Questions tagged [positivity]
The positivity tag has no usage guidance.
146
questions
7
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2
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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 ...
0
votes
0
answers
39
views
Find condition on signed matrix $A$ such that $\sum_jA_{ij}\cos(\theta_i-\theta_j)\geq 0$ implies $\sum_j\cos(\theta_i-\theta_j)\geq 0$?
As stated in the title, My question is:
Suppose $\sum_jA_{ij}\sin(\theta_i-\theta_j)=0,\forall i\in[n]$.
How to find condition on symmetric, diagonal-free, signed matrix $A$ ($A_{ij}=+1$ or $-1$ if $i\...
1
vote
0
answers
78
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
0
answers
193
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
313
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
296
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
267
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
491
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
301
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
143
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
191
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
65
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
111
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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:
$$
{}_{4}\phi_3\left(\begin{matrix} q^{-i_1},...
3
votes
3
answers
1k
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Positivity of a one-variable rational function
Let's consider the $1$-variable rational function
$$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
380
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
135
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
219
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
260
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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
390
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
207
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
577
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
311
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$$
where every entry of $A,b$ are non-negative and $A\in\{0,1\}^{m\times n}$ and there are $O(1)$ (say $\leq8$) non-negative entries per row of $A$ is it possible to ...
1
vote
0
answers
43
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Detecting non-negativity of a single constraint by polyhedral constraints - $II$
Let $$\langle a,x\rangle=b$$ be a linear constraint where $x\in\mathbb R^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\mathbb Z_{\...
0
votes
1
answer
108
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Detecting non-negativity of a single constraint by polyhedral constraints - $I$
We consider $$\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
294
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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
267
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
67
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
237
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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
378
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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
525
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
51
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
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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
589
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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
248
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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
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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
189
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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
147
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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
100
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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
308
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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
248
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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
321
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
154
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
301
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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....
2
votes
0
answers
171
views
Vanishing of a global sections space
Let $X\hookrightarrow\mathbb P^n_k$ be a projective variety over $k$ of degree $\delta$ with respect to $\mathcal O(1)$, $x\in X(k)$ be a regular rational point (I think a closed point is also OK). ...