Given integers $n > k > 0$, and a set of integers $L \subset [0, k-1]$, an $L$-system is a family of sets $\mathcal{F} \subset \binom{[n]}{k}$ such that $|F \cap F'| \in L$ for distinct $F, F'\in \mathcal{F}$. $L$-systems correspond to independent sets in a certain generalized Johnson graph $G(n, k, L)$, so that the maximum size of an $L$-system is equivalent to finding the independence number of the graph $G(n, k, L)$. The Lov\'asz number $\vartheta(G)$ is a semidefinite programming approximation of the independence number of a graph $G$. In this paper, we determine the order of magnitude of $\vartheta(G(n, k, L))$ of any generalized Johnson graph with $k$ and $L$ fixed and $n\rightarrow \infty$. As an application of this theorem, we give an explicit construction of a graph $G$ on $n$ vertices with large gap between the Lov\'asz number and the Shannon capacity $c(G)$. Specifically, we prove that for any $\epsilon > 0$, for infinitely many $n$ there is a generalized Johnson graph $G$ on $n$ vertices which has ratio $\vartheta(G)/c(G) = \Omega(n^{1-\epsilon})$, which greatly improves on the best known explicit construction.

Classical Krylov subspace projection methods for the solution of linear problem $Ax = b$ output an approximate solution $\widetilde{x}\simeq x$. Recently, it has been recognized that projection methods can be understood from a statistical perspective. These probabilistic projection methods return a distribution $p(\widetilde{x})$ in place of a point estimate $\widetilde{x}$. The resulting uncertainty, codified as a distribution, can, in theory, be meaningfully combined with other uncertainties, can be propagated through computational pipelines, and can be used in the framework of probabilistic decision theory. The problem we address is that the current probabilistic projection methods lead to the poorly calibrated posterior distribution. We improve the covariance matrix from previous works in a way that it does not contain such undesirable objects as $A^{-1}$ or $A^{-1}A^{-T}$, results in nontrivial uncertainty, and reproduces an arbitrary projection method as a mean of the posterior distribution. We also propose a variant that is numerically inexpensive in the case the uncertainty is calibrated a priori. Since it usually is not, we put forward a practical way to calibrate uncertainty that performs reasonably well, albeit at the expense of roughly doubling the numerical cost of the underlying projection method.

An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+\epsilon)$-spanner of a metric space is called (vertex) $f$-fault-tolerant ($f$-FT) if it remains a $(1+\epsilon)$-spanner (for the non-faulty points) when up to $f$ faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which na\"{\i}vely runs in $O(n^3) \cdot 2^{O(f)}$ time. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an $f$-FT spanner with lightness that depends solely on $f$ (even exponentially): all existing constructions have lightness $\Omega(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $\Omega(\log n)$. In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of $f$-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any $n$-point doubling metric, any $\epsilon > 0$, and any integer $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+\epsilon)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$.

Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.

We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations, which generalize the free Euler-Maruyama method introduced by Schl\"{u}chtermann and Wibmer [27]. Under some mild conditions, we prove the strong convergence and exponential stability in mean square of the numerical solution. The free stochastic theta method with $\theta=1$ can inherit the exponential stability of original equations for any given step size. Our method can offer better stability and efficiency than the free Euler-Maruyama method. Moreover, numerical results are reported to confirm these theoretical findings.

Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations which holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments including contact waves and magnetic reconnection.

Input-output conformance simulation (iocos) has been proposed by Gregorio-Rodr\'iguez, Llana and Mart\'inez-Torres as a simulation-based behavioural preorder underlying model-based testing. This relation is inspired by Tretmans' classic ioco relation, but has better worst-case complexity than ioco and supports stepwise refinement. The goal of this paper is to develop the theory of iocos by studying logical characterisations of this relation, rule formats for it and its compositionality. More specifically, this article presents characterisations of iocos in terms of modal logics and compares them with an existing logical characterisation for ioco proposed by Beohar and Mousavi. It also offers a characteristic-formula construction for iocos over finite processes in an extension of the proposed modal logics with greatest fixed points. A precongruence rule format for iocos and a rule format ensuring that operations take quiescence properly into account are also given. Both rule formats are based on the GSOS format by Bloom, Istrail and Meyer. The general modal decomposition methodology of Fokkink and van Glabbeek is used to show how to check the satisfaction of properties expressed in the logic for iocos in a compositional way for operations specified by rules in the precongruence rule format for iocos .

This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq-Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ODE. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data is not known.

Erd\H{o}s and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

We define game semantics for the constructive $\mu$-calculus and prove its equivalence to bi-relational semantics. We use the game semantics to prove that the $\mu$-calculus collapses to modal logic over constructive variants of $\mathsf{S5}$. Finally, we use the collapse to prove the completeness of constructive variants of $\mathsf{S5}$ with fixed-point operators.

Given integers $n > k > 0$, and a set of integers $L \subset [0, k-1]$, an $L$-system is a family of sets $\mathcal{F} \subset \binom{[n]}{k}$ such that $|F \cap F'| \in L$ for distinct $F, F'\in \mathcal{F}$. $L$-systems correspond to independent sets in a certain generalized Johnson graph $G(n, k, L)$, so that the maximum size of an $L$-system is equivalent to finding the independence number of the graph $G(n, k, L)$. The Lov\'asz number $\vartheta(G)$ is a semidefinite programming approximation of the independence number of a graph $G$. In this paper, we determine the order of magnitude of $\vartheta(G(n, k, L))$ of any generalized Johnson graph with $k$ and $L$ fixed and $n\rightarrow \infty$. As an application of this theorem, we give an explicit construction of a graph $G$ on $n$ vertices with large gap between the Lov\'asz number and the Shannon capacity $c(G)$. Specifically, we prove that for any $\epsilon > 0$, for infinitely many $n$ there is a generalized Johnson graph $G$ on $n$ vertices which has ratio $\vartheta(G)/c(G) = \Omega(n^{1-\epsilon})$, which greatly improves on the best known explicit construction.

Classical Krylov subspace projection methods for the solution of linear problem $Ax = b$ output an approximate solution $\widetilde{x}\simeq x$. Recently, it has been recognized that projection methods can be understood from a statistical perspective. These probabilistic projection methods return a distribution $p(\widetilde{x})$ in place of a point estimate $\widetilde{x}$. The resulting uncertainty, codified as a distribution, can, in theory, be meaningfully combined with other uncertainties, can be propagated through computational pipelines, and can be used in the framework of probabilistic decision theory. The problem we address is that the current probabilistic projection methods lead to the poorly calibrated posterior distribution. We improve the covariance matrix from previous works in a way that it does not contain such undesirable objects as $A^{-1}$ or $A^{-1}A^{-T}$, results in nontrivial uncertainty, and reproduces an arbitrary projection method as a mean of the posterior distribution. We also propose a variant that is numerically inexpensive in the case the uncertainty is calibrated a priori. Since it usually is not, we put forward a practical way to calibrate uncertainty that performs reasonably well, albeit at the expense of roughly doubling the numerical cost of the underlying projection method.

An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+\epsilon)$-spanner of a metric space is called (vertex) $f$-fault-tolerant ($f$-FT) if it remains a $(1+\epsilon)$-spanner (for the non-faulty points) when up to $f$ faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which na\"{\i}vely runs in $O(n^3) \cdot 2^{O(f)}$ time. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an $f$-FT spanner with lightness that depends solely on $f$ (even exponentially): all existing constructions have lightness $\Omega(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $\Omega(\log n)$. In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of $f$-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any $n$-point doubling metric, any $\epsilon > 0$, and any integer $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+\epsilon)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$.

Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.

We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations, which generalize the free Euler-Maruyama method introduced by Schl\"{u}chtermann and Wibmer [27]. Under some mild conditions, we prove the strong convergence and exponential stability in mean square of the numerical solution. The free stochastic theta method with $\theta=1$ can inherit the exponential stability of original equations for any given step size. Our method can offer better stability and efficiency than the free Euler-Maruyama method. Moreover, numerical results are reported to confirm these theoretical findings.

Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations which holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments including contact waves and magnetic reconnection.

Input-output conformance simulation (iocos) has been proposed by Gregorio-Rodr\'iguez, Llana and Mart\'inez-Torres as a simulation-based behavioural preorder underlying model-based testing. This relation is inspired by Tretmans' classic ioco relation, but has better worst-case complexity than ioco and supports stepwise refinement. The goal of this paper is to develop the theory of iocos by studying logical characterisations of this relation, rule formats for it and its compositionality. More specifically, this article presents characterisations of iocos in terms of modal logics and compares them with an existing logical characterisation for ioco proposed by Beohar and Mousavi. It also offers a characteristic-formula construction for iocos over finite processes in an extension of the proposed modal logics with greatest fixed points. A precongruence rule format for iocos and a rule format ensuring that operations take quiescence properly into account are also given. Both rule formats are based on the GSOS format by Bloom, Istrail and Meyer. The general modal decomposition methodology of Fokkink and van Glabbeek is used to show how to check the satisfaction of properties expressed in the logic for iocos in a compositional way for operations specified by rules in the precongruence rule format for iocos .

This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq-Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ODE. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data is not known.

Erd\H{o}s and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

We define game semantics for the constructive $\mu$-calculus and prove its equivalence to bi-relational semantics. We use the game semantics to prove that the $\mu$-calculus collapses to modal logic over constructive variants of $\mathsf{S5}$. Finally, we use the collapse to prove the completeness of constructive variants of $\mathsf{S5}$ with fixed-point operators.

Given integers $n > k > 0$, and a set of integers $L \subset [0, k-1]$, an $L$-system is a family of sets $\mathcal{F} \subset \binom{[n]}{k}$ such that $|F \cap F'| \in L$ for distinct $F, F'\in \mathcal{F}$. $L$-systems correspond to independent sets in a certain generalized Johnson graph $G(n, k, L)$, so that the maximum size of an $L$-system is equivalent to finding the independence number of the graph $G(n, k, L)$. The Lov\'asz number $\vartheta(G)$ is a semidefinite programming approximation of the independence number of a graph $G$. In this paper, we determine the order of magnitude of $\vartheta(G(n, k, L))$ of any generalized Johnson graph with $k$ and $L$ fixed and $n\rightarrow \infty$. As an application of this theorem, we give an explicit construction of a graph $G$ on $n$ vertices with large gap between the Lov\'asz number and the Shannon capacity $c(G)$. Specifically, we prove that for any $\epsilon > 0$, for infinitely many $n$ there is a generalized Johnson graph $G$ on $n$ vertices which has ratio $\vartheta(G)/c(G) = \Omega(n^{1-\epsilon})$, which greatly improves on the best known explicit construction.

Classical Krylov subspace projection methods for the solution of linear problem $Ax = b$ output an approximate solution $\widetilde{x}\simeq x$. Recently, it has been recognized that projection methods can be understood from a statistical perspective. These probabilistic projection methods return a distribution $p(\widetilde{x})$ in place of a point estimate $\widetilde{x}$. The resulting uncertainty, codified as a distribution, can, in theory, be meaningfully combined with other uncertainties, can be propagated through computational pipelines, and can be used in the framework of probabilistic decision theory. The problem we address is that the current probabilistic projection methods lead to the poorly calibrated posterior distribution. We improve the covariance matrix from previous works in a way that it does not contain such undesirable objects as $A^{-1}$ or $A^{-1}A^{-T}$, results in nontrivial uncertainty, and reproduces an arbitrary projection method as a mean of the posterior distribution. We also propose a variant that is numerically inexpensive in the case the uncertainty is calibrated a priori. Since it usually is not, we put forward a practical way to calibrate uncertainty that performs reasonably well, albeit at the expense of roughly doubling the numerical cost of the underlying projection method.

An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+\epsilon)$-spanner of a metric space is called (vertex) $f$-fault-tolerant ($f$-FT) if it remains a $(1+\epsilon)$-spanner (for the non-faulty points) when up to $f$ faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which na\"{\i}vely runs in $O(n^3) \cdot 2^{O(f)}$ time. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an $f$-FT spanner with lightness that depends solely on $f$ (even exponentially): all existing constructions have lightness $\Omega(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $\Omega(\log n)$. In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of $f$-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any $n$-point doubling metric, any $\epsilon > 0$, and any integer $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+\epsilon)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$.

Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.

We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations, which generalize the free Euler-Maruyama method introduced by Schl\"{u}chtermann and Wibmer [27]. Under some mild conditions, we prove the strong convergence and exponential stability in mean square of the numerical solution. The free stochastic theta method with $\theta=1$ can inherit the exponential stability of original equations for any given step size. Our method can offer better stability and efficiency than the free Euler-Maruyama method. Moreover, numerical results are reported to confirm these theoretical findings.

Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations which holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments including contact waves and magnetic reconnection.

Input-output conformance simulation (iocos) has been proposed by Gregorio-Rodr\'iguez, Llana and Mart\'inez-Torres as a simulation-based behavioural preorder underlying model-based testing. This relation is inspired by Tretmans' classic ioco relation, but has better worst-case complexity than ioco and supports stepwise refinement. The goal of this paper is to develop the theory of iocos by studying logical characterisations of this relation, rule formats for it and its compositionality. More specifically, this article presents characterisations of iocos in terms of modal logics and compares them with an existing logical characterisation for ioco proposed by Beohar and Mousavi. It also offers a characteristic-formula construction for iocos over finite processes in an extension of the proposed modal logics with greatest fixed points. A precongruence rule format for iocos and a rule format ensuring that operations take quiescence properly into account are also given. Both rule formats are based on the GSOS format by Bloom, Istrail and Meyer. The general modal decomposition methodology of Fokkink and van Glabbeek is used to show how to check the satisfaction of properties expressed in the logic for iocos in a compositional way for operations specified by rules in the precongruence rule format for iocos .

This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq-Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ODE. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data is not known.

Erd\H{o}s and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

We define game semantics for the constructive $\mu$-calculus and prove its equivalence to bi-relational semantics. We use the game semantics to prove that the $\mu$-calculus collapses to modal logic over constructive variants of $\mathsf{S5}$. Finally, we use the collapse to prove the completeness of constructive variants of $\mathsf{S5}$ with fixed-point operators.

Given integers $n > k > 0$, and a set of integers $L \subset [0, k-1]$, an $L$-system is a family of sets $\mathcal{F} \subset \binom{[n]}{k}$ such that $|F \cap F'| \in L$ for distinct $F, F'\in \mathcal{F}$. $L$-systems correspond to independent sets in a certain generalized Johnson graph $G(n, k, L)$, so that the maximum size of an $L$-system is equivalent to finding the independence number of the graph $G(n, k, L)$. The Lov\'asz number $\vartheta(G)$ is a semidefinite programming approximation of the independence number of a graph $G$. In this paper, we determine the order of magnitude of $\vartheta(G(n, k, L))$ of any generalized Johnson graph with $k$ and $L$ fixed and $n\rightarrow \infty$. As an application of this theorem, we give an explicit construction of a graph $G$ on $n$ vertices with large gap between the Lov\'asz number and the Shannon capacity $c(G)$. Specifically, we prove that for any $\epsilon > 0$, for infinitely many $n$ there is a generalized Johnson graph $G$ on $n$ vertices which has ratio $\vartheta(G)/c(G) = \Omega(n^{1-\epsilon})$, which greatly improves on the best known explicit construction.

Classical Krylov subspace projection methods for the solution of linear problem $Ax = b$ output an approximate solution $\widetilde{x}\simeq x$. Recently, it has been recognized that projection methods can be understood from a statistical perspective. These probabilistic projection methods return a distribution $p(\widetilde{x})$ in place of a point estimate $\widetilde{x}$. The resulting uncertainty, codified as a distribution, can, in theory, be meaningfully combined with other uncertainties, can be propagated through computational pipelines, and can be used in the framework of probabilistic decision theory. The problem we address is that the current probabilistic projection methods lead to the poorly calibrated posterior distribution. We improve the covariance matrix from previous works in a way that it does not contain such undesirable objects as $A^{-1}$ or $A^{-1}A^{-T}$, results in nontrivial uncertainty, and reproduces an arbitrary projection method as a mean of the posterior distribution. We also propose a variant that is numerically inexpensive in the case the uncertainty is calibrated a priori. Since it usually is not, we put forward a practical way to calibrate uncertainty that performs reasonably well, albeit at the expense of roughly doubling the numerical cost of the underlying projection method.

An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+\epsilon)$-spanner of a metric space is called (vertex) $f$-fault-tolerant ($f$-FT) if it remains a $(1+\epsilon)$-spanner (for the non-faulty points) when up to $f$ faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which na\"{\i}vely runs in $O(n^3) \cdot 2^{O(f)}$ time. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an $f$-FT spanner with lightness that depends solely on $f$ (even exponentially): all existing constructions have lightness $\Omega(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $\Omega(\log n)$. In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of $f$-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any $n$-point doubling metric, any $\epsilon > 0$, and any integer $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+\epsilon)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$.

Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.

We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations, which generalize the free Euler-Maruyama method introduced by Schl\"{u}chtermann and Wibmer [27]. Under some mild conditions, we prove the strong convergence and exponential stability in mean square of the numerical solution. The free stochastic theta method with $\theta=1$ can inherit the exponential stability of original equations for any given step size. Our method can offer better stability and efficiency than the free Euler-Maruyama method. Moreover, numerical results are reported to confirm these theoretical findings.

Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations which holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments including contact waves and magnetic reconnection.

Input-output conformance simulation (iocos) has been proposed by Gregorio-Rodr\'iguez, Llana and Mart\'inez-Torres as a simulation-based behavioural preorder underlying model-based testing. This relation is inspired by Tretmans' classic ioco relation, but has better worst-case complexity than ioco and supports stepwise refinement. The goal of this paper is to develop the theory of iocos by studying logical characterisations of this relation, rule formats for it and its compositionality. More specifically, this article presents characterisations of iocos in terms of modal logics and compares them with an existing logical characterisation for ioco proposed by Beohar and Mousavi. It also offers a characteristic-formula construction for iocos over finite processes in an extension of the proposed modal logics with greatest fixed points. A precongruence rule format for iocos and a rule format ensuring that operations take quiescence properly into account are also given. Both rule formats are based on the GSOS format by Bloom, Istrail and Meyer. The general modal decomposition methodology of Fokkink and van Glabbeek is used to show how to check the satisfaction of properties expressed in the logic for iocos in a compositional way for operations specified by rules in the precongruence rule format for iocos .

This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq-Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ODE. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data is not known.

Erd\H{o}s and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

We define game semantics for the constructive $\mu$-calculus and prove its equivalence to bi-relational semantics. We use the game semantics to prove that the $\mu$-calculus collapses to modal logic over constructive variants of $\mathsf{S5}$. Finally, we use the collapse to prove the completeness of constructive variants of $\mathsf{S5}$ with fixed-point operators.

Given integers $n > k > 0$, and a set of integers $L \subset [0, k-1]$, an $L$-system is a family of sets $\mathcal{F} \subset \binom{[n]}{k}$ such that $|F \cap F'| \in L$ for distinct $F, F'\in \mathcal{F}$. $L$-systems correspond to independent sets in a certain generalized Johnson graph $G(n, k, L)$, so that the maximum size of an $L$-system is equivalent to finding the independence number of the graph $G(n, k, L)$. The Lov\'asz number $\vartheta(G)$ is a semidefinite programming approximation of the independence number of a graph $G$. In this paper, we determine the order of magnitude of $\vartheta(G(n, k, L))$ of any generalized Johnson graph with $k$ and $L$ fixed and $n\rightarrow \infty$. As an application of this theorem, we give an explicit construction of a graph $G$ on $n$ vertices with large gap between the Lov\'asz number and the Shannon capacity $c(G)$. Specifically, we prove that for any $\epsilon > 0$, for infinitely many $n$ there is a generalized Johnson graph $G$ on $n$ vertices which has ratio $\vartheta(G)/c(G) = \Omega(n^{1-\epsilon})$, which greatly improves on the best known explicit construction.

Classical Krylov subspace projection methods for the solution of linear problem $Ax = b$ output an approximate solution $\widetilde{x}\simeq x$. Recently, it has been recognized that projection methods can be understood from a statistical perspective. These probabilistic projection methods return a distribution $p(\widetilde{x})$ in place of a point estimate $\widetilde{x}$. The resulting uncertainty, codified as a distribution, can, in theory, be meaningfully combined with other uncertainties, can be propagated through computational pipelines, and can be used in the framework of probabilistic decision theory. The problem we address is that the current probabilistic projection methods lead to the poorly calibrated posterior distribution. We improve the covariance matrix from previous works in a way that it does not contain such undesirable objects as $A^{-1}$ or $A^{-1}A^{-T}$, results in nontrivial uncertainty, and reproduces an arbitrary projection method as a mean of the posterior distribution. We also propose a variant that is numerically inexpensive in the case the uncertainty is calibrated a priori. Since it usually is not, we put forward a practical way to calibrate uncertainty that performs reasonably well, albeit at the expense of roughly doubling the numerical cost of the underlying projection method.

An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+\epsilon)$-spanner of a metric space is called (vertex) $f$-fault-tolerant ($f$-FT) if it remains a $(1+\epsilon)$-spanner (for the non-faulty points) when up to $f$ faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which na\"{\i}vely runs in $O(n^3) \cdot 2^{O(f)}$ time. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an $f$-FT spanner with lightness that depends solely on $f$ (even exponentially): all existing constructions have lightness $\Omega(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $\Omega(\log n)$. In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of $f$-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any $n$-point doubling metric, any $\epsilon > 0$, and any integer $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+\epsilon)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$.

Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.

We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations, which generalize the free Euler-Maruyama method introduced by Schl\"{u}chtermann and Wibmer [27]. Under some mild conditions, we prove the strong convergence and exponential stability in mean square of the numerical solution. The free stochastic theta method with $\theta=1$ can inherit the exponential stability of original equations for any given step size. Our method can offer better stability and efficiency than the free Euler-Maruyama method. Moreover, numerical results are reported to confirm these theoretical findings.

Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations which holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments including contact waves and magnetic reconnection.

Input-output conformance simulation (iocos) has been proposed by Gregorio-Rodr\'iguez, Llana and Mart\'inez-Torres as a simulation-based behavioural preorder underlying model-based testing. This relation is inspired by Tretmans' classic ioco relation, but has better worst-case complexity than ioco and supports stepwise refinement. The goal of this paper is to develop the theory of iocos by studying logical characterisations of this relation, rule formats for it and its compositionality. More specifically, this article presents characterisations of iocos in terms of modal logics and compares them with an existing logical characterisation for ioco proposed by Beohar and Mousavi. It also offers a characteristic-formula construction for iocos over finite processes in an extension of the proposed modal logics with greatest fixed points. A precongruence rule format for iocos and a rule format ensuring that operations take quiescence properly into account are also given. Both rule formats are based on the GSOS format by Bloom, Istrail and Meyer. The general modal decomposition methodology of Fokkink and van Glabbeek is used to show how to check the satisfaction of properties expressed in the logic for iocos in a compositional way for operations specified by rules in the precongruence rule format for iocos .

This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq-Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ODE. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data is not known.

Erd\H{o}s and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

We define game semantics for the constructive $\mu$-calculus and prove its equivalence to bi-relational semantics. We use the game semantics to prove that the $\mu$-calculus collapses to modal logic over constructive variants of $\mathsf{S5}$. Finally, we use the collapse to prove the completeness of constructive variants of $\mathsf{S5}$ with fixed-point operators.