In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in \cite{yuan}. The DG methods in \cite{yuan} can maintain positivity and high order accuracy, but they rely both on the scaling limiter in \cite{zhang1} and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax-Wendroff type theorem is proved in \cite{yuan}, guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $P^k$ of piecewise polynomials of degree at most $k$. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in \cite{zhang1} can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $P^k$ or $Q^k$ spaces (piecewise $k$-th degree polynomials or piecewise tensor-product $k$-th degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in \cite{zhang1}. Computational results are provided to demonstrate the good performance of our DG schemes.

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We propose an inverse iterative method for computing the Perron pair of an irreducible nonnegative third order tensor. The method involves the selection of a parameter $\theta_k$ in the $k$th iteration. For every positive starting vector, the method converges quadratically and is positivity preserving in the sense that the vectors approximating the Perron vector are strictly positive in each iteration. It is also shown that $\theta_k=1$ near convergence. The computational work for each iteration of the proposed method is less than four times (three times if the tensor is symmetric in modes two and three, and twice if we also take the parameter to be $1$ directly) that for each iteration of the Ng--Qi--Zhou algorithm, which is linearly convergent for essentially positive tensors.

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group $G$ and a 4-cocycle twist $\omega_4$ of $G$'s cohomology group $\mathcal{H}^4(G,\mathbb{R}/\mathbb{Z})$ in 3 dimensional space and 1 dimensional time (3+1D). We establish the topological spin and the spin-statistics relation for the closed strings, and their multi-string braiding statistics. The 3+1D twisted gauge theory can be characterized by a representation of a modular transformation group SL$(3,\mathbb{Z})$. We express the SL$(3,\mathbb{Z})$ generators $\mathsf{S}^{xyz}$ and $\mathsf{T}^{xy}$ in terms of the gauge group $G$ and the 4-cocycle $\omega_4$. As we compactify one of the spatial %3D's directions $z$ into a compact circle with a gauge flux $b$ inserted, we can use the generators $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$ of an SL$(2,\mathbb{Z})$ subgroup to study the dimensional reduction of the 3D topological order $\mathcal{C}^{3\text{D}}$ to a direct sum of degenerate states of 2D topological orders $\mathcal{C}_b^{2\text{D}}$ in different flux $b$ sectors: $\mathcal{C}^{3\text{D}} = \oplus_b \mathcal{C}_b^{2\text{D}}$. The 2D topological orders $\mathcal{C}_b^{2\text{D}}$ are described by 2D gauge theories of the group $G$ twisted by the 3-cocycles $\omega_{3(b)}$, dimensionally reduced from the 4-cocycle $\omega_4$. We show that the SL$(2,\mathbb{Z})$ generators, $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$, fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.