Lagrange proved that every (positive) rational integer is a sum of 4 squares. Are there general results like this for ring of integers of a number field? Is this class field theo …

For a research problem that I'm working on, I need to solve this Diophantine equation:- $a^3+b^3+c^3-3d=-83449$, where $a,b,c,d>0$ are all DISTINCT positive integers and$ a,b,c∉ $ …

Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain c …

I have the following, probably very elementary question: Let $Cl^{p,q}$ be the Clifford algebra on generators $e_i$, $i=1, \ldots, p+q$ with $e_i e_j = -e_j e_i$ and $e_{i}^{2}=-1$ …

This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/396217/7110/ The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann m …

M is an n-dim closed orientable manifold.I find in a book"Intuitively,the fundamental class can be thought of as the sum of the (top-dimension)simplices of a suitable triangulation …

Suppose we are given a commutative ring R with unit-element. Now we have a composition of R as the direct product of two rings $R\cong R_1\times R_2$. It is now straight forward, …

This question is inspired by Joseph O'Rourke's beautiful answer to my previous question. Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns hav …

Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question: Is there a …

Given $N$ and $a$ positive integers, with $a\ge 2$ is it possible to prove the inequality: $$\sum_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}\le\frac{N}{2}$$

Let $K$ be a local field with mix char, $k$ residue field. We have an exact sequence $0 \longrightarrow I \longrightarrow G_{K} \longrightarrow G_{k} \longrightarrow 0$ Then we o …

$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$ $$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) …

Suppose I have an affine subvariety $A \subset {\mathbb C}^N$ of dimension $n \geq 3$ which has an isolated singularity at $0$ (lets say for the sake of simplicity that it is non-s …

This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/397164/7110/ Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous …

For $n=1$ the answer is "yes." -- A group is abelian iff its generators commute. Let $G_0=G$ be a group and let it be generated by $X_0=X$. For each $n>0$ let $G_n=[G_{n-1},G_{n- …