Unanswered Questions
369,662 questions with no upvoted or accepted answers
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Confusions about $\mathrm{Spec}(R)$ and algebraic groups
So I am studying about classical algebraic geometry (the one working with affine varieties instead of schemes) and I suppose I'm lost in a number of points. Note that I'm trying to study affine ...
- 1,622
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Identity involving the polylogarithm and $\pi i/ 6$
Using the polylog multiplication formula, $\sum_{m=0}^{p-1} \operatorname{Li}_s(z e^{2\pi i m/p}) = p^{1-s} \operatorname{Li}_s(z^p)$
it's easy to see that the the real part of
$- \operatorname{Li}_2(...
- 390
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What is the number of such circular arrangements?
I have been reading "Graph Theory with Applications to Engineering and Computer Science" by Narsingh Deo. And I found this interesting seating problem therein.
Nine members of a new club ...
- 2,321
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Characterization of real numbers $ c $ for which $ \hat{\mu}(n) = e^{c i n^2}$ holds for a finite Borel measure $ \mu $ on $ \mathbb{T} $
Which real numbers $ c $ does the following hold for?
There exists a complex Borel measure $ \mu $ on the torus $ \mathbb{T} := \mathbb{R} / 2\pi \mathbb{Z} $ such that the Fourier coefficients of $ \...
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Question about singular matrix in thin-plate spline interpolation
I'm trying to interpolate a function $f:\mathbb{R}^2\to\mathbb{R}$ given known values $y_i=f(\mathbf{x}_i)$, for $i=1,\dots,m$, to new points $\mathbf{x}$ using thin plate splines as described in this ...
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Eigenvalues of $A+cd^T$ without determinants
Suppose that $\{\lambda_1, ..., \lambda_n\}$ are the eigenvalues for
$A_{n\times n}$ and let $(\lambda_k, c)$ be a particular eigenpair.
Let $d$ be a arbitrary vector, how to prove that the ...
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If $f(x) = \sum_{b \in B(x)}f_b(x)$ and $B(k) \subset B(k+1)$ while $\sum_{b \in B(k)} f_b(x) \geq \sum_{b \in B(k+1)} f_b(x)$ what is $f^{-1}(A)$?
I'm having real trouble with finding a formula for the inverse image of a sum of functions whose summation bound depends upon the input of the function $x$.
Specifically, let $\{f_i\}$ be a family of ...
- 22.1k
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Eigenvalues of $PA$ with $P$ a projection and $A$ having eigenvalues inside open unit disk
Given a projection matrix $P$ (not necessarily an orthogonal projection), and another matrix $A$ where $\mathrm{eig}(A)$ all have magnitude strictly less than $1$, and both $P,A$ are real-valued. Can ...
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Zeros of $f(z) = z^5 - ae^z$ in $B(0;1)$ for $a<0$
In this question, they wanted to find the zeros of $f(z) = z^n - ae^z$ in $B(0;1)$ for $n\in\mathbb{N}$ and $a>0$. My question is, what about when $a<0$.
The case I am dealing with is when $n = ...
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If ordinal $\alpha$ isomorphic to subset of ordinal $\beta$ then $\alpha \leq \beta$
If ordinal $\alpha$ isomorphic to subset of ordinal $\beta$ then
$\alpha \leq \beta$
My best attempt so far: Since any two well-ordered sets either two are isomorphic to each other or one is ...
- 578
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Show that $∫_{X_1(t)}^{X_2(t)}u(t,x)dx=∫_{y_1}^{y_2}u(0,x)dx, (t\in \mathbb R)$.
Let $a\in C^1(\mathbb R)$ be a bounded function and suppose a and its derivative are both bounded. Let $u\in C^1(\mathbb R^2)$ be a solution to the following equation:
$$∂_tu(t,x)+∂_x(a(x)u(t,x))=0,((...
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Evans' PDE: proof theorem 2 section 5.9
The book proves the following $u(t)=u(s)+\int_{s}^{t}u'(\tau)d\tau$ and says this equality implies "easily" that $\max_{0\leq t\leq T}\lVert u(t)\rVert\leq C \lVert u\rVert_{W^{1,p}(0,T;X)}$ ...
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Exact Triangle Sorting for Orthographic Rendering of a Triangulated Surface
I want a definitive front-to-back triangle drawing order under orthographic projection.
This is an X-post due to inactivity on the other: https://tex.stackexchange.com/q/735053/319072
I have a ...
- 239
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Null Space of Infinite Dimensional Matrix
Is there a general way to tell if the null space of an infinite dimensional matrix contains a vector which is not zero?
This question connects to the following problem:
If I know that
\begin{equation}
...
- 1
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$1/|x|$ interpreted as a distribution and Dirac delta distribution
From Tao's blog:
Let ${d=1}$. For any ${r > 0}$, show that the functional ${\lambda_r}$ defined by the formula
$$\displaystyle \langle f, \lambda_r \rangle := \int_{|x| < r} \frac{f(x)-f(0)}{|x|}...
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