Newest Questions
1,689,795 questions
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Specific Automorphism(?) of $\mathbb{H}$
In this pre-print, at the bottom of page 20, the authors write that
\begin{equation}
\phi_n(z)=\frac{nz-1}{z+n}
\end{equation}
is an elliptic automorphism of $\mathbb{H}:=\{\operatorname{Im}(z)>0\}$...
- 45
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Does $u\in L_{\text{loc}}^{1}$ automatically imply $f(u)\in L_{\text{loc}}^{1}$ when $f$ is only locally Lipschitz?
I am reading Bressan’s Hyperbolic Systems of Conservation Laws.
In Chapter 6 he defines an entropy solution of a 1-D conservation law
and says that we implicitly assume that both $u$ and $f(u)$
are ...
- 3
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Discrete cosinus transform of pde
i have pde system to solve it numerically :$f_t= \Delta f- \nabla( f* \nabla g) + f^3$,$g_t=\Delta g-c*g*f+a*g^2$ , so i do finite difference scheme and my aim is to write f and g using discrete ...
- 11
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Example of a bijective isometry where the inverse is not an isometry?
Is there an example of an isometry that is bijective, but where the inverse is not an isometry?
- 445
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The minimum degree Wiener index and the maximum degree Wiener index
The minimum degree Wiener index of the wheel graph $W_{n}, n>3$ is $n(n-2)$. And the minimum degree Wiener index for a complete graph $K_{n}$ is $\frac{n(n-1)}{2}$ (the source is: KG, S., & ...
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If $X_{k,n}=O_p(1/\sqrt{n})$ i.i.d., can we say that $S_n=\sum_{k=1}^nX_{k,n}/\sqrt{n}\to 0$ in probability granted $\mathbb{E}X_{k,n}=0$?
On a probability space $(\Omega,\mathscr{F},\mathbb{P})$, let $X_{1,n},...,X_{n,n}$ be a centered stationary triangular array such that $\sqrt{n}X_{k,n}$ is tight (that is $X_{k,n}=O_p(1/\sqrt{n})$, ...
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vector "norm" induced by a positive definite tensor
For a positive definite matrix $A\in\mathbb{R}^{n\times n}$, we know that it can induce an inner product in $\mathbb{R}^{n}$:
$$\left\langle x,x \right\rangle = x^TAy,$$
thus further inducing a norm $\...
- 127
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Shortest Sequence to Cover All Episode Orders and Pairs
I have two episodes, labeled 1 and 2. I want to watch them in such a way that all possible two-episode combinations (like 11, 12, 21, and 22) appear at least once as consecutive pairs. For example, ...
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What does the square of a line segment look like?
Suppose I have a closed line segment (inside the punctured closed unit disk) joining two fixed points of the unit circle of the complex plane. What is the image of this segment on the Riemann surface ...
- 18.8k
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Computing the work to bring two charged particles together using translated spherical coordinates
Consider a charge $ q_1 $ sitting at the point $ p_1 $. Take another charge $ q_2 $. I'm trying to compute the work needed to bring $ q_2 $ from "infinity" down to the position $ p_2 $, but ...
- 1,043
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A problem related to Coutinuous Function $f\left ( \xi +a \right ) = f\left ( \xi \right ) $
If $f\in C\left[0,1\right]$, $f\left(0\right)=f\left(1\right)$, show that $\forall a\in \left[0,\frac{1}{2}\right]$, $\exists\ \xi $ such that $f\left ( \xi +a \right ) = f\left ( \xi \right ) $.
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Find all the subsets from a set $\{1,2,3,\cdots,n\}$, which contain only coprime numbers.
Consider all subsets of $S=\{1,2,3,…,30\}$ so that each pair of
numbers in that subset are coprime. Find the subset $\Omega \in S$
satisfying the previous condition whose elements have the largest sum....
- 6,579
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Jacobi Field Growth and Compactness under Sectional Curvature Bounds
Let $(M^n, g)$ be a complete Riemannian manifold and $p\in M$.
(i) Suppose along any normalized (unit-speed) geodesic $\gamma$ with $\gamma(0) = p$, the sectional curvatures of $M$ in any plane $\...
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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(...
- 400