All Questions
Tagged with ra.rings-and-algebras ca.classical-analysis-and-odes
9 questions
8
votes
1
answer
436
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
13
votes
4
answers
1k
views
Showing that a family of polynomials has positive and real roots.
Hi everybody, for my research I am dealing with the following function:
$$\alpha_n(x):=\left.\frac{\partial^{2n+1}}{\partial z^{2n+1}}\frac{\sinh(z)}{\cosh(z)-1+x}\right|_{z=0},\quad n\in \mathbb{N},...
- 131
2
votes
1
answer
570
views
Is a polynomial positive on the sphere a sum of squares of spherical harmonic polynomials?
Let $p\in {\mathbb{R}}[x_1,\ldots, x_n]$ be a homogenous polynomial of even degree $2d$. If $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$, then does there exist some $m>0$ and ...
user2529
24
votes
12
answers
3k
views
Constructions unique up to non-unique isomorphism
1) Fields have algebraic closures unique up to a non-unique isomorphism.
2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism.
3) Modules have ...
3
votes
1
answer
671
views
Is the set of polynomial sum of squares closed under limits?
A real polynomial $f(x_1,\ldots, x_n)$ in several variables is a sum of squares if there are polynomials $g_1,\ldots, g_k$ such that $f=g_1^2+\cdots +g_k^2$.
Fix a positive number $d>0$. The ...
user2529
3
votes
1
answer
566
views
Is every polynomial a limit of polynomials in quadratic variables?
Let $d>0$ be even. Consider ${\mathbb{R}}[x_1,\ldots, x_n]_d$, i.e. polynomials of degree $d$.
Call a homogeneous polynomial $f$ of degree $d$ a polynomial in quadratic variables if it is of the ...
user2529
5
votes
1
answer
2k
views
Examples and importance of Embedding (and Non-Embedding) Theorems
An embedding is an injective map into a universal, simpler model object. Many embedding theorems are without obstruction, in the sense that every object which you wish to embed can be embedded. ...
0
votes
2
answers
259
views
Existence of an "anti-additive" (or "never linear") map?
(I've edited this question)
I'm searching for a continuously differentiable function $f:\mathbb R^2\to\mathbb R$ such that $f(x)+f(x+u+v)\neq f(x+u)+f(x+v)$ for all $x$ and all linearly independent $...
- 365
10
votes
1
answer
835
views
what was Hilbert's geometric construction in his 17th problem?
Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
user2529