Using the elementary axioms of special relativity and quantum mechanics we construct a wave equation which generalizes the Schrodinger equation. We also solve the general second and some higher order differential equations.

Developers often build software on top of third-party libraries (Libs) to improve productivity, but these libraries may contain vulnerabilities that enable supply chain attacks. Existing tools detect vulnerable dependencies, yet developers often distrust their reports without concrete exploit evidence. Manually crafting such demonstrations is costly, and tool support is lacking. To help developers enhance software security, in this study, we systematically explored the usage of a large language model (LLM) --ChatGPT-4.0--to generate security tests, which unit tests demonstrate how vulnerable library dependencies facilitate the supply chain attacks to given Apps. In our exploration, we defined prompt templates to take in the various vulnerability-relevant information we manually collected, and generated prompts from those templates to query ChatGPT for security test generation. We found that ChatGPT-generated tests demonstrated 24 pieces of evidence or proof of vulnerability for 49 Apps. To assess the consistency of test generation, we also evaluated another five state-of-the-art LLMs. All the models generated security tests for at least 17 cases that successfully demonstrate the vulnerabilities. We filed six reports for the newly revealed vulnerabilities in Apps, and got four Common Vulnerability Entries (CVEs) assigned. Our use of ChatGPT outperformed two state-of-the-art security test generators (TRANSFER and SIEGE), by generating a lot more tests and achieving more attacks.

A pair of tensors $(g,B)$ form the induced metric and shape operator of an immersion into hyperbolic space if and only if they satisfy the Gauss-Codazzi equations. Such a pair of tensors induce a pair $(\hat{g},\hat{B})$ related to the ideal boundary of hyperbolic space. Krasnov and Schlenker, and Bridgeman and Bromberg show in the surface case that there is a duality between $(g,B)$ and $(\hat{g},\hat{B})$. Moreover, $(g,B)$ solves the Gauss-Codazzi equations if and only if $(\hat{g},\hat{B})$ solve a corresponding set of equations. We show a similar duality exists and identify these corresponding equations for an arbitrary dimension, as well as show there exists a unique solution for $\hat{B}$ provided $\hat{g}$ is locally conformally flat. As an application, we offer a proof of the Weyl-Schouten theorem concerning locally conformally flat metrics that factors through hyperbolic geometry.

Adversarial training (AT) is a prominent technique employed by deep learning models to defend against adversarial attacks, and to some extent, enhance model robustness. However, there are three main drawbacks of the existing AT-based defense methods: expensive computational cost, low generalization ability, and the dilemma between the original model and the defense model. To this end, we propose a novel benchmark called meta adversarial defense (MAD). The MAD benchmark consists of two MAD datasets, along with a MAD evaluation protocol. The two large-scale MAD datasets were generated through experiments using 30 kinds of attacks on MNIST and CIFAR-10 datasets. In addition, we introduce a meta-learning based adversarial training (Meta-AT) algorithm as the baseline, which features high robustness to unseen adversarial attacks through few-shot learning. Experimental results demonstrate the effectiveness of our Meta-AT algorithm compared to the state-of-the-art methods. Furthermore, the model after Meta-AT maintains a relatively high clean-samples classification accuracy (CCA). It is worth noting that Meta-AT addresses all three aforementioned limitations, leading to substantial improvements. This benchmark ultimately achieved breakthroughs in investigating the transferability of adversarial defense methods to new attacks and the ability to learn from a limited number of adversarial examples. Our codes and attacked datasets address will be available at https://github.com/PXX1110/Meta_AT.

We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Lie group. The corresponding Gibbs measure is given by a Brownian motion on the Lie group, which plays a central role in stochastic geometry. Our main theorem is the almost sure global well-posedness and invariance of the Gibbs measure for the wave maps equation. It is the first result of this kind for any geometric wave equation. Our argument relies on a novel finite-dimensional approximation of the wave maps equation which involves the so-called Killing renormalization. The main part of this article then addresses the global convergence of our approximation and the almost invariance of the Gibbs measure under the corresponding flow. The proof of global convergence requires a carefully crafted Ansatz which includes modulated linear waves, modulated bilinear waves, and mixed modulated objects. The interactions between the different objects in our Ansatz are analyzed using an intricate combination of analytic, geometric, and probabilistic ingredients. In particular, geometric aspects of the wave maps equation are utilized via orthogonality, which has previously been used in the deterministic theory of wave maps at critical regularity. The proof of almost invariance of the Gibbs measure under our approximation relies on conservative structures, which are a new framework for the approximation of Hamiltonian equations, and delicate estimates of the energy increment.

Instrumental variable approaches have gained popularity for estimating causal effects in the presence of unmeasured confounders. However, the availability of instrumental variables in the primary dataset is often challenged due to stringent and untestable assumptions. This paper presents a novel method to identify and estimate causal effects by utilizing instrumental variables from the auxiliary dataset, incorporating a structural equation model, even in scenarios with nonlinear treatment effects. Our approach involves using two datasets: one called the primary dataset with joint observations of treatment and outcome, and another auxiliary dataset providing information about the instrument and treatment. Our strategy differs from most existing methods by not depending on the simultaneous measurements of instrument and outcome. The central idea for identifying causal effects is to establish a valid substitute through the auxiliary dataset, addressing unmeasured confounders. This is achieved by developing a control function and projecting it onto the function space spanned by the treatment variable. We then propose a three-step estimator for estimating causal effects and derive its asymptotic results. We illustrate the proposed estimator through simulation studies, and the results demonstrate favorable performance. We also conduct a real data analysis to evaluate the causal effect between vitamin D status and body mass index.

We investigate when an ordered abelian group $G$ is stably embedded in a given elementary extension $H$. We focus on a large class of ordered groups which includes maximal ordered groups with interpretable archimedean valuation. We give a complete answer for groups in this class which takes the form of a transfer principle for valued groups. It follows in particular that all types in the lexicographic product $\prod_{i\in ω} \mathbb{Z}$ are definable.

Randomly moving active particles can be herded into directed motion by asymmetric geometric structures. Although such a rectification process has been extensively studied due to its fundamental, biological, and technological relevance, a comprehensive understanding of active matter rectification based on single particle dynamics remains elusive. Here, by combining experiments, simulations, and theory, we study the directed transport and energetics of swimming bacteria navigating through funnel-shaped obstacles -- a paradigmatic model of rectification of living active matter. We develop a microscopic parameter-free model for bacterial rectification, which quantitatively explains experimental and numerical observations and predicts the optimal geometry for the maximum rectification efficiency. Furthermore, we quantify the degree of time irreversibility and measure the extractable work associated with bacterial rectification. Our study provides quantitative solutions to long-standing questions on bacterial rectification and establishes a generic relationship between time irreversibility, particle fluxes, and extractable work, shedding light on the energetics of non-equilibrium rectification processes in living systems.

Coalescing simple random walks in the plane form an infinite tree. A natural directed distance on this tree is given by the number of jumps between branches when one is only allowed to move in one direction. The Brownian web distance is the scale-invariant limit of this directed metric. It is integer-valued and has scaling exponents 0:1:2 as compared to 1:2:3 in the KPZ world. However, we show that the shear limit of the Brownian web distance is still given by the Airy process. We conjecture that our limit theorem can be extended to the full directed landscape.

High-quality low-dimensional layered and van der Waals materials are typically exfoliated, with sample cross sectional areas on the order of tens to hundreds of microns. The small size of flakes makes the experimental characterization of their dielectric properties unsuitable with conventional spectroscopic ellipsometry, due to beam-sample size mismatch and non-uniformities of the crystal axes. Previously, the experimental measurement of the dielectrirc permittivity of such microcrystals was carried out with near-field tip-based scanning probes. These measurements are sensitive to external conditions like vibrations and temperature, and require non-deterministic numerical fitting to some a priori known model. We present an alternative method to extract the in-plane dielectric permittivity of van der Waals microcrystals, based on identifying reflectance minima in spectroscopic measurements. Our method does not require complex fitting algorithms nor near field tip-based measurements and accommodates for small-area samples. We demonstrate the robustness of our method using hexagonal boron nitride and α-MoO3, and recover their dielectric permittivities that are close to literature values.

The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work, the long-time asymptotics for the defocusing nonlinear Schrödinger equation with general step-like initial data is investigated by the Whitham modulation theory and Riemann-Hilbert formulation. The Whitham modulation theory shows that there are six cases for the initial discontinuity problem according to the orders of the Riemann invariants. The leading-order terms and the corresponding error estimates for each region of the six cases are formulated by the Deift-Zhou nonlinear steepest descent method for oscillatory Riemann-Hilbert problems. It is demonstrated that the long-time asymptotic solutions match very well with the results from Whitham modulation theory and the numerical simulations.

The Barnett effect, discovered more than a century ago, describes how an inertial body with otherwise zero net magnetic moment acquires spontaneous magnetization when mechanically spinning. Breakthrough experiments have recently shown that an ultrashort laser pulse destroys the magnetization of an ordered ferromagnet within hundreds of femtoseconds, with the spins losing angular momentum to circularly-polarized optical phonons as part of the ultrafast Einstein-de Haas effect. However, the prospect of using such high-frequency vibrations of the lattice to reciprocally switch magnetization in a nearby magnetic medium has not yet been experimentally explored. Here we show that the spontaneous magnetization temporarily gained via the ultrafast Barnett effect, through the resonant excitation of circularly-polarized optical phonons in paramagnetic substrates, can be used to permanently reverse the magnetic state of the substrate-mounted heterostructure. With the handedness of the phonons steering the direction of magnetic switching, the ultrafast Barnett effect offers a selective and potentially universal method for exercising ultrafast non-local control over magnetic order.

Assessing causal effects in the presence of unmeasured confounding is challenging. Although auxiliary variables, such as instrumental variables, are commonly used to identify causal effects, they are often unavailable in practice due to stringent and untestable conditions. To address this issue, previous researches have utilized linear structural equation models to show that the causal effect is identifiable when noise variables of the treatment and outcome are both non-Gaussian. In this paper, we investigate the problem of identifying the causal effect using the auxiliary covariate and non-Gaussianity from the treatment. Our key idea is to characterize the impact of unmeasured confounders using an observed covariate, assuming they are all Gaussian. We demonstrate that the causal effect can be identified using a measured covariate, and then extend the identification results to the multi-treatment setting. We further develop a simple estimation procedure for estimating causal effects and derive a $\sqrt{n}$-consistent estimator. Finally, we evaluate the performance of our estimator through simulation studies and apply our method to investigate the effect of the trade on income.

Let $(\{f_j\}_{j=1}^n, \{τ_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^m, \{ω_k\}_{k=1}^m)$ be p-Schauder frames for a finite dimensional Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align} (1) \quad \|θ_f x\|_0^\frac{1}{p}\|θ_g x\|_0^\frac{1}{q} \geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(ω_k)|}\quad \text{and} \quad \|θ_g x\|_0^\frac{1}{p}\|θ_f x\|_0^\frac{1}{q}\geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|g_k(τ_j)|}. \end{align} where \begin{align*} θ_f: \mathcal{X} \ni x \mapsto (f_j(x) )_{j=1}^n \in \ell^p([n]); \quad θ_g: \mathcal{X} \ni x \mapsto (g_k(x) )_{k=1}^m \in \ell^p([m]) \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (1) as \textbf{Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle}. Inequality (1) improves Ricaud-Torrésani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, it improves Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.

Modern economies require increasingly diverse and specialized skills, many of which depend on the acquisition of other skills first. Here we analyse US survey data to reveal a nested structure within skill portfolios, where the direction of dependency is inferred from asymmetrical conditional probabilities-occupations require one skill conditional on another. This directional nature suggests that advanced, specific skills and knowledge are often built upon broader, fundamental ones. We examine 70 million job transitions to show that human capital development and career progression follow this structured pathway in which skills more aligned with the nested structure command higher wage premiums, require longer education and are less likely to be automated. These disparities are evident across genders and racial-ethnic groups, explaining long-term wage penalties. Finally, we find that this nested structure has become even more pronounced over the past two decades, indicating increased barriers to upward job mobility.

The multi-index model with sparse dimension reduction matrix is a popular approach to circumvent the curse of dimensionality in a high-dimensional regression setting. Building on the single-index analysis by Alquier, P. & Biau, G. (Journal of Machine Learning Research 14 (2013) 243-280), we develop a PAC-Bayesian estimation method for a possibly misspecified multi-index model with unknown active dimension and an orthogonal dimension reduction matrix. Our main result is a non-asymptotic oracle inequality, which shows that the estimation method adapts to the active dimension of the model, the sparsity of the dimension reduction matrix and the regularity of the link function. Under a Sobolev regularity assumption on the link function the estimator achieves the minimax rate of convergence (up to a logarithmic factor) and no additional price is paid for the unknown active dimension.

We provide a definitive classification of all finite sets of regular polygons that admit a tiling of the hyperbolic plane, thereby establishing the decidability of the Domino Problem for this class of prototiles. We show that admissibility is determined by a finite set of local and inductive combinatorial constraints. This classification further leads to the discovery of the first known examples of weakly aperiodic protosets consisting of regular polygons in $\mathbb{H}^2$.

Antiferromagnetic (AFM) materials possess a well-recognized potential for ultrafast data processing thanks to their intrinsic ultrafast spin dynamics, absence of stray fields, and large spin transport effects. The very same properties, however, make their manipulation difficult, requiring frequencies in THz range and magnetic fields of tens of Teslas. Switching of AFM order implies going into the nonlinear regime, a largely unexplored territory. Here we use THz light from a free electron laser to drive antiferromagnetic NiO into a highly nonlinear regime and steer it out of nonlinearity with magnetic field from a 33-Tesla Bitter magnet. This demonstration of large-amplitude dynamics represents a crucial step towards ultrafast resonant switching of AFM order.

Blackbody thermal emission is spatially diffuse. Achieving highly directional thermal emission typically requires nanostructuring the surface of the thermally emissive medium. The most common configuration is a subwavelength grating that scatters surface polaritonic modes from the near-field to the far-field and produces antenna-like lobes of thermal emission. This concept, however, is typically limited to a particular linear polarization, and requires sophisticated lithography. Here, we revisit the simple motif of a planar Salisbury screen. We show analytically how the interplay between the real and imaginary part of the dielectric permittivity of the emitter layer defines the directional characteristics of emission, which can range from diffuse to highly directional. We propose a realistic configuration and show that hexagonal Boron Nitride thin films can enable grating-like thermal emission lobes in a lithography-free platform.

We study the problem of learning multivariate dependencies in nonparametric and high-dimensional settings. This includes but is not limited to graphical models. Our approach effectively combines several features that are missing from previous work on this problem: We show how the entire dependence structure can be learned nonparametrically while simultaneously evading the curse of dimensionality and relaxing common assumptions such as faithfulness. To this end, we introduce and study the neighbourhood lattice decomposition of a distribution, which is a compact, non-graphical representation of conditional independence (CI) that is valid in the absence of a faithful graphical representation. We show that the neighbourhood lattice decomposition exists in any graphical model and can be computed efficiently, nonparametrically, and consistently in high-dimensions without paying the usual curse of dimensionality. This gives a way to learn all of the independence relations implied by any graphical model, without requiring a priori knowledge of the graph or even the graph type. As a special case, our results provide a general solution to the problem of nonparametric estimation of high-dimensional CI structures over any graphical model.

We present an algorithm for regular expression parsing and submatch extraction based on tagged deterministic finite automata. The algorithm works with different disambiguation policies. We give detailed pseudocode for the algorithm, covering important practical optimizations. All transformations from a regular expression to an optimized automaton are explained on a step-by-step example. We consider both ahead-of-time and just-in-time determinization and describe variants of the algorithm suited to each setting. We provide benchmarks showing that the algorithm is very fast in practice. Our research is based on two independent implementations: an open-source lexer generator RE2C and an experimental Java library.

We compare the Iwasawa invariants of fine Selmer groups of $p$-adic Galois representations over admissible $p$-adic Lie extensions of a number field $K$ to the Iwasawa invariants of ideal class groups along these Lie extensions. More precisely, let $K$ be a number field, let $V$ be a $p$-adic representation of the absolute Galois group $G_K$ of $K$, and choose a $G_K$-invariant lattice ${T \subseteq V}$. We study the fine Selmer groups of ${A = V/T}$ over suitable $p$-adic Lie extensions $K_\infty/K$, comparing their corank and $μ$-invariant to the corank and the $μ$-invariant of the Iwasawa module of ideal class groups in $K_\infty/K$. In the second part of the article, we compare the Iwasawa $μ$- and $l_0$-invariants of the fine Selmer groups of CM modular forms on the one hand and the Iwasawa invariants of ideal class groups on the other hand over trivialising multiple $\mathbb{Z}_p$-extensions of $K$.

This paper discusses an algorithmic tool introduced in the Basque Country (Spain) to assess the risk of intimate partner violence. The algorithm was introduced to address the lack of human experts by automatically calculating the level of violence based on psychometric features such as controlling or violent behaviour. Given that critical literature on risk assessment tools for domestic violence mainly focuses on English-speaking countries, this paper offers an algorithmic accountability analysis in a non-English speaking region. It investigates the algorithmic risks, harms, and limitations associated with the Basque tool. We propose a transdisciplinary approach from a critical statistical and legal perspective. This approach unveils issues and limitations that could lead to unexpected consequences for individuals suffering from partner violence. Moreover, our analysis suggests that the algorithmic tool has a high error rate on severe cases, i.e., cases where the aggressor could murder his partner -- 5 out of 10 high-risk cases are misclassified as low risk -- and that there is a lack of appropriate legal guidelines for judges, the end users of this tool. The paper concludes that this risk assessment tool needs to be urgently evaluated by independent and transdisciplinary experts to better mitigate algorithmic harms in the context of intimate partner violence.

Most of the exotic hidden-charm hadrons discovered over the last 20 years fit neatly into the quark model as normal mesons and baryons if the existence of a seventh flavor of quark is hypothesized. For the quark to reproduce the spectrum (mass, spin, parity) of exotic hadrons, it would have to have a mass of $\sim$2.9 GeV and a charge of $-\tfrac{1}{3}$. The observed production and decay modes of these hadrons can be reproduced by a model that also includes light scalar bosons.

In this paper, we consider a particular class of solutions to the Boltzmann equation which are referred to as homoenergetic solutions. They describe the dynamics of a dilute gas due to collisions and the action of either a shear, a dilation or a combination of both. We prove that solutions with initially high temperature remain close and converge to a Maxwellian distribution with temperature going to infinity. Furthermore, we give precise asymptotic formulas for the temperature. This local stability result is a consequence of a dominant shear and the homogeneity $ γ>0 $ of the collision operator with respect to relative velocities. The proof relies on an ansatz which is motivated by a Hilbert-type expansion. We consider both non-cutoff and cutoff kernels.

The fragility index is a clinically motivated metric designed to supplement the $p$ value during hypothesis testing. The measure relies on two pillars: selecting cases to have their outcome modified and modifying the outcomes. The measure is interesting but the case selection suffers from a drawback which can hamper its interpretation. This work presents the drawback and a method, the stochastic generalized fragility indices, designed to remedy it. Two examples concerning electoral outcomes and the causal effect of smoking cessation illustrate the method.

We consider a modified Boltzmann equation which contains, together with the collision operator, an additional drift term that is characterized by a matrix A. Furthermore, we consider a Maxwell gas, where the collision kernel has an angular singularity. Such an equation is used in the study of homoenergetic solutions to the Boltzmann equation. Our goal is to prove that, under smallness assumptions on the drift term, the longtime asymptotics is given by self-similar solutions. We work in the framework of measure-valued solutions with finite moments of order p > 2 and show existence, uniqueness and stability of these self-similar solutions for sufficiently small A. Furthermore, we prove that they have finite moments of arbitrary order if A is small enough. In addition, the singular collision operator allows to prove smoothness of these self-similar solutions. Finally, we study the asymptotics of particular homoenergetic solutions. This extends previous results from the cutoff case to non-cutoff Maxwell gases.

We show that the action of Hecke operators away from $p$ on the space of ($p$-adic) overconvergent modular forms is ($p$-adically) locally analytic in a certain sense. As a corollary, the action of the Hecke algebra can be extended naturally to an action of rigid functions on its generic fiber. This directly determines the Hodge-Tate-Sen weights of Galois representation associated to an overconvergent eigenform and confirms a conjecture of Gouvêa.

A numerical model is built, simulating the principles of kinetic gas theory, to predict pressures of molecules in a spherical pressure vessel; the model tracks a single particle and multiplies the force on the spherical walls by a mole of molecules to predict the net pressure. An intermolecular attractive force is added for high-density simulations, to replicate a real fluid; the force is chosen to ensure the fluid matches the Peng-Robinson equation of state as it is compressed to a near supercritical density. The standard deviations of the molecule velocity with respect to temperature and density is studied to define the entropy. A parametric study of a Stirling cycle heat engine utilizing near-supercritical densities is modeled, to study how the temperature dependence of the attractive intermolecular Van der Waal forces can affect the net total entropy change to the surrounding environment. A practical, macroscopic-scale piston-cylinder engine was then built and demonstrated, utilizing a novel thermodynamic cycle that closely resembles the Carnot heat engine cycle, utilizing an arrangement of valves and pneumatic air to replicate the isothermal and isentropic compression and expansion of the working fluid. This heat engine cycle could be built without requiring advanced manufacturing, and utilized non-ideal carbon dioxide as the working fluid, to take advantage of the entropy effects of the Van der Waals forces demonstrated in the Argon simulations to boost the thermodynamic efficiency. This engine demonstrated this capability in a practical, macroscopic heat engine, and offers great opportunities to practical energy generation.

We analyse supersymmetric near-horizon geometries of extremal black holes in $N=(1,0)$, $D=6$ supergravity with one tensor multiplet and $U(1)$ $R$-symmetry gauging. Assuming smooth bosonic fields and a compact, connected, boundaryless spatial horizon section $\mathcal{S}$, we solve the Killing spinor equations (KSEs) along the lightcone directions and identify the independent horizon system satisfied by the spinors $η_\pm$ on $\mathcal{S}$. We then prove generalized Lichnerowicz-type theorems for both lightcone chiralities, showing that the zero modes of the relevant horizon Dirac operators are in one-to-one correspondence with Killing spinors on $\mathcal{S}$. As a consequence, the supersymmetry-counting formula $N = 2N_{-} + \mathrm{Index}(D_E)$ holds for the class of regular horizons under consideration, where $D_E$ is the horizon Dirac operator twisted by the bundle naturally associated to the gauge structure of the theory. The $D=6$ case is distinguished from the previously analysed $D=11$ and type-IIA horizons because $\mathcal{S}$ is a compact four-manifold and the theory is chiral, so the relevant index need not vanish. In the ungauged case this reduces to the ordinary chiral Dirac index on $\mathcal{S}$, while in the gauged case the index is that of the corresponding twisted operator. We also analyse the map $η_- \mapsto Γ_+Θ_-η_-$. For non-trivial fluxes, the resulting spacetime $\mathfrak{sl}(2,\mathbb{R})$ symmetry is proved unconditionally in the ungauged theory. In the gauged theory the same conclusion follows provided one assumes $\mathrm{Ker}\,Θ_- = \{0\}$. We state this assumption explicitly and do not claim a full gauged symmetry-enhancement theorem without it.