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Questions tagged [positivity]

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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) =...
Malkoun's user avatar
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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 ...
Malkoun's user avatar
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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 ...
Valerio's user avatar
  • 397
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 ...
Daniel Goc's user avatar
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} \...
Malkoun's user avatar
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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)=\...
astana's user avatar
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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)...
Peter O.'s user avatar
  • 697
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 ...
Oromis's user avatar
  • 151
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. ...
James Propp's user avatar
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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 ...
Malkoun's user avatar
  • 5,041
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 ...
Valerio's user avatar
  • 397
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 ...
Bogdan's user avatar
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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 ...
Jack's user avatar
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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 ...
Nicolas Medina Sanchez's user avatar
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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)=\...
Nicolas Medina Sanchez's user avatar
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},...
Leonid Petrov's user avatar
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 ...
T. Amdeberhan's user avatar
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 $...
Malkoun's user avatar
  • 5,041
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 ...
Louis Deaett's user avatar
  • 1,513
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 ...
Denis Serre's user avatar
  • 51.8k
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-...
Denis Serre's user avatar
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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 ...
Andromeda's user avatar
  • 169
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 ...
Andromeda's user avatar
  • 169
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 ...
Paul B. Slater's user avatar
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,...
mathoverflowUser's user avatar
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.,...
Andromeda's user avatar
  • 169
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 ...
Turbo's user avatar
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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_{\...
Turbo's user avatar
  • 13.7k
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 $\...
Turbo's user avatar
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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 $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...
MBolin's user avatar
  • 139
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\...
Black's user avatar
  • 471
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 ...
Malkoun's user avatar
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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 ...
Geordie Williamson's user avatar
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 ...
AThomas's user avatar
  • 587
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 ...
saolof's user avatar
  • 1,833
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\...
Hvjurthuk's user avatar
  • 573
-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 ...
Quiet_waters's user avatar
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 ...
Quiet_waters's user avatar
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 $n$...
mathoverflowUser's user avatar
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}\...
Denis Serre's user avatar
  • 51.8k
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 ...
Giorgio Metafune's user avatar
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 \,\,...
Trusio's user avatar
  • 71
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 ...
Jochen Glueck's user avatar
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 ...
SophiaA's user avatar
  • 61
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 $...
S. Euler's user avatar
  • 285
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, ...
Per Alexandersson's user avatar
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, ...
Tobias Fritz's user avatar
  • 5,967
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(...
Zermelo-Fraenkel's user avatar
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 ...
Arnold Neumaier's user avatar
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....
Luis Ferroni's user avatar
  • 1,889