Newest Questions
1,693,640 questions
0
votes
1
answer
10
views
Is provability of existential arithmetic sentences/halting of Turing machines independent with theory
Let $D(x_1, x_2, …, x_n) = 0$ be a Diophantine equation of which coefficients are computable integers via Turing machines. Suppose we can prove in theory like ZFC+large cardinal that $D$ has a zero, ...
- 351
0
votes
0
answers
9
views
Rational points on this elliptic curve?
For a "wurzelnightrider" (fairy chess) tour, it would be interesting to know if $i^4+j^4-i^2\cdot j^2$ can be a (rational would suffice!) square. This is easy to transform to an elliptic ...
- 823
0
votes
0
answers
4
views
Codimension of singular points of hypersurface $Y\subset X$
Let $X$ be a complex manifold, and let $Y\subset X$ be an irreducible hypersurface (i.e. analytic subset of codimension 1). Lemma 2.3.22 in Huybrechts' Complex Geometry shows the sheaves $\mathcal{O}(-...
-3
votes
0
answers
24
views
nCr and perfect squares [closed]
Find the largest two digit $n$ such that, $$\binom{n}{3}\binom{n}{4}\binom{n}{5}\binom{n}{6}$$ is a perfect square.
- 68
0
votes
0
answers
5
views
Deriving the Jacobi equation as linearization of the geodesic equation
Consider the geodesic equation $D_{\gamma'}\gamma'=0$. In coordinates this reads as the non-linear ODE:
$$(1)\hspace{2cm}(\gamma^i)''+(\gamma^j)'(\gamma^k)'\Gamma_{jk}^i=0.\hspace{5cm}$$
In Eschenburg'...
- 1,165
0
votes
0
answers
9
views
Proof check: $\hat{\mathbb{C}}$-valued limit of holomorphic functions is holomorphic or identically $\infty$
Question: Let $d$ be the distance function induced on the Riemann sphere $\hat{\mathbb{C}}$ via stereographic projection,
$$ d(z,w) = \frac{|z-w|}{ \sqrt{1+|z|^2} \sqrt{1+|w|^2}}, \quad d(z,\infty) = \...
- 1,416
0
votes
0
answers
4
views
Deriving the inequality $|R(1,w)|\geq k$ via wall-crossing parity in Davis's Coxeter Groups book
Suppose $\mathbf{s}=(s_1,\dots,s_k)$ is a word in $S$. Define $w_i\in W$ by $w_0=1$ and $w_i=s_1\cdots s_i$, and $r_i\in R$ by $r_i=w_{i-1}s_iw_{i-1}^{-1}$. Set $\Phi(\mathbf{s}):=(r_1,\dots,r_k)$. ...
- 624
0
votes
0
answers
11
views
Positive solution of nonlinear elliptic equation
We consider the following problem
$$
\begin{cases}
- \Delta u = \lambda f(u), \quad u>0 \quad &\text{in} \ \Omega, \\
u=0\quad&\text{on }\partial\Omega.
\end{cases}
$$
where $\Omega \...
- 387
1
vote
1
answer
27
views
Expected value on a fair 6 sided die with coin flip
You are given a fair
6−sided die and play the following game: You receive the value of the face-up side on each roll. If you roll an odd number, the game ends. If you roll an even number, you flip a ...
- 88
0
votes
0
answers
10
views
Explanation of an assertion by Gauss on an embedding of a surface in space
[I ask this question after a previous question of mine, which deals with a related theme, was closed (because it was too broad and unfocused). I will try now to ask several much narrower questions ...
- 1,236
2
votes
0
answers
15
views
Identity in MCMC autocorrelation
I am reading an article on metrics regarding the samples of MCMC methods. It is this one: https://arxiv.org/abs/1903.08008 . On pg. 7 the authors define the autocorrelation of lag $t$ of a chain of ...
- 192
0
votes
0
answers
18
views
$f:U\to\mathbb{R}$ is $C^k$ at $p\in U$. Is this definition really correct? ("An Introduction to Manifolds Second Edition" by Loring W. Tu.)
I am reading "An Introduction to Manifolds Second Edition" by Loring W. Tu.
Definition 1.1. Let $k$ be a nonnegative integer. A real-valued function $f:U\to\mathbb{R}$ is said to be $C^k$ ...
- 9,896
0
votes
0
answers
15
views
How to endow a fiber of smooth map at a point with manifold structure? [closed]
Let $f: M \to N$ be a smooth map between smooth manifolds and $q \in N$ be a point.
I wonder when and how I can endow the fiber $f^{-1}(q) \subset M$ with structure of smooth manifold? What type of ...
- 381
0
votes
0
answers
8
views
Proof of a characterization of quasiconcave functions.
A function f is quasiconcave at x iff it is concave on the tangent to the level set of f through x. I am looking for a proof of this but can't find any. Can someone please point me to the right source?...
- 1
-1
votes
1
answer
26
views
Conditions on linear independance of a family of vectors defined as an infinite linear combination
Let $(E,\left<.,.\right>)$ a vector space over $\mathbb{R}$ with an inner product. I consider $x:= (x_n)_{n \in \mathbb{N}} \in E^{\mathbb{N}}$ a sequence of elements of $E$ such that any finite ...
