# Questions tagged [positivity]

The tag has no usage guidance.

132 questions
Filter by
Sorted by
Tagged with
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 ...
249 views

176 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.,...
46 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
42 views

179 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 $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...
237 views

48 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 ...
258 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 ...
177 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 $n$...
1k views

1 vote
123 views

239 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 ...
286 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 ...
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 ...