Newest Questions
166,651 questions
2
votes
0
answers
36
views
What are the terms in the Eagon Northcott complex?
Let $E \to F$ be a map of vector bundles on a scheme $X$ of ranks $e, f$ (actually, I hope $X$ may be a stack here). Suppose $e \leq f$. I want to describe the locus $D \subseteq X$ where $E \to F$ ...
- 1,154
4
votes
0
answers
37
views
Examples of cocomplete categories without equalizers
What are some examples of cocomplete categories without equalizers? And what are some examples of cocomplete categories without binary products?
They must exist, but at the moment I don't know any. Of ...
- 66.2k
0
votes
0
answers
30
views
Interpretations of the tangent Chern numbers of the complex projective spaces $CP^n$?
I've identified the generating functions for the tangent Chern numbers of the complex projective spaces $CP^n$ given in "Algebraic topology of the Lagrange inversion" by Victor Buchstaber ...
- 11.2k
1
vote
0
answers
56
views
Does a continuous variation through compact $n$ dim manifolds preserve topology?
I begin by writing the definition below that tries to capture what a continuous family /path of manifolds is. The underlying motivation behind the definition is that the transition maps should be ...
- 1,441
2
votes
0
answers
52
views
Triangulated manifolds with local orientation reversal
For a (smoothly) triangulated $n$ manifold $M$, I'll say that the triangulation is amphichiral if it admits an orientation-reversing automorphism. I'll say that the triangulation is locally ...
- 239
0
votes
0
answers
37
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Explosion of the moment of double stochastic exponential
We consider the stochastic system
$$\frac{dS_t}{S_t}=-R_t\,dW_t,$$
with
$$dR_t=-R_t\,dt-R_t\,dW_t, \quad R_0>0.$$
We conjecture, and would like to show that
$$\mathbb{E}[S_t^2]
=
S_0^2\,\mathbb{E}\...
- 696
14
votes
2
answers
836
views
Was Fermat's Last Theorem known for infinitely many primes before Wiles?
Before Andrew Wiles's 1997 proof of Fermat's Last Theorem, in 1985, Étienne Fouvry et al. proved that the first case of FLT holds for infinitely many primes $p$.
Is there any infinite class of primes ...
- 1,125
0
votes
0
answers
84
views
System of divergence free vector fields
Let $\Omega \subset \mathbb R^p$ be a convex, bounded domain with a smooth boundary. Let $a_{ij} : \Omega \to \mathbb R_+$ be a non-negative smooth function for $i, j \in \{1, 2\}.$ I am interested in ...
- 105
0
votes
0
answers
22
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Let G be a finite group such that for every natural n the number of solutions to x^n=e is smaller/equal to n. Prove G is cyclic [migrated]
I received this question as homework for a graduate-level course. I don't want the full answer, just a hint on how to proceed with my current direction.
I reduced the problem to the following case: ...
- 11
1
vote
0
answers
38
views
Number of connected acyclic gentle tree algebras
Let $X_n$ denote the number of acyclic connected gentle tree algebras (given by quiver and admissible relations over a field) with $n$ simple modules. Those are also exactly the connected quiver ...
- 28.5k
2
votes
1
answer
176
views
Identity using $\varphi(n), d(n)$ and $\sigma(n)$
Let
$\varphi(n)$ be the Euler totient function.
$d(n)$ be the number of divisors of $n$.
$\sigma(n)$ be the sum of the divisors of $n$.
$a(n)$ be A344598, i.e., an integer sequence such that $$ a(n) =...
5
votes
0
answers
82
views
Constructive status of the meta-theory of intuitionistic propositional logic
Intuitionistic propositional logic has several kinds of models. Bezhanishvili and Holliday [1] showed that these models form a neat hierarchy:
Kripke
Beth
Topological
Dragalin
Heyting
in the order ...
- 150
0
votes
0
answers
25
views
A problem about the vector-valued error estimate for multivariate Hermite interpolation
I am reading an Hermite interpolation method on manifold which is in Section4 in HERE, the core idea is as follows.
Set $dim(\mathcal{M})=m$. We construct an interpolation $\hat{f}_{\tan }: \mathbb{R}^...
- 1,051
-2
votes
0
answers
78
views
Why do continuous analytic tools fail against discrete arithmetic congruences? [closed]
I am investigating the $p$-adic analytic continuation of the discrete sum $\sum_{i=1}^n i^d \lfloor i^{p^k}/p^k \rfloor$. By writing the sum as $\sum_{m = 0}^n a_m\binom{n}{m}$, we obtain its explicit ...
- 579
-6
votes
0
answers
99
views
Is this proof that $A^3+B^3=C^n$ has no primitive solutions correct? [closed]
I am an independent researcher. This arose in the context of studying the Beal conjecture.
Setup: Factor $A^3+B^3=(A+B)(A^2-AB+B^2)=C^n$. For coprime $A,B$: $\gcd(A+B, A^2-AB+B^2)$ divides $3$. This ...
5
votes
1
answer
268
views
Books on complex dynamics that discuss polynomial mating?
Note that I’ve asked this question on mathstackexchange last month. I am looking for book recommendations on complex dynamics that include discussion of polynomial mating.
Ideally, the book would ...
- 51
10
votes
1
answer
221
views
Does the category of free abelian groups have sequential colimits?
Does the category $\mathbf{FreeAb}$ of free abelian groups have sequential colimits? I assume that the answer is No, but what is an explicit sequence of free abelian groups and homomorphisms
$$A_0 \to ...
- 66.2k
5
votes
1
answer
282
views
An intriguingly simple integral functional for star-shaped, planar, simple, closed, smooth curves
The following is a question that popped up in my research in geometric analysis some time ago and that I dropped and kept coming back to multiple times. I will first state the problem, or rather my ...
- 159
7
votes
1
answer
209
views
Non-constant maps between $T_2$-spaces
Are there $T_2$-spaces $X, Y$, each having more than $1$ point, such that every continuous map $f:X\to Y$ is constant, and every continuous map $g:Y\to X$ is constant?
1
vote
0
answers
40
views
Tail log-convexity of moments of an even Hermite polynomial of a Gaussian
Let $G\sim N(0,1)$ and let $\{\mathrm{He}_n\}_{n\ge 0}$ denote the probabilists' Hermite polynomials.
Let $H_n:=\mathrm{He}_n/\sqrt{n!}$ be the orthonormal version, so that
$\mathbb{E}[H_n(G)H_m(G)]=\...
- 73
7
votes
3
answers
292
views
$\mathbf{CRing}$ has no regular subobject classifier, right?
A regular subobject classifier in a category with finite limits is a morphism $1 \to \Omega$ such that every regular monomorphism $Y \hookrightarrow X$ is the pullback of $1 \to \Omega$ along some ...
- 66.2k
7
votes
1
answer
522
views
A basis-free formula for the determinant as a polynomial
Let $V$ be an $n$-dimensional $\mathbb{K}$-vector space. By a simple calculus trick (*) on homogeneous functions of degree $n$ the determinant is a linear map on the $n$-th symmetric power of the ...
- 1,248
6
votes
1
answer
231
views
Reference request: Moduli space of immersed polygons
I am looking for references describing the structure of the set of immersions of the disk $D^2$ in the Euclidean plane $\mathbb{R}^2$ or in the hyperbolic plane $\mathbb{H}^2$ such that the boundary ...
- 687
2
votes
0
answers
43
views
Non-admissible variation mixed Hodge structure
As is well-known, admissiblity plays an important role in mixed Hodge theory. I am wondering if there are some explicit examples of variation of mixed Hodge structure on a smooth open variety, even an ...
- 113
4
votes
0
answers
52
views
Structure of residually nilpotent groups with nilpotent quotients
Let $G$ be a finitely generated, residually finite, and residually nilpotent group. Suppose $G$ satisfies the following properties:
Every proper quotient of $G$ is virtually nilpotent with Hirsch ...
- 1,219
3
votes
0
answers
65
views
Computing all graph homomorphisms of two graphs
Is there any software that, given two graphs $G$ and $H$, can compute all graph homomorphisms from $G$ to $H$?
I found this rather old question, but it does not seem to answer my query.
It could be ...
- 1,434
7
votes
0
answers
69
views
Does the category of lucid presheaves form a cocompletion under a class of colimits?
Following Freyd in Several new concepts: Lucid and concordant functors, pre-limits, pre-completeness, the continuous and concordant completions of categories, call a presheaf $P : \mathbf A^{\text{op}}...
- 12.9k
5
votes
0
answers
88
views
Existence of minimizer of autoconvolution inequality
I was reading E. P. White's paper An optimal $L^2$ autoconvolution inequality, which is about $\text{inf}_{f\in \mathcal{F}}\|f*f\|^2$, where $\mathcal{F}$ denotes the family of nonnegative functions $...
6
votes
0
answers
203
views
Composite number $n$ with most $k \le n$ such that $n \mid \binom nk$
This problem arised in a local forum, proposed by a user named zxt.
Let $f(n)$ be the number of nonnegative integer $k$ not greater than $n$ such that $n \mid \binom{n}{k}$. If for each positive ...
- 221
5
votes
0
answers
107
views
Is there a natural metric on the space of proofs of a fixed proposition that captures "strategic similarity"?
Background
By the Curry–Howard correspondence, a proof of $A \to B$ is a term $p : A \to B$ in a suitable type theory. For a fixed pair of propositions $(A, B)$, there may be many distinct proof terms ...
- 111
-3
votes
0
answers
127
views
Are recursively defined sets countable? [closed]
Specifically are recursive sets of this form countable:
Base case: $x \in S \subseteq \mathbb{R}$
Recursive step: $A \subseteq S \Rightarrow \phi (A) \in S$
At any state of this set there are only a ...
- 3
5
votes
2
answers
187
views
On formal systems for $\mathsf{BISH}$, $\mathsf{INT}$, and $\mathsf{RUSS}$ in constructive reverse mathematics
In informal constructive reverse mathematics, one typically works in Bishop-style constructive mathematics ($\mathsf{BISH}$) as a base theory and studies the strength of various principles over it (...
- 1,361
-2
votes
0
answers
149
views
In the absence of choice, is there a definition of cardinality in terms of surjective relations? [duplicate]
It’s known that, assuming AD, there is no surjective mapping from $R/Q$ to $R$. This is because elements of $R/Q$ are so similar to each other that any function from $R/Q$ will return a constant with ...
- 97
4
votes
1
answer
183
views
What is the volume of $\Sigma(2, 3, 13)$ associated with its $\widetilde{\operatorname{SL}(2, \mathbb{R})}$ geometry?
I've been considering a research topic based on extending the material from Khoi's research paper concerning a Chern–Simons-type invariant for 3-manifolds, and I'm stuck on a specific problem ...
4
votes
1
answer
208
views
Non-increasing property of a norm-like function over matrices
Let $P,Q$ be two real orthogonal projections on $\mathbb R^n$, and assume that they are permutation similar. More specifically, assume that each of them is permutation similar to a block diagonal ...
- 857
4
votes
0
answers
92
views
Do K-well generated categories exist for any regular K?
A. Neeman has defined K-well generated triangulated categories for any infinite regular cardinal K; in the case $K=\aleph_0$ these are the compactly generated ones. He (and also H. Krause) has also ...
- 17.7k
0
votes
0
answers
37
views
Characterizations of Kalton's notion of $\beta$-complete bases via the associated sequence of coefficient functionals
In 1972, N. J. Kalton introduced the concept of $\beta$-complete bases to characterize the weak sequential completeness of a Banach space with a basis: A Banach space with a basis is weakly ...
- 3,553
8
votes
0
answers
84
views
chain level intersection product
Let $F$ be a field and $M$ an oriented smooth $n$-manifold.
Is it always possible to construct a differential graded $F$-vector space $(C_M,d)$ equipped with a strictly coassociative coproduct $\Delta ...
- 2,165
0
votes
0
answers
165
views
Finding all integer solutions to a family of elliptic curves depending on a parameter $n$
Consider this equation
\begin{equation}
y^2 = x^3 + (36n + 27)^2 \cdot x^2 + (15552 n^3 + 34992 n^2 + 26244 n + 6561) \cdot x + (46656 n^4 + 139968 n^3 + 157464 n^2 + 78713 n + 14748)
\end{equation}
...
- 139
13
votes
1
answer
478
views
Find the gcd of $n^{2} + 1$ and $n! + 1$
I want to find the $\gcd$ of $n^{2} + 1$ and $n! + 1$. I have verified the first $10,000$ $n$ using a Python program. Their results are all $1$. So is it true that for all $n \geq 2$ , $n^2 + 1$ ...
- 131
0
votes
0
answers
47
views
Is the composition of a logarithmic multiplier and a singular integral operator bounded at $L^\infty$?
I need to consider an operator of the form
$$
W=T\circ (\log|D|)^{-1}\chi(|D|) \quad \text{in } \mathbb{R}^2,
$$
where $\chi$ is a smooth cut-off function supported near zero and $T$ is an operator ...
6
votes
1
answer
111
views
The principal block of a Frobenius group
Let $G$ be a finite group. Fix a prime $p$. Let $P$ be a Sylow $p$-subgroup such that $P\cap P^x=1$ for all $x\not\in P$. (In other words, $P$ is a Frobenius complement.)
It follows from Frobenius' ...
- 2,603
5
votes
2
answers
425
views
Rigidifying $E_\infty$ structures
It is well-known that any $\infty$-category can be rigidified to a topologically-enriched category (with strictly associative composition of morphisms); alternatively, topologically-enriched ...
- 1,219
2
votes
0
answers
71
views
Isotopy to achieve finite intersections
I've asked this question on math.stackexchange a week ago, with no response. More context is available there.
Suppose that $\alpha$ and $\beta$ are closed curves on the $2$-manifold, say $F$ (possibly ...
0
votes
0
answers
114
views
symplectic resolution of K theoretic coulomb branch
Consider the type $A_n$ quiver with gauge group $G=\prod_i \mathrm{GL(V_i)}$ and representation $N=\oplus_i \mathrm{Hom(N_i, N_{i+1})}$, will the K-theoretic Coulomb branch $Spec(\mathrm{K}^{ G(\...
- 591
3
votes
0
answers
106
views
Optimal play in a splitting game on diamond-shaped polyominoes
When I was a child, my mother taught me a simple pencil-and-paper game. I would like to know whether this game, or an equivalent formulation of it, has already been studied.
Let the integer $n > 1$ ...
- 2,181
4
votes
1
answer
134
views
Is the $\Xi(n) = \prod_{k=1}^{\infty} \text{lcm}(1, 2, \dots, \lfloor n^{1/k} \rfloor)$ sequence a subset of the highly abundant numbers?
In this previous discussion, it was demonstrated that the standard Least Common Multiple sequence $\text{lcm}(1, 2, \dots, n)$ is not a subset of the highly abundant numbers.
In analytic number theory,...
- 1,287
2
votes
1
answer
125
views
Relation between $\sigma$ and some other ordinals
Let's define $\sigma$ as the supremum of OTM clockable ordinals. Also consider the ordinals defined in http://www.madore.org/%7Edavid/math/ordinal-zoo.pdf. As far as I can understand, it seems to me ...
- 923
2
votes
0
answers
80
views
Fallback for failure case in Galois factoring with units
Let $N = x^2 + 3y^2$ be a composite integer with a known representation of this form. Consider the cubic polynomial
$$
f(t) = 4t^3 - 3Nt - Nx,
$$
and let $K = \mathbb{Q}(\alpha)$ be the cubic number ...
- 427
2
votes
1
answer
94
views
relation between eventual coconnectiveness and tor amplitude of $f$ and cotangent complex of $f$ be perfect
I was currently reading derived geometry from Lurie's thesis and DAG's. I am wondering about the following.
Let $f:X \to Y$ be a morphism of derived Deligne-Mumford stacks. Let the cotangent complex $...
- 1,619