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Security attacks are rising, as evidenced by the number of reported vulnerabilities. Among them, unknown attacks, including new variants of existing attacks, technical blind spots or previously undiscovered attacks, challenge enduring security. This is due to the limited number of techniques that diagnose these attacks and enable the selection of adequate security controls. In this paper, we propose an automated technique that detects and diagnoses unknown attacks by identifying the class of attack and the violated security requirements, enabling the selection of adequate security controls. Our technique combines anomaly detection to detect unknown attacks with abductive reasoning to diagnose them. We first model the behaviour of the smart home and its requirements as a logic program in Answer Set Programming (ASP). We then apply Z-Score thresholding to the anomaly scores of an Isolation Forest trained using unlabeled data to simulate unknown attack scenarios. Finally, we encode the network anomaly in the logic program and perform abduction by refutation to identify the class of attack and the security requirements that this anomaly may violate. We demonstrate our technique using a smart home scenario, where we detect and diagnose anomalies in network traffic. We evaluate the precision, recall and F1-score of the anomaly detector and the diagnosis technique against 18 attacks from the ground truth labels provided by two datasets, CICIoT2023 and IoT-23. Our experiments show that the anomaly detector effectively identifies anomalies when the network traces are strong indicators of an attack. When provided with sufficient contextual data, the diagnosis logic effectively identifies true anomalies, and reduces the number of false positives reported by anomaly detectors. Finally, we discuss how our technique can support the selection of adequate security controls.

We consider pairs of anti-commuting $2p$-by-$2p$ Hermitian matrices that are chosen randomly with respect to a Gaussian measure. Generically such a pair decomposes into the direct sum of $2$-by-$2$ blocks on which the first matrix has eigenvalues $\pm x_j$ and the second has eigenvalues $\pm y_j$. We call $\{ (x_j, y_j) \}$ the skew spectrum of the pair. We derive a formula for the probability density of the skew spectrum, and show that the elements are repelling.

Connected and Automated Vehicles (CAVs) can have a profound influence on transport systems. However, most levels of automation and connectivity require support from the road infrastructure. Additional support such as Cooperative Intelligent Transport Systems (C-ITS) services can facilitate safe and efficient traffic, and alleviate the environmental impacts of road surface vehicles. However, due to the rapidly evolving technology, C-ITS service deployment requirements are not always clear. Furthermore, the costs and benefits of infrastructure investments are subject to tremendous uncertainty. This study articulates the requirements using a structured approach to propose a CAV-Readiness Framework (CRF). The main purpose of the CRF is allowing road authorities to assess their physical and digital infrastructure readiness, define requirements for C-ITS services, and identify future development paths to reach higher levels of readiness to support CAVs by enabling C-ITS services. The CRF is intended to guide and support road authorities' investment decisions on infrastructure.

With software systems permeating our lives, we are entitled to expect that such systems are secure by design, and that such security endures throughout the use of these systems and their subsequent evolution. Although adaptive security systems have been proposed to continuously protect assets from harm, they can only mitigate threats arising from changes foreseen at design time. In this paper, we propose the notion of Sustainable Adaptive Security (SAS) which reflects such enduring protection by augmenting adaptive security systems with the capability of mitigating newly discovered threats. To achieve this objective, a SAS system should be designed by combining automation (e.g., to discover and mitigate security threats) and human intervention (e.g., to resolve uncertainties during threat discovery and mitigation). In this paper, we use a smart home example to showcase how we can engineer the activities of the MAPE (Monitor, Analysis, Planning, and Execution) loop of systems satisfying sustainable adaptive security. We suggest that using anomaly detection together with abductive reasoning can help discover new threats and guide the evolution of security requirements and controls. We also exemplify situations when humans can be involved in the execution of the activities of the MAPE loop and discuss the requirements to engineer human interventions.

We define a random commuting $d$-tuple of $n$-by-$n$ matrices to be a random variable that takes values in the set of commuting $d$-tuples and has a distribution that is a rapidly decaying continuous weight on this algebraic set. In the Hermitian case, we characterize the eigenvalue distribution as $n$ tends to infinity. In the non-Hermitian case, we get a formula that holds if the set is irreducible. We show that there are qualitative differences between the single matrix case and the several commuting matrices case.

Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankha we consider the following problem: for a given inner function $\phi\not\in\mathsf{Aut}(\mathbb D)$, find a non-constant inner function $\Psi$ satisfying the functional equation $\Psi\circ\phi=\tau\Psi$, where $\tau$ is a unimodular constant. We prove that this problem has a solution if and only if $\phi$ is of positive hyperbolic step. More precisely, if this condition holds, we show that there is an infinite Blaschke product $B$ satisfying the equation for $\tau=1$. If in addition, $\phi$ is parabolic, we prove that the problem has a solution $\Psi$ for $any$ unimodular $\tau$. Finally, we show that if $\phi$ is of zero hyperbolic step, then no non-constant Bloch function $f$ and no unimodular constant $\tau$ satisfy $f\circ\phi=\tau f$.

In this thesis we study the principle that extremal objects in differential geometry correspond to stable objects in algebraic geometry. In our introduction we survey the most famous instances of this principle with a view towards the results and background needed in the later chapters. In Part I we discuss the notion of a $Z$-critical metric recently introduced in joint work with Ruadha\'i Dervan and Lars Martin Sektnan. We prove a correspondence for existence with an analogue of Bridgeland stability in the large volume limit, and study important properties of the subsolution condition away from this limit, including identifying the analogues of the Donaldson and Yang-Mills functionals for the equation. In Part II we study the recent theory of optimal symplectic connections on K\"ahler fibrations in the isotrivial case. We prove a correspondence with the existence of Hermite-Einstein metrics on holomorphic principal bundles., Comment: PhD Thesis

"When the liquid phase of a particle-laden droplet evaporates, a ring of solute is typically formed - what has become known as the "coffee ring effect". A key focus of recent work has been the suppression of the coffee-ring effect to leave behind more spatially-uniform particle coatings instead. Efforts to suppress the coffee ring effect often focus on physical effects such as Marangoni flows. Here we focus on a purely geometric effect - the effect of substrate curvature - by evaporating suspension droplets on spherical surfaces of different radius of curvature. We find that stains formed on more highly curved substrates are more uniform. To understand this, we propose a model of the evaporation-induced flow, combined with a detailed calculation of how curvature modifies the local evaporation rate. This model shows that evaporation in the centre of the droplet is enhanced, leading to increased concentration of solute there and a reduction in the propensity for ring-formation., Comment: 4 pages, 4 figures, Sup Info

I share my opinions on how to get the most out of conferences. I have some advice on giving talks, but most of the essay is devoted to the much larger share of the conference not spent giving talks: attending talks and socializing.

We define a monomial space to be a subspace of $\ltwo$ that can be approximated by spaces that are spanned by monomial functions. We describe the structure of monomial spaces.

We prove that the invariant subspaces of the Hardy operator on $L^2[0,1]$ are the spaces that are limits of sequences of finite dimensional spaces spanned by monomial functions.

We study the algebra $\mathcal{A}$ generated by the Hardy operator $H$ and the operator $M_x$ of multiplication by $x$ on $L^2[0,1]$. We call $\mathcal{A}$ the Hardy-Weyl algebra. We show that its quotient by the compact operators is isomorphic to the algebra of functions that are continuous on $\Lambda$ and analytic on the interior of $\Lambda$ for a planar set $\Lambda$ = $[-1,0] \cup \bar{ \mathbb{D}(1,1)}$, which we call the lollipop. We find a Toeplitz-like short exact sequence for the $C^*$-algebra generated by $\mathcal{A}$. We study the operator $Z = H - M_x$, show that its point spectrum is $(-1,0] \cup \mathbb{D}(1,1)$, and that the eigenvalues grow in multiplicity as the points move to $0$ from the left.

We study monomial operators on $ L^2[0,1]$, that is bounded linear operators that map each monomial $x^n$ to a multiple of $x^{p_n}$ for some $p_n$. We show that they are all unitarily equivalent to weighted composition operators on a Hardy space. We characterize what sequences $p_n$ can arise. In the case that $p_n$ is a fixed translation of $n$, we give a criterion for boundedness of the operator.

The Hardy operator has all the monomial functions as eigenvectors. We study bounded operators on L^2 that take monomial functions to multiples of other monomials, with a shifted exponent. We prove that they all leave the space of functions vanishing on [0,s] invariant. We prove an asymptotic Muntz-Szasz theorem, characterizing the set of functions that are limits of linear combinations of monomials with exponents between n and 2n.

We show that for every connected analytic subvariety $V$ there is a pseudoconvex set $\Omega$ such that every bounded matrix-valued holomorphic function on $V$ extends isometrically to $\Omega$. We prove that if $V$ is two analytic disks intersecting at one point, if every bounded scalar valued holomorphic function extends isometrically to $\Omega$, then so does every matrix-valued function. In the special case that $\Omega$ is the symmetrized bidisk, we show that this cannot be done by finding a linear isometric extension from the functions that vanish at one point.

If $V$ is an analytic set in a pseudoconvex domain $\Omega$, we show there is always a pseudoconvex domain $G \subseteq \Omega$ that contains $V$ and has the property that every bounded holomorphic function on $V$ extends to a bounded holomorphic function on $G$ with the same norm. We find such a $G$ for some particular analytic sets. When $\Omega$ is an operhedron we show there is a norm on holomorphic functions on $V$ that can always be preserved by extensions to $\Omega$.

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function $f$ in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type $f=\varphi g$, where $g$ is cyclic, $\varphi$ is a contractive multiplier, and $\|f\|=\|g\|$. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization., Comment: 63 pages

In this article we completely describe the existence of canonical metrics, known as optimal symplectic connections, on isotrivial K\"ahler fibrations. In this setting an optimal symplectic connection is induced from a Hermite--Einstein connection on the holomorphic principal bundle of relative automorphisms, and the Hitchin--Kobayashi correspondence asserts the existence of such a connection precisely when the principal bundle is polystable. Combined with results of Dervan and Sektnan this generates many new examples of cscK metrics on the total space of holomorphic fibre bundles. Our results indicate that in general the optimal symplectic connection equation should be viewed as a generalisation of the Hermite--Einstein equation to holomorphic fibrations where the complex structure of the fibres varies., Comment: 15 pages

We establish a theory of NC functions on a class of von Neumann algebras with a particular direct sum property, e.g. $B(\mathcal{H})$. In contrast to the theory's origins, we do not rely on appealing to results from the matricial case. We prove that the $k^{\text{th}}$ directional derivative of any NC function at a scalar point is a $k$-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g. the non-commutative polydisk and non-commutative row ball., Comment: 21 pages

We characterize the common range of the adjoints of cyclic multiplication operators on the Drury--Arveson space. We show that a function belongs to this common range if and only if its Taylor coefficients satisfy a simple decay condition. To achieve this, we introduce the uniform Smirnov class on the ball and determine its dual space. We show that the dual space of the uniform Smirnov class equals the dual space of the strictly smaller Smirnov class of the Drury-Arveson space, and that this in turn equals the common range of the adjoints of cyclic multiplication operators.

We associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. We call solutions to these equations $Z$-critical connections, with $Z$ a central charge. Deformed Hermitian Yang--Mills connections are a special case. We explain how our equations arise naturally through infinite dimensional moment maps. Our main result shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a $Z$-critical connection if and only if it is asymptotically $Z$-stable. Even for the deformed Hermitian Yang--Mills equation, this provides the first examples of solutions in higher rank., Comment: v3: 73 pages, published version

We prove a variant of the well-known result that intertwiners for the bilateral shift on `$\ell^2(Z)$ are unitarily equivalent to multiplication operators on $L^2(T)$. This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts., Comment: 25 pages, 7 figures

We compare the quantitative risk of infection from short range airborne transmission of SARS-CoV-2 to the long-range risk from aerosol transmission in an enclosed space.

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space., Comment: 40 pages; minor changes

We define by interpolation a scale analogous to the Hardy $H^p$ scale for complete Pick spaces, and establish some of the basic properties of the resulting spaces, which we call $\mathcal{H}^p$. In particular, we obtain an $\mathcal{H}^p-\mathcal{H}^q$ duality and establish sharp pointwise estimates for functions in $\mathcal{H}^p$.