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A two-dimensional model of wildfire spread and merger is presented. Three features affect the fire propagation: (i) a constant basic rate of spread term accounting for radiative and convective heat transfer, (ii) the unidirectional, constant ambient wind, and (iii) a fire-induced pyrogenic wind. Two numerical methods are proposed to solve for the harmonic pyrogenic potential. The first utilizes the conformal invariance of the Laplace equation, reducing the wildfire system to a single Polubarinova-Galin type equation. The second method uses a AAA-least squares method to find a rational approximation of the potential. Various wildfire scenarios are presented and the effects of the pyrogenic wind and the radiative/convective basic rate of spread terms investigated. Firebreaks such as roads and lakes are also included and solutions are found to match well with existing numerical and experimental results. The methods proposed in this work are suitably fast and accurate to be considered for operational use., Comment: 22 pages, 12 figures. Submitted to Elsevier Environmental Modelling and Software

We show that quantum graph parameters for finite, simple, undirected graphs encode winning strategies for all possible synchronous non-local games. Given a synchronous game $\mathcal{G}=(I,O,\lambda)$ with $|I|=n$ and $|O|=k$, we demonstrate what we call a weak $*$-equivalence between $\mathcal{G}$ and a $3$-coloring game on a graph with at most $3+n+9n(k-2)+6|\lambda^{-1}(\{0\})|$ vertices, strengthening and simplifying work implied by Z. Ji (arXiv:1310.3794) for winning quantum strategies for synchronous non-local games. As an application, we obtain a quantum version of L. Lov\'{a}sz's reduction (Proc. 4th SE Conf. on Comb., Graph Theory & Computing, 1973) of the $k$-coloring problem for a graph $G$ with $n$ vertices and $m$ edges to the $3$-coloring problem for a graph with $3+n+9n(k-2)+6mk$ vertices. Moreover, winning strategies for a synchronous game $\mathcal{G}$ can be transformed into winning strategies for an associated graph coloring game, where the strategies exhibit perfect zero knowledge for an honest verifier. We also show that, for ``graph of the game" $X(\mathcal{G})$ associated to $\mathcal{G}$ from A. Atserias et al (J. Comb. Theory Series B, Vol. 136, 2019), the independence number game $\text{Hom}(K_{|I|},\overline{X(\mathcal{G})})$ is hereditarily $*$-equivalent to $\mathcal{G}$, so that the possibility of winning strategies is the same in both games for all models, except the game algebra. Thus, the quantum versions of the chromatic number, independence number and clique number encode winning strategies for all synchronous games in all quantum models., Comment: 30 pages; 2 figures. v2 adds an application of graph coloring games to perfect zero knowledge

Penguins huddling in a cold wind are represented by a two-dimensional, continuum model. The huddle boundary evolves due to heat loss to the huddle exterior and through the reorganisation of penguins as they seek to regulate their heat production within the huddle. These two heat transfer mechanisms, along with area, or penguin number, conservation, gives a free boundary problem whose dynamics depend on both the dynamics interior and exterior to the huddle. Assuming the huddle shape evolves slowly compared to the advective timescale of the exterior wind, the interior temperature is governed by a Poisson equation and the exterior temperature by the steady advection-diffusion equation. The exterior, advective wind velocity is the gradient of a harmonic, scalar field. The conformal invariance of the exterior governing equations is used to convert the system to a Polubarinova-Galin type equation, with forcing depending on both the interior and exterior temperature gradients at the huddle boundary. The interior Poisson equation is not conformally invariant, so the interior temperature gradient is found numerically using a combined adaptive Antoulas-Anderson and least squares algorithm. The results show that, irrespective of the starting shape, penguin huddles evolve into an egg-like steady shape. This shape is dependent on the wind strength, parameterised by the P\'eclet number Pe, and a parameter \b{eta} which effectively measures the strength of the interior self-generation of heat by the penguins. The numerical method developed is applicable to a further five free boundary problems., Comment: 16 pages, 8 figures, submitted to Acta Applicandae Mathematicae

We introduce classical and quantum no-signalling bicorrelations and characterise the different types thereof in terms of states on operator system tensor products, exhibiting connections with bistochastic operator matrices and with dilations of quantum magic squares. We define concurrent bicorrelations as a quantum input-output generalisation of bisynchronous correlations. We show that concurrent bicorrelations of quantum commuting type correspond to tracial states on the universal C*-algebra of the projective free unitary quantum group, showing that in the quantum input-output setup, quantum permutations of finite sets must be replaced by quantum automorphisms of matrix algebras. We apply our results to study the quantum graph isomorphism game, describing the game C*-algebra in this case, and make precise connections with the algebraic notions of quantum graph isomorphism, existing presently in the literature., Comment: 76 pages

We compare the algebras of the quantum automorphism group of finite-dimensional C$^\ast$-algebra $B$, which includes the quantum permutation group $S_N^+$, where $N = \dim B$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $\Gamma$ lead to isomorphic $\ast$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $1$-boundedness between the various algebras associated with these quantum groups., Comment: 27 pages. Revised version to appear in IMRN

We establish several strong equivalences of synchronous non-local games, in the sense that the corresponding game algebras are $*$-isomorphic. We first show that the game algebra of any synchronous game on $n$ inputs and $k$ outputs is $*$-isomorphic to the game algebra of an associated bisynchronous game on $nk$ inputs and $nk$ outputs. As a result, we show that there are bisynchronous games with equal question and answer sets, whose optimal strategies only exist in the quantum commuting model, and not in the quantum approximate model. Moreover, we exhibit a bisynchronous game with $20$ questions and $20$ answers that has a non-zero game algebra, but no winning commuting strategy, resolving a problem of V.I. Paulsen and M. Rahaman. We also exhibit a $*$-isomorphism between any synchronous game algebra with $n$ questions and $k>3$ answers and a synchronous game algebra with $n(k-2)$ questions and $3$ answers., Comment: 20 pages

We introduce concurrent quantum non-local games, quantum output mirror games and concurrent classical-to-quantum non-local games, as quantum versions of synchronous non-local games, and provide tracial characterisations of their perfect strategies belonging to various correlation classes. We define *-algebras and C*-algebras of concurrent classical-to-quantum and concurrent quantum non-local games, and algebraic versions of the orthogonal rank of a graph. We show that quantum homomorphisms of quantum graphs can be viewed as entanglement assisted classical homomorphisms of the graphs, and give descriptions of the perfect quantum commuting and the perfect approximately quantum strategies for the quantum graph homomorphism game. We specialise the latter results to the case where the inputs of the game are based on a classical graph.

Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a "quantum-classical game"--that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analogue of the notion of non-commutative graph homomorphisms due to D. Stahlke [44]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by J.W. Helton, K. Meyer, V.I. Paulsen and M. Satriano [22]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the $4$-coloring game for a quantum graph is always non-trivial, extending a result of [22]., Comment: v2: fixed errors in proofs of the old Theorem 4.7 and 5.6; removed Lemma 4.8

We consider the tensor product of the completely depolarising channel on $d\times d$ matrices with the map of Schur multiplication by a $k \times k$ correlation matrix and characterise, via matrix theory methods, when such a map is a mixed (random) unitary channel. When $d=1$, this recovers a result of O'Meara and Pereira, and for larger $d$ is equivalent to a result of Haagerup and Musat that was originally obtained via the theory of factorisation through von Neumann algebras. We obtain a bound on the distance between a given correlation matrix for which this tensor product is nearly mixed unitary and a correlation matrix for which such a map is exactly mixed unitary. This bound allows us to give an elementary proof of another result of Haagerup and Musat about the closure of such correlation matrices without appealing to the theory of von Neumann algebras., Comment: 15 pages; new results in Section 3 (updated Propositions 3.2, 3.3)

We consider the matrix-valued generalizations of bipartite tensor product quantum correlations and bipartite infinite-dimensional tensor product quantum correlations, respectively. These sets are denoted by $C_q^{(n)}(m,k)$ and $C_{qs}^{(n)}(m,k)$, respectively, where $m$ is the number of inputs, $k$ is the number of outputs, and $n$ is the matrix size. We show that, for any $m,k \geq 2$ with $(m,k) \neq (2,2)$, there is an $n \leq 4$ for which we have the separation $C_q^{(n)}(m,k) \neq C_{qs}^{(n)}(m,k)$., Comment: 11 pages

In this paper we introduce the crossed product construction for a discrete group action on an operator system. In analogy to the work of E. Katsoulis and C. Ramsey, we describe three canonical crossed products arising from such a dynamical system. We describe how these crossed product constructions behave under $G$-equivariant maps, tensor products, and the canonical $C^*$-covers. We show that hyperrigidity is preserved under two of the three crossed products. Finally, using A. Kavruk's notion of an operator system that detects $C^*$-nuclearity, we give a negative answer to a question on operator algebra crossed products posed by Katsoulis and Ramsey., Comment: 31 pages. Final version to appear in the Journal of Functional Analysis

Let $\mathcal{B}_d$ be the unital $C^*$-algebra generated by the elements $u_{jk}, \, 0 \le i, j \le d-1$, satisfying the relations that $[u_{j,k}]$ is a unitary operator, and let $C^*(\mathbb{F}_{d^2})$ be the full group $C^*$-algebra of free group of $d^2$ generators. Based on the idea of teleportation and super-dense coding in quantum information theory, we exhibit the two $*$-isomorphisms $M_d(C^*(\mathbb{F}_{d^2}))\cong \mathcal{B}_d\rtimes \mathbb{Z}_d\rtimes \mathbb{Z}_d$ and $M_d(\mathcal{B}_d)\cong C^*(\mathbb{F}_{d^2})\rtimes \mathbb{Z}_d\rtimes \mathbb{Z}_d$, for certain actions of $\mathbb{Z}_d$. As an application, we show that for any $d,m\ge 2$ with $(d,m)\neq (2,2)$, the matrix-valued generalization of the (tensor product) quantum correlation set of $d$ inputs and $m$ outputs is not closed., Comment: 28pages. Comments are welcome!

We consider quantum XOR games, defined in [11], from the perspective of unitary correlations defined in [7]. We show that Connes' embedding problem has a positive answer if and only if every quantum XOR game has entanglement bias equal to the commuting bias. In particular, the embedding problem is equivalent to determining whether every quantum XOR game $G$ with a winning strategy in the commuting model also has a winning strategy in the approximate finite-dimensional model., Comment: 10 pages

The unitary correlation sets defined by the first author in conjunction with tensor products of $\mathcal{U}_{nc}(n)$ are further studied. We show that Connes' embedding problem is equivalent to deciding whether or not two smaller versions of the unitary correlation sets are equal. Moreover, we obtain the result that Connes' embedding problem is equivalent to deciding whether or not two cross norms on $M_n \otimes M_n$ are equal for all $n \geq 2$., Comment: 24 pages, fixed a small error in the proof of Theorem 5.3 (v2)

The $C^{\ast}$-algebra $\mathcal{U}_{nc}(n)$ is the universal $C^{\ast}$-algebra generated by $n^2$ generators $u_{ij}$ that make up a unitary matrix. We prove that Kirchberg's formulation of Connes' embedding problem has a positive answer if and only if $\mathcal{U}_{nc}(2) \otimes_{\min} \mathcal{U}_{nc}(2)=\mathcal{U}_{nc}(2) \otimes_{\max} \mathcal{U}_{nc}(2)$. Our results follow from properties of the finite-dimensional operator system $\mathcal{V}_n$ spanned by $1$ and the generators of $\mathcal{U}_{nc}(n)$. We show that $\mathcal{V}_n$ is an operator system quotient of $M_{2n}$ and has the OSLLP. We obtain necessary and sufficient conditions on $\mathcal{V}_n$ for there to be a positive answer to Kirchberg's problem. Finally, in analogy with recent results of Ozawa, we show that a form of Tsirelson's problem related to $\mathcal{V}_n$ is equivalent to Connes' Embedding problem., Comment: 29 pages, final version to appear in the Indiana University Mathematics Journal