We study the worst-case approximation of multivariate periodic functions from weighted Korobov spaces Hd,α,γ with smoothness α>1/2 and weights γ in Lp([0,1]d), 1≤p≤∞. We analyze a median lattice algorithm that reconstructs a truncated Fourier series by estimating coefficients on a hyperbolic-cross-type index set using R rank-1 lattice sampling rules with independent randomly chosen generating vectors, and then aggregating the coefficient estimators via the componentwise median. For odd R and odd prime lattice size N, where R depends logarithmically on N, we prove high-probability error bounds in both L∞ and L2. Interpolation then yields the result for all p∈[1,∞]. In particular, with high probability, the worst-case Lp-error is bounded, for any β>0, byCd,α,β,γ,τ,pR12+(12−1p)+N−α+(12−1p)++β, where (x)+=max{x,0}, and the constant Cd,α,β,γ,τ,p is independent of N and R. For p=∞, the constant is dimension-independent under the summability condition ∑j=1∞γj1/(2α)<∞.
The Review summarizes much of particle physics and cosmology. Using data from previous editions, plus 3,200 new measurements from 903 papers, we list, evaluate, and average measured properties of gauge bosons and the recently discovered Higgs boson, leptons, quarks, mesons, and baryons. We summarize searches for hypothetical particles such as supersymmetric particles, heavy bosons, axions, dark photons, etc. Particle properties and search limits are listed in Summary Tables. We give numerous tables, figures, formulae, and reviews of topics such as Higgs Boson Physics, Supersymmetry, Grand Unified Theories, Neutrino Mixing, Dark Energy, Dark Matter, Cosmology, Particle Detectors, Colliders, Probability and Statistics. Most of the 118 reviews are updated, including many that are heavily revised. The Review is divided into two volumes. Volume 1 includes the Summary Tables and 96 review articles. Volume 2 consists of the Particle Listings and contains also 22 reviews that address specific aspects of the data presented in the Listings. The complete Review (both volumes) is published online on the website of the Particle Data Group ( pdg.lbl.gov ) and in a journal. Volume 1 is available in print as the PDG Book. A Particle Physics Booklet with the Summary Tables and essential tables, figures, and equations from selected review articles is available in print, as a web version optimized for use on phones, and as an Android app. The PDG API (Application Programming Interface) provides access to the data published in the Review in machine-readable format.
The last two decades has seen quantum thermodynamics become a well established field of research in its own right. In that time, it has demonstrated a remarkably broad applicability, ranging from providing foundational advances in the understanding of how thermodynamic principles apply at the nano-scale and in the presence of quantum coherence, to providing a guiding framework for the development of efficient quantum devices. Exquisite levels of control have allowed state-of-the-art experimental platforms to explore energetics and thermodynamics at the smallest scales which has in turn helped to drive theoretical advances. This Roadmap provides an overview of the recent developments across many of the field's sub-disciplines, assessing the key challenges and future prospects, providing a guide for its near term progress.
We study an elastic version of the Calderón problem: determine the internal mass density ρ (x) from the Neumann-to-Dirichlet (N-D) map associated with the isotropic Lamé system ℒ_λ ,μu + ω ^2 ρ (x) u = 0 in a bounded elastic body Ω⊂ℝ^3 . To the best of our knowledge, this work provides the first constructive strategy, based on embedding resonant hard inclusions, for the Calderón-type inverse problem in the isotropic Lamé system to reconstruct the density ρ . The key to our strategy is to induce a uniform negative shift in the effective density (i.e., a negative effective density) by embedding a subwavelength periodic array of resonant high-density inclusions. We insert a periodic cluster of high-density inclusions of size a and density ρ _1 ≃ a^-2 into Ω , away from ∂Ω . For excitation frequencies ω tuned to a suitable eigenvalue of the elastic Newton operator (i.e., Kelvin operator) associated with a single inclusion, we show that the N-D map Λ _D of the composite medium converges, as a → 0 and the number M of inclusions tends to infinity, to an effective map Λ _𝒫 corresponding to an elastic medium with a uniform negative density shift -𝒫^2 . We prove an operator norm estimate ‖Λ _D - Λ _𝒫‖≤ C a^α𝒫^6, with α > 0 depending on the geometric scaling. We then derive a first-order linearization formula for Λ _𝒫 around this negative background, expressed in terms of ρ and the Newton volume potential for the shifted Lamé operator. By testing this linearized relation with suitable complex geometric optics solutions for the Lamé system, we obtain a reconstruction formula for the Fourier transform of ρ , and hence a global density recovery scheme. The method proposed in this paper demonstrates how metamaterial-inspired effective media can be exploited as an analytic tool for inverse coefficient problems in linear elasticity, enabling a tractable linearization around a negative background and an explicit global reconstruction procedure. This provides a novel strategy and paradigm for using nanoscale metamaterials to solve inverse problems.
High-entropy carbide ceramics (HECCs) possess promising properties for extreme high-temperature applications. Machine learning offers an effective pathway to accelerate the discovery of novel HECCs, but data imbalance poses challenges for predictive performance. Here, we integrate the Borderline-SMOTE with machine learning algorithms to address this issue. A dataset containing 251 samples was established from literature, experimental synthesis, and synthetic oversampling. Key features influencing phase formation were selected via a four-step feature selection strategy. Ten common machine learning models were trained and optimized, with the random forest (RF) model identified as the most suitable for predicting HECCs phase formation ability. Eight HECCs compositions with high uncertainty were experimentally validated, and the results were incorporated back into the dataset to iteratively improve model accuracy. This work provides an efficient strategy for predicting phase formation in HECCs, particularly for small or imbalanced datasets, facilitating the accelerated design and reliable prediction of new HECCs.
