All Questions
157,088
questions
-4
votes
0
answers
24
views
Math's secret sonnet
$\int_0^{\pi/2} \sin(x) \, dx = \heartsuit$
$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$, a symphony divine
$\sqrt{-1}$ dances with $i$, a mystery untold,
$\lim_{n \to \infty} \left(1+\frac{1}...
1
vote
0
answers
11
views
Zero dimensional complete intersection ring of length a power of $p$
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C$ be a $k$-algebra of finite dimension over $k$ such that $k[C^p]=k$. Under these hypothesis it is known by results of L. ...
- 627
-2
votes
0
answers
29
views
Representations of knot groups and galois groups on varieties over number fields
Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Consider a smooth projective variety $X$ defined over $K$, and let $\mathcal{L}$ be an ample line bundle on $X$. For each prime ideal $\...
- 1
1
vote
0
answers
10
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Characteristic function of transformed random variable
Consider a random variable $X$ and a function $g(\cdot)$. Let $Y:=g(X)$, and let $\phi_X(\cdot), \phi_Y(\cdot)$ be the characteristic function (cf) of $X,Y$, respectively. Suppose that $\phi_X$ is non-...
- 123
-1
votes
0
answers
31
views
Arithmetic deformation of motives
Let $X$ be a smooth projective variety defined over a number field $K$. We denote by $X_{\overline{K}}$ the base change of $X$ to an algebraic closure $\overline{K}$ of $K$. Consider a p-adic Galois ...
-1
votes
0
answers
19
views
What’s the Fractional Shortest Path Problem in a Fractional Graph?
Fractional Shortest Path Problem in a Fractional Graph
In a traditional graph, each edge between vertices has a weight, typically representing distance, time, or cost. Consider a fractional graph ...
-2
votes
0
answers
36
views
Exotic homotopy equivalences
Let $X$ and $Y$ be simply-connected, finite CW complexes with isomorphic integral cohomology rings, i.e., $H^*(X; \mathbb{Z}) \cong H^*(Y; \mathbb{Z})$ as graded rings. Suppose further that $f: X \to ...
1
vote
0
answers
37
views
Decay of fourier coefficients
Let $\mathbb{S}^1$ denote the unit circle in $\mathbb{R}^2$, and let $\mathcal{M}(\mathbb{S}^1)$ denote the space of finite Borel measures on $\mathbb{S}^1$. For $\mu \in \mathcal{M}(\mathbb{S}^1)$, ...
0
votes
0
answers
24
views
Existence of an ergodic subgroup of the full group
Let $X = \{0, 1\}^{\mathbb{N}}$ and $G \curvearrowright X$ be the group action that only permutes finitely many coordinates of elements in $X$. Let $X$ be endowed with $\mathcal{B}$ the $\sigma$-...
2
votes
0
answers
55
views
Homotopy of Brown-Gitler spectra
Let $A^\vee = \mathbb{F}_2[\bar\xi_1, \bar\xi_2, ...]$ be the mod-2 dual Steenrod algebra. One can define a weight filtration on $A^\vee$ by setting $wt(\bar\xi_i)=2^i$ and $wt(xy)=wt(x)wt(y)$. There ...
1
vote
1
answer
56
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PDE where the square of gradient of the unknown equals a given positive function
Let $V(x)$ be a non-negative smooth function defined in a open domain $U\subset\mathbb{R}^n$. Suppose that $V(x)=0$ only at a given point $x_0\in U$. Consider the PDE
$$|\nabla u|^2=V$$
with ...
- 13
0
votes
0
answers
54
views
Lebesgue measure of the level set of sum of two nonnegative functions
Let $f, g:\mathbb{R}^n\to \mathbb{R}$ be nonnegative functions such that $g$ is a strictly positive homogeneous function. As commented by Fedor Petrov below, one may not have that for any $\lambda>...
- 361
-2
votes
0
answers
35
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Can acyclicity be used to internalize functions within models of Z?
Suppose $M$ is a model of $\sf Z +\neg AC$ that is externally bijective to an element $k \in M$. Obviously if $j$ denotes such an external bijection, then it cannot be used in Separation within $M$, ...
- 10.5k
0
votes
1
answer
93
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Proof of a zeta function limit
At following page from Mathworld I found an interesting limit but I can not get its proof as it figures as a personal communication from B. Cloître (B. Cloître, pers. comm., Oct. 4, 2005):
$$\gamma=\...
- 111
0
votes
0
answers
53
views
Pull back of line bundle on an abelian variety
Let $(X,\mathcal L)$ be a polarized abelian variety of type $(1,\cdots,1,r,\cdots,r)$ where $r$ is repeated $k$ times. Let $(Y,\mathcal M)$ and $(Z,\mathcal N)$ be a complementary pair incide $X$ such ...
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