Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dirichlet polynomials and entropy   A Dirichlet polynomial $d$ in one variable ${\mathcal{y}}$ is a function of
the form $d({\mathcal{y}})=a_n
n^{\mathcal{y}}+\cdots+a_22^{\mathcal{y}}+a_11^{\mathcal{y}}+a_00^{\mathcal{y}}$
for some $n,a_0,\ldots,a_n\in\mathbb{N}$. We will show how to think of a
Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical
distribution. We can then consider the Shannon entropy $H(d)$ of the
corresponding probability distribution, and we define its length (or,
classically, its perplexity) by $L(d)=2^{H(d)}$. On the other hand, we will
define a rig homomorphism $h\colon\mathsf{Dir}\to\mathsf{Rect}$ from the rig of
Dirichlet polynomials to the so-called rectangle rig, whose underlying set is
$\mathbb{R}_{\geq0}\times\mathbb{R}_{\geq0}$ and whose additive structure
involves the weighted geometric mean; we write $h(d)=(A(d),W(d))$, and call the
two components area and width (respectively).
  The main result of this paper is the following: the rectangle-area formula
$A(d)=L(d)W(d)$ holds for any Dirichlet polynomial $d$. In other words, the
entropy of an empirical distribution can be calculated entirely in terms of the
homomorphism $h$ applied to its corresponding Dirichlet polynomial. We also
show that similar results hold for the cross entropy.

26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg