Questions tagged [positivity]

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12 views

Cyclic P\'olya frequency function or cyclic variation-diminishing kernel of even order?

A cyclic P\'olya frequency function $f:(0,2\pi)\mapsto \mathbb R$ of order $2r+1$ (CPF$_{2r+1}$) satisfies $$\det[f(x_j-y_k)]_{j,k=1}^n\ge 0$$ for $n=1,2,\ldots,2r+1$ and $$x_1<x_2<\ldots<x_n&...
2
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90 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
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44 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
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91 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
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1answer
173 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
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0answers
48 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
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144 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 ...
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1answer
39 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
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139 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
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216 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....
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134 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). ...
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82 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^\...
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104 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\...
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103 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
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1answer
227 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 $...
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199 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 ...
5
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1answer
213 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 ...
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1answer
238 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 ...
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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,+\...
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84 views

Positive splitting of Sobolev convergence

Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions wuch that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
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1answer
191 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 $L$ ...
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1answer
66 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 ...
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1answer
156 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 ...
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1answer
189 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 ...
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1answer
151 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-...
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161 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 $\...
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98 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 ...
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1answer
119 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 \...
11
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2answers
582 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 $p(x) \ge 0$ ...
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108 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 ...
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2answers
1k 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 ...
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143 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 ...
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1answer
426 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 $L$ ...
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0answers
26 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 ...
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189 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}...
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1answer
475 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?
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1answer
100 views

Specific quaternary quartic that is positive semi-definite but not sum of squares

Does there exist a quaternary quartic $f$ (a form in $\mathbb{R}[x_1,x_2,x_3,x_4]$ of degree $4$), which is positive semi-definite ($f \geq 0$ on $\mathbb{R}^4$) but not a sum of squares, such that ...
1
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1answer
264 views

Probability of positive definiteness of a random matrix [duplicate]

Given an $n \times n$ symmetric random matrix whose entries have distribution $N(0,1)$, how to calculate the probability of positive definiteness of this matrix?
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195 views

Can we extend c.p. normal maps on a finite von Neumann algebra $M$ to $L_0(M)_+$?

Suppose that $M$ is a von Neumann algebra with a finite, normal, faithful trace $\tau$. Let $T\colon M\to M$ be a completely positive, normal map. Can $T$ be extended to a `positively linear map' ...
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3answers
2k views

Wanted: Positivity certificate for the AM-GM inequality in low dimension

I'm seeking for a Certificate of Positivity for the AM-GM inequality in five variables $$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$ Can one write the LHS as a sum $\,\...
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2answers
207 views

Positivity of coefficients of a polynomial derived from Schubert polynomials

Let $W=\bigcup_{n=1}^\infty S_n$ be the union of all symmetric groups $S_n$. For an element $w\in W$, denote by $\mathfrak{S}_w$ the Schubert polynomial associated to $w$, and by $\partial_w$ the ...
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2answers
483 views

What techniques are there to prove Schur positivity?

As the title says, what methods exists for proving that a symmetric polynomial (or function) is Schur positive, perhaps involving extra parameters, in which case coefficients should be polynomials in ...
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1answer
121 views

Does every locally positive-definite function have a positive-definite extension?

Let $B$ denote the unit ball in $\mathbb{R}^d$, and suppose $f\colon B\rightarrow\mathbb{C}$ has the property that for every $n\geq1$ and $x_1,\ldots,x_n\in\mathbb{R}^d$ with $\|x_i-x_j\|<1$, the $...
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0answers
25 views

Extending the projective action of several positive linear maps to a complex neighbourhood

I am currently reading a paper which, somewhat indirectly, asserts the following result: Lemma: Let $\Delta \subset \mathbb{R}^d$ denote the simplex $\{(x_1,\ldots,x_d):\sum_{i=1}^d x_i=1\}$, let $...
4
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1answer
224 views

A discrete operator begets even/odd polynomials

Given a function $f(x)$ define the forward shift operator by $Ef(x)=f(x+1)$ and the discrete derivative $\delta f(x)=(E-1)f(x)=f(x+1)-f(x)$. Given a partition $\lambda=(\lambda_1,\lambda_2,\dots,\...
8
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0answers
164 views

Hilbert spaces over the semi-field $\mathbb R_+$

Let $\mathbb R_+$ be the semi-field of non-negative real numbers. Definition (preliminary): A Hilbert space over $\mathbb R_+$ is a pair $(H,P)$, where $H$ is a complex Hilbert space, and $P\subset ...
1
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1answer
89 views

Is it true that $B \leq I \rightarrow B^2 \leq I$ (or higher powers) for positive semidefinite $B$, mimicking positive scalars? [closed]

If we know that $B \geq 0 $ (positive semidefinite) and that $I-B \geq 0$, is it necessarily true that $I-B^2 \geq 0$?
2
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1answer
165 views

Perturbation of linear system of equations: Is the solution still non-negative?

Let $A = (a_{ij})_{i,j=1,\dots,n}$ be a matrix such that $a_{ij} \ge 0$ for all $i,j = 1, \dots, n,$ and $A$ is positive definite. Let $I$ be the identity matrix, and $\pmb{1}$ the vector ...
2
votes
1answer
95 views

Interplay between the trace operator and the positive part of a Sobolev function

I consider the Sobolev trace $\gamma \colon H^1 (\mathbb{R}^{N+1}_{+}) \to H^{1/2}(\mathbb{R}^N)$. Let $U^{+}$ denote the positive part of a function $U$. It is correct, or only meaningful, to say ...
2
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0answers
80 views

Reference request: Positive solution of positive system of linear equations

Let $A \in \mathbb{R}^{n\times n}$ be an invertible matrix with positive entries, and $b \in \mathbb{R}^n$ a vector with positive entries. When does $A^{-1}b$ have all positive entries? I am looking ...