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Widening and increasing participation in advanced mathematics studies at upper-secondary level (age 16–18) is a significant challenge for most education systems. Policy makers in Ireland have attempted to address this challenge over the past decade by introducing an incentive to encourage students to study advanced mathematics. This study examines the reasons why students, who would appear to have sufficient prior achievement to enable them to engage in advanced mathematics studies at upper-secondary level, opted not to do so even with the presence of this incentive. Responses to questionnaires completed by 183 students in 10 secondary schools across Ireland were analysed. This analysis indicated that these students tended to avoid engaging in advanced mathematics study at upper-secondary level for a range of reasons. Most cited the expectation that they would struggle or had struggled too much with advanced mathematics. Other commonly cited reasons included the amount of time and effort required to engage effectively in the study of advanced mathematics and the impact this would have on time available to study other subjects.

The one-dimensional transient solidification of a binary alloy undergoing shrinkage is well-known as an invaluable benchmark for the testing of numerical codes that model macrosegregation. Here, recent work that considered the small-time behaviour of this problem is extended until complete solidification, thereby determining the solute profile across the entire solidified domain. The small-time solution is used as the initial condition for the numerical integration of a problem having three moving boundaries. Of particular significance is the so-called inverse segregation that is observed at the start of solidification, and the extreme segregation that is observed at the end; in the case of the example presented, which is for the often-cited Al–Cu system, the macrosegregation is found to be positive or negative, depending on whether Scheil’s equation or the lever rule is assumed at the microscale, respectively. The relevance of these results for the modelling of steady-state continuous casting processes – in particular, the phenomenon of centreline segregation – is also discussed.

We study the Turán number of long cycles in random and pseudo‐random graphs. Denote by ex ( G ( n , p ) , H ) $\operatorname{ex}(G(n,p),H)$ the random variable counting the number of edges in a largest subgraph of G ( n , p ) $G(n,p)$ without a copy of H $H$ . We determine the asymptotic value of ex ( G ( n , p ) , C t ) $\operatorname{ex}(G(n,p), C_t)$ , where C t $C_t$ is a cycle of length t $t$ , for p ⩾ C n $p\geqslant \frac{C}{n}$ and A log n ⩽ t ⩽ ( 1 − ε ) n $A \log n \leqslant t \leqslant (1 - \varepsilon )n$ . The typical behaviour of ex ( G ( n , p ) , C t ) $\operatorname{ex}(G(n,p), C_t)$ depends substantially on the parity of t $t$ . In particular, our results match the classical result of Woodall on the Turán number of long cycles, and can be seen as its random version, showing that the transference principle holds here as well. In fact, our techniques apply in a more general sparse pseudo‐random setting. We also prove a robustness‐type result, showing the likely existence of cycles of prescribed lengths in a random subgraph of a graph with a nearly optimal density. Finally, we also present further applications of our main tool (the Key Lemma) for proving results on Ramsey‐type problems about cycles in sparse random graphs.

The Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring in the image of a word map on a finite nilpotent group G is either 0, or at least 1/|G|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit–Ashurst conjecture and show that the Amit–Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup.

A compressible, viscous and heat conducting fluid is confined between two parallel plates maintained at a constant temperature and subject to a strong stratification due to the gravitational force. We consider the asymptotic limit, where the Mach number and the Froude number are of the same order proportional to a small parameter. We show the limit problem can be identified with Majda's model of layered ``stack-of-pancake'' flow.

General anaesthetics are widely used for their analgesic, immobilising, and hypnotic effects. The mechanisms underlying these effects remain unclear, but likely arise from alterations to cell microstructure, and potentially mechanics. Here we investigate this hypothesis using a custom experimental setup combining calcium imaging and nanoindentation to quantify the firing activity and mechanical properties of dorsal root ganglion-derived neurons exposed to a clinical concentration of 1% isoflurane gas, a halogenated ether commonly used in general anaesthesia. We found that cell viscoelasticity and functional activity are simultaneously and dynamically altered by isoflurane at different stages of exposure. Particularly, cell firing count correlated linearly with the neuronal loss tangent, the ratio of mechanical energy dissipation and storage by the cell. Our results demonstrate that anaesthetics affect cells as a whole, reconciling seemingly contradictory theories of how anaesthetics operate, and highlight the importance of considering cell mechanics in neuronal functions, anaesthesia, and clinical neuroscience in general.

A search for the lepton-flavour violating decays B0 → K*0μ±e∓ and $$ {B}_s^0 $$ B s 0 → ϕμ±e∓ is presented, using proton-proton collision data collected by the LHCb detector at the LHC, corresponding to an integrated luminosity of 9 fb−1. No significant signals are observed and upper limits of$$ {\displaystyle \begin{array}{c}\mathcal{B}\left({B}^0\to {K}^{\ast 0}{\mu}^{+}{e}^{-}\right) B B 0 → K ∗ 0 μ + e − < 5.7 × 10 − 9 6.9 × 10 − 9 , B B 0 → K ∗ 0 μ − e + < 6.8 × 10 − 9 7.9 × 10 − 9 , B B 0 → K ∗ 0 μ ± e ∓ < 10.1 × 10 − 9 11.7 × 10 − 9 , B B s 0 → ϕ μ ± e ∓ < 16.0 × 10 − 9 19.8 × 10 − 9 are set at 90% (95%) confidence level. These results constitute the world’s most stringent limits to date, with the limit on the decay $$ {B}_s^0 $$ B s 0 → ϕμ±e∓ the first being set. In addition, limits are reported for scalar and left-handed lepton-flavour violating New Physics scenarios.

An amplitude analysis of the D+→ π−π+π+ decay is performed with a sample corresponding to 1.5 fb−1 of integrated luminosity of pp collisions at a centre-of-mass energy $$ \sqrt{s} $$ s = 8 TeV collected by the LHCb detector in 2012. The sample contains approximately six hundred thousand candidates with a signal purity of 95%. The resonant structure is studied through a fit to the Dalitz plot where the π−π+ S-wave amplitude is extracted as a function of π−π+ mass, and spin-1 and spin-2 resonances are included coherently through an isobar model. The S-wave component is found to be dominant, followed by the ρ(770)0π+ and f2(1270)π+ components. A small contribution from the ω(782) → π−π+ decay is seen for the first time in the D+→ π−π+π+ decay.

A search for the electroweak production of pairs of charged sleptons or charginos decaying into two-lepton final states with missing transverse momentum is presented. Two simplified models of R-parity-conserving supersymmetry are considered: direct pair-production of sleptons $$ \left(\overset{\sim }{\ell}\overset{\sim }{\ell}\right) $$ ℓ ~ ℓ ~ , with each decaying into a charged lepton and a $$ {\overset{\sim }{\chi}}_1^0 $$ χ ~ 1 0 neutralino, and direct pair-production of the lightest charginos $$ \left({\overset{\sim }{\chi}}_1^{\pm }{\overset{\sim }{\chi}}_1^{\mp}\right) $$ χ ~ 1 ± χ ~ 1 ∓ , with each decaying into a W-boson and a $$ {\overset{\sim }{\chi}}_1^0 $$ χ ~ 1 0 . The lightest neutralino $$ \left({\overset{\sim }{\chi}}_1^0\right) $$ χ ~ 1 0 is assumed to be the lightest supersymmetric particle (LSP). The analyses target the experimentally challenging mass regions where $$ m\left(\overset{\sim }{\ell}\right)-m\left({\overset{\sim }{\chi}}_1^0\right) $$ m ℓ ~ − m χ ~ 1 0 and $$ m\left({\overset{\sim }{\chi}}_1^{\pm}\right)-m\left({\overset{\sim }{\chi}}_1^0\right) $$ m χ ~ 1 ± − m χ ~ 1 0 are close to the W-boson mass (‘moderately compressed’ regions). The search uses 139 fb−1 of $$ \sqrt{s} $$ s = 13 TeV proton-proton collisions recorded by the ATLAS detector at the Large Hadron Collider. No significant excesses over the expected background are observed. Exclusion limits on the simplified models under study are reported in the $$ \left(\overset{\sim }{\ell },{\overset{\sim }{\chi}}_1^0\right) $$ ℓ ~ χ ~ 1 0 and $$ \left({\overset{\sim }{\chi}}_1^{\pm }{\overset{\sim }{\chi}}_1^0\right) $$ χ ~ 1 ± χ ~ 1 0 mass planes at 95% confidence level (CL). Sleptons with masses up to 150 GeV are excluded at 95% CL for the case of a mass-splitting between sleptons and the LSP of 50 GeV. Chargino masses up to 140 GeV are excluded at 95% CL for the case of a mass-splitting between the chargino and the LSP down to about 100 GeV.

This article presents a search for new resonances decaying into a Z or W boson and a 125 GeV Higgs boson h, and it targets the $$ \nu \overline{\nu}b\overline{b} $$ ν ν ¯ b b ¯ , $$ {\ell}^{+}{\ell}^{-}b\overline{b} $$ ℓ + ℓ − b b ¯ , or $$ {\ell}^{\pm}\nu b\overline{b} $$ ℓ ± νb b ¯ final states, where ℓ = e or μ, in proton-proton collisions at $$ \sqrt{s} $$ s = 13 TeV. The data used correspond to a total integrated luminosity of 139 fb−1 collected by the ATLAS detector during Run 2 of the LHC at CERN. The search is conducted by examining the reconstructed invariant or transverse mass distributions of Zh or Wh candidates for evidence of a localised excess in the mass range from 220 GeV to 5 TeV. No significant excess is observed and 95% confidence-level upper limits between 1.3 pb and 0.3 fb are placed on the production cross section times branching fraction of neutral and charged spin-1 resonances and CP-odd scalar bosons. These limits are converted into constraints on the parameter space of the Heavy Vector Triplet model and the two-Higgs-doublet model.

This article presents the first measurement of the differential Z-boson production cross-section in the forward region using proton-lead collisions with the LHCb detector. The dataset was collected at a nucleon-nucleon centre-of-mass energy of $$ \sqrt{s_{\textrm{NN}}} $$ s NN = 8.16 TeV in 2016, corresponding to an integrated luminosity of 30.8 nb−1. The forward-backward ratio and the nuclear modification factors are measured together with the differential cross-section as functions of the Z boson rapidity in the centre-of-mass frame, the transverse momentum of the Z boson and a geometric variable ϕ*. The results are in good agreement with the predictions from nuclear parton distribution functions, providing strong constraining power at small Bjorken-x.

A measurement of the top-quark mass (mt) in the $$ t\overline{t} $$ t t ¯ → lepton + jets channel is presented, with an experimental technique which exploits semileptonic decays of b-hadrons produced in the top-quark decay chain. The distribution of the invariant mass mℓμ of the lepton, ℓ (with ℓ = e, μ), from the W-boson decay and the muon, μ, originating from the b-hadron decay is reconstructed, and a binned-template profile likelihood fit is performed to extract mt. The measurement is based on data corresponding to an integrated luminosity of 36.1 fb−1 of $$ \sqrt{s} $$ s = 13 TeV pp collisions provided by the Large Hadron Collider and recorded by the ATLAS detector. The measured value of the top-quark mass is mt = 174.41 ± 0.39 (stat.) ± 0.66 (syst.) ± 0.25 (recoil) GeV, where the third uncertainty arises from changing the Pythia8 parton shower gluon-recoil scheme, used in top-quark decays, to a recently developed setup.

The quantum partition function for Abelian Higgs vortices at high density—the regime of dissolving vortices—is calculated explicitly, using spectral data for the Beltrami Laplacian on the N-vortex moduli space C P N with a scaled Fubini–Study metric. From the partition function, the pressure of the vortex gas is derived. There are three asymptotic regimes—High, Intermediate and Low Temperature. The phase crossover from Intermediate to Low Temperature is modelled by a Bessel function. In the Low Temperature regime the free energy is not extensive but is proportional to N 2.

Conservation laws have many applications in numerical relativity. However, it is not straightforward to define local conservation laws for general dynamic spacetimes due the lack of coordinate translation symmetries. In flat space, the rate of change of energy-momentum within a finite spacelike volume is equivalent to the flux integrated over the surface of this volume; for general spacetimes it is necessary to include a volume integral of a source term arising from spacetime curvature. In this work a study of continuity of matter in general relativity is extended to include angular momentum of matter and Noether currents associated with gauge symmetries. Expressions for the Noether charge and flux of complex scalar fields and complex Proca fields are found using this formalism. Expressions for the angular momentum density, flux and source are also derived which are then applied to a numerical relativity collision of boson stars in 3D with non-zero impact parameter as an illustration of the methods., Comment: 14 pages, 9 figures

The MaxCut problem asks for the size mc ( G ) ${\rm mc}(G)$ of a largest cut in a graph G $G$ . It is well known that mc ( G ) ⩾ m / 2 ${\rm mc}(G)\geqslant m/2$ for any m $m$ ‐edge graph G $G$ , and the difference mc ( G ) − m / 2 ${\rm mc}(G)-m/2$ is called the surplus of G $G$ . The study of the surplus of H $H$ ‐free graphs was initiated by Erdős and Lovász in the 70s, who, in particular, asked what happens for triangle‐free graphs. This was famously resolved by Alon, who showed that in the triangle‐free case the surplus is Ω ( m 4 / 5 ) $\Omega (m^{4/5})$ , and found constructions matching this bound. We prove several new results in this area. (i) Confirming a conjecture of Alon, Krivelevich and Sudakov, we show that for every fixed odd r ⩾ 3 $r\geqslant 3$ , any C r $C_r$ ‐free graph with m $m$ edges has surplus Ω r ( m r + 1 r + 2 ) $\Omega _r(m^{\frac{r+1}{r+2}})$ . This is tight, as is shown by a construction of pseudo‐random C r $C_r$ ‐free graphs due to Alon and Kahale. It improves previous results of several researchers, and complements a result of Alon, Krivelevich and Sudakov which is the same bound when r $r$ is even. (ii) Generalising the result of Alon, we allow the graph to have triangles, and show that if the number of triangles is a bit less than in a random graph with the same density, then the graph has large surplus. For regular graphs, our bounds on the surplus are sharp. (iii) We prove that an n $n$ ‐vertex graph with few copies of K r $K_r$ and average degree d $d$ has surplus Ω r ( d r − 1 / n r − 3 ) $\Omega _r(d^{r-1}/n^{r-3})$ , which is tight when d $d$ is close to n $n$ provided that a conjectured dense pseudo‐random K r $K_r$ ‐free graph exists. This result is used to improve the best known lower bound (as a function of m $m$ ) on the surplus of K r $K_r$ ‐free graphs. Our proofs combine techniques from semi‐definite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.

This article studies the design of contracts involving a single retailer and multiple competing manufacturers who supply substitutable products. We consider a retail context in which contracts with manufacturers are negotiated relatively infrequently and signed before the demand environment is known, and the retail prices are determined when the demand is known. We develop a Stackelberg model to study the retailer's product selection and pricing decisions and the manufacturers' contract design decisions. We show that it is optimal for each manufacturer to offer a contract with nonlinear prices so that total payments are the total production cost plus a fixed additional cost. In the case of two manufacturers this result allows us to characterize an equilibrium in which the retailer's choice maximizes the supply chain profit, each manufacturer makes a profit equal to its marginal contribution to the supply chain, and the retailer takes the remaining profit. We also find that while increasing demand correlation always benefits the retailer, it benefits the manufacturers only when the production costs are convex. In an extension it is found that our equilibrium continues to hold when the retailer's reservation profit is below a threshold, but the competition dynamics may change when the reservation profit is above the threshold. Finally, we show that the equilibrium results remain true for the case with more than two manufacturers under a submodularity property, which holds in the case of quadratic costs and linear demand.

We study slowly rotating black hole solutions in the six independent theories of Quartic Generalized Quasi-topological Gravity in four dimensions. Unlike in the static case for which all six theories yield the same solution, for rotating black holes we obtain distinct results for five out of the six theories. Working to leading order in the rotation parameter, we find that the equations characterizing these black holes can be reduced to second order for each theory, similar to what has already been done for Einstein Cubic Gravity. We construct approximate and numerical solutions to these equations, and study how physical properties of the solutions such as the angular velocity, photon sphere, black hole shadow, and innermost stable circular orbit are modified, working to leading order in the coupling constant.

This article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator (- Δ) 2. These fractional p -biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional p -biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties, monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces Ht,p for any t∈ R, 1 ≤ p< ∞ and s∈ R+\ N: If u∈ Ht,p(Rn) satisfies (- Δ) su= u= 0 in a nonempty open set V, then u≡ 0 in Rn. This property of the fractional Laplacian is then used to obtain a UCP for the fractional p -biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli–Silvestre extension., Calculus of Variations and Partial Differential Equations, 62 (4), ISSN:0944-2669, ISSN:1432-0835

It is proved that every finitely generated profinite group with fewer than $2^{\aleph_0}$ conjugacy classes of elements of infinite order is finite, Comment: 6 pages

Searches for CP violation in the decays $$ {D}_{(s)}^{+}\to \eta {\pi}^{+} $$ D s + → η π + and $$ {D}_{(s)}^{+}\to {\eta}^{\prime }{\pi}^{+} $$ D s + → η ′ π + are performed using pp collision data corresponding to 6 fb−1 of integrated luminosity collected by the LHCb experiment. The calibration channels $$ {D}_{(s)}^{+}\to \phi {\pi}^{+} $$ D s + → ϕ π + are used to remove production and detection asymmetries. The resulting CP-violating asymmetries are$$ {\displaystyle \begin{array}{l}{\mathcal{A}}^{CP}=\left({D}^{+}\to \eta {\pi}^{+}\right)=\left(0.34\pm 0.66\pm 0.16\pm 0.05\right)\%,\\ {}{\mathcal{A}}^{CP}=\left({D}_s^{+}\to \eta {\pi}^{+}\right)=\left(0.32\pm 0.51\pm 0.12\right)\%,\\ {}\begin{array}{l}{\mathcal{A}}^{CP}=\left({D}^{+}\to {\eta}^{\prime }{\pi}^{+}\right)=\left(0.49\pm 0.18\pm 0.06\pm 0.05\right)\%,\\ {}{\mathcal{A}}^{CP}=\left({D}_s^{+}\to {\eta}^{\prime }{\pi}^{+}\right)=\left(0.01\pm 0.12\pm 0.08\right)\%,\end{array}\end{array}} $$ A CP = D + → η π + = 0.34 ± 0.66 ± 0.16 ± 0.05 % , A CP = D s + → η π + = 0.32 ± 0.51 ± 0.12 % , A CP = D + → η ′ π + = 0.49 ± 0.18 ± 0.06 ± 0.05 % , A CP = D s + → η ′ π + = 0.01 ± 0.12 ± 0.08 % , where the first uncertainty is statistical, the second is systematic and the third, relevant for the D+ channels, is due to the uncertainty on $$ {\mathcal{A}}^{CP}=\left({D}^{+}\to \phi {\pi}^{+}\right) $$ A CP = D + → ϕ π + . These measurements, currently the most precise for three of the four channels considered, are consistent with the absence of CP violation. A combination of these results with previous LHCb measurements is presented.

Measurements of single-, double-, and triple-differential cross-sections are presented for boosted top-quark pair-production in 13 TeV proton–proton collisions recorded by the ATLAS detector at the LHC. The top quarks are observed through their hadronic decay and reconstructed as large-radius jets with the leading jet having transverse momentum (pT) greater than 500 GeV. The observed data are unfolded to remove detector effects. The particle-level cross-section, multiplied by the $$ t\overline{t}\to WWb\overline{b} $$ t t ¯ → WWb b ¯ branching fraction and measured in a fiducial phase space defined by requiring the leading and second-leading jets to have pT> 500 GeV and pT> 350 GeV, respectively, is 331 ± 3(stat.) ± 39(syst.) fb. This is approximately 20% lower than the prediction of $$ {398}_{-49}^{+48} $$ 398 − 49 + 48 fb by Powheg+Pythia 8 with next-to-leading-order (NLO) accuracy but consistent within the theoretical uncertainties. Results are also presented at the parton level, where the effects of top-quark decay, parton showering, and hadronization are removed such that they can be compared with fixed-order next-to-next-to-leading-order (NNLO) calculations. The parton-level cross-section, measured in a fiducial phase space similar to that at particle level, is 1.94 ± 0.02(stat.) ± 0.25(syst.) pb. This agrees with the NNLO prediction of $$ {1.96}_{-0.17}^{+0.02} $$ 1.96 − 0.17 + 0.02 pb. Reasonable agreement with the differential cross-sections is found for most NLO models, while the NNLO calculations are generally in better agreement with the data. The differential cross-sections are interpreted using a Standard Model effective field-theory formalism and limits are set on Wilson coefficients of several four-fermion operators.

Acknowledgements: The authors would like to thank their PhD supervisor Béla Bollobás for his valuable comments., Funder: Engineering and Physical Sciences Research Council; doi: http://dx.doi.org/10.13039/501100000266, Funder: Cambridge Commonwealth, European and International Trust; doi: http://dx.doi.org/10.13039/501100003343, Petruševski and Škrekovski recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph G is said to be odd if for each non-isolated vertex $$x \in V(G)$$ x ∈ V ( G ) there exists a colour c appearing an odd number of times in its neighbourhood N(x). Petruševski and Škrekovski proved that for any planar graph G there is an odd colouring using at most 9 colours and, together with Caro, showed that 8 colours are enough for a significant family of planar graphs. We show that 8 colours suffice for all planar graphs.

Probabilistic sensitivity analysis identifies the influential uncertain input to guide decision-making. We propose a general sensitivity framework with respect to the input distribution parameters that unifies a wide range of sensitivity measures, including information theoretical metrics such as the Fisher information. The framework is derived analytically via a constrained maximization and the sensitivity analysis is reformulated into an eigenvalue problem. There are only two main steps to implement the sensitivity framework utilising the likelihood ratio/score function method, a Monte Carlo type sampling followed by solving an eigenvalue equation. The resulting eigenvectors then provide the directions for simultaneous variations of the input parameters and guide the focus to perturb uncertainty the most. Not only is it conceptually simple, but numerical examples demonstrate that the proposed framework also provides new sensitivity insights, such as the combined sensitivity of multiple correlated uncertainty metrics, robust sensitivity analysis with an entropic constraint, and approximation of deterministic sensitivities. Three different examples, ranging from a simple cantilever beam to an offshore marine riser, are used to demonstrate the potential applications of the proposed sensitivity framework to applied mechanics problems., The datasets generated during and/or analysed during the current study are available in the GitHub repository: https://github.com/longitude-jyang/Probabilistic-sensitivity-framework; Accepted for publication at Probabilistic Engineering Mechanics

We present an interpretation of the phase diagram for a binary alloy from the point of view of reaction dynamics. We consider a model system in which the two-component liquid phase is an ideal solution, but the solid phase can be non-ideal, with its non-ideality increasing with reducing temperature. We show how a ‘batch’ model for the evolution of the two-component solid–liquid system, in which the interfacial growth rates are proportional to free energy differences, leads to a set of four differential equations, whose equilibria correspond to the familiar solidus and liquidus curves. In addition, we explain how the transitions between them depend on the bifurcation structure of the solutions. We show that locally stable mixed-phase solutions can exist below the eutectic temperature, thus providing an alternative explanation for the observation of super-cooled liquids, and we also explain why complete solidification ‘normally’ occurs below the eutectic temperature, despite the fact that this temperature has no intrinsic dynamic significance, being simply a value at which two completely different equilibrium states happen to share a common liquid concentration.

For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal A_{k})$. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured that $\text{sat}^*(n, \mathcal A_{k})=(k-1)n(1+o(1))$, and proved this for $k\leq 4$. In this paper we prove this conjecture for $k=5$ and $k=6$. Moreover, we give the exact value for $\text{sat}^*(n, \mathcal A_5)$ and $\text{sat}^*(n, \mathcal A_6)$. We also give some open problems inspired by our analysis., Comment: 8 pages

The black hole rigidity theorem asserts that a rotating stationary black hole must be axisymmetric. This theorem holds for General Relativity with suitable matter fields, in four or more dimensions. We show that the theorem can be extended to any diffeomorphism invariant theory of vacuum gravity, assuming that this is interpreted in the sense of effective field theory, with coupling constants determined in terms of a ``UV scale'', and that the black hole solution can locally be expanded as a power series in this scale., 37pp, 1 figure

Peer reviewed: True, A half-space x 2 > 0 is occupied by a two-component statistically uniform random composite with specified volume fractions and two-point correlation. It is bonded to a uniform half-space x 2 < 0 from which a plane wave is incident. The problem requires the specification of the properties of three media: those of the two constituents of the composite and those of the homogeneous half-space. The complexity of the problem is minimized by considering a model acoustic-wave problem in which the two media comprising the composite have the same modulus but different densities. The homogeneous half-space can have any chosen modulus and density. An approximate formulation based on a stochastic variational principle is formulated and solved explicitly in the particular case of an exponentially decaying correlation function, generalizing a previous solution in which the homogeneous medium had the same modulus as the composite. The main novelty of the present work is that the mean energy fluxes in the composite and reflected from the boundary are calculated, demonstrating explicitly the large backscatter associated with the mean-zero component of the reflected signal and the systematic transfer of energy from the decaying mean transmitted waves into the mean-zero disturbance as distance from the boundary increases.

A half-space x 2 > 0 is occupied by a two-component statistically uniform random composite with specified volume fractions and two-point correlation. It is bonded to a uniform half-space x 2 < 0 from which a plane wave is incident. The problem requires the specification of the properties of three media: those of the two constituents of the composite and those of the homogeneous half-space. The complexity of the problem is minimized by considering a model acoustic-wave problem in which the two media comprising the composite have the same modulus but different densities. The homogeneous half-space can have any chosen modulus and density. An approximate formulation based on a stochastic variational principle is formulated and solved explicitly in the particular case of an exponentially decaying correlation function, generalizing a previous solution in which the homogeneous medium had the same modulus as the composite. The main novelty of the present work is that the mean energy fluxes in the composite and reflected from the boundary are calculated, demonstrating explicitly the large backscatter associated with the mean-zero component of the reflected signal and the systematic transfer of energy from the decaying mean transmitted waves into the mean-zero disturbance as distance from the boundary increases.

Motivated by the results of Anastasiou et al., we consider the convolutional double copy for BRST and anti-BRST covariant formulations of gravitational and gauge theories in more detail. We give a general BRST and anti-BRST invariant formulation of linearised $\mathcal{N}=0$ supergravity using superspace methods and show how this may be obtained from the square of linearised Yang-Mills theories. We demonstrate this relation for the Schwarzschild black hole and the ten-dimensional black string solution as two concrete examples., Comment: 35 pages

We develop a constructive theory of finite multisets in Homotopy Type Theory, defining them as free commutative monoids. After recalling basic structural properties of the free commutative-monoid construction, we formalise and establish the categorical universal property of two, necessarily equivalent, algebraic presentations of free commutative monoids using 1-HITs. These presentations correspond to two different equational theories invariably including commutation axioms. In this setting, we prove important structural combinatorial properties of finite multisets. These properties are established in full generality without assuming decidable equality on the carrier set. As an application, we present a constructive formalisation of the relational model of classical linear logic and its differential structure. This leads to constructively establishing that free commutative monoids are conical refinement monoids. Thereon we obtain a characterisation of the equality type of finite multisets and a new presentation of the free commutative-monoid construction as a set-quotient of the list construction. These developments crucially rely on the commutation relation of creation/annihilation operators associated with the free commutative-monoid construction seen as a combinatorial Fock space., Comment: Appeared in MFPS'22

The paper applies synthetic instruments, initially developed for cross-sectional regression, to estimate dynamic spatial panel data models. These have two main advantages. First, instruments correlated with endogenous variables and yet independent of the errors are difficult to find. Not only are synthetic instruments normally exogenous, but they are usually strongly correlated with endogenous variables, and thus help to avoid the problem of weak instruments. Secondly, they help to reduce instrumental variables proliferation, which is a common result of standard methods of avoiding endogeneity bias. As demonstrated by Monte Carlo simulation, instrument proliferation causes bias in the Sargan–Hansen J test statistic, which is an important indicator of instrument validity and hence estimation consistency. It is also associated with a downward bias in parameter standard error estimates. The paper shows the results of applying synthetic instruments across a variety of different specifications and data generating processes, and it illustrates the method with real data leading to more reliable inference of causal impacts on the level of employment across London districts.

Predicting the risk of, and time to biochemical recurrence (BCR) in prostate cancer patients post-operatively is critical in patient treatment decision pathways following surgical intervention. This study aimed to investigate the predictive potential of mRNA information to improve upon reference nomograms and clinical-only models, using a dataset of 187 patients that includes over 20,000 features. Several machine learning methodologies were implemented for the analysis of censored patient follow-up information with such high-dimensional genomic data. Our findings demonstrated the potential of inclusion of mRNA information for BCR-free survival prediction. A random survival forest pipeline was found to achieve high predictive performance with respect to discrimination, calibration, and net benefit. Two mRNA variables, namely ESM1 and DHAH8, were identified as consistently strong predictors with this dataset.

Seese’s conjecture for finite graphs states that monadic second-order logic (MSO) is undecidable on all graph classes of unbounded clique-width. We show that to establish this it would suffice to show that grids of unbounded size can be inter- preted in two families of graph classes: minimal hereditary classes of unbounded clique-width; and antichains of unbounded clique-width under the induced sub- graph relation. We explore all the currently known classes of the former category and establish that grids of unbounded size can indeed be interpreted in them.

All models may be wrong -- but that is not necessarily a problem for inference. Consider the standard $t$-test for the significance of a variable $X$ for predicting response $Y$ whilst controlling for $p$ other covariates $Z$ in a random design linear model. This yields correct asymptotic type~I error control for the null hypothesis that $X$ is conditionally independent of $Y$ given $Z$ under an \emph{arbitrary} regression model of $Y$ on $(X, Z)$, provided that a linear regression model for $X$ on $Z$ holds. An analogous robustness to misspecification, which we term the "double-estimation-friendly" (DEF) property, also holds for Wald tests in generalised linear models, with some small modifications. In this expository paper we explore this phenomenon, and propose methodology for high-dimensional regression settings that respects the DEF property. We advocate specifying (sparse) generalised linear regression models for both $Y$ and the covariate of interest $X$; our framework gives valid inference for the conditional independence null if either of these hold. In the special case where both specifications are linear, our proposal amounts to a small modification of the popular debiased Lasso test. We also investigate constructing confidence intervals for the regression coefficient of $X$ via inverting our tests; these have coverage guarantees even in partially linear models where the contribution of $Z$ to $Y$ can be arbitrary. Numerical experiments demonstrate the effectiveness of the methodology., To appear in Statistical Science

We derive parity-even graviton bispectra in the Effective Field Theory of Inflation (EFToI) to all orders in derivatives. Working in perturbation theory, we construct all cubic interactions that can contribute to tree-level graviton bispectra, showing that they all come from EFToI operators containing two or three powers of the extrinsic curvature and its covariant derivatives: all other operators can be removed by field redefinitions or start at higher-order in perturbations. For operators cubic in the extrinsic curvature, where the single-clock consistency relations are satisfied without a correction to the graviton two-point function, we use the Manifestly Local Test (MLT) to efficiently extract the effects of evolving graviton fluctuations to the end of inflation. Despite the somewhat complicated nature of the bulk interactions, the final boundary correlators take a very compact form. For operators quadratic in the extrinsic curvature, the leading order bispectra are a sum of contact and single exchange diagrams, which are tied together by spatial diffeomorphisms, and to all orders in derivatives we derive these bispectra by computing the necessary bulk time integrals. For single exchange diagrams we exploit factorisation properties of the bulk-bulk propagator for massless gravitons and write the result as a finite sum over residues. Perhaps surprisingly, we show these single exchange contributions have only total-energy poles and also satisfy the MLT.

Fundamental properties of the multicomponent diffuse-interface model (DIM), such as the maximum entropy principle and conservation laws, are used to explore the basic interfacial dynamics and phase transitions in fluids. Flat interfaces with monotonically changing densities of the components are proved to be stable. A liquid layer in contact with oversaturated but stable vapour is shown to either fully evaporate or eternally expand (depending on the initial perturbation), whereas a liquid in contact with saturated vapour always evaporates. If vapour is bounded by a solid wall with a sufficiently large contact angle, spontaneous condensation occurs in the vapour. The external parameters of the multicomponent DIM – e.g. the Korteweg matrix describing the long-range intermolecular forces – are determined for the water/air combination. The Soret and Dufour effects are shown to be negligible in this case, and the interfacial flow, close to isothermal.

To generalise the theory of well-bounded operators, Ashton and Doust defined the Banach algebras BV (σ) and AC(σ) of functions of bounded variation and absolutely continuous functions defined on any non-empty compact subset σ of the plane. The main aims of this thesis are to explore the properties of the spaces BV (σ) and AC(σ), and to extend the representation theory of well-bounded operators in this setting. Well-bounded operators have a functional calculus for the absolutely continuous functions on an interval. When this functional calculus is weakly compact, a well-bounded operator is of type (B), and can be represented as an integral with respect to a spectral family of projections. Every well-bounded operator on a reflexive space is of type (B) and it is conjectured that these are the only spaces on which this is the case. We will examine the current status of this conjecture and produce further evidence for it by ruling out some suggested counterexamples, a family of Banach spaces introduced by Pisier in 1983. Moreover, we show that the conjecture holds for Banach lattices. All well-bounded operators have real spectrum, and so it is natural to ask whether there is a generalisation of normal operators in this context. Earlier attempts at such a generalisation had many shortcomings, and this led to Ashton and Doust’s definition of variation in the plane in 2000. Their original definitions were challenging to work with, and in this thesis we rework the important parts of the theory using a simplified, but equivalent definition of variation in the plane. The goal in such a generalisation would be to develop a spectral integration theory so that an operator has a weakly compact AC(σ) functional calculus if and only if it can be represented as an integral with respect to a family of spectral projections. One challenge is the lack of an analogue of the Tietze extension theorem for absolutely continuous functions. Such a result is needed for showing that T having an AC(σ) functional calculus implies T has an AC(σ(T)) functional calculus. We will show that on a polygonally inscribed curve, or PIC set, introduced by Al-shakarchi and Doust, any absolutely continuous function always has an absolutely continuous extension to any larger compact set. In the final chapter, we develop a new version of the integration theory for well-bounded operators of type (B) and show that this generalises to a spectral theorem for operators with a weakly compact AC(σ) functional calculus with σ in PIC, extending the theories of well-bounded and trigonometrically well-bounded operators.

In this work, we explore degree assortativity in complex networks, and extend its usual definition beyond that of nearest neighbours. We apply this definition to model networks, and describe a rewiring algorithm that induces assortativity. We compare these results to real networks. Social networks in particular tend to be assortatively mixed by degree in contrast to many other types of complex networks. However, we show here that these positive correlations diminish after one step and in most of the empirical networks analysed. Properties besides degree support this, such as the number of papers in scientific coauthorship networks, with no correlations beyond nearest neighbours. Beyond next-nearest neighbours we also observe a disassortative tendency for nodes three steps away indicating that nodes at that distance are more likely different than similar

While major strides have been made towards gender equality in public life, serious inequality remains in the domestic sphere, especially around parenting. The present study analyses discussions about parenting on Reddit (i.e., a content aggregation website) to explore audience effects and gender stereotypes. It suggests a novel method to study topical variation in individuals’ language when interacting with different audiences. Comments posted in 2020 were collected from three parenting subreddits (i.e., topical communities), described as being for fathers (r/Daddit), mothers (r/Mommit), and all parents (r/Parenting). Users posting on r/Parenting and r/Daddit or on r/Parenting and r/Mommit were assumed to identify as fathers or mothers, respectively, allowing gender comparison. Users’ comments on r/Parenting (to a mixed-gender audience) were compared with their comments to single-gender audiences on r/Daddit or r/Mommit using Latent Dirichlet Allocation (LDA) topic modelling. Results show that the most discussed topic among parents is about education and family advice, a topic mainly discussed in the mixed-gender subreddit and more by fathers than mothers. The topic model also indicates that, when it comes to the basic needs of children (sleep, food, and medical care), mothers seem to be more concerned regardless of the audience. In contrast, topics such as birth and pregnancy announcements and physical appearance are more discussed by fathers in the father-centric subreddit. Overall, findings seem to show that mothers are generally more concerned about the practical sides of parenting while fathers’ expressed concerns are more contextual: with other fathers, there seems to be a desire to show their fatherhood and be recognized for it while they discuss education with mothers. These results demonstrate that concerns expressed by parents on Reddit are context-sensitive but also consistent with gender stereotypes, potentially reflecting a persistent gendered and unequal division of labour in parenting.

Modern variable selection procedures make use of penalization methods to execute simultaneous model selection and estimation. A popular method is the LASSO (least absolute shrinkage and selection operator), the use of which requires selecting the value of a tuning parameter. This parameter is typically tuned by minimizing the cross-validation error or Bayesian information criterion (BIC) but this can be computationally intensive as it involves fitting an array of different models and selecting the best one. In contrast with this standard approach, we have developed a procedure based on the so-called "smooth IC" (SIC) in which the tuning parameter is automatically selected in one step. We also extend this model selection procedure to the distributional regression framework, which is more flexible than classical regression modelling. Distributional regression, also known as multiparameter regression (MPR), introduces flexibility by taking account of the effect of covariates through multiple distributional parameters simultaneously, e.g., mean and variance. These models are useful in the context of normal linear regression when the process under study exhibits heteroscedastic behaviour. Reformulating the distributional regression estimation problem in terms of penalized likelihood enables us to take advantage of the close relationship between model selection criteria and penalization. Utilizing the SIC is computationally advantageous, as it obviates the issue of having to choose multiple tuning parameters.

In its usual form, Freiman's 3 k − 4 $3k-4$ theorem states that if A $A$ and B $B$ are subsets of Z ${\mathbb {Z}}$ of size k $k$ with small sumset (of size close to 2 k $2k$ ), then they are very close to arithmetic progressions. Our aim in this paper is to strengthen this by allowing only a bounded number of possible summands from one of the sets. We show that if A $A$ and B $B$ are subsets of Z ${\mathbb {Z}}$ of size k $k$ such that for any four‐element subset X $X$ of B $B$ the sumset A + X $A+X$ has size not much more than 2 k $2k$ , then already this implies that A $A$ and B $B$ are very close to arithmetic progressions.

Acknowledgements: I am grateful to Maciej Dunajski, Piotr Chruściel, Greg Galloway, Roger Penrose, Harvey Reall, Paul Tod, Eric Woolgar, Gary Gibbons and Claude Warnick for useful discussions, as well as to the anonymous referee for their extremely helpful comments. I am also grateful to St. John’s College, Cambridge for their support through a College Scholarship from the Todd/Goddard Fund. Some of this work was carried out while I was visiting Institut Mittag-Leffler in Djursholm, Sweden, where I was supported by the Swedish Research Council under Grant No. 2016-06596. Funding was provided by Science and Technology Facilities Council (Grant No. ST/P000681/1, ST/T000694/1), St John’s College, University of Cambridge (Todd/Goddard Fund), Vetenskapsr’det (Grant No. 2016-06596)., The positive mass theorem in higher dimensions is proved using causality arguments inspired by those of Penrose et al. (a positive mass theorem based on the focusing and retardation of null geodesics, 1993) in $$3+1$$ 3 + 1 dimensions.

This study presents the convergence and stability analysis of a recently developed fixed pivot technique for fragmentation equations (Liao et al. in Int J Numer Methods Fluids 87(4):202–215, 2018). The approach is based on preserving two integral moments of the distribution, namely (a) the zeroth-order moment, which defines the number of particles, and (b) the first-order moment, which describes the total mass in the system. The present methodology differs mathematically in a way that it delivers the total breakage rate between a mother and a daughter particle immediately, whereas existing numerical techniques provide the partial breakup rate of a mother and daughter particle. This affects the computational efficiency and makes the current model reliable for CFD simulations. The consistency and unconditional second-order convergence of the method are proved. This demonstrates efficiency of the method over the fixed pivot technique (Kumar and Warnecke in Numer Math 110(4):539–559, 2008) and the cell average technique (Kumar and Warnecke in Numer Math 111(1):81–108, 2008). Numerical results are compared against the cell average technique and the experimental order of convergence is calculated to confirm the theoretical order of convergence.

We consider the analysis of temporal data arising from online interactive social experiments, which is complicated by the fact that classical independence assumptions about the observations are not satisfed. Therefore, we propose an approach that compares the output of a ftted (linear) model from the observed interaction data to that generated by an assumed agent-based null model. This allows us to discover, for example, the extent to which the structure of social interactions difers from that of random interactions. Moreover, we provide network visualisations that identify the extent of ingroup favouritism and reciprocity as well as particular individuals whose behaviour difers markedly from the norm. We specifcally consider experimental data collected via the novel Virtual Interaction APPLication (VIAPPL). We fnd that ingroup favouritism and reciprocity are present in social interactions observed on this platform, and that these behaviours strengthen over time. Note that, while our proposed methodology was developed with VIAPPL in mind, its potential usage extends to any type of social interaction data.

We obtain Rosenthal-type inequalities with sharp constants for moments of sums of independent random variables which are mixtures of a fixed distribution. We also identify extremisers in log-concave settings when the moments of summands are individually constrained., Comment: 15 pages. Final version (referee comments incorporated). To appear in Bull. Lond. Math. Soc

Generalized Linear Models (GLMs) and XGBoost are widely used in insurance risk pricing and claims prediction, with GLMs dominant in the insurance industry. The increasing prevalence of connected car data usage in insurance requires highly accurate and interpretable models. Deep learning (DL) models have outperformed traditional Machine Learning (ML) models in multiple domains; despite this, they are underutilized in insurance risk pricing. This study introduces an alternative DL architecture, TabNet, suitable for insurance telematics datasets and claim prediction. This approach compares the TabNet DL model against XGBoost and Logistic Regression on the task of claim prediction on a synthetic telematics dataset. TabNet outperformed these models, providing highly interpretable results and capturing the sparsity of the claims data with high accuracy. However, TabNet requires considerable running time and effort in hyperparameter tuning to achieve these results. Despite these limitations, TabNet provides better pricing models for interpretable models in insurance when compared to XGBoost and Logistic Regression models.