Atmospheric systems incorporating thermal dynamics must be stable with respect to both energy and entropy. While energy conservation can be enforced via the preservation of the skew-symmetric structure of the Hamiltonian form of the equations of motion, entropy conservation is typically derived as an additional invariant of the Hamiltonian system, and satisfied via the exact preservation of the chain rule. This is particularly challenging since the function spaces used to represent the thermodynamic variables in compatible finite element discretisations are typically discontinuous at element boundaries. In the present work we negate this problem by constructing our equations of motion via weighted averages of skew-symmetric formulations using both flux form and material form advection of thermodynamic variables, which allow for the necessary cancellations required to conserve entropy without the chain rule. We show that such formulations allow for stable simulations of both the thermal shallow water and 3D compressible Euler equations on the sphere using mixed compatible finite elements without entropy damping.
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We introduce an information-theoretic method for quantifying causality in chaotic systems. The approach, referred to as IT-causality, quantifies causality by measuring the information gained about future events conditioned on the knowledge of past events. The causal interactions are classified into redundant, unique, and synergistic contributions depending on their nature. The formulation is non-intrusive, invariance under invertible transformations of the variables, and provides the missing causality due to unobserved variables. The method only requires pairs of past-future events of the quantities of interest, making it convenient for both computational simulations and experimental investigations. IT-causality is validated in four scenarios representing basic causal interactions among variables: mediator, confounder, redundant collider, and synergistic collider. The approach is leveraged to address two questions relevant to turbulence research: i) the scale locality of the energy cascade in isotropic turbulence, and ii) the interactions between inner and outer layer flow motions in wall-bounded turbulence. In the former case, we demonstrate that causality in the energy cascade flows sequentially from larger to smaller scales without requiring intermediate scales. Conversely, the flow of information from small to large scales is shown to be redundant. In the second problem, we observe a unidirectional causality flow, with causality predominantly originating from the outer layer and propagating towards the inner layer, but not vice versa. The decomposition of IT-causality into intensities also reveals that the causality is primarily associated with high-velocity streaks.
Common-signal-induced synchronization of semiconductor lasers with optical feedback inspired a promising physical key distribution with information-theoretic security and potential in high rate. A significant challenge is the requirement to shorten the synchronization recovery time for increasing key rate without sacrificing operation parameter space for security. Here, open-loop synchronization of wavelength-tunable multi-section distributed Bragg reflector (DBR) lasers is proposed as a solution for physical-layer key distribution. Experiments show that the synchronization is sensitive to two operation parameters, i.e., currents of grating section and phase section. Furthermore, fast wavelength-shift keying synchronization can be achieved by direct modulation on one of the two currents. The synchronization recovery time is shortened by one order of magnitude compared to close-loop synchronization. An experimental implementation is demonstrated with a final key rate of 5.98 Mbit/s over 160 km optical fiber distance. It is thus believed that fast-tunable multi-section semiconductor lasers opens a new avenue of high-rate physical-layer key distribution using laser synchronization.
A new numerical domain decomposition method is proposed for solving elliptic equations on compact Riemannian manifolds. The advantage of this method is to avoid global triangulations or grids on manifolds. Our method is numerically tested on some $4$-dimensional manifolds such as the unit sphere $S^{4}$, the complex projective space $\mathbb{CP}^{2}$ and the product manifold $S^{2} \times S^{2}$.
The reverse engineering of a complex mixture, regardless of its nature, has become significant today. Being able to quickly assess the potential toxicity of new commercial products in relation to the environment presents a genuine analytical challenge. The development of digital tools (databases, chemometrics, machine learning, etc.) and analytical techniques (Raman spectroscopy, NIR spectroscopy, mass spectrometry, etc.) will allow for the identification of potential toxic molecules. In this article, we use the example of detergent products, whose composition can prove dangerous to humans or the environment, necessitating precise identification and quantification for quality control and regulation purposes. The combination of various digital tools (spectral database, mixture database, experimental design, Chemometrics / Machine Learning algorithm{\ldots}) together with different sample preparation methods (raw sample, or several concentrated / diluted samples) Raman spectroscopy, has enabled the identification of the mixture's constituents and an estimation of its composition. Implementing such strategies across different analytical tools can result in time savings for pollutant identification and contamination assessment in various matrices. This strategy is also applicable in the industrial sector for product or raw material control, as well as for quality control purposes.
Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their possible applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown [1]. Additionally, in a recent manuscript by Pal and Stanica [20], a very interesting connection between the $c$-differential uniformity and boomerang uniformity when $c=-1$ was pointed out, showing that that they are the same for an odd APN permutations. This makes the construction of functions with low $c$-differential uniformity an intriguing problem. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic~$3$. As a byproduct, our proofs shows the permutation property of these classes. To solve the involved equations over finite fields, we use various techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest.
The most fundamental model of a molecule is a cloud of unordered atoms, even without chemical bonds that can depend on thresholds for distances and angles. The strongest equivalence between clouds of atoms is rigid motion, which is a composition of translations and rotations. The existing datasets of experimental and simulated molecules require a continuous quantification of similarity in terms of a distance metric. While clouds of m ordered points were continuously classified by Lagrange's quadratic forms (distance matrices or Gram matrices), their extensions to m unordered points are impractical due to the exponential number of m! permutations. We propose new metrics that are continuous in general position and are computable in a polynomial time in the number m of unordered points in any Euclidean space of a fixed dimension n.
We investigate the properties of the high-order discontinuous Galerkin spectral element method (DGSEM) with implicit backward-Euler time stepping for the approximation of hyperbolic linear scalar conservation equation in multiple space dimensions. We first prove that the DGSEM scheme in one space dimension preserves a maximum principle for the cell-averaged solution when the time step is large enough. This property however no longer holds in multiple space dimensions and we propose to use the flux-corrected transport limiting [Boris and Book, J. Comput. Phys., 11 (1973)] based on a low-order approximation using graph viscosity to impose a maximum principle on the cell-averaged solution. These results allow to use a linear scaling limiter [Zhang and Shu, J. Comput. Phys., 229 (2010)] in order to impose a maximum principle at nodal values within elements. Then, we investigate the inversion of the linear systems resulting from the time implicit discretization at each time step. We prove that the diagonal blocks are invertible and provide efficient algorithms for their inversion. Numerical experiments in one and two space dimensions are presented to illustrate the conclusions of the present analyses.
A formulation of the asymptotically exact first-order shear deformation theory for linear-elastic homogeneous plates in the rescaled coordinates and angles of rotation is considered. This allows the development of its asymptotically accurate and shear-locking-free finite element implementation. As applications, numerical simulations are performed for circular and rectangular plates, showing complete agreement between the analytical solution and the numerical solutions based on two-dimensional theory and three-dimensional elasticity theory.
Survey-based measurements of the spectral energy distributions (SEDs) of galaxies have flux density estimates on badly misaligned grids in rest-frame wavelength. The shift to rest frame wavelength also causes estimated SEDs to have differing support. For many galaxies, there are sizeable wavelength regions with missing data. Finally, dim galaxies dominate typical samples and have noisy SED measurements, many near the limiting signal-to-noise level of the survey. These limitations of SED measurements shifted to the rest frame complicate downstream analysis tasks, particularly tasks requiring computation of functionals (e.g., weighted integrals) of the SEDs, such as synthetic photometry, quantifying SED similarity, and using SED measurements for photometric redshift estimation. We describe a hierarchical Bayesian framework, drawing on tools from functional data analysis, that models SEDs as a random superposition of smooth continuum basis functions (B-splines) and line features, comprising a finite-rank, nonstationary Gaussian process, measured with additive Gaussian noise. We apply this *Splines 'n Lines* (SnL) model to a collection of 678,239 galaxy SED measurements comprising the Main Galaxy Sample from the Sloan Digital Sky Survey, Data Release 17, demonstrating capability to provide continuous estimated SEDs that reliably denoise, interpolate, and extrapolate, with quantified uncertainty, including the ability to predict line features where there is missing data by leveraging correlations between line features and the entire continuum.
This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic field along the boundary. Based on time-dependent jump conditions of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretising the nonlinear system of boundary integral equations with Runge--Kutta based convolution quadrature in time and Raviart--Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates.
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