# Questions tagged [positivity]

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### 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 ...
199 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. ...
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### 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 ...
378 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 ...
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### 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
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### 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
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### 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 ...
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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 ...
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### 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.,...
49 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
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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 $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...
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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 ...
436 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 ...
190 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 $n$...
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1 vote
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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 $\mathbb{S}_{n}... 9 votes 1 answer 274 views ### Nonnegative coefficients of a product of polynomials Let$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]$such that the product$...
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 ...
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 ...