Questions tagged [positivity]
The positivity tag has no usage guidance.
147
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Ampleness detected by the first Chern class?
Let $M$ be a compact complex manifold. By Kodaira's embedding theorem, a line bundle $L$ on $M$ is ample if and only if its holomorphic de Rham first Chern class $c_1(L)\in H_{\operatorname{dR}}^2(M)$ ...
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A conjectured generalization of Oppenheim's inequality, inspired by Horn-Yang's theorem
In this post, $A$ and $B$ are hermitian $n \times n$ positive semidefinite matrices.
It is well known that if $A$ has rank $n$ and if $B$ has only positive entries on its diagonal, then the rank of ...
- 5,118
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Polynomials such that $|p(z)|\leq p(|z|)$
Let $p(x)=1+p_1x+p_2x^2+\cdots+p_nx^n$ be a polynomial with real coefficients and no positive zeros. Define
$$\mu(x)=\frac{xp'(x)}{p(x)}, \hspace{3ex} \sigma(x)=x \mu'(x).$$
Many years ago, as part of ...
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Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$
Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...
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Do these cousins of permanents satisfy the following inequality?
Let $H$ denote an $n$ by $n$ hermitian positive semidefinite matrix. Let $G$ and $K$ be two subgroups of the symmetric group $\Sigma_n$. Define
$$ f_{G, K}(H) = \sum_{(\sigma, \tau) \in G \times K} \...
- 5,118
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Surfaces with $\Omega_X$ big are of general type
Given a complete algebraic variety $X$ over $\mathbb{C}$ and a vector bundle $E$ of rank $r$, let $\Omega(X,E)$ denote the graded ring $\bigoplus_{m\ge 0}H^0(X,S^mE)$, and define $$\lambda(E,X)=\...
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Bounds on the coin-flipping degree
Let $p(\lambda)$ be a polynomial that maps the closed unit interval to itself and satisfies $0\lt p(\lambda)\lt 1$ whenever $0\lt\lambda\lt 1$.
The polynomial can be written in power form:
$$p(\lambda)...
- 697
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One-point compactification of ample line bundle
Given a smooth complex projective variety with an ample line bundle $L$, it seems to be folklore that one can get a one-point compactification of the total space $\mathbb{V}(L)$ of $L$ such that ...
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Solving systems of linear equations without introducing negative numbers
Consider a system of $n$ linear equations with $n$ unknowns, all of whose coefficients and right hand sides are nonnegative integers, with a unique solution consisting of nonnegative rational numbers. ...
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Questions about hermitian positive semidefinite matrices
Motivation: I am faced with a $5 \times 5$ hermitian positive semidefinite matrix, depending on parameters, and I wish to show that it is positive definite, for any points in the parameter space (I ...
- 5,118
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509
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Conditions for a power of a polynomial to have no negative coefficients
Consider a polynomial in one variable $p(x)$ with $p(0)>0$, and that is not a polynomial in $x^m$ for any $m>1$ (that is, the $gcd$ of the exponents appearing in $p(x)$ is 1). I would like to ...
- 397
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Strong positivity of Neumann Laplacian
There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...
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Approximating a strictly increasing non-negative function on a non-negative domain by polynomials with non-negative coefficients
Let $f:[0,2]\rightarrow [0,\infty)$ be a strictly increasing smooth function. The Weierstrass approximation theorem says that we can uniformly approximate $f$ by polynomials. But my concern is
...
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Semisimple Lie algebra and convexity
There is any relation between semisimple lie algebras and symmetric cones? I'm saying this because the classification of the euclidean Jordan algebras, dual by Koecher-Vinberg theorem to homogeneous ...
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Non-proper orthant automorphisms
Given a real vector space $V$ of dimension $d$, it is known that the automorphism Lie group of the nonnegative orthant $\mathcal{O}^+_d$ can be described just as $$\mathrm{Aut}(\mathcal{O}^+_d)=\...