The classical trisecant lemma says that a general chord of a non-degenerate space curve is not a trisecant; that is, the chord only meets the curve in two points. The generalized trisecant lemma extends the result to higher-dimensional varieties. It states that the linear space spanned by general points on a projective variety intersects the variety in exactly these points, provided the dimension of the linear space is smaller than the codimension of the variety and that the variety is irreducible, reduced, and non-degenerate. We prove a real analogue of the generalized trisecant lemma, which takes the form of a trichotomy. Along the way, we characterize the possible numbers of real intersection points between a real projective variety and a complimentary dimension real linear space. We show that any integer of correct parity between a minimum and a maximum number can be achieved. We then specialize to Segre-Veronese varieties, where our results apply to the identifiability of independent component analysis, tensor decomposition and to typical tensor ranks.

BGK (Bhatnagar-Gross-Krook) model is a relaxation-type model of the Boltzmann equation, which is popularly used in place of the Boltzmann equation in physics and engineering. In this paper, we address the ill-posedness problem for the BGK model, in which the solution instantly escapes the initial solution space. For this, we propose two ill-posedness scenarios, namely, the homogeneous and the inhomogeneous ill-posedness mechanisms. In the former case, we find a class of spatially homogeneous solutions to the BGK model, where removing the small velocity part of the initial data triggers ill-posedness by increasing temperature. For the latter, we construct a spatially inhomogeneous solution to the BGK model such that the local temperature constructed from the solution has a polynomial growth in spatial variable. These ill-posedness properties for the BGK model pose a stark contrast with the Boltzmann equation for which the solution map is, at least for a finite time, stable in the corresponding solution spaces.

Optical Ising machines have emerged as a promising dynamical hardware solver for computational hard optimization problems. These Ising machines typically require an optical modulator to represent the analog spin variables of these problems. However, modern day optical modulators have a relatively low modulation resolution. We therefore investigate how the low bit-resolution of optical hardware influences the performance of this type of novel computing platform. Based on numerical simulations, we determine the minimum required bit-resolution of an optical Ising machine for different benchmark problems of different sizes. Our study shows that a limited bit-resolution of 8bit is sufficient for the optical modulator. Surprisingly, we also observe that the use of a 1bit-resolution modulator significantly improves the performance of the Ising machine across all considered benchmark problems.

We classify deformation quantizations of the symplectic supervarieties that are smooth and admissible. This generalizes the corresponding result of Bezrukavnikov and Kaledin to the super case. We relate the equivalence classes of quantizations of supervarieties with that of their even reduced symplectic varieties. Finally, we prove that certain nilpotent orbits of basic Lie superalgebras are admissible and split, and classify their deformation quantizations.

Space-time finite-element discretizations are well-developed in many areas of science and engineering, but much work remains within the development of specialized solvers for the resulting linear and nonlinear systems. In this work, we consider the all-at-once solution of the discretized Navier-Stokes equations over a space-time domain using waveform relaxation multigrid methods. In particular, we show how to extend the efficient spatial multigrid relaxation methods from [37] to a waveform relaxation method, and demonstrate the efficiency of the resulting monolithic Newton-Krylov-multigrid solver. Numerical results demonstrate the scalability of the solver for varying discretization order and physical parameters.

The longstanding issue of general covariance in effective models of quantum gravity is addressed, which arises when canonical quantum gravity leads to a semiclassical model described by an effective Hamiltonian constraint. In the context of spherically symmetric models, general covariance is precisely formulated into a set of equations, leading to the necessary and sufficient conditions for ensuring covariance. With the aid of these conditions, we derive the equations for the effective Hamiltonian constraint. The equations yield two candidates for effective Hamiltonian constraints dependent on a quantum parameter. The resulting quantum modified black hole spacetimes are analyzed. Our models show improvement by casting off the known limitations of previous works with similar results.

We study some interesting aspects of the spectral properties of SU(3) gauge theory, both with and without dynamical quarks (QCD) at thermal equilibrium using lattice gauge theory techniques. By calculating the eigenstates of a massless overlap Dirac operator on the gauge configurations, we implement a gauge-invariant method to study spectral properties of non-Abelian gauge theories. We have unambiguously categorized Dirac eigenvalues into different regimes based on a quantity defined in terms of the ratios of nearest neighbor spacings. While majority of these eigenstates below the magnetic scale are similar to those of random matrices belonging to the Gaussian Unitary ensemble at temperatures much higher than the chiral crossover transition in QCD, a few among them start to become prominent only near the crossover. These form fractal-like clusters with the median value for their fractal dimensions hinting at the universality class of the chiral transition in QCD. We further demonstrate that momentum modes below the magnetic scale in a particular non-equilibrium state of QCD are classically chaotic and estimate an upper bound on the thermalization time $\sim 1.44$ fm/c by matching this magnetic scale with that of a thermal state at $\sim 600$ MeV.

This research chapter provides a comprehensive overview of ferromagnetic gyroidal nanostructures, combining a review of state-of-the-art research with our new findings on their implications for 3D magnonics. Both static and dynamic magnetization studies show that non-trivial shape anisotropy, chirality, and inhomogeneous demagnetization fields influenced by specific crystallographic arrangements lead to multiple low-energy state magnetization textures, spin-wave mode localization, and controllable spin-wave propagation, highlighting the substantial potential of gyroidal nanostructures. The review integrates insights into micro/nano texturing to elucidate the intricate relationships between gyroidal geometry, chirality, and their effective magnetic properties, especially at microwave frequencies. Our study of resonance frequencies in gyroid samples under rotational field manipulation further reveals the significant influence of geometric anisotropy on ferromagnetic resonance signal strength. This chapter establishes a fundamental understanding of ferromagnetic gyroidal nanostructures, paving the way for their future investigation in 3D magnonics.

Unit testing is essential for software reliability, yet manual test creation is time-consuming and often neglected. Search-based software testing improves efficiency but produces tests with poor readability and maintainability, while LLMs show promise but lack comprehensive evaluation across reasoning-based prompting and real-world scenarios. This study presents the first large-scale empirical evaluation of LLM-generated unit tests at the full class level, analyzing four models (GPT-3.5, GPT-4, Mistral 7B, and Mixtral 8x7B) against EvoSuite across 216,300 test cases targeting Defects4J, SF110, and CMD. We evaluate five prompting techniques, ZSL, FSL, CoT, ToT, and GToT, assessing compilability, hallucination-driven failures, readability, coverage, and test smells. Reasoning-based prompting, particularly GToT, significantly enhances reliability and compilability, yet hallucination-driven failures remain persistent, with compilation failure rates reaching 86%. While LLM-generated tests are generally more readable than SBST outputs, recurring issues such as Magic Number Tests and Assertion Roulette hinder maintainability. These findings suggest that hybrid approaches combining LLM-based generation with automated validation and search-based refinement are necessary for production-ready results.

We argue that a general system of scalar fields and gauge fields manifesting vacuum degeneracy induces a principal groupoid bundle over spacetime and that the pattern of spontaneous symmetry breaking and the Higgs mechanism are encoded by the singular foliation canonically induced on the moduli space of scalar vacuum expectation values by the Lie groupoid structure. Recent mathematical results in the classification of singular foliations then provide a qualitative classification of the possible patterns of vacuum degeneracy.

We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of $p$ for which tempered distributions in homogeneous Hardy--Sobolev spaces $\dot{H}^{s,p}$ with regularity index $s \in (-1,1)$ are initial data. Source terms of Lions' type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$.

Noninformative priors constructed for estimation purposes are usually not appropriate for model selection and testing. The methodology of integral priors was developed to get prior distributions for Bayesian model selection when comparing two models, modifying initial improper reference priors. We propose a generalization of this methodology to more than two models. Our approach adds an artificial copy of each model under comparison by compactifying the parametric space and creating an ergodic Markov chain across all models that returns the integral priors as marginals of the stationary distribution. Besides the guarantee of their existence and the lack of paradoxes attached to estimation reference priors, an additional advantage of this methodology is that the simulation of this Markov chain is straightforward as it only requires simulations of imaginary training samples for all models and from the corresponding posterior distributions. We present some examples, including situations where other methodologies need specific adjustments or do not produce a satisfactory answer.

This paper delves into the late-time accelerated expansion of the universe and the evolution of cosmic structures within the context of a specific \( f(R, L_m) \) gravity model, formulated as \( f(R, L_m) = λR + βL_m^α+ η\). To study the cosmological viability of the model, we employed the latest cosmic measurement datasets: i) 57 observational Hubble parameter data points (\texttt{OHD}); ii) 1048 distance moduli data points (\texttt{SNIa}); iii) a combined dataset (\texttt{OHD+SNIa}); and large scale structure datasets, including iv) 14 growth rate data points (\texttt{f}); and v) 30 redshift space distortion data points (\texttt{f}$σ_8$). These datasets facilitated the constraint of the \( f(R, L_m) \)-gravity model via MCMC simulations, followed by a comparative analysis with the \(Λ\)CDM model. A comprehensive statistical analysis has been conducted to evaluate the \( f(R, L_m) \)-gravity model's efficacy in explaining both the accelerated expansion of the universe and the growth of cosmic structures.

Spherical spin random fields are used to model the Cosmic Microwave Background polarization, the study of which is at the heart of modern Cosmology and will be the subject of the LITEBIRD mission, in the 2030s. Its scope is to collect datas to test the theoretical predictions of the Cosmic Inflation model. In particular, the Minkowski functionals, or the Lipschitz-Killing curvatures, of excursion sets can be used to detect deviations from Gaussianity and anisotropies of random fields, being fine descriptors of their geometry and topology. In this paper we give an explicit, non-asymptotic, formula for the expectation of the Lipshitz-Killing curvatures of the excursion set of the real part of an arbitrary left-invariant Gaussian spin spherical random field, seen as a field on $SO(3)$. Our findings are coherent with the asymptotic ones presented in Carrón Duque, J. et al. "Minkowski Functionals in $SO(3)$ for the spin-2 CMB polarisation field", Journal of Cosmology and Astroparticle Physics (2024). We also give explicit expressions for the Adler-Taylor metric, and its curvature. We obtain such result as an application of a general formula that applies to any nondegenerate Gaussian random field defined on an arbitrary three dimensional compact Riemannian manifold. The novelty is that the Lipschitz-Killing curvatures are computed with respect to an arbitrary metric, possibly different than the Adler-Taylor metric of the field.

We extend Bobkov and Chistyakov's (2015) upper bounds on concentration functions of sums of independent random variables to a multivariate entropic setting. The approach is based on pointwise estimates on densities of sums of independent random vectors uniform on centred Euclidean balls. In this vein, we also obtain sharp bounds on volumes of noncentral sections of isotropic convex bodies.

Linear non-Gaussian causal models postulate that each random variable is a linear function of parent variables and non-Gaussian exogenous error terms. We study identification of the linear coefficients when such models contain latent variables. Our focus is on the commonly studied acyclic setting, where each model corresponds to a directed acyclic graph (DAG). For this case, prior literature has demonstrated that connections to overcomplete independent component analysis yield effective criteria to decide parameter identifiability in latent variable models. However, this connection is based on the assumption that the observed variables linearly depend on the latent variables. Departing from this assumption, we treat models that allow for arbitrary non-linear latent confounding. Our main result is a graphical criterion that is necessary and sufficient for deciding the generic identifiability of direct causal effects. Moreover, we provide an algorithmic implementation of the criterion with a run time that is polynomial in the number of observed variables. Finally, we report on estimation heuristics based on the identification result and explore a generalization to models with feedback loops.

Insufficient reference data is ubiquitous in data-driven computational fluid dynamics, as it is usually too expensive to compute or impossible to observe over long enough times needed for data-driven methods. The lack of data can significantly compromise the fidelity of results computed with data-driven methods or render them inapplicable. To challenge this problem, we propose a probabilistic reconstruction method that enhances the hyper-parameterisation (HP) approach with ideas underlying the probabilistic-evolutionary approach. We offer to use the HP method ``Advection of the image point'' on data sampled from the joint probability distribution of the reference dataset. The HP method has been tested regionally on the sea surface temperature and surface relative vorticity computed with the global 1/4-deg and 1/12-deg resolution NEMO model. Our results show that the HP solution (the solution computed with the HP method) in the probabilistically-reconstructed and reduced (in terms of dimensionality) phase space at 1/4-deg resolution is more accurate than the 1/4-deg-solution computed with NEMO. Additionally, the HP solution is several orders of magnitude faster to compute than the 1/4-deg NEMO solution. The proposed method shows encouraging results for the NEMO model and the potential for the use in other operational ocean and ocean-atmospheric models for both deterministic and probabilistic predictions. Furthermore, the method can be used as a fast reanalysis tool allowing the complex dynamics of a comprehensive ocean model to be approximated by the HP solution. It can also function as a dynamic interpolation method to fill gaps in observational data.

Liquid scintillator (LS) is commonly utilized in experiments seeking rare events due to its high light yield, transparency, and radiopurity. The concentration of $^{238}$U and $^{232}$Th in LS consistently remains below 1 ppq (10$^{-15}$ g/g), and the current screening result is based on a minimum 20-ton detector. Inductively coupled plasma mass (ICP-MS) spectroscopy is well-regarded for its high sensitivity to trace $^{238}$U and $^{232}$Th. This study outlines a method for detecting $^{238}$U and $^{232}$Th in LS at the sub-ppq level using ICP-MS, involving the enrichment of $^{238}$U/$^{232}$Th from the LS through acid extraction. With meticulous cleanliness control, $^{238}$U/$^{232}$Th in approximately 2 kg of LS is concentrated by acid extraction with 0.4 (0.3) pg $^{238}$U ($^{232}$Th) contamination. Three standard adding methods are employed to assess recovery efficiency, including radon daughter, 2,5-diphenyloxazole (PPO), and natural non-existent $^{233}$U/$^{229}$Th. The method detection limit at a 99% confidence level of this approach can reach approximately 0.2-0.3 ppq for $^{238}$U/$^{232}$Th with nearly 100% recovery efficiency.

Social agents with finitely nested opponent models are vulnerable to manipulation by agents with deeper recursive capabilities. This imbalance, rooted in logic and the theory of recursive modelling frameworks, cannot be solved directly. We propose a computational framework called $\aleph$-IPOMDP, which augments the Bayesian inference of model-based RL agents with an anomaly detection algorithm and an out-of-belief policy. Our mechanism allows agents to realize that they are being deceived, even if they cannot understand how, and to deter opponents via a credible threat. We test this framework in both a mixed-motive and a zero-sum game. Our results demonstrate the $\aleph$-mechanism's effectiveness, leading to more equitable outcomes and less exploitation by more sophisticated agents. We discuss implications for AI safety, cybersecurity, cognitive science, and psychiatry.

Our goal is to study $p$-adic local systems on a rigid-analytic variety with semistable formal model. We prove that such a local system is semistable if and only if so are its restrictions to the points corresponding to the irreducible components of the special fiber. For this, the main body of the paper concerns analytic prismatic $F$-crystals on the absolute logarithmic prismatic site of a semistable $p$-adic log formal scheme. Analyzing Breuil-Kisin log prisms, we obtain a prismatic purity theorem and deduce the above purity theorem for semistable local systems.

In their seminal paper introducing the theory of random graphs, Erdős and Rényi considered the evolution of the structure of a random subgraph of $K_n$ as the density increases from $0$ to $1$, identifying two key points in this evolution -- the \emph{percolation threshold}, where the order of the largest component seemingly jumps from logarithmic to linear in size, and the \emph{connectivity threshold}, where the subgraph becomes connected. Similar phenomena have been observed in many other random graph models, and in particular, works of Ajtai, Komlós and Szemerédi and of Spencer and Erdős determine corresponding thresholds for random subgraphs of the hypercube. We study similar questions on the \emph{permutahedron}. The permutahedron, like the hypercube, has many different equivalent representations, and arises as a natural object of study in many areas of combinatorics. In particular, as a highly-symmetric simple polytope, like the $n$-simplex and $n$-cube, this percolation model naturally generalises the Erdős-Rényi random graph and the percolated hypercube. We determine the percolation threshold and the connectivity threshold for random subgraphs of the permutahedron. Along the way we develop a novel graph exploration technique which can be used to find exponentially large clusters after percolation in high-dimensional geometric graphs and we initiate the study of the isoperimetric properties of the permutahedron.

In [8,9], the authors developed a nice formula to compute the Nielsen number of a self-map on an infra-nilmanifold. For the case of nilmanifolds this formula was extended to $n$-valued maps in [4]. In this paper, we extend these results further and establish the averaging formula to compute the Nielsen number of any $n$-valued affine map on an infra-nilmanifold.

The classical theory of efficient allocations of an aggregate endowment in a pure-exchange economy has hitherto primarily focused on the Pareto-efficiency of allocations, under the implicit assumption that transfers between agents are frictionless, and hence costless to the economy. In this paper, we argue that certain transfers cause frictions that result in costs to the economy. We show that these frictional costs are tantamount to a form of subadditivity of the cost of transferring endowments between agents. We suggest an axiomatic study of allocation mechanisms, that is, the mechanisms that transform feasible allocations into other feasible allocations, in the presence of such transfer costs. Among other results, we provide an axiomatic characterization of those allocation mechanisms that admit representations as robust (worst-case) linear allocation mechanisms, as well as those mechanisms that admit representations as worst-case conditional expectations. We call the latter Robust Conditional Mean Allocation mechanisms, and we relate our results to the literature on (decentralized) risk sharing within a pool of agents.

Polynomials commute under composition are referred to as commuting polynomials. In this paper, we study division properties for commuting polynomials with rational (and integer) coefficients. As a consequence, we show an algebraic particularity of the commuting polynomials coming from weighted sums for cycle graphs with pendant edges (arXiv:2402.07209v1.). We also discuss a set of commuting polynomials over a field of positive characteristic.

The following refinement of the Higman embedding theorem is proved: A finitely generated group $R$ is recursively presented if and only if there exists a quasi-isometric malnormal embedding of $R$ into a finitely presented group $H$ such that the image of the embedding enjoys the congruence extension property. Moreover, it is shown that the finitely presented group $H$ can be constructed to have decidable Word Problem if and only if the Word Problem for $R$ is decidable, yielding a refinement of a theorem of Clapham. Finally, given a countable group $G$ and a computable function $\ell:G\to\mathbb{N}$ satisfying some necessary requirements, it is proved that there exists a malnormal embedding of $G$ into a finitely presented group $H$ such that the restriction of $|\cdot|_H$ to $G$ is equivalent to $\ell$, producing a refinement of a theorem of Ol'shanskii.

In this work we develop an algorithmic procedure for associating a function defined on the Riemann surface of the $\log$ to given asymptotic data from a function at an essential singularity. We do this by means of rational approximations (Padé approximants) used in tandem with Borel-Écalle summation. Our method is capable of handling situations where classical methods either do not work or converge very slowly eg. We provide a general outline of the procedure and then apply it to generating approximate tritronquée solutions to Painlevé's first equation ($\text{P}_\text{I}$). Our approximations (including $\text{P}_\text{I}$) are written as a finite linear combination of exponential integrals $\text{Ei}^+$. Furthermore, we have explicit rational approximations for each $\text{Ei}^+$ and thus for the approximation as a whole. In addition to rational approximations of $\text{P}_\text{I}$, we provide the first hundred or so poles of a tritronquée solution with essentially arbitrary accuracy which is dependent upon the order of Padé used.

When estimating area means, direct estimators based on area-specific data, are usually consistent under the sampling design without model assumptions. However, they are inefficient if the area sample size is small. In small area estimation, model assumptions linking the areas are used to "borrow strength" from other areas. The basic area-level model provides design-consistent estimators but error variances are assumed to be known. In practice, they are estimated with the (scarce) area-specific data. These estimators are inefficient, and their error is not accounted for in the associated mean squared error estimators. Unit-level models do not require to know the error variances but do not account for the survey design. Here we describe a unified estimator of an area mean that may be obtained both from an area-level model or a unit-level model and based on consistent estimators of the model error variances as the number of areas increases. We propose bootstrap mean squared error estimators that account for the uncertainty due to the estimation of the error variances. We show a better performance of the new small area estimators and our bootstrap estimators of the mean squared error. We apply the results to education data from Colombia.

We study the interplay between magic and entanglement in quantum many-body systems. We show that non-local magic, which is supported by the quantum correlations is lower bounded by the non-flatness of entanglement spectrum and upper bounded by the amount of entanglement in the system. We then argue that a smoothed version of non-local magic bounds the hardness of classical simulations for incompressible states. In conformal field theories, we conjecture that the non-local magic should scale linearly with entanglement entropy but sublinearly when an approximation of the state is allowed. We support the conjectures using both analytical arguments based on unitary distillation and numerical data from an Ising CFT. If the CFT has a holographic dual, then we prove that the non-local magic vanishes if and only if there is no gravitational back-reaction. Furthermore, we show that non-local magic is approximately equal to the rate of change of the minimal surface area in response to the change of cosmic brane tension in the bulk.

Fix an integer $p\geq 1$ and refer to it as the number of growing domains. For each $i\in\{1,\ldots,p\}$, fix a compact subset $D_i\subseteq\mathbb R^{d_i}$ where $d_1,\ldots,d_p\ge 1$. Let $d= d_1+\dots+d_{p}$ be the total underlying dimension. Consider a continuous, stationary, centered Gaussian field $B=(B_x)_{x\in \mathbb R^d}$ with unit variance. Finally, let $φ:\mathbb R \rightarrow \mathbb R$ be a measurable function such that $\mathrm E[φ(N)^2]<\infty$ for $N\sim N(0,1)$. In this paper, we investigate central and non-central limit theorems as $t_1,\ldots,t_p\to\infty$ for functionals of the form \[ Y(t_1,\dots,t_p):=\int_{t_1D_1\times\dots \times t_pD_p}φ(B_{x})dx. \] Firstly, we assume that the covariance function $C$ of $B$ is {\it separable} (that is, $C=C_1\otimes\ldots\otimes C_{p}$ with $C_i:\mathbb R^{d_i}\to\mathbb R$), and thoroughly investigate under what condition $Y(t_1,\dots,t_p)$ satisfies a central or non-central limit theorem when the same holds for $\int_{t_iD_i}φ(B^{(i)}_{x_i})dx_i$ for at least one (resp. for all) $i\in \{1,\ldots,p\}$, where $B^{(i)}$ stands for a stationary, centered, Gaussian field on $\mathbb R^{d_i}$ admitting $C_i$ for covariance function. When $φ$ is an Hermite polynomial, we also provide a quantitative version of the previous result, which improves some bounds from A. Reveillac, M. Stauch, and C. A. Tudor, Hermite variations of the fractional brownian sheet, Stochastics and Dynamics 12 (2012). Secondly, we extend our study beyond the separable case, examining what can be inferred when the covariance function is either in the Gneiting class or is additively separable.

In an attempt to show that the acceptance probability of a quantum query algorithm making $q$ queries can be well-approximated almost everywhere by a classical decision tree of depth $\leq \text{poly}(q)$, Aaronson and Ambainis proposed the following conjecture: let $f: \{ \pm 1\}^n \rightarrow [0,1]$ be a degree $d$ polynomial with variance $\geq ε$. Then, there exists a coordinate of $f$ with influence $\geq \text{poly} (ε, 1/d)$. We show that for any polynomial $f: \{ \pm 1\}^n \rightarrow [0,1]$ of degree $d$ $(d \geq 2)$ and variance $\text{Var}[f] \geq 1/d$, if $ρ$ denotes a random restriction with survival probability $\dfrac{\log(d)}{C_1 d}$, $$ \text{Pr} \left[f_ρ \text{ has a coordinate with influence} \geq \dfrac{\text{Var}[f]^2 }{d^{C_2}} \right] \geq \dfrac{\text{Var}[f] \log(d)}{50C_1 d}$$ where $C_1, C_2>0$ are universal constants. Thus, Aaronson-Ambainis conjecture is true for a non-negligible fraction of random restrictions of the given polynomial assuming its variance is not too low.