Inspired by a remark of Serre, we extend the search for primes $p$ such that the maximum Hasse bound for the number of points on an elliptic curve over $\mathbb{F}_{p^5}$ is not achieved. We then give a list of all $q<10^{70}$ such that the Hasse bound is not achieved over $\mathbb{F}_{q}$. We explore the heuristics for how many such numbers should exist in each case. Finally, look at similar criteria for genus $2$ and $3$ curves.

Suppose that \[ \vecγ(t) := (γ_1(t),\dots,γ_n(t)) = (a_1 t^{d_1},\dots,a_n t^{d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\] is a homogeneous polynomial curve. We prove that whenever $p_1,\dots,p_n > 1$ and $\frac{1}{p} = \sum_{j=1}^n \frac{1}{p_j} \leq 1$, there exists an absolute constant $0 < C = C_{p_1,\dots,p_n;\vecγ} < \infty$ so that \[ \| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-γ_i(t))| \ dt \|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})}. \] Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright.

Understanding the role of high-redshift protoclusters in cosmic reionization is essential to unveiling the early stages of structure formation. Using deep imaging and spectroscopy from the James Webb Space Telescope (JWST) JADES Deep Survey in GOODS-South, we identify two prominent protoclusters at z>7 and investigate their environmental properties in comparison to field galaxies. Protocluster members exhibit systematically higher ionizing photon production efficiency ($ξ_{\text{ion}}$) and inflated [OIII]/H$β$ ratios at fixed UV magnitude or stellar mass, likely driven by young, metal-poor stellar populations and intense star formation. Despite these properties, their Ly$α$ emission is weak or absent, and they show high proximate neutral hydrogen column densities, suggesting insufficient ionizing output to maintain ionized bubbles. We also find that a strong Ly$α$ emitter (LAE), JADES-GS-z7-LA, may lie within the same ionized region as one protocluster. Although their Lyman continuum escape fractions ($f_{\mathrm{esc}} \sim 0.1$) are comparable to those of LAEs, individual protocluster galaxies are faint ($M_{\mathrm{UV}} > -19$) and low-mass ($\log(M_*/M_\odot) \sim 8.5$). The enhanced number density within protoclusters boosts the local UV luminosity density by nearly 1 dex. The surrounding gas remains largely neutral, suggesting that reionization was highly patchy and modulated by environment. The protocluster galaxies likely host ionization-bounded nebulae with holes, suppressing Ly$α$ visibility, in contrast to field galaxies that are more consistent with density-bounded nebulae.

We discuss the exclusive photoproduction of $π^{+}π^{-}$ pairs in photon-proton and in proton-proton collisions at high energies. The $ρ^{0}$, $ω$, $f_{2}(1270)$, and non-resonant (Drell-Söding) contributions are considered. The calculation is based on the tensor-pomeron model. In the Drell-Söding contribution we have different subenergies for the $π^{+}p$ and $π^{-}p$ systems. In the method which we propose now we take this into account. Respecting the gauge-invariance constraints is then a nontrivial problem for which, however, we present a solution here. In the present paper we give in this way a substantial improvement of the calculations for real photoproduction of $π^{+}π^{-}$ from JHEP 01, 151 (2015), and we extend the calculations to low $Q^{2}$ electroproduction, $0 \leqslant Q^{2} \leqslant 0.5$ GeV$^{2}$. The photo- and electroproduction amplitudes are then the basis for the calculation of central exclusive production (CEP) of $π^{+}π^{-}$ pairs in $pp$ collisions, where at least one proton participates in the CEP via a virtual-photon emission. The revised model leads to enhanced cross sections and gives an increased skewing of the $ρ^{0}$ spectral shape. For the $pp \to pp π^{+}π^{-}$ reaction, we calculate differential cross sections as function of the two-pion invariant mass, pion transverse momentum and pion pseudorapidity. This research is relevant in the context of ALICE, ATLAS, CMS, and LHCb measurements in $pp$ collisions. Our results can also serve as basis for the description of coherent $π^{+}π^{-}$ production in ultra-peripheral $p$A and AA collisions at the LHC. The formulas given in our paper can be used for the analysis of photoproduction and small-$Q^{2}$ electroproduction in $ep$ collisions at high energies. Such data exist from the HERA experiments and will be obtained in the future at the electron-ion colliders.

In this paper, we first investigate weighted Minkowski type inequalities for nearly spherical sets in space forms, focusing on the sets that are $C^1$-close to geodesic spheres. Our results generalize the work of \cite{G22} by incorporating broader geometric settings and convex weight functions. Additionally, we establish quantitative stability estimates for weighted Alexandrov-Fenchel type inequalities in $\mathbb{R}^{n+1}$ and $\mathbb{H}^{n+1}$, extending the earlier results of \cite{VW24} and \cite{ZZ23}. These inequalities hold for nearly spherical sets that are $W^{2,\infty}$-close to geodesic spheres coupled with general convex weights.

The Planck measurement of the cosmic microwave background (CMB) has established the $Λ$-cold-dark-matter ($Λ$CDM) model as the concordant model along with other observations. However, recent measurements of baryon acoustic oscillations (BAO) from the Dark Energy Spectroscopic Instrument (DESI) have renewed the matter fraction $Ω_\mathrm{m}$ tension between Planck-$Λ$CDM and DESI-$Λ$CDM. Directly reconciling this CMB-BAO tension with a dynamical DE in Chevallier-Polarski-Linder (CPL) parametrization seems to imply a crossing of the equation-of-state (EOS) through $w=-1$ at low redshifts. In this paper, we resolve this $Ω_\mathrm{m}$ tension by allowing for the DM nonminimally coupled to gravity via a quintessence field. This non-minimal coupling is preferred over $3σ$ confidence level. Consequently, even though the usual effective EOS of the coupled quintessence apart from the standard CDM part never crosses but always is above $w=-1$, a misidentification with the $w_0w_a$CDM model would exactly fake such a crossing behavior, and the tensions on neutrino mass and growth rate in the $Λ$CDM model are also relieved in our model as a result of the resolved $Ω_\mathrm{m}$ tension.

Very recently, Sharoglazova et al. performed an experiment measuring the energy-velocity relationship and Bohmian velocity in coupled waveguides. Their data show a discrepancy between the semi-classical `speed' $v=\sqrt{2|Δ|/m}$ and Bohmian velocity $v_s$ for $Δ<-\hbar J_0$, leading them to claim a challenge to Bohmian mechanics. Here, we definitively demonstrate this experiment poses no challenge to Bohmian mechanics. First, $v$ and $v_S$ represent fundamentally distinct physical quantities -- comparing them is physically unjustified and cannot adjudicate between Copenhagen and Bohmian interpretations. Second, we rigorously show that both interpretations predict identical photon tunneling dynamics in coupled waveguides.

For upper semi-continuous potentials defined on shifts over countable alphabets, this paper ensures sufficient conditions for the existence of a maximizing measure. We resort to the concept of blur shift, introduced by T. Almeida and M. Sobottka as a compactification method for countable alphabet shifts consisting of adding new symbols given by blurred subsets of the alphabet. Our approach extends beyond the Markovian case to encompass more general countable alphabet shifts. In particular, we guarantee a convex characterization and compactness for the set of blur invariant probabilities with respect to the discontinuous shift map.

The Unified Gas-Kinetic Wave-Particle (UGKWP) method, developed for multiscale simulation of partially ionized plasmas, has been extended to unstructured meshes, enabling the modeling of electromagnetic flows around a hemisphere across near-continuum to rarefied regimes. To the best of our knowledge, this work represents the first application of a multiscale plasma solver to this problem. In our approach, neutrals, ions, and electrons are treated as distinct species, with electrons modeled beyond the conventional fluid approximation. The numerical implementation is validated through comparison with reference solutions for neutral hypersonic flow around a sphere, as well as benchmarking against experimental data for a Mach 4.75 pre-ionized argon flow. In both cases, the UGKWP results show good agreement with the reference and experimental data. The findings reveal that rarefied effects play a significant role in the prediction of electromagnetic flow control, underscoring the necessity of multiscale modeling in plasma flow applications.

Emergent modified gravity has shown that the canonical formulation of general relativity gives rise to a larger class of covariant modifications than action-based approaches, so far in symmetry-reduced models. This outcome is made possible by distinguishing between the space-time metric on a given solution, and the basic field degrees of freedom in which equations of motion are formulated. In this general treatment, the metric is no longer fundamental but emerges after field equations and covariance conditions are solved. Here, the results are extended to perturbative inhomogeneity on a spatially flat cosmological background, showing that new modifications are possible while maintaining the classical derivative order and setting the stage for dynamical equations suitable for detailed studies of early-universe models.

We present a methodology for accelerating the estimation of the free energy from path integral Monte Carlo simulations by considering an intermediate artificial reference system where interactions are inexpensive to evaluate numerically. Using the spherically averaged Ewald interaction as this intermediate reference system for the uniform electron gas, the interaction contribution for the free energy was evaluated up to 18 times faster than the Ewald-only method. Furthermore, a $ξ$-extrapolation technique was tested and applied to alleviate the fermion sign problem and to resolve the sign for large particle numbers. Combining these two techniques enabled the evaluation of the free energy for a system of 1000 electrons, where both finite-size and statistical errors are below chemical accuracy. The general procedure can be applied to systems relevant for planetary and inertial confinement fusion modeling with low to moderate levels of quantum degeneracy.

Since the public release of ChatGPT in November 2022, the AI landscape is undergoing a rapid transformation. Currently, the use of AI chatbots by consumers has largely been limited to image generation or question-answering language models. The next generation of AI systems, AI agents that can plan and execute complex tasks with only limited human involvement, will be capable of a much broader range of actions. In particular, consumers could soon be able to delegate purchasing decisions to AI agents acting as Custobots. Against this background, the Article explores whether EU consumer law, as it currently stands, is ready for the rise of the Custobot Economy. In doing so, the Article makes three contributions. First, it outlines how the advent of AI agents could change the existing e-commerce landscape. Second, it explains how AI agents challenge the premises of a human-centric consumer law which is based on the assumption that consumption decisions are made by humans. Third, the Article presents some initial considerations how a future consumer law could look like that works for both humans and machines.

Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every complete geometric graph on $n$ vertices can be partitioned into $O(n^{3/2})$ crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set $A$ of $n$ \emph{randomly} selected points, uniformly distributed in $[0,1]^2$, with probability tending to $1$ as $n\rightarrow\infty$, the edge set of $K_n[A]$ can be covered by $O(n\log n)$ crossing-free paths and by $O(n\sqrt{\log n})$ crossing-free matchings. On the other hand, we construct $n$-element point sets such that covering the edge set of $K_n[A]$ requires a quadratic number of monotone paths.

The Hubbard model in the $U\to\infty$ limit has been known to have resonating valence bond (RVB) ground states on certain corner-sharing simplex lattices. Examples include both the quasi-1D sawtooth lattice with open boundary and a larger class of higher dimensional lattices without boundaries. The two types of results were obtained by different approaches which do not apply to one another. In the second class of lattices, the simplest simplex is a tetrahedron. We hereby generalize both results by studying the singly hole-doped system on the quasi-1D lattice of a tetrahedron chain, which can be considered a stripe of the pyrochlore or checkerboard lattices. The energy level ordering of irreducible representations of each tetrahedron shows that a chain of them has exponentially degenerate partial RVB or dimer-monomer ground states where each tetrahedron hosts one spin-$1/2$ monomer and one spin-$0$ dimer. The exact ground states in the infinitely long chain limit are analytically solved by introducing basis transformations between local Hilbert spaces of neighboring tetrahedra, and its energy agrees with the extrapolation of numerical exact diagonalization results of finite sized systems.

We present a formalism that allows for the direct manipulation and optimization of subspaces, circumventing the need to optimize individual states when using subspace methods. Using the determinant state mapping, we can naturally extend notions such as distance and energy to subspaces, as well as Monte Carlo estimators, recovering the excited states estimation method proposed by Pfau et al. As a practical application, we then introduce Bridge, a method that improves the performance of variational dynamics by extracting linear combinations of variational time-evolved states. We find that Bridge is both computationally inexpensive and capable of significantly mitigating the errors that arise from discretizing the dynamics, and can thus be systematically used as a post-processing tool for variational dynamics.

Quantum effects are expected to modify the cosmological dynamics of the early universe while maintaining some (potentially discrete) notion of space-time structure. In one approach, loop quantum cosmology, current models are shown here to either be incompatible with a consistent space-time structure, or to have physical singularities. The latter happens in spite of a non-zero scale factor in the isotropic background dynamics. A new effective Friedmann equation shows that a bounce is obtained at sub-Planckian densities, preceded by a physical singularity at infinite scale factor that resembles a time-reversed big rip. The entire phase is accompanied by rapid changes of the Hubble radius. In addition, a new version of perturbative inhomogeneity in loop quantum cosmology is introduced that maintains a consistent space-time structure and has a non-singular background dynamics.

The Euclid space telescope of the European Space Agency (ESA) is designed to provide sensitive and accurate measurements of weak gravitational lensing distortions over wide areas on the sky. Here we present a weak gravitational lensing analysis of early Euclid observations obtained for the field around the massive galaxy cluster Abell 2390 as part of the Euclid Early Release Observations programme. We conduct galaxy shape measurements using three independent algorithms (LensMC, KSB+, and SourceXtractor++). Incorporating multi-band photometry from Euclid and Subaru/Suprime-Cam, we estimate photometric redshifts to preferentially select background sources from tomographic redshift bins, for which we calibrate the redshift distributions using the self-organising map approach and data from the Cosmic Evolution Survey (COSMOS). We quantify the residual cluster member contamination and correct for it in bins of photometric redshift and magnitude using their source density profiles, including corrections for source obscuration and magnification. We reconstruct the cluster mass distribution and jointly fit the tangential reduced shear profiles of the different tomographic bins with spherical Navarro--Frenk--White profile predictions to constrain the cluster mass, finding consistent results for the three shape catalogues and good agreement with earlier measurements. As an important validation test we compare these joint constraints to mass measurements obtained individually for the different tomographic bins, finding good consistency. More detailed constraints on the cluster properties are presented in a companion paper that additionally incorporates strong lensing measurements. Our analysis provides a first demonstration of the outstanding capabilities of Euclid for tomographic weak lensing measurements.

We compute the intersections between the automorphism strata and the pullback by the Torelli map of the Ekedahl-Oort strata inside the moduli space of genus two curves. We first describe explicitly which possible automorphism groups a genus two curve can have over a field of positive characteristic, and parametrise the families of curves with a prescribed automorphism group. Then, we describe an algorithm to compute the strata of genus two curves whose Jacobian variety has a fixed Ekedahl-Oort type. Finally, we compute the dimension and number of irreducible components of the intersections between the strata.

In unitary circuit games, two competing parties, an "entangler" and a "disentangler", can induce an entanglement phase transition in a quantum many-body system. The transition occurs at a certain rate at which the disentangler acts. We analyze such games within the context of matchgate dynamics, which equivalently corresponds to evolutions of non-interacting fermions. We first investigate general entanglement properties of fermionic Gaussian states (FGS). We introduce a representation of FGS using a minimal matchgate circuit capable of preparing the state and derive an algorithm based on a generalized Yang-Baxter relation for updating this representation as unitary operations are applied. This representation enables us to define a natural disentangling procedure that reduces the number of gates in the circuit, thereby decreasing the entanglement contained in the system. We then explore different strategies to disentangle the systems and study the unitary circuit game in two different scenarios: with braiding gates, i.e., the intersection of Clifford gates and matchgates, and with generic matchgates. For each model, we observe qualitatively different entanglement transitions, which we characterize both numerically and analytically.

The set of hook lengths of an integer partition $λ$ is the complement of some numerical semigroup $S$. There has been recent interest in studying the number of partitions with a given set of hook lengths. Very little is known about the distribution of sizes of this finite set of partitions. We focus on the problem of determining the size of the smallest partition with its set of hook lengths equal to $\mathbb{N}\setminus S$.

We study toposes satisfying De Morgan's law, in particular we give characterizations of geometric theories whose classifying topos is De Morgan, clarifying the link with the amalgamation property of the category of models of such theory. We then give several ways of turning a topos into a De Morgan topos.

We derive a thermodynamically consistent model, which describes the time evolution of a two-phase flow in an evolving domain. The movement of the free boundary of the domain is driven by the velocity field of the mixture in the bulk, which is determined by a Navier--Stokes equation. In order to take interactions between bulk and boundary into account, we further consider two materials on the boundary, which may be the same or different materials as those in the bulk. The bulk and the surface materials are represented by respective phase-fields, whose time evolution is described by a bulk-surface convective Cahn--Hilliard equation. This approach allows for a transfer of material between bulk and surface as well as variable contact angles between the diffuse interface in the bulk and the boundary of the domain. To provide a more accurate description of the corresponding contact line motion, we include a generalized Navier slip boundary condition on the velocity field. Based on local mass balance laws, we derive our model from scratch in two different ways: by the Lagrange Multiplier Approach and (in the case of matched densities and no mass flux between bulk and surface) by the Energetic Variational Approach. We further show that our model generalizes previous models from the literature, which can be recovered from our system by either dropping the dynamic boundary conditions or assuming a static boundary of the domain.

Many modern areas have not learned their lessons and often hope for the wisdom of later generations, resulting in them only possessing modern technology and difficult to iterate ancient civilizations. At present, there is no way to tell how we should learn from history and promote the gradual upgrading of civilization. Therefore, we must tell the history of civilization's progress and the means of governance, learn from experience to improve the comprehensive strength and survival ability of civilization, and achieve an optimal solution for the tempering brought by conflicts and the reduction of internal conflicts. Firstly, we must follow the footsteps of history and explore the reasons for the long-term stability of each country in conflict, including providing economic benefits to the people and means of suppressing them; then, use mathematical methods to demonstrate how we can achieve the optimal solution at the current stage. After analysis, we can conclude that the civilization transformed from human plowing to horse plowing can easily suppress the resistance of the people and provide them with the ability to resist; The selection of rulers should consider multiple institutional aspects, such as exams, elections, and drawing lots; Economic development follows a lognormal distribution and can be adjusted by population mean and standard deviation. Using a lognormal distribution with the maximum value to divide equity can adjust the wealth gap.

The center of our galaxy hosts Sagittarius~A*, a supermassive compact object of $\sim 4.3\times 10^6$ solar masses, usually associated with a black hole. Nevertheless, black holes possess a central singularity, considered unphysical, and an event horizon, which leads to loss of unitarity in a quantum description of the system. To address these theoretical inconsistencies, alternative models, collectively known as exotic compact objects, have been proposed. In this paper, we investigate the potential detectability of signatures associated with non-rotating exotic compact objects within the Sgr~A* polarized flares dataset, as observed through GRAVITY and future instruments. We examine a total of eight distinct metrics, originating from four different categories of static and spherically symmetric compact objects: Black Holes, Boson stars, Fluid spheres, and Gravastars. Our approach involves utilizing a toy model that orbits the compact object in the equatorial plane. Using simulated astrometric and polarimetric data with present GRAVITY and future GRAVITY+ uncertainties, we fit the datasets across all metrics examined. We evaluated the detectability of the metric for each dataset based on the resulting $χ^2_\mathrm{red}$ and BIC-based Bayes factors. Plunge-through images of ECOs affect polarization and astrometry. With GRAVITY's present uncertainties, none of the metric model is discernible. GRAVITY+'s improved sensitivity allows detection of some exotic compact object models. However, enhancing the astrophysical complexity of the hot spot model diminishes these outcomes. Presently, GRAVITY's uncertainties do not allow us to detect exotic compact object metric. With GRAVITY+'s enhanced sensitivity, we can expect to uncover additional exotic compact object models and use Sgr~A* as a laboratory for fundamental physics.

An $L$-space knot is a knot that admits a positive Dehn surgery yielding an $L$-space. Many known hyperbolic $L$-space knots are braid positive, meaning they can be represented as the closure of a positive braid. Recently, Baker and Kegel showed that the hyperbolic $L$-space knot $o9\_30634$ from Dunfield's census is not braid positive, and they constructed infinitely many candidates for hyperbolic $L$-space knots that may not be braid positive. However, it remains unproven whether their examples are genuinely non-braid positive. In this paper, we construct infinitely many hyperbolic $L$-space knots that are not braid positive, and our examples are distinct from those considered by Baker and Kegel.

We present the first lattice simulations of the nonlinear evolution after axion inflation by self-consistently incorporating currents arising from Schwinger pair production. The tachyonically amplified gauge fields trigger the growth of Schwinger currents, leading to universal values for the conductivity and magnetic field at the onset of strong backreaction and subsequent quenching of gauge field production. We show that the Schwinger effect (prematurely) saturates gauge field production, thereby diminishing the prospects of high scale axion inflation magnetogenesis as a viable solution for blazar observations.

High-efficiency single-photon detection in the microwave domain is a key enabling technology for quantum sensing, communication, and information processing. However, the extremely low energy of microwave photons (~μeV) presents a fundamental challenge, preventing direct photon-to-charge conversion as achieved in optical systems using semiconductors. Semiconductor quantum dot (QD) charge qubits offer a compelling solution due to their highly tunable energy levels in the microwave regime, enabling coherent coupling with single photons. In this work, we demonstrate microwave photon detection with an efficiency approaching 70% in the single-photon regime. We use a hybrid system comprising a double quantum dot (DQD) charge qubit electrostatically defined in a GaAs/AlGaAs heterostructure, coupled to a high-impedance Josephson junction (JJ) array cavity. We systematically optimize the hybrid device architecture to maximize the conversion efficiency, leveraging the strong charge-photon coupling and the tunable DQD tunnel coupling rates. Incoming cavity photons coherently excite the DQD qubit, which in turn generates a measurable electrical current, realizing deterministic photon-to-charge conversion. Moreover, by exploiting the independent tunability of both the DQD transition energy and the cavity resonance frequency, we characterize the system efficiency over a range of 3-5.2 GHz. Our results establish semiconductor-based cavity-QED architectures as a scalable and versatile platform for efficient microwave photon detection, opening new avenues for quantum microwave optics and hybrid quantum information technologies.

We offer a view of mathematics as an experimental science where axioms play the role of foundational theories like general relativity and quantum mechanics in physics. Under this view, axioms are provisional and inferred from experience with the experiental substrate of mathematics which we locate within computation rather than encoding intuitive and absolute truths. This offers a reframing of Godel's theorem, placing its impact sharply upon the incompleteness rather than the potentially contradictory nature of any computational set of axioms. The essay originated in an attempt to make precise the nature of mathematics in order to estimate how AI might impact it. This exploration is continued in the paired essay "The mechanical creation of mathematical concepts."

We demonstrate that the local nonfreeness, an unbiased measure of correlation between electrons at a single lattice site, can be computed as the mutual information between local natural spin orbitals. This leads us to prove a general result: local electron correlations in Hubbard-type models that conserve the orbital- and spin-resolved electron number are fully classical, since the local reduced density matrix is separable in the natural basis and no quantum correlations beyond entanglement are present. Finally, we compare different theoretical descriptions of magnetic and nonmagnetic states, showing that local classical correlations are drastically influenced by nonlocal processes. These results confirm the relation between local classical correlations within an open system and nonlocal entanglement and they provide a clear path for the study of the relationship between traditional quantum resources and the nonfreeness in terms of experimentally accessible quantities.

We present an improvement of a recent funnel controller design for uncertain nonlinear multi-input, multi-output systems modeled by higher order functional differential equations in the presence of input constraints. The objective is to guarantee the evolution of the tracking error within a performance funnel with prescribed desired shape for the case of inactive saturation. Compared to its precursor, controller complexity is significantly reduced, much fewer design parameters are involved and simulations exhibit a superior performance.