In this paper we introduce the notion of a relative volutive (higher) category, specializing to the notion of a lax volutive (higher) category. Our primary motivation to study these objects is the following: while any rigid symmetric monoidal category admits a volutive structure, any closed symmetric monoidal category admits a lax volutive structure. We develop some of the basic theory of relative volutive categories and provide several equivalent formulations of lax volutive categories. We then study examples of interest, including categories of complete bornological vector spaces and modules over star-rings. We will also separately discuss unbounded operators between Hilbert spaces and Morita 2-categories, the latter of which in the context of fully closed symmetric monoidal 2-categories.

In this article, we establish pinching conditions under which all weakly stable Yang-Mills connections on compact manifolds are flat. As a corollary, we provide a dimension-dependent constant $δ(n)$ and prove that there exist no non-flat weakly stable Yang-Mills connections on $δ(n)$-pinched compact simply-connected Riemannian manifolds.

In this paper we present Hardy type inequalities for magnetic Dirichlet forms with singular integral weights. We analyze the local and global optimality of the integral weight and discuss several examples in details. An application of our results to spectral estimates for magnetic Schrödinger operators is provided as well.

We develop a geometric framework for three-dimensional quantum Hall fluids of extended objects (quasi-strings) in the presence of a strong three-form background field associated with a bundle gerbe. In the strong-field regime, fast internal dynamics is frozen and the low-energy kinematics is governed by generalized guiding-center variables consisting of vectorial and tensorial coordinates. We show that these guiding-center variables obey a noncommutative geometry giving rise to a three-dimensional generalization of the Girvin-MacDonald-Platzman (GMP) algebra for projected density operators. Moreover, we relate this algebra to the canonical quantization of a topological BF+BB theory whose level is identified with the Dixmier-Douady invariant. Our results clarify the structure of incompressible quantum Hall-type phases and their geometric and topological features in three spatial dimensions.

Emergent gauge fields and Coulomb liquids have long been central to the physics of frustrated pyrochlore magnets, yet their realization beyond conventional, i.e. rank-1 $U(1)$, spin ice and into fully quantum higher-rank regimes has remained elusive. Here we provide a controlled demonstration of this physics in the spin-$\tfrac{1}{2}$ quantum Heisenberg antiferromagnet on the breathing pyrochlore lattice with symmetry-allowed Dzyaloshinskii--Moriya interactions, using the pseudofermion functional renormalization group. We show that tuning the breathing asymmetry stabilizes extended quantum analogues of both rank-1 and rank-2 $U(1)$ Coulomb liquids within a single microscopic model, directly distinguished by their characteristic pinch-point morphologies in momentum space. This provides the first controlled quantum realization in three dimensions where gauge theories of different rank emerge within a single microscopic spin Hamiltonian. In addition, quantum fluctuations qualitatively reshape the classical nearest-neighbor atlas of phases, causing an incommensurate spiral instability and an extended quantum-disordered regime without dipolar order, both absent from the classical model. Our results establish the breathing pyrochlore as a timely and experimentally relevant platform where higher-rank gauge constraints, conventional magnetic order, and fluctuation-driven quantum phases compete on equal footing, opening a direct route to diagnosing emergent gauge structure in three-dimensional quantum magnets.

We prove a precompactness theorem for invariant metrics on compact homogeneous spaces without injectivity radius bounds, assuming uniform bounds on the diameter and on all derivatives of the curvature tensor. As a consequence, we prove that every ancient homogeneous Ricci flow on a compact manifold admits a blow-down sequence that converges to a gradient shrinking Ricci soliton.

We analyze decay phase observations of the GOES class C6.7 flare SOL2022-08-19T20:31 by the Visible Spectropolarimeter (ViSP) on the National Science Foundation's Daniel K. Inouye Solar Telescope (DKIST). The data include the first flare-time DKIST observations of the chromospheric Ca II H 396.8 nm and H-epsilon 397.0 nm spectral lines. These diagnostics have rarely been studied together during the modern era of high-resolution solar flare observations, and never at the spectral and spatial resolution of the DKIST. We directly compare DKIST spectra to state-of-the-art RADYN+RH simulations, including one heated by a nonthermal electron beam and one by in-situ thermal conduction. While certain salient properties of the spectra such as the width of H-epsilon are reproduced, the models severely underestimate the width of Ca II H in the red wing and fail to reproduce the exact relative intensity of Ca II H to H-epsilon. The models exhibit a range of condensation electron densities spanning over an order of magnitude. Unlike the modeled lower-order Balmer-series lines, we find that the width of H-epsilon is not solely related to the condensation properties; the widths and intensities are also sensitive to the deeper flare layers. We outline possible avenues towards improvement of flare models, such as a comprehensive evaluation of flare heating mechanisms in the context of both impulsive and decay phase high-resolution data.

We study a simple model for a particle that is active due to self-phoresis and that has been proposed to model symmetric camphor grains. The particle generates a concentration field through the continuous emission of a chemical substance, and its motion is driven by gradients of this field as it diffuses within a confined channel whose ends perfectly reflect the chemical. The reflection of the chemical field leads to an effective confinement of the particle, which itself is reflected before encountering the channel ends. The system displays a transition from a passive state, where the particle rests at the channel midpoint, to an active state characterized by highly regular, non-chaotic oscillations. We analytically construct the phase diagram and derive the oscillation frequency and amplitude in the vicinity of the transition. A perturbative analysis perfectly describes the dynamics of the particle even for oscillations as large as half the channel size. Furthermore, we develop an analysis which explains the mechanism of particle reflection close to the channel edges in the regime of large activity.

The COHERENT experiment measures neutrino-induced recoils from coherent elastic neutrino-nucleus scattering (CEvNS) with multiple nuclear targets at the Spallation Neutron Source (SNS) at the Oak Ridge National Laboratory (ORNL), USA. Several successful CEvNS measurements have been achieved in recent years with tens-of-kg detector masses, with a CsI scintillating crystal, a liquid argon single-phase detector, and high-purity germanium spectrometers. For the next phase, COHERENT aims at high-statistics detection of CEvNS events for precision tests of the standard model of particle physics, and to probe new physics beyond-the-standard model. Percent-level precision can be achieved by lowering thresholds, reducing backgrounds, and by scaling up the detector masses. It goes hand in hand with benchmarking the neutrino flux from the SNS. Further detectors will measure CEvNS in additional nuclei, including lighter target nuclei such as sodium and neon, to continue to test the expected neutron-number-squared dependence of the cross section. COHERENT can furthermore study charged-current and neutral-current inelastic neutrino-nucleus cross sections on various nuclei at neutrino energies below $\sim$50 MeV. Many of these cross sections have never been measured before, but are critical input for the interpretation of core-collapse supernova detection in large-scale neutrino experiments such as DUNE, Super-K, Hyper-K, and HALO.

This paper complements the existing theory developed in [5] for the Dirichlet and Neumann problems for the Laplace equation, in multiply connected domains. Within the framework of layer potential methods, we study the Laplace equation under Robin boundary conditions, representing the solutions by means of a double layer potential. We observe that the classical approach searches the solutions in terms of a single layer potential.

This study investigates the structural stability of ionized gold clusters of sizes ranging from 22 to 100 atoms, contrasting compact, cage and planar structures. While it is well known that neutral clusters in the upper part of this size range predominantly favor compact structures, our results reveal that positively ionized gold clusters exhibit structural transitions in which planar structures become energetically preferred once the charge is sufficiently large. In addition, we study the finite-temperature stability of the structures and find that thermodynamic effects further stabilize planar configurations relative to their compact counterparts. To explore the potential energy surface, we use the Minima Hopping algorithm combined with a machine-learned potential. Since the machine-learned potential does not apply to ionized clusters, we introduce a charge-correction term to incorporate Coulomb interactions and charge screening.

Multimessenger observation of the Seyfert-starburst composite galaxies NGC 1068 and NGC 7469 indicate a characteristic feature in the radio band (the so-called mm-bump) as well as indication of high-energy neutrinos by the AGN corona. Moreover, also the starburst ring of these sources is bright in the radio and hence, a potential source of $γ$-rays and neutrinos. We aim to explain the non-thermal features of these two sources with our homogeneous steady-state Seyfert-starburst composite model, which we refined in this work. Hereby, we account for stochastic diffuse acceleration and energy losses within the corona and $γγ$-pair attenuation of the escaping $γ$-rays. Since the non-thermal features of Seyfert sources contribute only marginally to the electromagnetic spectrum, only few data points can be assigned to the starburst ring or the AGN corona. Hence, prior information on the physical parameters is incorporated within a Markov Chain Monte Carlo approach to avoid overfitting. Based on this Bayesian parameter study we show, that the non-thermal features of NGC 1068 can be explained well. Still a more detailed treatment of the spatial inhomogeneities in the central region of the AGN could further improve the fit results. This manifests itself even more clearly in the case of NGC 7469, where the mm-bump needs to emerge from a coronal size $R_{\rm c}>100\,\mathcal{R}_{\rm s}$, whereas (TeV-PeV)-neutrino emission requires $R_{\rm c}< 10\,\mathcal{R}_{\rm s}$. Similar to what has previously been shown in other wavebands, our analysis highlights that the spatial extension of the so-called AGN corona depends the considered energy of the messenger. Hence, it seems that there is not a unique edge of the corona and a substantial progress in the understanding of these phenomena is expected if future analysis account for these spatial inhomogeneities.

We investigate an entropy-regularized reinforcement learning (RL) approach to optimal stopping problems motivated by real option models. Classical stopping rules are strict and non-randomized, limiting natural exploration in RL settings. To address this, we introduce entropy regularization, allowing randomized stopping policies that balance exploitation and exploration. We derive an explicit analytical solution to the regularized problem and prove convergence of the associated free boundary to the classical stopping threshold as the entropy vanishes. The regularized problem admits a natural formulation as a singular stochastic control problem. Building on this structure, we propose both model-based and model-free policy iteration algorithms to learn the optimal boundary. The model-free method operates without knowledge of system dynamics, using only trajectories from the stochastic environment. We establish convergence guarantees and illustrate strong numerical performance. This framework provides a principled and tractable approach for data-driven stopping problems under uncertainty.

Let $f(x) = (x^{2}+1)^{n} - a x^{n} \in \mathbb{Z}[x]$ and assume $f(x)$ is irreducible. Let $θ$ be a root of $f(x)$, set $K= \mathbb{Q}(θ)$, and denote by $\mathbb{Z}_{K}$ the ring of integers of $K$. The index of $f$, denoted $\operatorname{ind}(f)$, is the index of $\mathbb{Z}[θ]$ in $\mathbb{Z}_{K}$. A polynomial $f(x)$ is said to be monogenic if $\operatorname{ind}(f) = 1$. In this article, we explicitly compute the discriminant of the polynomial $f(x)$, and then derive necessary and sufficient conditions on the parameters $a$ and $n$ for $f(x)$ to be monogenic. Furthermore, we provide a complete description of the primes that divide $\operatorname{ind}(f)$.

We adapt the microcanonical framework of equilibrium statistical mechanics to predict the statistics of short waves in inhomogeneous moving media. For steady inhomogeneities and background flow, we compute the wave spectrum at any location in the domain based on an ergodic prescription for the action density in phase space, constrained by conservation of absolute frequency. We illustrate the method for shallow-water waves subject to a background flow or to topographic inhomogeneities, and for deep-water surface capillary waves over a background flow, validating the predicted maps of rms surface elevation and interfacial slope against numerical simulations.

Business plan (BP) writing plays a key role in entrepreneurship education by helping learners construct, evaluate, and iteratively refine their ideas. However, conventional BP writing remains a rigid, linear process that often fails to reflect the dynamic and recursive nature of entrepreneurial ideation. This mismatch is particularly challenging for novice entrepreneurial students, who struggle with the substantial cognitive demands of developing and refining ideas. While reflection and meta-reflection are critical strategies for fostering divergent and convergent thinking, existing writing tools rarely scaffold these higher-order processes. To address this gap, we present the Meflex System, a large language model (LLM)-based writing tool that integrates BP writing scaffolding with a nonlinear idea canvas to support iterative ideation through reflection and meta-reflection. We report findings from an exploratory user study with 30 participants that examined the system's usability and cognitive impact. Results show that Meflex effectively scaffolds BP writing, promotes divergent thinking through LLM-supported reflection, and enhances meta-reflective awareness while reducing cognitive load during complex idea development. These findings highlight the potential of non-linear LLM-based writing tools to foster deeper and coherent entrepreneurial thinking.

This expository article elaborates upon my talk at the 2025 AMS Summer Institute on Algebraic Geometry. It gives an introduction to a conjecture from Tate's 1966 Séminaire Bourbaki report, predicting the existence of a symplectic form on Brauer groups of surfaces over finite fields, and then an informal tour of the proof in \cite{Feng20} and \cite{CF}.

We study analytically and numerically the mean fastest first-passage time (fFPT) to an immobile target for an ensemble of $N$ independent finite-speed random searchers driven by dichotomous noise and described by the telegrapher's equation. In stark contrast to the well-studied case of Brownian particles -- for which the mean fFPT vanishes logarithmically with $N$ -- we uncover that the mean fFPT is bounded from below by the minimal ballistic travel time, with an exponentially fast convergence to this bound as $N \to \infty$. This behavior reveals a dramatic efficiency advantage of physically realistic, finite-speed searchers over Brownian ones and illustrates how diffusive macroscopic models may be conceptually misleading in predicting the short-time behavior of a physical system. We extend our analysis to anomalous diffusion generated by Riemann-Liouville-type dichotomous noises and find that target detection is more efficient in the superdiffusive regime, followed by normal and then subdiffusive regimes, in agreement with physical intuition and contrary to earlier predictions.

We develop a perturbation theory to study the shape and the orientation of an initially spherical capsule of radius R with a viscosity contrast, a surface tension σ and a bending rigidity $κ$ in linear flows. The elastic mechanical response of membrane to deformations is described by three elastic constitutive law which are either Hookean, Neohookean or Skalak type leading to the introduction of a surface shear elastic modulus $G_s$ and the Poisson ratio (or analog quantities). At the leading order, the deformation, i.e. the so-called Taylor parameter is proportional to the elastic capillary number Ca which evaluates the ratio between the external viscous stress and the elastic membrane response. In this linear regime, the results do not depend on the elastic constitutive law as expected. Without surface tension and bending rigidity, we recover the results of Barthes-Biesel & Rallison (1981) and notably the fact that the Taylor parameter does not depend on the viscosity contrast $λ$ contrary to the case of a viscous droplet. In our more general model, the deformation does no longer depend on $λ$ at the upper order. Now, the Taylor parameter also depends on two other dimensionless numbers: the surface elastocapillary ratio $σ/G_s$ and the dimensionless bending rigidity $B= κ/G_sR^2$. At the further order, the angle of inclination of the capsule with the direction of the shear flow, the analog of the Chaffey and Brenner equation for droplets is determined in each case. The results are in excellent agreement with the numerical ones performed with a code based on the boundary integral method providing an useful method to valid numerical developments.

A new algorithm based on a deep neural network, DeepSets, for tagging the production flavour of neutral $B^0$ and $B^0_s$ mesons in proton-proton collisions is presented. Exploiting a comprehensive set of tracks associated with the hadronization process, the algorithm is calibrated on data collected by the LHCb experiment at a centre-of-mass energy of $13$ TeV. This inclusive approach enhances the flavour tagging performance beyond the established same-side and opposite-side tagging methods. The observed gains in tagging power of $35\%$ for $B^0$ mesons and $20\%$ for $B_s^0$ mesons relative to the combined performance of the existing LHCb flavour-tagging algorithms offer significant benefits for precision measurements of $C\!P$ violation and mixing in the neutral $B$ meson systems.

In this work we introduce a generalized flavor, in the sense of generalized Kohn-Sham density functional theory (gKS-DFT), of the recently derived local potential functional embedding theory (LPFET) [J. Chem. Theory Comput. 2025, 21, 20, 10293], where the in-principle exact formalism of DFT is combined with that of density matrix embedding theory (DMET). In generalized LPFET (gLPFET), the embedding clusters are designed from a full-size gKS system where the (in-principle non-local) Hartree-Fock exchange potential is combined with a local (in the localized orbital representation) correlation potential. The latter is optimized self-consistently such that gKS and local embedding cluster's densities match. Unlike in DMET, which uses the same (global) chemical potential value in all clusters, each embedded orbital has its own chemical potential in gLPFET. We show analytically that, when electron correlation is strongly local, the latter potential becomes a simple functional of the correlation potential. Numerical calculations on model systems confirm the high accuracy of gLPFET in this regime, in contrast to DMET. Moreover, we show that gLPFET completely fixes the flaw of LPFET in weaker correlation regimes, through its appropriate description of the Hartree-exchange potential.

It is known that waves generated by ambient noise sources and recorded by passive receivers can be used to image the reflectivities of an unknown medium. However, reconstructing the reflectivity of the medium from partial boundary measurements remains a challenging problem, particularly when the background wave speed is unknown. In this paper, we investigate passive correlation-based imaging in the daylight configuration, where uncontrolled noise sources illuminate the medium and only ambient fields are recorded by a sensor array. We first analyze daylight migration for a point reflector embedded in a homogeneous background. By introducing a searching wave speed into the migration functional, we derive an explicit characterization of the deterministic shift and defocusing effects induced by wave-speed mismatch. We show that the maximum of the envelope of the resulting functional provides a reliable estimator of the true wave speed. We then extend the analysis to a random medium with correlation length smaller than the wavelength. Leveraging the shift formula obtained in the homogeneous case, we introduce a virtual guide star that remains fixed under migration with different searching speeds. This property enables an effective wave-speed estimation strategy based on spatial averaging around the virtual guide star. For both homogeneous and random media, we establish resolution analyses for the proposed wave-speed estimators. Numerical experiments are conducted to validate the theoretical result.

The alternating (zigzag) numbers $A_n$, counting the ascending alternating permutations of $\left\{1,\cdots,n\right\}$ and defined by the exponential generating function $\tan x+\sec x$, admit several classical combinatorial and analytic representations. In this work we unify and extend three complementary structures of $A_n$. First, starting from the Stirling number expansion of zigzag numbers, we derive a contour integral representation, as well as a positive Laplace-type integral representation $$ A_n = 2^n \int_0^\infty e^{-y} f_n(y)\, dy, \qquad f_n(y) := \sum_{k=0}^{n} (-1)^k S(n,k) \left(\frac{y}{2}\right)^k, $$ where the kernel $f_n(y)$ is the polynomial generating function of Stirling numbers. A continuous interpolation of the discrete product (falling factorial) is introduced subsequently. This provides a direct analytic bridge between set partitions and Laplace asymptotics. Second, using the partial fraction expansion of $\tan$, we obtain the well-known hyperbolic integral representation $$ A_{2n+1}=\frac{1}π\int_0^\infty\frac{y^{2n+1}}{\sinh(y/2)}\,dy, $$ equivalently expressed in classical $\cosh$ form for $A_{2n}$. This representation interprets zigzag numbers as spectral moments associated with half-integer poles. The connection with Fourier analysis and Mellin transforms is also outlined. Finally, combining spectral expansions with Stirling identities, we derive congruence relations modulo primes for $A_n$. These results exhibit a dual analytic-combinatorial structure of zigzag numbers, linking partition expansions, trigonometric spectra, and arithmetic properties.

Let $\{v_α\}$ be a system of polynomial solutions of the parabolic equation $a_{hk}\partial_{x_{h}x_{k}}u - \partial_t u =0$ in a bounded $C^1$-cylinder $Ω_{T}$ contained in $\mathbb{R}^{n+1}$. Here $a_{hk}\partial_{x_{h}x_{k}}$ is an elliptic operator with real constant coefficients. We prove that $\{v_α\}$ is complete in $L^{p}(Σ')$, where $Σ'$ is the parabolic boundary of $Ω_{T}$. Similar results are proved for the adjoint equation $a_{hk}\partial_{x_{h}x_{k}} u+ \partial_t u =0$.

Nonlinear phase gates are essential to achieve the universality of continuous-variable quantum processing and its applications. We present a deterministic protocol for generating nonlinear phase gates in trapped ion systems using simultaneous two-tone sideband drives beyond the Lamb-Dicke regime. Our approach harnesses higher-order interaction terms typically neglected or suppressed to construct nonlinear phase gates. This methodology enables high-fidelity gate engineering with a near three-fold reduction in control pulses compared to state-of-the-art theoretical proposals.

In this paper we consider physics-informed detection of terrain material change in radar imagery (e.g., shifts in permittivity, roughness or moisture). We propose a lightweight electromagnetic (EM) forward model to simulate bi-temporal single-look complex (SLC) images from labelled material maps. On these data, we derive physics-aware feature stacks that include interferometric coherence, and evaluate unsupervised detectors: Reed-Xiaoli (RX)/Local-RX with robust scatter (Tyler's M-estimator), Coherent Change Detection (CCD), and a compact convolutional auto-encoder. Monte Carlo experiments sweep dielectric/roughness/moisture changes, number of looks and clutter regimes (gamma vs K-family) at fixed probability of false alarm. Results on synthetic but physically grounded scenes show that coherence and robust covariance markedly improve anomaly detection of material changes; a simple score-level fusion achieves the best F1 in heavy-tailed clutter.

In this paper, we investigate how attackers can discover sensitive information embedded within databases by exploiting inference rules. We demonstrate the inadequacy of naively applied existing state of the art differential privacy (DP) models in safeguarding against such attacks. We introduce ontology aware differential privacy (Onto-DP), a novel extension of differential privacy paradigms built on top of any classical DP model by enriching it with semantic awareness. We show that this extension is a sufficient condition to adequately protect against attackers aware of inference rules.

The application of operator overloading algorithmic differentiation (AD) to computer programs in order to compute the derivative is quite common. But, the replacement of the underlying computational floating point type with the specialized type of an AD tool has two problems. First, the memory structure of the program is changed and floating-point data is interleaved with identifiers from AD. This prevents the compiler from performing optimizations such as SIMD optimizations. Second, the AD tool does not see any domain-specific operations, e.,g. linear algebra operations, that the program uses. This prevents the AD tool from using specialized algorithms in such places. We propose a new AD tool that is tailored to such situations. The memory structure of the primal data is retained by associating an identifier with each entity, e.,g. matrix, and not with each floating point value, e.,g. element of the matrix. Operations on such entities can then be annotated and a generator is used to create the AD overloads. We demonstrate that this approach provides performance comparable to that of other specializations. In addition, the run-time factor is below the theoretical 4.5 of reverse AD for programs that are written purely with linear algebra entities and operations.

Axion-like particles (ALPs) can induce a birefringence effect that rotates the polarization angle of light, offering a probe of ultralight dark matter. We analyze archival near-infrared polarimetric data of the protoplanetary disk (PPD) around HD 163296. Whereas previous studies considered only single-epoch snapshots, we perform a consistent multi-epoch time-series analysis, extracting the polarization angle and its uncertainty from the polarized images. The resulting six-epoch time series is consistent with a constant polarization angle within the measurement uncertainties, while being sensitive to timescales of $\sim 170-400$ days. The typical polarization angle uncertainties are $1.6$--$6.4$ degrees, partly driven by multiple scattering in the optically thick disk, which broadens the intrinsic polarization angle distribution and introduces additional dispersion in the representative polarization angle. Based on these data, we derive the first upper limits on the ALP-photon coupling from PPD polarization variability, $g_{aγ} \lesssim 7.5 \times 10^{-12} (m_a / 10^{-22}\,{\rm eV})\,{\rm GeV}^{-1}$. Furthermore, we forecast that achieving a polarization angle uncertainty of $σ\sim 0.1$ degrees would enable world-leading sensitivity to ALP-induced birefringence.

We report an x-ray photoemission spectroscopy study of Ca$_2$RuO$_4$ surface-dosed with Cs alkali atoms and C$_{60}$ molecules. Due to its small ionization energy (large electron affinity), deposited Cs atoms (C$_{60}$ molecules) are expected to provide a solid surface with electrons (holes). Upon dosing the dopants to Mott-insulating Ca$_2$RuO$_4$, we found a new Ru $3d$ photoemission peak emerging on the lower binding-energy side, suggesting the creation of a core-hole screening channel associated with coherent Ru $4d$ states around the Fermi level. For both the Cs and C$_{60}$ dosing, this change occurred without an appreciable chemical potential jump. The coherent state, therefore, develops within the Mott gap through hybridization with the impurity level of the dopants. The present work highlights the flexibility of Mott-insulator surfaces as a playground for metal-insulator transitions.