I am studying manifold. For comprehension, I read the site http://en.wikipedia.org/wiki/Manifold, and there is some information about infinite dimensional manifold. Now I have two questions or requests: (1) When was infinite dimensional manifold introduced? I guess this may be related to Functional Analysis. But I want more details about its history. (2) I am still […]

I want to understand sum of binomials better in terms of ideals such as binomial ideals, normal ideals and so by toric ideals. Examples about toric ideals contain $$\sum x^\alpha+\sum x^\beta\in\mathbb R$$ where $\alpha,\beta$ are multidegrees. I have a gut feeling that toric varieties and this Sparse elimination theory is crucial to understand features of […]

The following is a proposition in Artin’s Algebra: Proposition 11.5.5 Let $R$ be a ring, and let $f(x)$ be a monic polynomial of positive degree $n$ with coeeficients in $R$. Let $R[\alpha]$ denote the ring $R[x]/(f)$ obtained by adjoining an element satisfying the relation $f(\alpha)=0$. Then the set $(1,\alpha,\cdots, \alpha^{n-1})$ is a basis of $R[\alpha]$ […]

Recently, I am thinking about the question in spectral theory. And I finally found that I need help with the properties of unitary operator. Its a consequence of the spectral theorem for the normal operator, as the wiki says. The spectral theorem in the book I have ever read is only for self-adjoint operator and […]

What are some braid invariants (analogous to the idea of knot invariants) or a resource where I can find them?

I’m looking for a translation to either English, French or German of Kolmogorov’s Russian paper Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi de Gauss. Bull. Acad. Sci. URSS Ser. Math. 6, 3–32. This seminal, oft-cited paper (e.g. in Lehmann and Romano’s Testing Statistical Hypotheses (2008) p. 20 and in Blackwell and […]

Many (most?) number theory proofs employing the method of infinite descent proceed something like this: Assume a given [Diophantine] equation (e.g., $x^3+y^3=z^3$) has solutions in positive integers. Manipulate the equation until you find an equation of the same form which is an implication/consequence of the first (e.g., $u^3+v^3=w^3$) . Show that $0<u<x$ [or $0<v<y$ or […]

Consider the “function” (more precisely it is a tempered distribution) given by $f : \mathbb{R}^n \to \mathbb{R}$, $f(x) = \frac{1}{|x|^p}$, where $0 < p < n$. It can be calculated that the Fourier transform of $f$ is given (upto a constant) by $\hat{f}(\xi) = \frac{1}{|\xi|^{n – p}}$. Now, I am trying to prove that the […]

For example, you approximate structure functions of finite simple graphs in cases where only cut sets of the systems are known. The inverse problem means to build possible scenarios in underdetermined system. Gröbner bases provides a way to express a set of structure functions — however how can you know that the set of structure […]

In the very commonly used J. Silverman’s AEC an elliptic curve is defined as a genus 1 projective curve with a fixed point 0. In all the other books I looked at it is defined to be (also) smooth. By the way in AEC it is given a proof of the fact that a genus […]

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