# Questions tagged [positivity]

The positivity tag has no usage guidance.

147
questions

0
votes

0
answers

90
views

### Sufficient conditions for the rank of the Hadamard product to be as large as possible

Let $A$ and $B$ be hermitian $n \times n$ positive semidefinite matrices. Assume that the rank of $A$ is 2. We know that
$$ \operatorname{rk}(A \circ B) \leq \operatorname{rk}(A) \operatorname{rk}(B) =...

0
votes

0
answers

80
views

### 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 ...

7
votes

2
answers

735
views

### 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 ...

1
vote

0
answers

82
views

### 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 ...

3
votes

0
answers

209
views

### 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} \...

2
votes

0
answers

136
views

### 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)=\...

2
votes

1
answer

219
views

### 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)...

4
votes

1
answer

331
views

### 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 ...

9
votes

2
answers

315
views

### 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. ...

5
votes

1
answer

288
views

### 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 ...

10
votes

1
answer

506
views

### 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 ...

4
votes

1
answer

328
views

### 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 ...

1
vote

1
answer

157
views

### 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
...

1
vote

1
answer

195
views

### 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 ...

0
votes

0
answers

67
views

### 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)=\...

3
votes

1
answer

115
views

### Nonnegativity of q-hypergeometric series

What are methods for proving nonnegativity of q-hypergeometric functions? Specifically, I have a function of the type 4-phi-3, it is a terminating series:
$$
{}_{4}\phi_3\left(\begin{matrix} q^{-i_1},...

3
votes

3
answers

1k
views

### Positivity of a one-variable rational function

Let's consider the $1$-variable rational function
$$F(z):=\frac{1-z}{(z^3 - z^2 + 2z - 1)\,(z^3 + z^2 + z - 1)}.$$
Numerical evidence convinces me of the truth of the following.
QUESTION. Can you ...

3
votes

0
answers

259
views

### Inequalities involving traces of products of hermitian positive semidefinite matrices

$\DeclareMathOperator{\tr}{tr}$
Fix an integer $n \geq 2$. Let $A_1, \dotsc, A_n$ be hermitian positive semidefinite matrices, with each $A_i$ being $m$ by $m$.
Consider the symmetric group $S_n$ on $...

10
votes

1
answer

411
views

### Symmetric polynomials that detect positivity

Imagine there are numbers $a_1,\ldots,a_n \in \mathbb R$ and you want to know whether they are all positive. You cannot access the numbers themselves, but you can choose any symmetric polynomials you ...

1
vote

1
answer

146
views

### Cofactor an geometrical mean in $\mathit{SPD}_3$: a Gårding-like inequality

The cofactor map $A\mapsto\widehat A$ is polynomial homogeneous of degree $n-1$ over $\mathbf M_n({\mathbb R})$. It can be polarized into an $(n-1)$-linear symmetric map. When $n=3$, this provides a ...

3
votes

1
answer

242
views

### Cofactor matrices and positive semi-definiteness

If $A$ is an $n\times n$ matrix with real entries, let me write $\widehat A$ its cofactor matrix. Since the map $A\mapsto\widehat A$ is polynomial, homogeneous of degree $n-1$, it can be multi-...

6
votes

2
answers

263
views

### Minimal injective extension is rigid

Let $V$ be an operator system.
Definition 1: A pair $(W, \kappa)$ is called extension of $V$ if $W$ is an operator system and $\kappa: V \to W$ is a unital complete isometry.
Definition 2: An ...

4
votes

1
answer

422
views

### If a completely positive unital map admits a completely positive unital left inverse, it is a complete isometry

Let $T$ be an injective operator system and $U$ be an arbitrary operator system. Let $\varphi: T \to U$ be a unital completely positive map and $\psi: U \to T$ be a unital completely positive map with ...

0
votes

1
answer

230
views

### Does positivity of the n(n-1)/2 principal minors formed from 2 x 2 submatrices ensure positive-definiteness of the n x n matrix itself?

I am interested in conditions under which an $n \times n$ matrix ($\rho$) is positive definite. Of course, one necessary and sufficient set of conditions is that the $n$ leading minors of $\rho$ each ...

3
votes

1
answer

635
views

### Is there a feature mapping for this kernel $k(x,y) = (\frac{\min(x,y)}{\max(x,y)})^2$?

In the following paper:
https://perso.crans.org/besson/publis/mva-2016/MVA_2015-16__Kernel_Methods__Homework__Besson_Clement_Zerbib.en.pdf
problem 2, Kernel 9. it is shown that $K(x,y) = \frac{\min(x,...

4
votes

1
answer

338
views

### Tensor product of positive linear maps is positive

Let $\pi_1: A_1 \to B_1$ and $\pi_2: A_2 \to B_2$ be positive linear maps between complex $*$-algebras. Is the mapping
$$\pi_1 \otimes \pi_2: A_1 \otimes A_2 \to B_1 \otimes B_2$$
again positive?
I.e.,...

3
votes

0
answers

50
views

### Modular counting of integral points under sparse non-negativity

Given a polyhedron
$$Ax\geq b$$
where every entry of $A,b$ are non-negative and $A\in\{0,1\}^{m\times n}$ and there are $O(1)$ (say $\leq8$) non-negative entries per row of $A$ is it possible to ...

1
vote

0
answers

43
views

### Detecting non-negativity of a single constraint by polyhedral constraints - $II$

Let $$\langle a,x\rangle=b$$ be a linear constraint where $x\in\mathbb R^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\mathbb Z_{\...

0
votes

1
answer

108
views

### Detecting non-negativity of a single constraint by polyhedral constraints - $I$

We consider $$\langle a,x\rangle=b$$ (linear constraints) where $x\in\mathbb R_{\geq0}^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\...

2
votes

1
answer

311
views

### How to check positive-definiteness of this function?

Consider a real random vector $\vec X =(X_1, X_2)$ with characteristic function $\phi(\vec t) \equiv \mathbb{E} \big[ e^{i \vec t \cdot \vec X} \big]$ (where $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...

5
votes

1
answer

278
views

### Non-unital Russo-Dye Theorem

Let $A$ be a C$^*$-algebra and let $\phi$ be a positive linear map from $A$ to $B(H)$ (bounded linear operators on Hilbert's
space). If $A$ is unital, then the Russo-Dye Theorem implies that $\|\phi\...

0
votes

0
answers

68
views

### Formulas involving traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices

While working on the Atiyah problem on configurations of points, I came across formulas involving products of traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices.
To ...

7
votes

0
answers

242
views

### Ample divisors on $T$-varieties

Question: how does one use a torus action to help decide whether a divisor or line bundle is ample?
In more detail: I have a (normal, but usually rather singular) $T$-variety $X$, where $T$ is an ...

10
votes

3
answers

397
views

### Positivity of Iwahori–Hecke algebra characters on the Kazhdan-Lusztig basis

$\DeclareMathOperator\tr{tr}$I'm interested in the irreducible characters of a finite Iwahori–Hecke algebra evaluated at the Kazhdan–Lusztig basis. These are Laurent polynomials.
Are the coefficients ...

12
votes

1
answer

548
views

### Is there a straightforward generalization of min(x,y) to positive-semidefinite Hermitian matrices?

This is an open-ended question I have. Is there a function of two positive-semidefinite hermitian operators $\min(A,B)$ returning another positive-semidefinite Hermitian operator such that:
If A and ...

2
votes

0
answers

54
views

### Does there always exist a(n uniform) polynomial that makes a positive definite symmetric matrix with polynomial entries into a sum of squares?

Suppose that I have a square and positive definite for every evaluation $x\in\mathbb{R}^{n}$ symmetric matrix $M(x)\in(\mathbb{R}[x])^{s\times s}.$
Does there always exist a polynomial $p(x)\in\...

-1
votes

1
answer

58
views

### Seek help to formalize an argument to positiveness of function defined inductively by integral [closed]

I have on domain $[0,\infty)$ a known and positive function $f(x)$ and two unknown functions $g(x), h(x)$ that start positive when $x=0$.
I also know that if $h(x)$ is positive, then $g(x)$ is also ...

3
votes

1
answer

642
views

### Is inverse Laplace Transform of a power of $s$ a positive function?

It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace ...

0
votes

1
answer

278
views

### The d(abc)-theorem, the abc-conjecture and positive definite kernels over the natural numbers?

I noticed in the work of Hector Pasten, Th.1.11 the $d(abc)$ theorem and have a question, after doing some experiments with sagemath.
Let $s_k(n) = \sum_{d|n}{ d^k }$ be the sum of divisiors $d$ of $n$...

15
votes

2
answers

1k
views

### Square root of doubly positive symmetric matrices

I wonder whether the following property holds true: For every real symmetric matrix $S$, which is positive in both senses:
$$\forall x\in{\mathbb R}^n,\,x^TSx\ge0,\qquad\forall 1\le i,j\le n,\,s_{ij}\...

6
votes

0
answers

213
views

### Extension of positive functionals

Let $X$ be a function space as $C(K)$ or $L^p$, with its usual norm and order, that is $f \le g$ if and only if $f(x) \le g(x)$ for a.e. $x$. If $M$ is a subspace of $X$ and $L:M \to \bf R$ is a ...

2
votes

0
answers

150
views

### Structure of the big cone and Seshadri constant on Fano manifolds

I would like to know something about the following two questions.
Given $X$ Fano manifold and $L$ an ample line bundle on $X$, we define
\begin{gather}
\sigma(L,x):=\sup\{t>0\, :\, \mu^{*}L-tE \,\,...

5
votes

0
answers

105
views

### Reference request: a survey of (linear) Krein-Rutman theory

I'm looking for a survey article or book chapter where a rather exhaustive treatment of the Krein-Rutman theory of positive linear operators an ordered Banach spaces is given.
Motivation. Some ...

6
votes

1
answer

333
views

### On strict positivity and Schmüdgen's Positivstellensatz

Schmüdgen's Positivstellensatz requires the polynomial to be strictly positive on a semialgebraic set. While trying to understand it, I am just wondering if the strictly positive condition can be ...

8
votes

1
answer

258
views

### Positive solutions for semilinear parabolic equations

Let $X$ be a Banach lattice. Consider the system
$$y'(t)=Ay(t)+f(t,y(t)) \qquad \text{in } (0,T) , \qquad y(0)=y_0, \qquad (*)$$
where $T>0$, $A$ generates an analytic positive semigroup $S(t)$ on $...

3
votes

0
answers

70
views

### Schur positive expression involving border-strip tableaux

Recall the power-sum expansion of Schur functions,
$$
s_\lambda = \sum_\mu \chi^{\lambda}(\mu) \frac{p_\mu}{z_\mu},
$$
in terms of Sn-character. These can be calculated
by the Murnaghan-Nakayama rule, ...

9
votes

1
answer

324
views

### Hahn's approach to Hilbert's 17th problem?

The Wikipedia article on Hahn Series mentions mentioned that these were studied by Hahn "in his approach to Hilbert's seventeenth problem".
Is this correct? If so, what was this approach, ...

1
vote

1
answer

64
views

### Positivity in extensions of ordered fields

Let $F$ be an ordered field and $f\in F[X]$ be a polynomial such that $f(x)>0$ for all $x\in K$. Is it possible that there is an extension $L\supseteq K$ of ordered fields and $y\in L$ such that $f(...

2
votes

0
answers

156
views

### Formula for a completely positive map

Is there a family of completely positive maps $L(A,B)$ depending continuously on two nonzero, symmetric positive semidefinite $n\times n$ matrices $A$ and $B$, such that $L(A,B)$ maps $A$ to $B$ and ...

4
votes

0
answers

304
views

### Positivity of a finite sum involving Stirling numbers of the first kind

Past days I've been trying to prove that certain polynomials have positive coefficients. After a lot of thinking, I came up with a formula for each coefficient individually, and they are not that ugly....