Agustín-Aquino solved, in terms of the table of marks of $\Aff(\mathbb{Z}/2k\mathbb{Z})$, the problem of enumerating the classes of bicolour self-complementary and rigid patterns in $\mathbb{Z}/2k\mathbb{Z}$ (also known as \emph{strong dichotomy classes}). In particular, the rigid pattern-inventory polynomial appeared, for odd $k$, to yield the number of strong classes with negative sign when evaluated in $-1$, and it was conjectured that this is true for $k$ a power of an odd prime. Here we prove the conjecture is true for $k$ odd in general.

The experimental pursuit of neutrinoless double-beta decay ($0νββ$) constitutes one of the most compelling avenues for probing lepton-number violation and exploring physics beyond the Standard Model. Within this landscape, $^{76}$Ge has consistently ranked among the most promising isotopes for current and next-generation bolometric and liquid-scintillator experiments, notably GERDA and LEGEND. In the present work, we adapt a rigorous statistical protocol previously established for $^{48}$Ca~\cite{Horoi-prc22} and $^{136}$Xe~\cite{Horoi-Xe-2023} to the $^{76}$Ge system, utilizing a valence configuration that aligns with our recent investigation of $^{82}$Se~\cite{Neacsu-Symmetry-2024}. Our methodology introduces systematic, bounded fluctuations to the two-body matrix elements of established effective interactions, subsequently monitoring how these perturbations propagate through a suite of low-energy nuclear observables. Special emphasis is placed on the $0νββ$ nuclear matrix element (NME), whose theoretical uncertainty currently dominates the interpretation of experimental half-life limits. By integrating these simulated variations into a Bayesian Model Averaging framework and benchmarking against empirical spectroscopic data, we derive a constrained probability distribution for the NME. The resulting analysis yields a central value of 2.46 with an associated standard deviation of 0.25, thereby quantifying the intrinsic theoretical spread within the interacting shell model approach. Furthermore, we perform a comprehensive correlation analysis across all computed observables to evaluate internal consistency, identify non-trivial structural dependencies, and establish benchmarks that may guide the refinement of future effective interactions.

Why do societies remain stuck in inferior institutions even when superior al ternatives are widely recognized? This paper develops a model in which agents choose not only actions within a game but also transformations of the game it self. Transformations may be soft, changing payoffs through taxes or subsidies, or hard, changing feasibility through deletion or replacement of actions. Within a coordination model with status-quo bias (switching cost) and boundedly rational play (logit quantal response), we show that these interventions are qualitatively different: finite taxes shift behavior continuously but cannot eliminate residual use of the inherited action, whereas deletion bypasses inertia by removing the action from the feasible set. We further characterize how antagonistic social preferences at the meta level can block reforms that are individually beneficial for every player. The framework provides a formal rationale for why hard feasibility restrictions of ten dominate soft price incentives under inertia, with direct applications to climate transition (carbon tax vs. fossil-fuel phase-out) and platform regulation (fines vs. deletion of addictive features).

Let $A$ be a simple separable exact $C^*$-algebra that has traces. We show the following existed regularity properties are equivalent: \quad(1) $l^\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace kernel ideal. \quad(2) $A$ is tracially almost divisible. \quad(3) $A$ is tracially $m$-almost divisible for some $m\in\N\cup\{0\}.$ \quad(4) $A$ has tracial approximate oscillation zero. \quad(5) $A$ has Property (TM). We also show that for an algebraically simple separable stable rank one \CA\ $B$ with non-empty compact ${\rm T}(B)$ and locally finite nuclear dimension, its uniform tracial completion $(\ol B^{\rT(B)}, \rT(B))$ is hyperfinite, type ${\rm II_1},$ and isomorphic to $({\cal R}_{\rT(B)},\rT(B))$. Furthermore, $\ol{B}^{{\rm T}(B)}$ is pure, has real rank zero and stable rank one, and satisfies $\rT (\ol B^{\rT(B)} )= \rT(B).$ Consequently, every simple separable unital diagonal AH-algebra $V$ (e.g. Villadsen algebras of the first type) has the following tracial strict comparison: For every $a,b\in V_+,$ if $d_τ(a)<d_τ(b)$ holds for all traces $τ\in\rT(V),$ then there is a sequence $\{r_n\}\subset V$ such that $\lim_n\|a-r_n^*br_n\|_{2,\rT(V)}=0.$

Recent research has led to the development of MCMC algorithms with likelihood-informed proposals when targeting posterior distributions supported on discrete state spaces. Our work is placed within this field and puts forward a new MCMC methodology based upon similarity-driven proposals. Such proposals sway transitions towards states favored by the posterior via use of a data-driven measure of discrepancy between observations and the proposed model. Our approach can naturally cover classes of hierarchical models that involve both discrete variables and additional latent ones, without a requirement of integrating our the latter, in contrast to previous works in this field. The new algorithms are illustrated in simulation settings and in a involved real data scenario with a Dirichlet-Multinomial regression model.

Lead-free halide double perovskites are promising alternatives to Pb-based semiconductors, but their discovery is challenging because structural formability, thermodynamic stability, band-gap placement, optical-transition strength, dielectric screening, and carrier transport must all be satisfied within the vast A2BB'X6 space. We present a backward-mapping, genome-guided framework linking device-level targets to chemically interpretable descriptor families for Pb-free double-perovskite discovery. From 13,088 charge-balanced compositions, we apply a halide-aware workflow integrating geometric formability filtering, six-family chemical-genome descriptor encoding, evolutionary-optimized machine learning surrogates, SHAP-based interpretation, and DFT phenotype closure. Stability is modeled using Ehull-derived labels, while a band-gap surrogate predicts scalar-relativistic PBE Eg for target-driven selection. The funnel reduces the search space to seven DFT-validated candidates: K2BePdF6, K2MnCdCl6, Rb2TeCuBr6, Cs2SnGeBr6, Cs2GeSrBr6, Cs2NiBaI6, and Cs2AgInCl6, all verified for structural assignability, band-edge character, effective masses, dielectric response, optical absorption, conductivity, reflectivity, energy-loss spectra, and XRD fingerprints. Functional rules emerge from stability-function coupling rather than band-gap optimization alone, providing an interpretable inverse-design paradigm to accelerate Pb-free double-perovskite discovery.

Why do inefficient practices, technologies, or institutions persist even when su perior alternatives are available? This paper introduces a quantal response equilib rium with status-quo bias (QRE-SB) in which each player incurs a fixed switching cost when deviating from an inherited default action. In a binary coordination game, we compare two policy interventions: a tax on the default action (price-only reform) versus deleting the default action entirely (ban). We prove that there exists a threshold tax below which the status quo persists and above which a transition occurs; notably, this threshold does not depend on the degree of bounded ratio nality. Deleting the default action always forces play to the superior equilibrium, irrespective of switching costs or rationality. Moreover, when the superior equilib rium is Pareto-dominant, deletion yields strictly higher expected welfare than any finite tax that leaves the old action feasible. Numerical simulations illustrate the theoretical predictions. The framework provides a formal foundation for the policy principle that sometimes you must ban, not just tax, with direct applications to climate policy, social media regulation, and international sanctions.

Recovering probability measures from moments is a central theme in statistics and optimization. In particular, we focus on the recovery of measures from moments and pseudo-moments, which may come from solving the moment-SOS hierarchy in one dimension. A typical strategy when recovering a measure from moments is to verify the flat-extension property, which certifies that the underlying measure is finitely atomic and ultimately leads to recovery. For many classes of measures, however, the flat extension never occurs and thus if one aims to recover the measure corresponding to the moments, assumptions need to be made. We formulate a new kind of recovery problem, where one assumes that the measure has compact support and a fulfills a mild separation criterion. The key feature of this recovery problem formulation is that it covers not only finitely atomic measures, but also measures with continuous components. We study this new problem and describe three situations in which different guarantees can be proven. These guarantees are developed by studying the spectral representation of the Gelfand-Naimark-Segal construction and its connection to orthogonal polynomials, which ultimately allows us to provide several additional insights, which apply to algorithms widely used for the recovery of atomic measures from moments. Furthermore, the statements proven lead to novel algorithms, which we benchmark, further confirming the theoretical findings.

Spin squeezing in atomic ensembles enables atom interferometry with sensitivities below the shot-noise limit, but the associated entanglement is highly susceptible to loss, making imperfections in atom optics a central limitation. Bragg diffraction is an established technique for driving transitions between atomic momentum states and enables large-momentum transfer through higher-order diffraction while preserving the internal state. However, it is intrinsically limited by two competing mechanisms: short light pulses induce parasitic diffraction into off-resonant orders beyond an effective two-level description, while long pulses face velocity selectivity. We derive analytical expressions in a second-quantized framework for the atom optics and phase uncertainty of a Mach-Zehnder interferometer including these effects. We demonstrate that sub-shot-noise scaling is achieved only in a regime of intermediate pulse duration. Furthermore, we show that deleterious effects of higher-order diffraction are partially mitigated by optimizing the input quantum state.

Close hyperbolic encounters between black holes produce distinctive bursts of gravitational radiation with a time-frequency morphology that is qualitatively different from that of quasi-circular inspirals. Expected to arise in dense stellar environments through dynamical interactions, these encounters probe formation channels and mass ranges inaccessible to isolated binary evolution, making them a compelling target for current and next-generation detectors. In this work, we reanalyze \totalevents high-mass events from the LIGO-Virgo-KAGRA catalogs using the hyperbolic configuration of the~\dali~waveform model. We compare these with analyses using the quasi-circular, precessing configuration of the same model, computing Bayes factors to evaluate which description is favored by the data. We find that most events strongly to mildly favor the quasi-circular, precessing scenario, except for GW190521. For this event, we find that the signal is best fit by a dynamical capture waveform, with Bayes factor $\ln \mathcal{B}^{\rm hyp}_{\rm prec}=3.71^{+0.11}_{-0.11}$. We confirm this preference via further analyses with~\dali~in different configurations (quasi-circular, non-precessing; eccentric, non-precessing; and eccentric, precessing), as well as one using the quasi-circular, precessing numerical relativity surrogate model \nrsur. We also highlight the results we obtain for GW231123, another high-mass signal linked to evidence of strong precession, for which we find strong preference for the quasi-circular, precessing scenario, with $\ln \mathcal{B}^{\rm hyp}_{\rm prec}=-15.80^{+0.24}_{-0.24}$. The analysis of mock signals generated with the best fitting waveforms for GW190521 and GW231123 suggest that the former might belong to a region of parameter space where high-mass, bound, precessing signals can be hard to distinguish from dynamical captures in parameter estimation.

Understanding ideal points in the character varieties of knot complements has led to a number of important invariants for 3-manifolds. Ohtsuki (1994) counted the ideal points for character varieties of 2-bridge knot complements, and he made his techniques more concrete in an ensuing paper (1996). Drawing on these ideas, for all 2-bridge knots $K$, we explicitly determine the structure of a Serre tree for each essential surface in the knot complement directly from the knot diagram. Using these trees, we derive a formula for the number of ideal points associated to each incompressible surface.

We prove exponential decay of pair correlations for 1D stationary point processes when spacings satisfy a Markov condition, geometric ergodicity, and a condition on exponential moments. The conditions are phrased for stationary sequences of spacings (intervals between consecutive points) whose law comes from the Palm distribution of the point process. The key technical ingredient is a Markov renewal theorem with exponential convergence rate. The proofs combine classical regeneration techniques with the notion of geometric ergodicity for Markov chains with general state space. We apply the result to two models from statistical mechanics: (1) Gibbs point processes with a hard-core, finite-range pair potentials and (2) a harmonic chain of atoms, related to an autoregressive Gaussian process.

A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an $\mathbb{R}$-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an $\mathbb{R}^2$-valued function, also satisfying an analogous weak equivalence. For a point cloud $X \subset \mathbb{R}^d$, our trifiltration has size $O\bigl(|X|^{\lceil(d+1)/2\rceil+1}\bigr)$. We present an algorithm that computes this trifiltration in time $O\bigl(|X|^{\lceil d/2\rceil+2}\bigr)$, together with an implementation. Our experiments demonstrate that implementation can handle thousands of points in $\mathbb{R}^3$, with memory growth that is nearly linear.

Polygenic risk scores (PRSs) aggregate genetic effect estimates to predict disease susceptibility, yet clinical deployment often exposes raw genotype data to third-party compute infrastructure. Prior homomorphic-encryption approaches, still require trust in a designated evaluator. We present bioETH-PRS, a protocol that replaces that evaluator role with immutable smart contracts on a blockchain supporting Fully Homomorphic Encryption (fhEVM). Using the integer-exact TFHE scheme, bioETH-PRS computes the PRS dot product entirely within the encrypted domain, keeping both genotype dosage vectors and GWAS weight vectors hidden from external parties throughout execution. We introduce a three-step fixed-point quantisation scheme for representing signed GWAS weights as unsigned 64-bit integers, achieving machine-epsilon reconstruction accuracy on validated fixtures. A four-contract architecture separates data custody, model publication, computation, and output release, and supports both a classic chunked path and a streaming path, with the latter reducing mock-measured gas by 37%. An on-chain noisy output oracle emits an encrypted noisy-score handle and a publicly decryptable ternary category, reducing raw score exposure and probing risk. Prototype evaluation on real GWAS fixtures confirms linear gas scaling and suggests that the approach may be cost-competitive in low-gas deployment environments.

We develop a direct diagrammatic Monte Carlo framework for the Renyi entanglement entropy of interacting lattice fermions. The method starts from the fermionic graded-swap representation of Z_n[A]=Tr_Aρ_A^n, which converts the entropy problem into a replicated path integral with mixed temporal boundary conditions on the entangling region. In this representation the replica momenta are half-shifted, q_m=(2m+1)π/n, and the interaction expansion has a determinant form suitable for connected-determinant summation. We combine this expansion with a many-configuration Markov-chain Monte Carlo sampler to obtain order-by-order corrections for very large systems to very high orders. As a benchmark, we compare the order-by-order coefficients of a 3*3 Hubbard cluster with exact diagonalization. We then report a production calculation for a large periodic lattice with a square subregions. The dominant system-size limitation is therefore memory rather than a conventional auxiliary-field sign problem. The results provide a step toward diagrammatic calculations of fermionic entanglement observables in regimes where direct quantum Monte Carlo sampling is costly or sign-problem limited.

Tunneling spectroscopy and its evolution are crucial for elucidating the intricate electronic structure and emergent phenomena in quantum materials.Nevertheless, high-quality measurements -- specifically those tracking evolution across temperature and external fields -- remain a formidable challenge. We have fabricated a high-quality $\mathrm{TaO}_x$-based planar tunneling junction by using magnetron sputtering for van der Waals (vdW) superconductors. Using the vdW superconductor $\mathrm{Bi}_2\mathrm{Sr}_2\mathrm{CaCu}_2\mathrm{O}_{8+δ}$ (Bi2212) as a benchmark, this platform yields high-quality tunneling spectra, reproducing the electronic signatures obtained from scanning tunneling spectra acquired from atomically clean surfaces under ultra-high vacuum conditions. This architecture enables high-precision spectroscopy across extensive temperature and magnetic field ranges, offering a universal strategy for probing the electronic structures of diverse two-dimensional systems and facilitating future explorations of material properties.

Dissipative quantum chaos is an emerging theory that is expected to extend the ideas, concepts, and methodology of conventional Hamiltonian quantum chaos from coherent evolution to open quantum dynamics. The new theory should provide a set of tools to distinguish chaotic open quantum systems from integrable ones, as well as quantitative measures of their chaoticity (or, conversely, integrability). The foundations of this theory were laid in the late 1980s, and from the very start it was clear that, like its Hamiltonian predecessor, it had to be based on the spectral properties of the operators governing open quantum evolution. After these first steps, the field remained relatively quiet for many years and it is only over the last decade that the development of dissipative quantum chaos has received a strong boost, as confirmed by a large number of publications on this topic and, very recently, the first experiments performed to test its theoretical predictions. In this chapter, we review these recent developments and outline the basic foundations of dissipative quantum chaos.

We study distribution-free root cause analysis in multi-stream data, where an evolving underlying system is observed through multiple data streams that may each undergo distributional changes at unknown timepoints. In such settings, the stream exhibiting the earliest change provides a natural starting point for investigating the underlying cause, which we refer to as the root-cause index. Leveraging conformal $p$-values, we propose a novel framework, Conformal Root Cause Analysis (CROC), which constructs finite-sample valid confidence sets for the root-cause index under minimal assumptions: the data streams are independent, and within each stream the pre- and post-change observations are sampled exchangeably from arbitrary and unknown distributions. We further establish a universality property, showing that any distribution-free method for root cause localization can be represented within the CROC framework. In addition, under mild regularity conditions and principled score design, our method yields asymptotically sharp confidence sets that efficiently isolate the root cause. We further extend CROC to efficiently handle cross-stream dependence when present. Extensive simulations demonstrate accurate localization of the root stream, supporting our theoretical guarantees.

Feshbach resonances, first studied in the context of nuclear reactions, have since become a cornerstone of modern atomic physics. They offer a remarkable degree of control over interatomic (and even intermolecular) interactions by tuning external magnetic fields. This tunability arises from the interplay between quantum scattering and the internal structure of atoms or nuclei. In this work, we explore the essential physics of Feshbach resonances using only basic quantum mechanics, aiming to make this powerful concept accessible to educators and students alike.

Delay and Disruption Tolerant Networks (DTN) are critical for reliable communications in challenged network environments, particularly for space systems where end-to-end connectivity cannot be guaranteed. We present an open-source, full-stack implementation of the Bundle protocol for communicating with the International Space Station (ISS), with complete security features including Bundle Authentication Block (BAB), Payload Integrity Block (PIB), and Payload Confidentiality Block (PCB) using HMAC-SHA256 and AES-256-CBC encryption. The system includes bundle fragmentation and reassembly, priority-based queuing, custody transfer with ACK/NAK mechanisms, and automatic retransmission. Our system also includes a frontend facilitated by a modern responsive web interface. We consider this work highly relevant in the context of computer networking because: i) it demonstrates a full stack, open-source, freely available implementation of this critical and reliable protocol; and ii) it offers an interactive educational and learning framework in the field of computer networks and communications.

We prove a variant of Rauch's hot spots conjecture for hyperbolic planar domains with small Neumann or mixed Dirichlet-Neumann eigenvalues. We conclude, for instance, that on bounded convex domains in the hyperbolic plane with sufficiently large area, second Neumann Laplace eigenfunctions have no interior critical points.

Optimization is widely used to determine the physical and financial exchange of wholesale electricity in organized markets. Guarantees of solution optimality and feasibility rest largely on convexity, which is not in general a characteristic of the governing equations for power grid and gas pipeline networks. Policy decisions that base the scheduling and locational pricing of electricity transactions on optimization rely on the guarantee of revenue adequacy, which ensures that the market administrator will collect enough payments in congestion rents to settle financial transmission rights. Developing a similar mechanism for locational trade valuation of natural gas also requires assurance that pricing outcomes are revenue adequate, and also cover the costs of gas compressor operation. However, it has been shown that the AC power flow equations are in general non-convex and hence conditions for guaranteeing revenue adequacy in optimal power flow solutions are challenging to generalize. In this study, we develop a general formal mathematical setting for nonlinear physical network flows and examine the conditions for revenue adequacy. The result is verified for DC and AC power flow as well as steady-state gas flow in a pipeline network.

Managing the emission and absorption of thermal radiation is crucial for a wide range of technologies, from radiative cooling of buildings and vehicles to thermal regulation of satellites and future lunar and Mars habitats. Despite this universal and critical need, thermal emitters capable of adaptively modulating emissivity in a broadband, high-contrast, and fully solid-state manner remain elusive. Here, we leverage neural-network-guided photonic design to enable adaptive, solid-state thermal emitters based on chalcogenide phase-change materials capable of emissivity switching with extreme spectral contrast and bandwidth. These engineered nanophotonic emitters operate over a broad spectrum$-$from solar through thermal infrared$-$providing very low solar absorptivity while enabling switchable thermal infrared emissivity with high contrast. We experimentally demonstrate the core functionality of our approach in the space-like radiative environment in the stratosphere, observing a 31.5 °C temperature differential between the two solid-state phases of a simplified chalcogenide GeSbTe-225 thermal emitter. Our results point to even more significant capabilities, such as the potential to modulate >600 W/m$^2$ of radiative heat (at 100 °C) with minimal solar heating in the vacuum of space. The proposed nanophotonic solid-state adaptive emitter could provide high-power and high-speed heat modulation while requiring no power to maintain state, offering transformative capabilities for thermal control in dynamic radiative environments on Earth and in space.

We investigate chiral superconductivity emerging from parent electronic states with non-uniform Berry curvature, motivated by recent experiments in rhombohedral graphene multilayers. Using the continuum $λ_N$-model-a tunable platform with independently controllable Berry curvature profiles-we solve the full BCS gap equation on a continuum Chern band beyond the weak-coupling limit. We find that a non-uniform Berry curvature of the parent band enriches the superconducting order parameter, leading to the formation of momentum-space vortices in the gap function away from high-symmetry points. By tuning the Berry curvature profile, we identify distinct regimes associated with vortex nucleation and vortex number saturation, and show that the nucleation of momentum-space vortices tends to lower the condensation energy. We then show analytically that the parent band Chern number constrains the number of momentum-space vortices that can nucleate in the gap-independent of details of the $λ_N$-model. We also provide a gauge-invariant formulation for computing the Bogoliubov-de Gennes (BdG) Berry curvature for continuum models, and find that it is determined by a momentum-space phase current. The winding of this current around vortices in the occupied region in turn determines the BdG Chern number. Finally, we discuss how thermal Hall measurements can be used to probe the formation of momentum-space vortices. Our results highlight the crucial role of Berry curvature in shaping chiral superconductivity, and offer guiding principles for its identification in systems such as rhombohedral graphene.

Altermagnetic order, characterized by the Néel vector, breaks time-reversal symmetry (TRS) even in the nonrelativistic limit. Although spin-polarized and anomalous transport phenomena emerge with this order, they are mutually compensated by TRS-connected antiphase domains with opposite Néel vectors. Here we employ polarized neutron diffraction to directly probe the altermagnetic order in MnTe. Pronounced nuclear-magnetic interference terms were observed, providing direct evidence of a net Néel vector in the bulk crystal. Moreover, a weak ferromagnetic moment (WFM), originating from relativistic spin-orbit coupling, was found to be coupled with the altermagnetic order. Both the altermagnetic order and the WFM can be switched by milli-Tesla-scale magnetic field cooling.

High entropy oxides (HEOs) are a class of materials with vast compositional space and tunable properties, making them attractive for applications in thermoelectrics, magnetism, ionic conduction, and beyond. However, their metastable nature makes the local structure, and consequently their properties, highly sensitive to growth conditions. It is therefore essential to probe the local modulations in atomic, chemical, and electronic structure as a function of growth conditions. Here, advanced S/TEM techniques, including 4D-STEM combined with electron energy loss spectroscopy and energy-dispersive X-ray spectroscopy are used to investigate the effect of substrate temperature on structure and strain at the nanoscale regime in HEO thin films. We quantify how nanoscale strain variations correlate with Co valence and subtle chemical differences in the films with the same nominal composition but different growth temperatures. Our results demonstrate that identical HEO compositions can accommodate distinct strain and defect states in thin film form and highlight how synthesis conditions can be leveraged to manipulate strain and Co valence. These findings establish a framework to tailor functional properties via strain and valence control in high entropy oxide thin films.

Accurately estimating the point spread function (PSF) of an optical system requires solving free-space wave propagation, which entails evaluating a diffraction integral. This integral is traditionally computed numerically using Fast Fourier Transform (FFT) or Hankel Transform, as it lacks a closed-form solution. We show that, under defocus and spherical aberration, the diffraction integral admits an approximate closed-form solution by combining a piecewise Bessel approximation with Gaussian-type integrals. Based on this result, we develop a fast wave-based PSF simulator with linear complexity in the radial resolution. The proposed, un-optimized simulator achieves up to a 2x speedup over Hankel-based integration and a 4x speedup over FFT while closely matching wave-optical PSFs, enabling efficient large-scale depth-of-field synthesis.

In 1982, S.-T. Yau conjectured that there exist four distinct embedded minimal two-spheres in any manifold diffeomorphic to $S^3$. Wang-Zhou confirmed this conjecture for Riemannian three-spheres when the metric is bumpy or has positive Ricci curvature. We prove the following quantitative version of their theorem. Suppose that $(S^3,g)$ has positive Ricci curvature and scalar curvature $R_g\ge Λ_0>0$. Then there exist four distinct embedded minimal two-spheres $Σ_1,\ldots,Σ_4\subset (S^3,g)$ such that $\operatorname{area}_{g}(Σ_i)\le 12π(i+1)/Λ_0$ for every $i=1,\ldots,4$. We apply this result to a problem posed by S.-T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered at the origin in $\mathbb R^4$. Haslhofer-Ketover proved that ellipsoids with one sufficiently large semi-axis contain at least one non-planar embedded minimal two-sphere. We prove that such ellipsoids contain at least three non-planar embedded minimal two-spheres.

Email importance labeling has long been a critical yet challenging problem for businesses and individuals. Traditional approaches; such as keyword matching, user-defined rules, and sender-based heuristics; demand extensive manual feature engineering and fail to scale effectively or generalize. Recent advances in large language models (LLMs) demonstrate strong potential and a natural fit for this task, offering deep contextual understanding and superior labeling quality. However, using LLM models like GPT-4.1 at enterprise email volumes incurs prohibitive computational costs and hinders real-world deployment. We explore the trade-off space of using alternative labeling schemes as opposed to GPT4.1 scale LLMs, with the goal of achieving near GPT level labeling quality with significantly lower cost. We develop Argo, an enterprise email labeling framework, where we construct a profiler to efficiently search the cost quality trade-off space of labeling and identify cost-efficient alternatives to labeling emails. Additionally, we design an on-demand provisioning scheme to intelligently scale Argo with real time load, to minimize cost increases during peak load inference. Over 3 open-source email datasets, Argo achieves 148-167X inference cost reduction with negligible quality degradation and 20-640000X lower profiling costs, making large-scale, context-aware email labeling practical for enterprises.

Intra-device parallelism addresses resource under-utilization in ML inference and training by overlapping the execution of operators with different resource usage. However, its wide adoption is hindered by a fundamental conflict with the static, sequential programming model of existing frameworks. Integrating these strategies requires invasive, model-specific code overhauls, representing an intractable engineering cost. This is further amplified by the high sensitivity of strategies to execution contexts (e.g., workload, model architecture, hardware), forcing developers to implement and maintain multiple specialized solutions. To address this, we propose DynaFlow, a framework that enables the transparent and flexible integration of intra-device parallelism by decoupling the logical model definition from the physical execution schedule. DynaFlow introduces a flexible frontend with annotations for graph partitioning and a programmable interface for defining custom intra-device parallelism strategies. Its efficient backend manages complex control/data-flow asynchronously, uses custom memory management to eliminate copy overheads, and preserves compatibility with optimizations like CUDA Graphs and TorchInductor. We demonstrate that DynaFlow can integrate representative parallelism strategies into 6 state-of-the-art ML systems with minimal code changes, achieving up to a 1.29x throughput improvement. DynaFlow is publicly available at https://github.com/uw-syfi/DynaFlow.