Proceedings of the Royal Society A

We demonstrate that a pair consisting of a second-order homogeneous Hamiltonian structure in N components and its associated system of conservation laws is in bijective correspondence with an alternating three-form on a N+2-dimensional vector space. Additionally, we show that the three-form offers N+2 linear equations in the Plücker coordinates that define the associated line congruence. We utilize these results to characterize systems of conservation laws with second-order structure for N≤4. We finally comment on how to extend this result for N=6.

Barycentric and pairwise quantum Rényi leakages are proposed as two measures of information leakage for privacy and security analysis in quantum computing and communication systems. These quantities both require minimal assumptions on the eavesdropper, i.e. they do not make any assumptions on the eavesdropper’s attack strategy or the statistical prior on the secret or private classical data encoded in the quantum system. They also satisfy important properties of positivity, independence, post-processing inequality and unitary invariance. The barycentric quantum Rényi leakage can be computed by solving a semi-definite program; the pairwise quantum Rényi leakage possesses an explicit formula. The barycentric and pairwise quantum Rényi leakages form upper bounds on the maximal quantum leakage, the sandwiched quantum α-mutual information, the accessible information and the Holevo’s information. Furthermore, differentially private quantum channels are shown to bound these measures of information leakage. Global and local depolarizing channels, that are common models of noise in quantum computing and communication, restrict private or secure information leakage. Finally, a privacy-utility trade-off formula in quantum machine learning using variational circuits is developed. The privacy guarantees can only be strengthened, i.e. information leakage can only be reduced, if the performance degradation grows larger and vice versa.

Self-dual electromagnetic fields are shown to be maximally chiral. Monochromatic self-dual beams have electric and magnetic fields that are eigenstates of curl. The chiral density and chiral current of self-dual monochromatic beams are proportional to the energy and momentum densities. Their energy and momentum densities do not oscillate in time. Self-dual pulses based on an oscillatory solution of the wave equation lack the oscillations of non-self-dual pulses based on the same wave function. All of these properties could be significant in physical applications, but few self-dual fields seem to have been studied experimentally.

We present a method to obtain microstructural information from macroscopic boundary measurements exploiting scattering governed by the wave equation in a bounded linearly elastic domain in the long-wavelength regime. Applying a force to the outer boundary of the body on the macroscopic scale while measuring the resulting boundary displacement, we solve the inverse problem of identifying the geometry of the microstructure in the context of periodic homogenization minimizing a tracking-type objective functional as long as the geometry of the microstructure is parameterized by a finite set of parameters. Shape calculus is used to characterize the Gâteaux derivative of the objective function facilitating the use of gradient-based algorithms, and we present numerical experiments for a generic non-destructive testing problem for ellipsoidal microstructures showcasing the functioning of the identification method.

We present a novel multiscale formalism to characterize complex phenomena in electromagnetic media featuring micro-structures. By contrast to the conventional homogenization approaches, the proposed model is based on the variational framework given by the Method of Multiscale Virtual Power applied to Maxwell’s equations, which ensures energetic consistency between the macro- and micro-scales. From a set of well-defined minimal assumptions on the primal variables, the electromagnetic field and its curl, the formulation yields not only the micro-cell problem to be solved but also the homogenization formulae of the dual entities, as well as the homogenization of tangent operators that characterize, for instance, the effective electric permittivity and magnetic permeability. The proposed multiscale model is not restricted to periodic media, although it accounts for the periodicity assumption as a specific case. We apply the method in different cases of interest that range from low-frequency to high-frequency regimes to demonstrate the predictive and descriptive capabilities of the model in terms of effective electromagnetic properties.

How can scientific progress be conceived best? Does science mainly undergo revolutionary paradigm shifts? Or is the evolution of science mainly cumulative? Understanding whether science advances through cumulative evolution or through paradigm shifts can influence how we approach scientific research, education and policy. The most influential and cited account of science was put forth in Thomas Kuhn’s seminal book The structure of scientific revolutions. Kuhn argues that science does not advance cumulatively but goes through fundamental paradigm changes in the theories of a scientific field. There is no consensus yet on this core question of the nature and advancement of science that has since been debated across science. Examining over 750 major scientific discoveries (all Nobel Prize and major non-Nobel Prize discoveries), we systematically test this fundamental question about scientific progress here. We find that three key measures of scientific progress—major discoveries, methods and fields—each demonstrate that science evolves cumulatively. First, we show that no major scientific methods or instruments used across fields (such as statistical methods, X-ray methods or chromatography) have been completely abandoned, i.e. subject to paradigm shifts. Second, no major scientific fields (such as biomedicine, chemistry or computer science) have been completely abandoned. Rather, they have all continuously expanded over time, often over centuries, accumulating extensive bodies of knowledge. Third, scientific discoveries including theoretical discoveries are also predominately cumulative, with only 1% of over 750 major discoveries having been abandoned. The continuity of science is most compellingly evidenced by our methods and instruments, which enable the creation of discoveries and fields. We thus offer here a new perspective and answer to this classic question in science and the philosophy and history of science by utilizing methods from statistics and empirical sciences.

The interplay between the algebraic structure (operator algebras) for the quantum observables and the convex structure of the state space has been explored for a long time, and most advanced results are due to Alfsen and Shultz. Here we present a more elementary approach with a more general structure for the observables, which focuses on the transition probability of the quantum logical atoms. The binary case gives rise to the generalized qubit models and was fully developed in a preceding paper. Here we consider any case with finite information capacity (binary means that the information capacity is 2). A novel geometric property that makes any compact convex set a matching state space is presented. Generally, the transition probability is not symmetric; if it is symmetric, we get an inner product and a self-dual cone. The emerging mathematical structure comes close to the Euclidean Jordan algebras and becomes a new mathematical model for a potential extension of quantum theory.

In this paper, we study slow travelling wave solutions of the mixed local–non-local Fisher–KPP equation, which satisfy DUzz+cUz+U{1−αU−(1−α)ϕ∗U}=0, with D≪1 and c≪1. Here ϕ∗u≡∫−∞∞ϕ(z−y)U(y)dy is the convolution of the dimensionless population density, U(z), with a continuous, symmetric, strictly positive kernel, ϕ(y), which is decreasing for y>0 and has a finite derivative as y→0+, normalized so that ∫−∞∞ϕ(y)dy=1. The dimensionless parameter α, with 0≤α≤1, measures the strength of local competition relative to that of non-local competition. Through numerical and asymptotic analysis when c≪1, we show that as α, and hence local competition, increases, the spikes that characterize the solution of the non-local travelling wave equation are successively suppressed. We also show that when α=O(c), the spikes are affected at leading order by the local interaction and develop a sharp wavefront when c≪α. In this latter case, for example, with the Gaussian kernel, ϕ(y)=12πe−14y2, the travelling wave solution can contain multiple sharp wavefronts, all propagating at the same speed, qualitatively similar in form to a roll wave in fluid mechanics, although the underlying physics is very different.

Nature is a blossoming of regular structures, signature of self-organization of the underlying microscopic interacting agents. Turing theory of pattern formation is one of the most studied mechanisms to address such phenomena and has been applied to a widespread gallery of disciplines. Turing himself used a spatial discretization of the hosting support to eventually deal with a set of ODEs. Such an idea contained the seeds of the theory on discrete support, which has been fully acknowledged with the birth of network science in the early 2000s. This approach allows us to tackle several settings not displaying a trivial continuous embedding, such as multiplex, temporal networks and, recently, higher-order structures. This line of research has been mostly confined within the network science community, despite its inherent potential to transcend the conventional boundaries of the PDE-based approach to Turing patterns. Moreover, network topology allows for novel dynamics to be generated via a universal formalism that can be readily extended to account for higher-order structures. The interplay between continuous and discrete settings can pave the way for further developments in the field.

We consider a model thermokinetic reaction that is both autocatalytic and temperature sensitive. The mechanism consists of two reactions. The first is a reversible reaction with a temperature-dependent rate, and the second involves a cubically autocatalytic process. We demonstrate the presence of up to five steady states in the system, dependent on parameters, and we illustrate the fact that the system can exhibit tipping-point behaviour, in which small changes to initial conditions in an experiment could lead to radically different long-term outcomes. We prove that oscillatory behaviour is not possible over large regions in the parameter space; outside these regions, however, oscillations have been found, and we present a simple and robust method for computing them. Small-amplitude oscillations can arise through Hopf bifurcations at an equilibrium point. In addition, large-amplitude oscillations can bifurcate directly from global structures that are associated with equilibria behaving as saddles. This is illustrated with some numerical solutions of these highly nonlinear equations.

This work reviews the semi-analytical technique (SAT) and proposes a unique SAT based on the homotopy analysis method (HAM), called accelerated HAM (AHAM) (recently proposed in Hussain et al. (Hussain S, Arora G, Kumar R. 2024 Semi-analytical methods for solving non-linear differential equations: a review. J. Math. Anal. Appl. 531, 127821 (doi:10.1016/j.jmaa.2023.127821))) to solve pure aggregation and coupled aggregation–fragmentation population balance equations (PBEs), which are nonlinear integro-partial differential equations. The novelty of this approach is demonstrated through a comparative analysis of numerical results against existing findings from the Adomian decomposition (Singh R, Saha J, Kumar J. 2015 Adomian decomposition method for solving fragmentation and aggregation population balance equations. J. Appl. Math. Comput. 48, 265–292 (doi:10.1007/s12190-014-0802-5)), homotopy analysis (Kaur G, Singh R, Briesen H. 2022 Approximate solutions of aggregation and breakage population balance equations. J. Math. Anal. Appl. 512, 126166 (doi:10.1016/j.jmaa.2022.126166)), homotopy perturbation (Kaur G, Singh R, Singh M, Kumar J, Matsoukas T. 2019 Analytical approach for solving population balances: a homotopy perturbation method. J. Phys. A Math. Theor. 52, 385201 (doi:10.1088/1751-8121/ab2cf5)) and optimized decomposition (Kaushik S, Kumar R. 2023 A novel optimized decomposition method for Smoluchowski’s aggregation equation. J. Comput. Appl. Math. 419, 114710 (doi:10.1016/j.cam.2022.114710)) methods for these models. In addition, the theoretical convergence analysis and error estimation of the proposed method are also investigated. To validate the scheme, we analyse several numerical test cases and the results illustrate that the suggested technique offers the most accurate estimates for the two models under consideration.

The planform is a defining feature of mantle convection. It can be gleaned from the stratigraphic record by mapping the continent-scale distribution of hiatus and no hiatus surfaces serving as a proxy for high and low dynamic topography. We carry this out for all continents apart from Antarctica for eight geological series since the Upper Jurassic, showing that: (i) the planform as indicated by our maps contains wavelengths of the order of 1000 km, smaller than the convective scales implied by the geoid. (ii) The planform changes on timescales of geological series (10–20 Myrs), smaller than the mantle transit time. (iii) Flood basalt eruptions are frequently preceded by hiatus surfaces. (iv) Some hiatus surfaces are not linked to any known plume, potentially reflecting the lateral transport of material in the asthenosphere. Our results reveal the importance of mantle viscosity stratification in shaping the convective planform and the resulting dynamic topography. Geodynamic Earth models should aim to reproduce the global characteristics of our maps, as well as specific regional events identified in this work. Finally, we separate the effects of sea-level variation from regional changes in base level induced by dynamic topography by contrasting the stratigraphic evolution of different regions.

Voluntary vaccination is crucial for public health. The likelihood that the vaccinated get fully immune is typically referred to as the vaccine efficacy. In addition to this, the efficacy of vaccines also depends on inhibiting the transmission rates and protecting patients from critical symptoms. It has yet to be answered how the multi-efficacy of vaccines affects the vaccination behaviour. Here, we propose a game theoretical model of vaccination behaviour with a heterogeneous-transmission epidemiological process. Intuitively, individuals are likely to take vaccination for either high-efficient vaccines or epidemics with a large basic reproductive number. We find, however, a low economic discount rate can promote the uptake level for low-effectiveness vaccines and epidemics with small basic reproductive numbers. This result suggests that the perceived cost is more important than the risk of infection and the effectiveness of vaccines. In addition, we develop an analytical approximation to address the challenge arising from the coupled dynamics of vaccination and disease spread. This study shows how the interplay among three types of vaccine efficacies (transmission blocking, symptom mitigating and full immunity protecting) affects individual vaccination decisions and epidemic outcomes, and the model can be insightful for epidemic control.

We introduce a deterministic set of equations, the Lotka–Volterra (LV) generalized equations, to explain or predict the temporal dynamics and competition between land use and land cover (LULC) classes. We evaluated this new approach in two regions of Uruguay where human activities have reduced natural grasslands. We infer the interaction coefficients from the LV model through the pairwise maximum entropy method using the spatial covariance matrices obtained from annual remote-sensing LULC maps. The model performed well in predicting short-term temporal changes, with errors similar to those obtained using traditional Markov chains. This model introduces a novel avenue for assisting LULC modelling in improving the temporal dynamics.

A micro-hydrodynamics model based on elastic collisions of light point solvent particles with a heavy solute particle is investigated in the setting where the light particles have velocity distribution corresponding to a background flow. Considering a range of stationary background flows and distributions for the solvent particle velocities, the macroscopic Langevin-type description of the behaviour of the heavy particle is derived in the form of a generalized Ornstein–Uhlenbeck process. At leading order, the drift term in this process depends upon both the geometric structure of the background flow and the size of the heavy particle, while both drift and diffusion terms scale with moments of the light particle velocity distribution. Computational methods for simulating the micro-hydrodynamics model are then designed to confirm the theoretical results. To enable long-time calculations, simulations are performed in a frame co-moving with the heavy particle. Efficient methods for sampling the position and velocity distributions of incoming solvent particles at the boundaries of the co-moving frame are derived for a range of distributions of solvent particles. The simulations show good agreement with the theoretical results.

The homogenization procedure developed here is conducted on a laminate with periodic space–time modulation on the fine scale: at the leading order, this modulation creates convection in the low-wavelength regime if both parameters are modulated. However, if only one parameter is modulated, which is more realistic, this convective term disappears and one recovers a standard diffusion equation with effective homogeneous parameters; this does not describe the non-reciprocity and the propagation of the field observed from exact dispersion diagrams. This inconsistency is corrected here by considering second-order homogenization that results in a non-reciprocal propagation term that is proved to be non-zero for any laminate and verified via numerical simulation. The same methodology is also applied to the case when the density is modulated in the heat equation, leading, therefore, to a corrective advective term that cancels out non-reciprocity at the leading order but not at the second order.

Vertical axis turbines have received great attention in both offshore wind and tidal current energy communities considering their advantages of economic design and unidirectional operation. However, their commercialization process is rather slow compared with the development of horizontal axis turbines due to technical challenges resulting from higher loads from unsteady aero/hydrodynamic forces, centrifugal forces and gravity of the structures. These are mainly because, while the inherent unsteady tangential pulls and fatigue loads intensify the structural vibration, aggravating structural safety and fatigue life, the relationship between the geometric parameters and the structural responses of blades remains unclear. Therefore, in order to clarify this issue, in this study, we focus on the modal and transient analysis of vertical axis turbines using multi-body dynamics, geometrically exact beam theory and detached eddy simulation. We find that the natural frequency of the rotor is directly related to the ratio of blade length and arm length. The effect of arms cannot be ignored in the structural analysis of the turbine. Moreover, an increase of blade length intensifies the deformation and vibration of the turbine’s structure, making the turbine more prone to fatigue failure. This article is expected to extend the existing knowledge of wind and tidal current turbines and provide a reference for choosing proper design parameters and developing suitable-capacity vertical axis turbines.

This perspective is not meant to be a complete summary of the full series of our studies over decades, rather it aims to look back across the entire process to outline: What inspired this work at its beginning? How did the mindset gradually evolve from a solely engineering study to the new concept of Mesoscience? And where is the research currently being focused and to be extended? This approach elucidates the central importance of correlating precise engineering problems to be resolved with associated missing links in the underlying, fundamental science while exploring a notable commonality, namely, the universality of Mesoscience. Inspired by the physical phenomenon of the coexistence and interaction between a gas-rich dilute phase and a solid-rich dense phase with different physical mechanisms dominating the behaviour for each phase in gas–solid fluidization, it was recognized that an underlying compromise-in-competition (CIC) exists between these two physically dominant mechanisms. This, then, is the CIC principle. We believe that this is the origin of inherent structural complexity in these and other systems. We have shown that this CIC principle can be formulated as a multi-objective optimization problem through the so-called energy-minimization multi-scale model, leading to a massive improvement in both the accuracy and scalability of computation and its wide application in both academia and industries. Furthermore, this approach was extended to formulate and understand other complex systems in many apparently disparate fields, indicating the wide applicability of the CIC principle and then resulting in the proposition and advancement of the concept of Mesoscience. Furthermore, within the main body of this perspective article, future directions for identifying, developing and applying the concept of Mesoscience will be prospected. This includes extending its possible generality and applications in many different disciplines and fields, even to analysing global challenging issues and promoting CIC-informed AI. In this way, many challenging problems in engineering—identified through their underlying mesoscale complexity—will be closely correlated with the development of future fundamental science. Through this approach, we hope to illustrate that fundamental research can now tackle the intrinsic complexity existing in a multitude of engineering problems. Through the concept of Mesoscience, we aim to explore the possible commonality from complexity. The deduction of commonality from diversity is possible in resolving global challenges, shifting paradigms in science and filling in the existing gaps at the mesoscales of different levels of our knowledge, though there are difficulties and uncertainty coming from diversity.

Dynamic topography refers to vertical deflections of Earth’s surface from viscous flow within the mantle. Here we investigate how past subduction history affects present dynamic topography. We assimilate two plate reconstructions into TERRA forward mantle convection models to calculate past mantle states and predict Earth’s present dynamic topography; a comparison is made with a database of observed oceanic residual topography. The two assimilated plate reconstructions ‘Earthbyte’ and ‘Tomopac’ show divergent subduction histories across an extensive deep-time interval within Pacific-Panthalassa. We find that introducing an alternative subduction history perturbs our modelled present-day dynamic topography on the same order as the choice of radial viscosity. Additional circum-Pacific intra-oceanic subduction in Tomopac consistently produces higher correlations to the geoid (more than 20% improvement). At spherical harmonic degrees 1–40, dynamic topography models with intra-oceanic subduction produce universally higher correlations with observations and improve fit by up to 37%. In northeast Asia, Tomopac models show higher correlations (0.46 versus 0.18) to observed residual topography and more accurately predict approximately 1 km of dynamic subsidence within the Philippine Sea plate. We demonstrate that regional deep-time changes in subduction history have widespread impacts on the spatial distribution and magnitude of present-day dynamic topography. Specifically, we find that local changes to plate motion histories can induce dynamic topography changes in faraway regions located thousands of kilometres away. Our results affirm that present-day residual topography observations provide a powerful, additional constraint for reconstructing ancient subduction histories.

This research article explores the need to adapt the weighting system of a life cycle assessment (LCA) framework to a wide range of socio-technical and environmental considerations, which are complex and sensitive to external stressors. The study demonstrates the potential of agent-based systems to adapt a weighting system to dynamic conditions, and suggests integrating an agent-based model to handle uncertainties of weighting in environmental impact assessment. Generative adversarial networks (GANs) and multi-agent systems (MAS) are utilized to address data limitations and simulate diverse scenarios. We confirm the potential of agent-based systems to analyse the effective management of uncertainties and customization of weighting systems in environmental impact assessments to improve decision making. Furthermore, the study emphasizes the importance of continuous adaptation and recalibration to ensure the system remains relevant in dynamic environments. The results confirm that MAS is a powerful tool for managing uncertainty, customizing weighting systems and improving decision making in environmental assessments. Moreover, the study acknowledges challenges and sets the groundwork for future research.

Guided wave tomography plays a significantly important role in quantitatively mapping thickness variations in plate-like structures and pipelines in the petrochemical industry for accurate measurement of corrosion, as guided waves enable rapid screening of large areas without needing direct access. Several inverse algorithms have been implemented in guided wave tomography; however, almost all of them use a primary assumption: the three-dimensional (3D) guided wave thickness reconstruction is simplified as a two-dimensional (2D) acoustic wave velocity inversion, which is then mapped to thickness variation using the dispersive nature of guided waves. Although this assumption simplifies the inverse procedure, it makes it impossible to use non-dispersive modes in reconstructions. Geometrical full waveform inversion (GFWI) is promising to overcome this limitation since it can reconstruct corrosion profiles through geometry optimization using a data-fitting procedure, where velocity-to-thickness mapping is not needed. In this work, GFWI-based guided wave tomography is developed in plate-like structures, and is applied to reconstruct thickness maps in a series of corrosion defects using the non-dispersive SH0 mode, demonstrating high performance and achieving an improved reconstruction resolution.

Rayleigh–Bloch waves are modes localized to periodic arrays of scatterers with unbounded unit cells. Here, Rayleigh–Bloch waves are studied for line arrays of sound-hard circular scatterers embedded in a two-dimensional acoustic medium, for which it has recently been shown that Rayleigh–Bloch waves exist for higher frequencies than previously thought. Moreover, it was shown that Rayleigh–Bloch waves can cut off (disappear) and cut on (reappear), and additional Rayleigh–Bloch waves can cut on and interact with the existing ones. These complicated behaviours are reconsidered using a family of quasi-periodic Green’s functions that allow particular plane-wave components to become unbounded away from the array. The Green’s function formulation is combined with the block Sakurai–Sugiura method to trace the trajectories of the Rayleigh–Bloch wavenumbers as they swap between Riemann sheets that are categorized according to the unbounded plane wave(s). A detailed analysis is presented for three different scatterer radius values, and contrasting qualitative behaviours are identified. The findings are consistent with those published previously, extend to higher frequencies than allowed by the previous approach, and provide new understanding of Rayleigh–Bloch waves around the critical frequency intervals where they cut on/cut off/interact.

Swarmalators are oscillators that swarm through space as they synchronize in time. Introduced a few years ago to model many systems that mix synchrony with self-assembly, they remain poorly understood theoretically. Here, we obtain the first analytic results on swarmalators moving in two spatial dimensions by introducing a simplified model where the swarmalators have no hard-shell interaction terms and move on a periodic plane. These simplifications allow expressions for order parameters, stabilities and bifurcations to be derived exactly.

Bohr’s complementarity principle has long been a fundamental concept in quantum mechanics, positing that, within a given experimental set-up, a quantum system (or quanton) can exhibit either its wave-like character, denoted as W, or its particle-like character, denoted as P, but not both simultaneously. Modern interpretations of Bohr’s complementarity principle acknowledge the coexistence of these aspects in the same experiment while introducing the constraint W+P≤α. Notably, estimations of W or P frequently rely on indirect retrodiction methods, a practice that has led to the claim of the violation of Bohr’s complementarity principle. By taking a different route, recent advancements demonstrate that quantum complementarity relations can be rigorously derived from the axioms of quantum mechanics. To reconcile these observations and eliminate potential paradoxes or violations, we propose an updated formulation for the quantum complementarity principle, which is stated as follows: For a given quantum state preparation ρt at a specific instant of time t, the wave and particle behaviours of a quanton are constrained by a complementarity relation 𝔚(ρt)+𝔓(ρt)≤α(d), which is derived directly from the axioms of quantum mechanics.

To model dynamical systems on networks with higher-order (non-pairwise) interactions, we recently introduced a new class of ordinary differential equations (ODEs) on hypernetworks. Here, we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady-state branch ‘reluctant’ if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady-state branch.