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8 votes

When are the chirp signals orthogonal?

just for the record, the distance function has a complicated expression in terms of the imaginary error function erfi: $$d(x,y)=2T+\operatorname{Re}\,\left\{(\alpha+\alpha')^{-1/2}(-1)^{3/4} \left[\...
Carlo Beenakker's user avatar
7 votes

If $f$ is a derivative and $f=g$ a.e. for some Riemman integrable function $g$, then can we obtain the Riemann integrability of $f$?

Since $f$ is equal a.e. to $g$, $f$ is Lebesgue integrable. Now for $a < x < y < b$, $$F(y) - F(x) = \int_x^y f(t)\; dt = \int_x^y g(t)\; dt$$ (see e.g. Rudin, Real and Complex Analysis, ...
Robert Israel's user avatar
8 votes

Are “most” bounded derivatives not Riemann integrable?

In 1977 Clifford E. Weil showed that $A$ is a first Baire category set (i.e. a meager set) in $X$ (sup norm) -- see The space of bounded derivatives. So the situation, at least with respect to one ...
Dave L Renfro's user avatar
1 vote

Average distance between points of lower dimensional simplices in $\mathbb R^n$

It is highly unlikely that an explicit expression exists. Even the calculation of the volume of a polytope is a nontrivial problem, solved by Lawrence for simple polytopes. One can possibly use ...
Iosif Pinelis's user avatar
5 votes

On the Riemannian integrability of the bounded derivative

The answer to the question in the body of your post is no, for the reason that if $f' = g$ almost every where and $g$ is Riemann integrable and such that (*) holds, then $f'$ must be Riemann ...
Willie Wong's user avatar
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3 votes

Can we integrate arbitrary rational functions of Jacobian elliptic functions?

The answer is positive. Rational function of Jacobi elliptic function is elliptic, and for every elliptic function $f$, $f(z)dz$ is an Abelian differential, and integral of it is an Abelian integral. ...
Alexandre Eremenko's user avatar

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