Questions tagged [positivity]

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4
votes
1answer
92 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.,...
0
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1answer
84 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
1answer
138 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 $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...
4
votes
1answer
207 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
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0answers
54 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
0answers
188 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
3answers
268 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
1answer
334 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
0answers
34 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
1answer
44 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 ...
2
votes
1answer
108 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
0answers
156 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 $n$...
0
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0answers
31 views

Random stationary set with prescribed variance

Let $\Psi$ be a non-vanishing continuous function $\mathbb{R}_+\to\mathbb{R}_+$ such that $\Psi(R)\leq R^{2d}$. Is it always possible to find $X$ a random stationary set of $\mathbb R^d$ (for ...
14
votes
2answers
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}\...
5
votes
0answers
92 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
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0answers
118 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
0answers
59 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 ...
5
votes
0answers
105 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
1answer
185 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 $...
2
votes
0answers
52 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, ...
6
votes
0answers
149 views

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

The Wikipedia article on Hahn Series mentions that these were studied by Hahn "in his approach to Hilbert's seventeenth problem". Is this correct? If so, what was this approach, and where can I ...
1
vote
1answer
45 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
0answers
141 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
0answers
235 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....
2
votes
0answers
137 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). ...
6
votes
0answers
91 views

$\ell^\infty / ces_0$ as an ordered Banach space

Let $ces_0:=\{\xi\in\ell^\infty : \lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}\xi_k=0\}$ and $q:\ell^\infty \to \ell^\infty/ces_0$ be the usual quotient map. The space $ces_0$ is closed in $\ell^\...
4
votes
0answers
109 views

Positivity of q-analogs of central binomial coefficients?

With the usual $q-$notations $[n]_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q},$ $[n]_q!=[1]_q[2]_q\cdots[n]_q$ and $\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$ let $$b(n,k,r,q)=\det\left(q^{r\...
1
vote
0answers
119 views

Preserving the strictly total positivity of special bases by using radial basis functions

Let $\left[\alpha,\beta\right]$ be a non-empty interval and consider an $\left(n+1\right)$-dimensional subspace $\mathbb{S}_{n}$ of $C^n\left(\left[\alpha,\beta\right]\right)$ such that $\mathbb{S}_{n}...
9
votes
1answer
242 views

Nonnegative coefficients of a product of polynomials

Let $P(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n]$. Does there exist an algorithm to decide whether there is a nonzero polynomial $Q(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n]$ such that the product $...
3
votes
0answers
227 views

On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...
6
votes
1answer
244 views

Positive-definiteness of radial sinc function in three dimensions

In dimension one, it is well known that $\mathcal{F}\chi_{(-1,1)}=\frac{\sin{x}}{x}$. This implies, in particular, that $\frac{\sin{x}}{x}$ is a definite positive function. I wonder if a similar ...
8
votes
1answer
270 views

Maps which are both completely positive and positive

Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...
36
votes
2answers
2k views

When can a function be made positive by averaging?

Let $f: {\bf Z} \to {\bf R}$ be a finitely supported function on the integers ${\bf Z}$. I am interested in knowing when there exists a finitely supported non-negative function $g: {\bf Z} \to [0,+\...
6
votes
0answers
103 views

Positive splitting of Sobolev convergence

Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
0
votes
1answer
214 views

Dual of a stable locally free subsheaf is a locally free quotient sheaf

Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$. By [1] definition 1.2: A line bundle $L$ ...
1
vote
1answer
75 views

Completeness of Lowner order in separable Hilbert space

Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the interesection of positive operator and ...
1
vote
1answer
169 views

positive real matrix-valued function as linear combination of positive-real functions

In my question I am considering $z$ as the complex variable in the $z$-transform $X(z)$ of a discrete-time sequence $x[n]$: I have $M$ square complex matrices $\mathbf{R}_m$ of size $N\times N$. I ...
1
vote
1answer
217 views

Positive Solutions of second-order ODE

Consider second-order ordinary differential equations of the form $u''(t)=a(t)u(t)-2$ I'm interested in general criteria on the function $a(t)$, which guarantee respectively rule out the existence ...
0
votes
1answer
188 views

Stability of a matrix product

Motivation: I am working on a research problem and have been stuck for a while. I hope someone can help, as it requires only linear algebra. :) Let $H$ be a real, invertible and positive semi-...
7
votes
0answers
170 views

Positivity of certain polynomial coefficients

Consider the rational functions (in fact, polynomials) $$F_n(q)=\frac1{(1-q)^{2n}}\sum_{k=0}^n(-q)^k\frac{2k+1}{n+k+1}\binom{2n}{n-k} \prod_{j=0,\,j\neq k}^n\frac{1+q^{2j+1}}{1+q}.$$ The numbers $\...
0
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0answers
106 views

Spectral projection of a positive operator

Let $(X,K)$ be a partially ordered Banach space where the cone $K$ is generating and normal. Suppose $B$ is a bounded operator on $X$ such that $B(K)\subseteq K$, the spectral radius $\rho(B)=1$ and ...
0
votes
1answer
122 views

Polynomials $p$ such that $p$ and $p'$ preserve nonnegative numbers

Expanding on a previous post I made recently, let $$ \mathscr{P}:= \{ p(x) \in \mathbb{R}[x] \mid p(x) \ge 0,~\forall x\ge 0\}. $$ The Pòlya-Szegö theorem (see Theorem 3.21 here) asserts that $p \in \...
12
votes
2answers
679 views

Polynomials that preserve nonnegativity

A polynomial $p \in \mathbb{R}[x_1,\dots,x_n]$ is said to be positive on a subset $S$ of $\mathbb{R}^n$ if $p(x) > 0$ for every $x \in S$. The polynomial $p$ is called nonnegative if $p(x) \ge 0$ ...
2
votes
0answers
110 views

Do integral curves on simple abelian surfaces define big line bundles?

Let $A$ be a simple abelian surface over $\mathbb{C}$. Let $C\subset A$ be an irreducible and reduced one-dimensional closed subscheme. Since $A$ is simple, the normalization of $C$ is of genus at ...
33
votes
2answers
2k views

Is this proof of Perron's theorem correct, and if so is it original?

A few years ago, I came up with this proof of Perron's theorem for a class presentation: http://www.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf I've written an outline of it below ...
0
votes
0answers
144 views

Continuity under various topologies for positive linear functionals

It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is ...
7
votes
1answer
457 views

nef vs. 1-nef vector bundles

Let $X$ be a compact, connected, Kähler manifold, of dimension $d$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$. By [1] definition 3.1.2: A line bundle $L$ ...
1
vote
0answers
27 views

Non Negative Tensor Tucker Decomposition Error Degradation

I have been working on iterative decomposition methods of tensors with non negativity constraints. I have noticed that $\textbf{N}$on negative $\textbf{T}$ensor $\textbf{F}$actorization "NTF" which is ...
3
votes
0answers
197 views

When is the strict transform of very ample divisor ample?

Let $X$ be a projective variety and $Y \subset X$ is a very ample divisor on $X$. Let $Z \subset X$ be a regular subvariety (but $X$ need not be regular along $Z$) of codimension $2$ and $\pi:\tilde{X}...
11
votes
1answer
564 views

Why do we care about Schur Positivity

Some of the most important open problems in Algebraic Combinatorics concern the Schur positivity of classes of symmetric functions. Why is this an important property to have?