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Christophe Leuridan's user avatar
Christophe Leuridan's user avatar
Christophe Leuridan's user avatar
Christophe Leuridan
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A curious family of integrals that give $\pi$
Yes, but very few rational combinations of arctan of rational numbers give rational multiples of $\pi$.
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Maximum of a function
@Wikkie Wong. Yes there is probably no formula covering all cases. But, to avoid anyconfusion with homework, Zoran Vicovic should mention simple cases, e.g. p odd and q even, where the answer is immediatly 2.
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Periodic Orbit without Complex Eigenvalues
Is $I_\mathrm{app}$ a constant?
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A question about convergence of stochastic processes converging to a random walk
The question is not clear at all. First, the sequences should be indexed by $\mathbf{Z}_+$. Indexing by $\mathbf{Z}$ is not always possible. Next, you should say that $y_0,u_1,u_2\ldots$ are independent. Last which kind of convergence do you expect? For example, under your assumptions, the sequence of random variables $(Y^n_0,Y^n_1)$ does not converge in distribution to $(Y_0,Y_1)$.
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Geometric realization of a poset
If $j \ge n-k$, the condition can be written EITHER $\{n-k,...,j\} \not\subset U$ OR [$\{n-k,...,j\} \subset U$ and $\{1,...,n-k-1\} \not\subset U$ and $\{j+1,...,n\} \not\subset U$]. Does it help?
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If $f : [0,1] \to H$ has $t$-derivative with respect to the norm of $H$, and $H=L^2[0,1]$ itself, does the $t$-derivative exist in ordinary sense?
See reformulation in the answer. By the definition of the limit, for every $\epsilon>0$, one can find $\eta>0$ such that for all $t \in [-\eta,\eta] \setminus \{0\}$, $\Vert [f(t,\cdot)-f(0,\cdot)]/t - g \Vert \le \epsilon$. Apply this with $\epsilon_n := n^{-2}$ to get a corresponding $\eta_n$ and set $t_n = \min(\eta_n,n^{-1})$.
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Isomorphism between subgroups by preserving index
You should clarify the assumptions. Assume explicitly that $C$ and $D$ have index $2$ in $A$ and $B$. Isomorphism is not sufficient: for example, the quotient of $\mathbb{U} = \{z \in \mathbb{C} : |z|=1\}$ by $\{-1,1\}$ is isomorphic to $\mathbb{U}$ itself. Do you assume that $\pi$ is an isomorphism? If you have only an homomorphism, for example the constant homomorphism which sends $A$ on $1_B$, it will not help.
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