Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Analogue of the Hilton-Milner Theorem for weak compositions   Let $\mathbb N_0$ be the set of non-negative integers, and let $P(n,l)$
denote the set of all weak compositions of $n$ with $l$ parts, i.e., $P(n,l)=\{
(x_1,x_2,\dots, x_l)\in\mathbb N_0^l\ :\ x_1+x_2+\cdots+x_l=n\}$. For any
element $\mathbf u=(u_1,u_2,\dots, u_l)\in P(n,l)$, denote its $i$th-coordinate
by $\mathbf u(i)$, i.e., $\mathbf u(i)=u_i$. A family $\mathcal A\subseteq
P(n,l)$ is said to be $t$-intersecting if $\vert \{ i \ :\ \mathbf u(i)=\mathbf
v(i)\} \vert\geq t$ for all $\mathbf u,\mathbf v\in \mathcal A$. A family
$\mathcal A\subseteq P(n,l)$ is said to be trivially $t$-intersecting if there
is a $t$-set $T$ of $\{1,2,\dots,l\}$ and elements $y_s\in \mathbb N_0$ ($s\in
T$) such that $\mathcal{A}= \{\mathbf u\in P(n,l)\ :\ \mathbf u(j)=y_j\ {\rm
for all}\ j\in T\}$. We prove that given any positive integers $l,t$ with
$l\geq 2t+3$, there exists a constant $n_0(l,t)$ depending only on $l$ and $t$,
such that for all $n\geq n_0(l,t)$, if $\mathcal{A} \subseteq P(n,l)$ is
non-trivially $t$-intersecting then \begin{equation} \vert \mathcal{A}
\vert\leq {n+l-t-1 \choose l-t-1}-{n-1 \choose l-t-1}+t.\notag \end{equation}
Moreover, equality holds if and only if there is a $t$-set $T$ of
$\{1,2,\dots,l\}$ such that \begin{equation} \mathcal A=\bigcup_{s\in
\{1,2,\dots, l\}\setminus T} \mathcal A_s\cup \left\{ \mathbf q_i\ :\ i\in T
\right\},\notag \end{equation} where \begin{align} \mathcal{A}_s & =\{\mathbf
u\in P(n,l)\ :\ \mathbf u(j)=0\ {\rm for all}\ j\in T\ {\rm and}\ \mathbf
u(s)=0\}\notag \end{align} and $\mathbf q_i\in P(n,l)$ with $\mathbf q_i(j)=0$
for all $j\in \{1,2,\dots, l\}\setminus \{i\}$ and $\mathbf q_i(i)=n$.

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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn