0
votes
3answers
67 views
Does this ODE system have solution?
Let $A(t)$ and $B(t)$ be matrices with each element in $L^\infty(0,T).$ Let $A(t)$ have an inverse. I know nothing else about this inverse.
Let $c(t)$ be a vector in $L^2(0,T).$ …
1
vote
1answer
122 views
Definition of the homological Chern character
There is a homological Chern character $ch_\ast \colon K_\ast(X) \to H_\ast(X)$ for $X$ a smooth, compact manifold.
I found only one definition of it (in the paper "K-Homology and …
0
votes
1answer
118 views
Accessible problems on classical groups over complex or real numbers.
I am a undergraduate student doing project with my professor in group theory. I am Looking for some accessible problems for undergraduate on Classical groups over complex or real n …
5
votes
1answer
319 views
+100
Effective Chebotarev without Artin’s conjecture
Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form
of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Arti …
1
vote
1answer
83 views
Nested Sequence of Integers
In some combinatorial research I came across the following nested sequence:
$${a_n}={1,1,3,1,7,3,17,1,35,7,77,3,157,17,331,1,663,35,1361,7,2729,77,5535,3,11073, \dots}$$
which is n …
18
votes
2answers
287 views
Order type of the smallest set containing the identity function and closed under exponentiation
Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapst …
0
votes
0answers
46 views
Can Hartogs' extension theorem be used to prove there’s no naked singularity?
Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal ana …
0
votes
0answers
104 views
Reference request for differential geometry [closed]
I have a somewhat bizzare question . I'm studying differential geometry for it's applications in physics . I'm interested in applying my knowledge to concrete objects . For example …
1
vote
2answers
174 views
A machine learning application question
I am familiar with basic probabilities, random processes but not so much of machine learning methods. This is the problem I am trying to solve.
I want to predict the nature of use …
1
vote
1answer
233 views
Help me on proof of an equation.
I wanna prove following equation
$ \sum_{i=1}^n \prod_{k=1,k\neq i}^n \prod_{j=1,j\neq k}^{n+1}(x_j - x_k) = -\prod_{i=1}^n \prod_{j=1,j\neq i}^n (x_j - x_i) $
I have verified sev …
0
votes
1answer
115 views
When is an ample line bundle on an abelian variety base point free?
So, any line bundle $L$ on an abelian variety $X$ determines a type $(d_1,\ldots,d_g)$ where $d_i|d_{i+1}$. It's well known that if $d_1\geq 3$ then $L$ defines an embedding, that …
3
votes
1answer
88 views
Families of Hurwitz Curves
Hurwitz's theorem on automorphisms tells us that the group of automorphisms of a nonsingular complex algebraic curve of genus at least 2 is bounded above by $84(g-1)$ where $g$ is …
1
vote
2answers
210 views
Hyperbolic pair of pants.
Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential …
18
votes
1answer
610 views
Can the fact that the square of an integer is a natural number be categorified?
If $a$ and $b$ are natural numbers, then $a-b$ is an integer and so the square $(a-b)^2$ is a natural number. In particular
$$ (a-b)^2 \geq 0. \qquad (1)$$
Combining this fact …
6
votes
3answers
338 views
Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$
Hi.
Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I …
