Humans are exposed to complex mixtures of environmental pollutants rather than single chemicals, necessitating methods to quantify the health effects of such mixtures. Research on environmental mixtures provides insights into realistic exposure scenarios, informing regulatory policies that better protect public health. However, statistical challenges, including complex correlations among pollutants and nonlinear multivariate exposure-response relationships, complicate such analyses. A popular Bayesian semi-parametric Gaussian process regression framework (Coull et al., 2015) addresses these challenges by modeling exposure-response functions with Gaussian processes and performing feature selection to manage high-dimensional exposures while accounting for confounders. Originally designed for small to moderate-sized cohort studies, this framework does not scale well to massive datasets. To address this, we propose a divide-and-conquer strategy, partitioning data, computing posterior distributions in parallel, and combining results using the generalized median. While we focus on Gaussian process models for environmental mixtures, the proposed distributed computing strategy is broadly applicable to other Bayesian models with computationally prohibitive full-sample Markov Chain Monte Carlo fitting. We provide theoretical guarantees for the convergence of the proposed posterior distributions to those derived from the full sample. We apply this method to estimate associations between a mixture of ambient air pollutants and ~650,000 birthweights recorded in Massachusetts during 2001-2012. Our results reveal negative associations between birthweight and traffic pollution markers, including elemental and organic carbon and PM2.5, and positive associations with ozone and vegetation greenness.

Let $G$ be a simple finite connected graph of order $n$ greater than or equal to $3$. We obtain the following results: (1). We apply a result of Hamada and Yoshimura from 1976 and some recent results of Alikhani and Soltani (2020) and Kalinowski and Pilsniak (2015) to determine the distinguishing chromatic number of the middle graph $M(G)$ of the graph $G$. In particular, the distinguishing chromatic number $χ_{D}(M(G))$ of the middle graph $M(G)$ of the graph $G$ is $Δ(G)+1$ except for four small graphs $C_{4}, K_{4}, C_{6}$, and $K_{3,3}$, and $Δ(G)+2$ otherwise. (2). In 2016, Kalinowski, Pilsniak, and Wozniak introduced the total distinguishing number $D''(G)$ of $G$. Inspired by a recent result of Mirafzal (2024), we show that the distinguishing number $D(S(G))$ of the subdivision graph $S(G)$ of $G$ is $D''(G)$. Consequently, $D(S(G))$ is at most $\lceil \sqrt{Δ(G)}\rceil$. (3). We obtain a sharp upper bound for the distinguishing chromatic number of the subdivision graph $S(G)$ of $G$ in terms of the distinguishing number of $G$.

We study concavity properties of positive solutions to the Logarithmic Schrödinger equation $-Δu=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems $-Δu = σ\, (u^q-u)$ and build, for any $σ>0$ and $q>1$, solutions $u_q$ such that $u_q^{(1-q)/2}$ is convex. By choosing $σ_q=2/(q-1)$ and letting $q \to 1^+$ we eventually construct a solution $u$ of the Logarithmic Schrödinger equation such that $\log u$ is concave. This seems to be one of the few attempts at studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks.

Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.

This paper explores the relationship between Cartan symmetries, dynamical similarities, and dynamical symmetries in contact Hamiltonian mechanics. By introducing an alternative decomposition of vector fields, we characterize these symmetries and present a novel description in terms of tensor densities. Furthermore, we demonstrate that this framework allows, under specific conditions, for the recovery of integrals of motion. We also establish new criteria to assess their independence.

In the strong-coupling regime, the interaction between light and matter reaches a hybridization state where the photonic and material components are inseparably linked. Using tailored states of light to break symmetries in such systems can facilitate the development of novel non-equilibrium quantum materials. Chiral optical cavities offer a promising approach for this, enabling either temporal or spatial symmetry-breaking, both of which are unachievable with conventional mirror cavities. For spatial symmetry-breaking, a cavity must discriminate the handedness of circularly polarized light, a functionality uniquely provided by chiral metamaterials. Here, we propose and demonstrate experimentally a chiral transition metal dichalcogenide (TMDC) metasurface with broken out-of-plane symmetry, allowing for the selective formation of self-hybridized exciton-polaritons with specific handedness. Our metasurface maintains maximal chirality for oblique incidence up to 20°, significantly outperforming all previously known designs, thereby transforming the angle of incidence from a constraint into a new degree of freedom for sub-nanometer-precise tuning of the cavity's resonant wavelength. Moreover, we study the chiral strong-coupling regime in nonlinear experiments and reveal the polariton-driven nature of chiral third-harmonic generation. Our results demonstrate a clear pathway towards van der Waals (vdW) metasurfaces as a novel and potent platform for chiral polaritonics with implications in a wide range of photonics research, such as non-reciprocal photonic devices and valleytronics.

High precision measurements are essential to solve major scientific and technological challenges, from gravitational wave detection to healthcare diagnostics. Quantum sensing delivers greater precision, but an in-depth optimisation of measurement procedures has been overlooked. Here we present a systematic strategy for parameter estimation in the low-data limit that integrates experimental control parameters and natural symmetries. The method is guided by a Bayesian quantifier of precision gain, enabling adaptive optimisation tailored to the experiment. We provide general expressions for optimal estimators for any parameter. The strategy's power is demonstrated in a quantum technology experiment, in which ultracold caesium atoms are confined in a micromachined hole in an optical fibre. We find a five-fold reduction in the fractional variance of the estimated parameter, compared to the standard measurement procedure. Equivalently, our strategy achieves a target precision with a third of the data points previously required. Such enhanced device performance and accelerated data collection will be essential for applications in quantum computing, communication, metrology, and the wider quantum technology sector.

In response to the growing need for generating real-world evidence from multi-site collaborative studies, we introduce an efficient collaborative learning approach to evaluate average treatment effect (ECO-ATE) in a multi-site setting under data sharing constraints. Specifically, ECO-ATE operates in a federated manner, using individual-level data from a user-defined target population and summary statistics from other source populations, to construct efficient estimator for the average treatment effect on the target population of interest. Our federated approach does not require iterative communications between sites, making it particularly suitable for research consortia with limited resources for developing automated data-sharing infrastructures. Compared to existing work data integration methods in causal inference, ECO-ATE allows distributional shifts in outcomes, treatments and baseline covariates distributions, and achieves semiparametric efficiency bound under appropriate conditions. We conduct simulation studies to demonstrate the extent of efficiency gains achieved by incorporating additional data sources, as well as the robustness of our approach against varying levels of distributional shifts and overparameterization, compared to existing benchmarks. We apply ECO-ATE to a case study examining the effect of insulin vs. non-insulin treatments on heart failure for patients with type II diabetes using electronic health record data collected from the All of Us program.

Consciousness science faces the challenge of bridging first-person experience with third-person empirical measurements. Neurophenomenology aims to build such `generative passages' connecting the content of experience with behavioural and neuroscientific data. However, the mathematical machinery for such bridges remains underdeveloped. Here we develop a Rosetta Stone hypothesis from predictive processing, where beliefs serve as a central hub connecting phenomenology, behaviour, and neural dynamics. This hinges on a central technical assumption that phenomenology is a function of beliefs. We pursue a conditional approach: if this assumption holds, then certain predictions mathematically follow. We derive predictions for subjective similarity judgements, cognitive metabolic cost, subjective cognitive effort, and time perception. We review the connection between beliefs and neural dynamics to complete the generative passage for neurophenomenology, omitting the connection between beliefs and behaviour as this is already well-documented elsewhere. Testing our predictions will inform the validity of the central assumption connecting beliefs and phenomenology, and advance the neurophenomenology research programme.

We study the universal structure of late-time ensembles obtained from unitary dynamics in quantum chaotic systems with symmetries, such as charge or energy conservation. We find that although quantum states do not ergodically explore the entire Hilbert space at late times, the late-time ensemble typically becomes indistinguishable from Haar-random states in the thermodynamic limit at the level of finite statistical moments. Importantly, our results apply to initial states easy to prepare in ongoing experiments -- specifically, product states -- that lie in the middle of the spectrum of quantum chaotic systems. We show that these states typically exhibit not only the same late-time ensemble average as Haar-random states, but also the same state-to-state fluctuations and higher statistical moments. In other words, there is no measurement -- whether local or nonlocal -- at the level of finite statistical moments that can tell that the states are not exploring the entire Hilbert space. Interestingly, within the class of low-entanglement initial states, we also find atypical initial conditions in the middle of the spectrum of Hamiltonians known to be "maximally chaotic". Such atypical states have smaller variance of the symmetry operator than Haar-random states and evolve into non-universal ensembles that can be distinguished from the Haar ensemble by simple measurements or subsystem properties. In the limiting case of initial states with negligible variance of the symmetry operator (e.g., states with fixed particle number or energy eigenstates), the late-time ensemble has universal behavior captured by constrained random-state ensembles. Our results reveal that an extremely high level of quantum state randomness can still be achieved even when dynamics is constrained by symmetries.

We study the geometry of spaces of fitrations on a Noetherian local domain. We introduce a metric $d_1$ on the space of saturated filtrations, inspired by the Darvas metric in complex geometry, such that it is a geodesic metric space. In the toric case, using Newton-Okounkov bodies, we identify the space of saturated monomial filtrations with a subspace of $L^1_\mathrm{loc}$. We also consider several other topologies on such spaces and study the semi-continuity of the log canonical threshold function in the spirit of Kollár-Demailly. Moreover, there is a natural lattice structure on the space of saturated filtrations, which is a generalization of the classical result that the ideals of a ring form a lattice.

This paper is concerned with the construction of discrete counterparts of the Laplace-Beltrami operator on Riemannian manifolds that can be effectively used in the numerical solution of partial differential equations. Since existing constructions often lack rigorous convergence guarantees or imply a significant computational effort, we focus on designing operators that are both computationally feasible and supported by convergence results. We consider as a starting point the two-dimensional hyperbolic space $\mathbb{H}^2$, one of the simplest non-Euclidean settings, and develop two variants of discrete finite-difference operator tailored to this constant negatively curved space, both serving as approximations to the (continuous) Laplace-Beltrami operator within the $\mathrm{L}^2$ framework. We prove that the discrete heat equation associated with both operators mentioned above exhibits stability and converges towards the continuous heat-Beltrami Cauchy problem on $\mathbb{H}^2$. Moreover, using techniques inspired from the sharp analysis of discrete functional inequalities, we prove that the solutions of the discrete heat equations corresponding to both variants of discrete Laplacian exhibit an exponential decay asymptotically equal to the one induced by the Poincaré inequality on $\mathbb{H}^2$. Eventually, we illustrate that a discrete Laplacian specifically designed for the geometry of the hyperbolic space yields a more precise approximation and offers advantages from both theoretical and computational perspectives. Furthermore, this discrete operator can be effectively generalized to the three-dimensional hyperbolic space.

In most cases the semigroup at infinity $S$ of a curve $C$ with only one place at infinity is generated by a $δ$-sequence. This sequence provides geometrical information on $C$ such as the dual graph of the resolution of the singularity of $C$ at infinity. Since different $δ$-sequences can generate the same semigroup, it is an interesting problem to know the geometrical behaviour of curves $C$ sharing the same semigroup $S$. An analogous problem arises in a more general context when considering surfaces at infinity and their $δ$-semigroups. We show how to construct $δ$-sequences, and how to obtain different families that generate the same semigroup $S$, allowing us to study the geometrical content encoded by $S$.

Modern optomechanical systems employ increasingly sophisticated quantum-mechanical states of light to probe and manipulate mechanical motion. Squeezed states are now used routinely to enhance the sensitivity of gravitational-wave interferometers to small external forces, and they are also used in feedback-based trapping and damping experiments on the same interferometers to enhance the achievable cooling of fluctuations in the differential test mass mode (arXiv:2102.12665). In this latter context, an accurate accounting of the true test mass motion, incorporating all sources of loss, the effect of feedback control, and the influence of classical force and sensing noises, is paramount. We work within the two-photon formalism to provide such an accounting, which extends a previously described decomposition of the quantum-mechanical noise of the light field (arxiv:2105.12052). This decomposition provides insight, rooted in physically motivated parameters, into the optimal squeezed state and feedback control configuration that should be employed to achieve the lowest fluctuations. We apply this formalism to feedback damping experiments in current and possible future gravitational-wave interferometers -- LIGO A+, LIGO Voyager, Cosmic Explorer (CE), and CE Voyager -- and discuss how these multi-degree-of-freedom systems might be compared to a single degree-of-freedom oscillator. We find that, for the oscillator definition used most commonly in the literature so far, occupation numbers below 1 are possible in these interferometers over a frequency range comparable to the bandwidth of the trapped and cooled oscillator. We also discuss several technical issues in cooling experiments with gravitational-wave detectors

We study the Activated Random Walk model on the one-dimensional ring, in the high density regime. We develop a toppling procedure that gradually builds an environment that can be used to show that activity will be sustained for a long time. This yields a self-contained and relatively short proof of existence of a slow phase for arbitrarily large sleep rates.

Consider a binary mixture model of the form $F_θ= (1-θ)F_0 + θF_1$, where $F_0$ is standard Gaussian and $F_1$ is a completely specified heavy-tailed distribution with the same support. For a sample of $n$ independent and identically distributed values $X_i \sim F_θ$, the maximum likelihood estimator $\hatθ_n$ is asymptotically normal provided that $0 < θ< 1$ is an interior point. This paper investigates the large-sample behaviour for boundary points, which is entirely different and strikingly asymmetric for $θ=0$ and $θ=1$. The reason for the asymmetry has to do with typical choices such that $F_0$ is an extreme boundary point and $F_1$ is usually not extreme. On the right boundary, well known results on boundary parameter problems are recovered, giving $\lim \mathbb{P}_1(\hatθ_n < 1)=1/2$. On the left boundary, $\lim\mathbb{P}_0(\hatθ_n > 0)=1-1/α$, where $1\leq α\leq 2$ indexes the domain of attraction of the density ratio $f_1(X)/f_0(X)$ when $X\sim F_0$. For $α=1$, which is the most important case in practice, we show how the tail behaviour of $F_1$ governs the rate at which $\mathbb{P}_0(\hatθ_n > 0)$ tends to zero. A new limit theorem for the joint distribution of the sample maximum and sample mean conditional on positivity establishes multiple inferential anomalies. Most notably, given $\hatθ_n > 0$, the likelihood ratio statistic has a conditional null limit distribution $G\neqχ^2_1$ determined by the joint limit theorem. We show through this route that no advantage is gained by extending the single distribution $F_1$ to the nonparametric composite mixture generated by the same tail-equivalence class.

We propose a scalable framework for deciding, proving, and explaining (in-)equivalence of context-free grammars. We present an implementation of the framework and evaluate it on large data sets collected within educational support systems. Even though the equivalence problem for context-free languages is undecidable in general, the framework is able to handle a large portion of these datasets. It introduces and combines techniques from several areas, such as an abstract grammar transformation language to identify equivalent grammars as well as sufficiently similar inequivalent grammars, theory-based comparison algorithms for a large class of context-free languages, and a graph-theory-inspired grammar canonization that allows to efficiently identify isomorphic grammars.

With the escalating proliferation of artificial intelligence technologies, AI-generated content (AIGC) has progressively permeated across diverse domains. However, this explosive application has also sparked widespread public discussion about the copyright of AIGC. Existing copyright legal frameworks, originally designed around human creators, now face a paradigm shift. As human involvement in the generation of AIGC diminishes, where creative expression increasingly hinges on AI. This discrepancy has introduced multifaceted complexities and challenges in determining the copyright ownership of AIGC within established legal boundaries. Given this, meticulous recording and auditing of contributions from all parties in AIGC generation becomes imperative. Blockchain, with its decentralized storage, offers a robust technical foundation for AIGC copyright management. Yet existing blockchain-based solutions have clear limitations: most only focus on certifying final generated products, ignoring the management of critical intermediate data across the full lifecycle, thus failing to meet the needs of core scenarios like copyright confirmation and multi-party profit distribution. For this purpose, this paper introduces AIGC-Chain, a trustworthy AIGC copyright management system. It conducts a comprehensive recording of intermediate data generated across the full lifecycle of AIGC. Such data is deposited into a decentralized blockchain for secure multi-party auditing, thereby constructing a trustworthy management for AIGC copyright. In copyright dispute scenarios, auditors can retrieve critical proof from the blockchain, facilitating precise determination of the copyright ownership of AIGC products. Both theoretical and experimental analyses confirm that this scheme shows exceptional performance and security in AIGC copyright management.

We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length $N$ taking values of size close to $N^{3/4}$, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate $N(σ,T)\le T^{30(1-σ)/13+o(1)}$ and asymptotics for primes in short intervals of length $x^{17/30+o(1)}$.

This paper is concerned with the inverse problem of retrieving the initial value of a time-fractional fourth order parabolic equation from source and final time observation. The considered problem is an {\it ill-posed problem.} We obtain regularized approximations for the sought initial value by employing the quasi-boundary value method, its modified version and by Fourier truncation method(FTM). We provide both the apriori and aposteriori parameter choice strategies and derive the error estimates for all these methods under some {\it source conditions} involving some Sobolev smoothness. As an important implication of the obtained rates, we observe that for both the apriori and aposteriori cases, the rates obtained by all these three methods are same for some source sets. Moreover, we observe that in both the apriori and aposteriori cases, the FTM is free from the so-called {\it saturation effect}, whereas both the quasi-boundary value method and its generalizations possesses the saturation effect for both the cases. Further, we observe that the rates obtained by the FTM is always order optimal for all the considered source sets.

In this paper, we explore lifting Markov Decision Processes (MDPs) to the space of probability measures and consider the so-called measurized MDPs: deterministic processes where states are probability measures on the original state space, and actions are stochastic kernels on the original action space. We show that measurized MDPs are a generalization of stochastic MDPs, thus the measurized framework can be deployed without loss of fidelity. Bertsekas and Shreve studied similar deterministic MDPs under the discounted infinite-horizon criterion in the context of universally measurable policies. Here, we also consider the long-run average reward case, but we cast lifted MDPs within the semicontinuous-semicompact framework of Hernández-Lerma and Lasserre. This makes the lifted framework more accessible as it entails (i) optimal Borel-measurable value functions and policies, (ii) reasonably mild assumptions that are easier to verify than those in the universally-measurable framework, and (iii) simpler proofs. In addition, we showcase the untapped potential of lifted MDPs by demonstrating how the measurized framework enables the incorporation of constraints and value function approximations that are not available from the standard MDP setting. Furthermore, we introduce a novel algebraic lifting procedure for any MDP, showing that non-deterministic measure-valued MDPs can emerge from lifting MDPs impacted by external random shocks.

Andoni, Naor and Neiman (2018) established a family of quadratic metric inequalities that hold true in every CAT(0) space. As stated in their paper, this family seems to include all previously used quadratic metric inequalities that hold true in every CAT(0) space. We prove that there exists a metric space that satisfies all inequalities in this family but does not admit an isometric embedding into any CAT(0) space. More precisely, we prove that the 6-point metric space constructed by Nina Lebedeva, which does not admit an isometric embedding into any CAT(0) space, satisfies all inequalities in this family.

The propensity score is widely used for causal inference in observational studies, but common parametric estimators can produce biased and inefficient effect estimates when model assumptions are violated. Nonparametric approaches reduce sensitivity to misspecification but often yield unstable weights and inadequate covariate balance. We propose Local Balance with Calibration, implemented by Neural Networks, a weighting method that combines flexible function approximation with the explicit enforcement of covariate balance and calibration. When used with inverse probability weighting, the proposed estimator produces more stable weights, improved covariate balance, and reduced bias in average treatment effect estimation compared with existing approaches. We further develop an influence-function-based variance estimator that provides accurate uncertainty quantification for the resulting weighted estimators. Numerical studies demonstrate improved efficiency and reliable variance estimation across a range of data-generating scenarios. The method is implemented using the publicly available R package LBCNet.

We construct families of rational functions $f \colon \bP^1_k \to \bP^1_k$ of degree $d \geq 2$ over a perfect field $k$ whose associated fixed-point processes fail to be martingales. Conversely, for any normal variety $X \subset \bP^N_{\overline{k}}$ and a finite, generically étale morphism $f \colon X \to X$, we establish geometric conditions on the critical orbits of $f$ that guarantee the fixed-point process is a martingale. Our constructions answer a question of Bridy, Jones, Kelsey, and Lodge \cite{iterated} regarding the existence of non-martingale behaviour in arboreal Galois representations, and extend their martingale criteria to higher-dimensional dynamical systems. In particular, we exhibit infinitely many postcritically finite maps with non-martingale fixed-point processes and characterize the group-theoretic obstructions to the martingale property in the genus-zero case. Furthermore, we prove that despite the failure of the martingale property, the fixed-point proportion still vanishes with a quantifiable convergence rate.

The large-matrix limit laws of the rescaled largest eigenvalue of the orthogonal, unitary, and symplectic $n$-dimensional Gaussian ensembles -- and of the corresponding Laguerre ensembles (Wishart distributions) for various regimes of the parameter $α$ (degrees of freedom $p$) -- are known to be the Tracy-Widom distributions $F_β$ ($β=1,2,4$). We establish (paying particular attention to large or small ratios $p/n$) that, with careful choices of the rescaling constants and of the expansion parameter $h$, the limit laws embed into asymptotic expansions in powers of $h$, where $h \asymp n^{-2/3}$ resp. $h \asymp (n\,\wedge\,p)^{-2/3}$. We find explicit analytic expressions of the first few expansion terms as linear combinations of higher-order derivatives of the limit law $F_β$ with rational polynomial coefficients. The parametrizations are fine-tuned so that the expansion coefficients in the Gaussian cases are, for given $n$, the limits $p\to\infty$ of those of the Laguerre cases. Whereas the results for $β=2$ are presented with proof, the discussion of the cases $β=1,4$ is based on some hypotheses, focusing on the algebraic aspects of actually computing the polynomial coefficients. For the purposes of illustration and validation, the various results are checked against simulation data with large sample sizes.

We introduce a new invariant of fields that refines their real spectrum and is related to their absolute Galois group: the Artin-Schreier quandle. For formally real number fields, it is freely generated in its variety by a Cantor space of indeterminates. For Laurent series fields, we compute it in terms of the Artin-Schreier quandle of the coefficient field. This result and other examples show that, in general, there are relations.

Exploring the dependence between covariates across distributions is crucial for many applications. Copulas serve as a powerful tool for modeling joint variable dependencies and have been effectively applied in various practical contexts due to their intuitive properties. However, existing computational methods lack the capability for feasible inference and sampling of any copula, preventing their widespread use. This paper introduces an innovative quasi-random sampling approach for copulas, utilizing generative adversarial networks (GANs) and space-filling designs. The proposed framework constructs a direct mapping from low-dimensional uniform distributions to high-dimensional copula structures using GANs, and generates quasi-random samples for any copula structure from points set of space-filling designs. In the high-dimensional situations with limited data, the proposed approach significantly enhances sampling accuracy and computational efficiency compared to existing methods. Additionally, we develop convergence rate theory for quasi-Monte Carlo estimators, providing rigorous upper bounds for bias and variance. Both simulated experiments and practical implementations, particularly in risk management, validate the proposed method and showcase its superiority over existing alternatives.

Sparse signal recovery deals with finding the sparsest solution of an under-determined linear system $\vx = \mQ\vs$. In this paper, we propose a novel greedy approach to addressing the challenges from such a problem. Such an approach is based on a characterization of solutions to the system, which allows us to work on the sparse recovery in the $\vs$-space directly with a given measure. With $l_2$-based measure, an orthogonal matching pursuit (OMP)-type algorithm is proposed, which significantly outperforms the classical OMP algorithm in terms of recovery accuracy while maintaining comparable computational complexity. An $l_1$-based algorithm, denoted as $\text{Alg}_{GL1}$, is derived. Such an algorithm significantly outperforms the classical basis pursuit (BP) algorithm. Combining with the CoSaMP-strategy for selecting atoms, a class of high performance greedy algorithms is also derived. Extensive numerical simulations on both synthetic and image data are carried out, with which the superior performance of our proposed algorithms is demonstrated in terms of sparse recovery accuracy and robustness against numerical instability of the system matrix $\mQ$ and disturbance in the measurement $\vx$.

Crystallographic defects in a topological semimetal can result in charge doping, and the scattering due to the defects may mask its exotic transport properties. Here, we investigate the possible crystallographic defects including vacancy and antisite in Weyl semimetal LaAlGe using hybrid-density-functional theory calculations. We show that a considerable concentration of Al- and Ge-related defects naturally form during growth due to their low formation enthalpy. Specifically, Al can be easily replaced by Ge in the $I4_1md$ phase of LaAlGe, forming the Ge-on-Al antisite, Ge$_{\rm Al}$. The counterpart, Al-on-Ge (Al$_{\rm Ge}$), is also probable. The most abundant defect Ge$_{\rm Al}$ is donor-like, effectively electron-doping, and these defects are therefore not only scattering centers in the electronic transport process but may also induce the substantial vertical shift of the chemical potential. The results imply that the naturally occurring defects hinder both spectroscopic and transport features arising from the Weyl physics in LaAlGe. Our work can be applied to the $R$AlGe family ($R$=rare earth) and help improve the quality of single-crystal magnetic Weyl semimetal.

This paper develops an Afriat-type characterization of concave rationalization with an unknown ideal point. We show that a system of Afriat inequalities - where the unknown peak enters as a virtual observation with the highest utility - is necessary and sufficient for the existence of a continuous concave utility with an ideal point that rationalizes choices from linear budget sets anchored at different corners of the choice space. A stronger characterization adds the requirement that supergradients at observed choices point coordinatewise toward the peak, a necessary condition for single-peaked rationalizability. The resulting peak-oriented Afriat system provides the basis for a Houtman--Maks consistency index that measures the largest fraction of observations jointly rationalizable with a common ideal point. This characterization provides the theoretical foundation for the Priced Survey Methodology (PSM), in which respondents complete the same survey under different linear constraints. A parametric single-peaked specification then sharpens identification into estimates of ideal answers and importance weights. We apply the PSM to study political preferences in a sample of French respondents.