The distribution relation and inverse function theorem in arithmetic geometry

On the Inverse Function Theorem

The convolution inverse of an arithmetic function

Good, but does an inverse g of f have to exist? A necessary condition is that f(1) 6= 0. Indeed, if g is the inverse of f , then 1 = I(1) = (f ∗ g)(1) = f(1)g(1). We now show that this necessary condition is also sufficient. We assume that f(1) 6= 0 and we try and solve for g. What this means is that we are solving for infinitely many unknowns: g(1), g(2), . . . . From the above, we see that th...

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue’s dominated convergence theorem and Ekeland’s variational principle. As a consequence, the assumptions are substantially weakened: th...

identifying the strategies persian efl learners use in reading an expository text in english and examining its relation to reading-proficiency and motivation: a think-aloud study

هدف اصلی از این مطالعه بررسی نوع و میزان استراتژی هایی بود که دانشجویان فارسی زبان رشته ی زبان انگلیسی در حین خواندن یک متن انگلیسی به کار گرفتند. این مطالعه همچنین به بررسی تفاوت های استراتژی های مورد استفاده بین دارندگان سطح بالا و پایین درک مطلب پرداخت. نوع همبستگی بین استراتژی به کار گرفته و درک مطلب از یک سو و استراتژی به کار گرفته و انگیزه از سوی دیگر نیز در این تحقیق مورد آزمایش قرار گرف...

Polylogarithms in Arithmetic and Geometry

After a century of neglect the dilogarithm appeared twenty years ago in works of Gabrielov-Gelfand-Losik [GGL] on a combinatorial formula for the first Pontryagin class, Bloch on K-theory and regulators [Bl1] and Wigner on Lie groups. The dilogarithm has a single-valued cousin : the Bloch Wigner function L2(z) := ImLi2(z) + arg(1− z) log |z|. Let r(x1, ..., x4) be the cross-ratio of 4 distinct ...