Newest Questions
1,689,785 questions
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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 ...
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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} $ with finite total variation (i.e., $ |\mu|...
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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 ...
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Evaluate $I\left(n\right)=\int_{0}^{\infty}\frac{1}{\left(e^{x}+e^{-x}\right)^{n}}\,\mathrm dx$
Evaluate $I\left(n\right)=\int_{0}^{\infty}\frac{1}{\left(e^{x}+e^{-x}\right)^{n}}\,\mathrm dx$.
I tried substitution $t=e^{x}$, which led to the integral $$\int_{1}^{\infty}\frac{t^{n-1}}{\left(1+t^{...
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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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Find $O(b)$ given that $a^5=e$ and $ab^{-1}a=b^2$
In a mathematics group chat, somebody asked this question.
If in a Group G, $a^5=e$ and $ab^{-1}a=b^2$ then find O(b)
I believe he made an error, as this problem is similar to Problem 37 on p.49 in ...
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does a 0 in the basic solution mean there's positive slack in the corresponding equation
enter image description here
This question has the solution to Y* as being (0,1), but from my understanding of the Complementary Slackness Theorem, there should be positive slack in the first primal ...
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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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Show that A is equicontinuos on [0,1]. [closed]
Let $\big(C([0,1]), ||\cdot||_{\infty}\big)$ the metric space of continuos functions on $[0,1]$ taking values in $\mathbb{R}$. Let $M>0$. How can i proof that $A\subset C([0,1])$ is equicontinuos ...
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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 ...
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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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