, Fisher information, and metric slope for non-convex domains
$\newcommand{\R}{\mathbb R}$ If $\Omega\subset \R^d$ is a convex domain it is well-known that the relative entropy
$$
\mathcal H(\rho)=
\int_{\Omega}\rho\log\rho \ \mathrm{d}x
\qquad \mbox{for }\rho=...
- 5,380
2
votes
2
answers
681
views
Sphere tessellation with congruent regular hexagons except finitely many
Let $S^2$ be the 2 dimensional sphere embedded in $\mathbb{R}^3$. It is well known, using the Euler characteristic, that it is not possible to tessellate $S^2$ with all congruent regular hexagons.
My ...
- 131
1
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0
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102
views
Question on definition of inverse number theoretic transformation
In the paper Porkodi and Arumuganathan - Public key cryptosystem based on number theoretic transforms I found the following statement on the second page regarding the Inverse Number Theoretic ...
- 11
3
votes
0
answers
113
views
On the "Peano phenomenon" in higher dimensions
The following result in one-dimensional differential equations is sometimes referred to as "Peano phenomenon" (see e.g. here).
If $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, the ...
- 63
13
votes
2
answers
1k
views
Optimal search puzzle
Consider the following puzzle: On the integer line from 1 to $t$ (top, let's say 1000 for this example), you have two operators: uniform random on 1 to $t$, and subtract 1. What is the optimal ...
- 265
0
votes
1
answer
210
views
Centers of universal enveloping algebra of complex Lie algebras
Let $\mathfrak{g}$ and $\mathfrak{g'}$ be complex Lie algebras such that $\mathfrak{g}$ is a subalgebra of $\mathfrak{g'}$. Let $Z(\mathfrak{g})$ and $Z(\mathfrak{g'})$ be the centers of the universal ...
- 833
1
vote
0
answers
91
views
Functions that take quadratic residues to non quadratic residues
Let $p$ be prime and $Q$ be the set of integers $x$ mod $p$ so that $x^2-1$ is a quadratic residue. Let $Q^c$ be the complement of $Q$. If we don't consider $x = 1$ then these two sets have the same ...
- 591
5
votes
1
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376
views
How did the term "space" in mathematics started to be understood as a set with a structure?
In mathematical literature, the term 'space' is often used to describe a set endowed with additional structure, such as a metric space or a vector space. What is the historical evolution of the ...
- 879
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0
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263
views
Cosine Modulo $p$?
Consider the integers modulo a prime $p$. I'm looking for a nice polynomial function that acts as a sort of "cosine" on the integers modulo $p$. Specifically, I'm looking for solutions to ...
- 591
1
vote
1
answer
102
views
Normal modal Logic with finite proposition letters
Assume our modal language $L$ has only diamonds, and the set of proposition letters $Prop$ is finite. The deduction rules are the same as normal modal logic. Now consider $M$ is a finite model of this ...
- 11
0
votes
1
answer
166
views
Does this inequality hold for the cumulant generating function?
Suppose a random variable $X$ is zero-mean and the cumulant generating function is
$$
K\left( t \right) =\log \mathbb{E}[e^{tX}].
$$
Given any positive constant $\tau > 0$, does this inequality
$$
\...
- 49
9
votes
1
answer
291
views
A question related to Jordan's theorem on subgroups of $\mathrm{GL}_n(\mathbb{C})$
$\newcommand{\C}{\mathbb{C}}$
$\newcommand{\mr}{\mathrm}$
For any positive integer $n$, let $f(n)$ be the minimal integer with the following
property:
For any finite subgroup $G < \mr{GL}_n(\C)$ ...
- 10.5k
3
votes
1
answer
198
views
Do radially bounded sets form a bornology?
We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, ...
- 5,529
5
votes
1
answer
235
views
Hammock localization and free adjoints
The Hammock localization $L^H \mathcal{C}$ of a relative category $(\mathcal{C},\mathcal
{W})$ is a simplicial category defined by Dwyer and Kan as a way to compute the $\infty$-categorical ...
- 42.4k
7
votes
3
answers
598
views
Are these two representations of $SU(2)$ equivalent?
Consider $\Lambda^2(\mathbb{C}^n)$. It is a vector space of dimension $n$ choose $2$. On the other hand, consider $Sym^2(\mathbb{C}^{n-1})$. It so happens that this is also a vector space of dimension ...
- 5,215
25
votes
1
answer
513
views
Is there an inventory of closed billiard paths in a regular tetrahedron?
Conway found a closed billiard-ball trajectory in a regular tetrahedron:
Image: Izidor Hafner
Since then Bedaride and Rao
Bedaride, Nicolas, and Michael Rao. "Regular simplices and periodic ...
- 151k
4
votes
2
answers
673
views
Determine monodromy representation from local system
Let $X$ be a path-connected manifold nice enough such it's universal covering
space $p:\widetilde{X} \to X$ exists, $k$ a field. Then there exist a wellknown
correspondence
$$
\{\textit{linear}\text{ ...
- 619
4
votes
1
answer
181
views
Denominators of rational polytopes in terms of hyperplane coefficients
Let $\mathcal{P}$ be a convex polytope in $\mathbb{R}^n$ given in the form $\mathcal{P} = \{ x \in \mathbb{R}^n\colon A x\leq b \}$. Suppose that the entries of $A$ and $b$ are integers. Then it is ...
- 24.2k
7
votes
1
answer
322
views
Strength of Borel determinacy
In this blog post by Gowers on Borel determinacy, Andres Caicedo says the following in a comment (slightly rephrased).
Let $\mathsf{ZFC^-}$ be $\mathsf{ZFC}$ without power set and $\mathsf{ZC^-}$ be $...
- 949
1
vote
0
answers
134
views
Matrix valued word embeddings for natural language processing
In natural language processing, an area of machine learning, one would like to represent words as objects that can easily be understood and manipulated using machine learning. A word embedding is a ...
28
votes
1
answer
2k
views
Useful ideas in category theory which violate the principle of equivalence
Or an alternate title: using evil for the greater good.
In category theory, the principle of equivalence says that statements about things should be invariant under the appropriate notion of thing-...