We study Gromov's problem concerning minimal normal curvature immersions in the unit ball. In particular, we determine the minimal possible value of the normal curvature of an $S^n\times S^1$. We also prove a differentiable sphere theorem and an existence result for minimizers in this context.

Nonconvexities in markets with discrete decisions and nonlinear constraints make efficient pricing challenging, often necessitating subsidies. A prime example is the unit commitment (UC) problem in electricity markets, where costly subsidies are commonly required. We propose a new pricing scheme for nonconvex markets with both discreteness and nonlinearity, by convexifying nonconvex structures through a semidefinite programming (SDP) relaxation and deriving prices from the relaxation's dual variables. When the choice set is bounded, we establish strong duality for the SDP, which allows us to extend the envelope theorem to the value function of the relaxation. This extension yields a marginal price signal for demand, which we use as our pricing mechanism. We demonstrate that under certain conditions-for instance, when the relaxation's right hand sides are linear in demand-the resulting lost opportunity cost is bounded by the relaxation's optimality gap. This result highlights the importance of achieving tight relaxations. The proposed framework applies to nonconvex electricity market problems, including for both direct current and alternating current UC. Our numerical experiments indicate that the SDP relaxations are often tight, reinforcing the effectiveness of the proposed pricing scheme. Across a suite of IEEE benchmark instances, the lost opportunity cost under our pricing scheme is, on average, 46% lower than that of the commonly used fixed-binary pricing scheme.

Let $(ϕ_t)$ be an area-preserving smooth flow on a compact, connected, orientable surface $\mathcal M$ with at least one but finitely many fixed points. Assume that $(ϕ_t)$ is analytic (up to a canonical change of coordinates) in the neighborhood of each saddle fixed point. We show that the flow $(ϕ_t)$ is weakly mixing on each of its (finitely many) quasi-minimal components.

It is well known that, given two curves $\mathcal{X}: y^p+cy=x^m$ and $\mathcal{Y}:y^p+cy=x^n$, defined over $\F_p$, if $n$ divides $m$ then there exists a nonconstant morphism $\mathcal{X} \longrightarrow \mathcal{Y}$. In this paper we are interested in studying whether the converse of this statement is true, i.e., if there exists a morphism $\mathcal{X} \longrightarrow\mathcal{Y}$ then must it be true that $n$ divides $m$? In particular, we consider the case when $m=p^{k}+1$ and $n=p^\ell+1$. We prove that the converse is true under certain hypotheses. We deal with both the cases of Galois morphisms and non-Galois morphisms.

In this paper, we derive several regularity results for harmonic mappings into Euclidean spheres associated with rather general energies related to fractional Sobolev spaces. These maps generalize families of maps introduced by Da Lio, Rivière and Schikorra and are related to harmonic maps with free boundaries. In our context, there is in general no monotonicity formula, which prevents the use of some classical methods. Despite this limitation, under natural assumptions on a Gagliardo-type energy, we succeed in proving a variety of small energy regularity results and improve on known results, even in the isotropic case for which some monotonicity formula is available. To this end, we exploit recent developments in the regularity theory of nonlocal equations and as a by-product, we explain how these results apply to classes of harmonic maps with free boundary and lead to new potential-theoretic estimates. As another application, we obtain higher differentiability results for the fractional harmonic map heat flow.

This article provides a systematic investigation of the minimum modulus of dual truncated Toeplitz operators (DTTOs) $D_φ$ acting on the orthogonal complement of the model space $\mathcal{K}_u^{\perp}$, where $u$ is a nonconstant inner function and $φ\in L^\infty(\T)$. We first establish an explicit formula for the minimum modulus of the compressed shift $S_u$ and its dual $D_u$ in terms of $|u(0)|$, and prove that the minimum is always attained. For normal DTTOs, we derive sharp spectral bounds utilizing the essential range of the symbol and characterize the conditions under which $m(D_φ)$ coincides with the essential infimum of $|φ|$. In the general setting, for unimodular $\vp$, we obtain exact formulas and two sided estimates for $m(D_φ)$ by analyzing the norms of associated Toeplitz and Hankel operators restricted to the model space. Finally, we provide several concrete examples to illustrate our results.

We study Bregman proximal augmented Lagrangian methods with second-order oracles for convex convex-composite optimization problems. The outer loop is an instance of the Bregman proximal point algorithm with relative errors in the sense of Solodov and Svaiter, applied to the KKT operator associated with the problem. Akin to classical Lagrange-Newton methods, including primal-dual interior point methods the Bregman proximal point algorithm repeatedly solves regularized KKT inclusions by minimizing a smooth Bregman augmented Lagrangian function, obtained after marginalizing out the multiplier variables. Thanks to non-Euclidean geometries the marginal function is generalized self-concordant and therefore within the regime of Newton's method which converges quadratically if the step-size in the outer proximal point loop is chosen carefully. The operator-theoretic viewpoint allows us to employ the framework of metric subregularity to derive fast rates for the outer loop, and eventually state a joint complexity bound. Important special cases of our framework are a proximal variant of the exponential multiplier method due to Tseng and Bertsekas and interior-point proximal augmented Lagrangian schemes closely related to those of Pougkakiotis and Gondzio.

The depth-weighted tree DWT($f$) with weight function $f:\{0,1,2,\ldots\}\to (0,\infty)$ is a dynamic random tree grown from a root $r$ where vertices arrive consecutively and every new vertex attaches to a parent $u$ with probability proportional to $f$(distance between $u$ and $r$). This work is dedicated to a systematic analysis of the depth of DWT($f$). Namely, we provide precise analytic expressions of the typical depth of DWT($f$) for convergent, periodic, slowly growing, and (super-)exponentially growing weight functions. Furthermore, for bounded or exponentially growing $f$, we determine the typical depth up to a multiplicative constant, thus confirming and strengthening a conjecture of Leckey, Mitsche and Wormald.

We explore the industrial and scientific applicability of the VQE-LSTM framework by integrating meta-learning with GPU accelerated quantum simulation using NVIDIA's CUDA-Q (CUDAQ) platform. This work demonstrates how an LSTM-FC meta-initialization module can extend the practical reach of the Variational Quantum Eigensolver (VQE) in both chemistry and physics domains. In the chemical regime, the framework predicts ground-state energies of molecular Hamiltonians derived from PySCF, achieving near FCI accuracy while maintaining favorable O(N^2) scaling with molecular size. In the physical counterpart, we applied the same model to quantized Simple Harmonic Motion systems (SHM), successfully reproducing its ground and excited states through VQE and Variational Quantum Deflation (VQD) methods. Benchmark results on NVIDIA GPUs reveal significant speedups over CPU-based implementations, validating CUDAQ's capability to handle large-scale variational workloads efficiently. Overall, this study establishes VQE-LSTM as a viable and scalable approach for GPU accelerated quantum simulation, bridging quantum chemistry and condensed-matter physics through a unified, meta-learned initialization strategy.

We give generators for the nef cone and the cone of curves of rational surfaces obtained by blowing-up the complex projective plane at a set of points $\mathcal{B} \cup \mathcal{D}$, where $\mathcal{B}$ is the set of (proper and infinitely near) base points of a pencil associated with a curve having one place at infinity, and $\mathcal{D}$ is a set of finitely many infinitely near free points on the strict transforms of curves of the pencil. We also prove that, when the pencil is given by an AMS-type curve and $\mathcal{D}$ contains at most two free points on any curve considered, the Cox ring of the obtained surface is finitely generated.

Given two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ over the same ground set, the matroid intersection problem is to find the maximum cardinality common independent set. In the weighted version of the problem, the goal is to find a maximum weight common independent set. It has been a matter of interest to find efficient approximation algorithms for this problem in various settings. In many of these models, there is a gap between the best known results for the unweighted and weighted versions. In this work, we address the question of closing this gap. Our main result is a reduction which converts any $α$-approximate unweighted matroid intersection algorithm into an $α(1-\varepsilon)$-approximate weighted matroid intersection algorithm, while increasing the runtime of the algorithm by a $\log W$ factor, where $W$ is the aspect ratio. Our framework is versatile and translates to settings such as streaming and one-way communication complexity where matroid intersection is well-studied. As a by-product of our techniques, we derive new results for weighted matroid intersection in these models.

Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincaré inequality, establishing existence of solutions to Dirichlet problem in a bounded domain in such a metric space is accomplished via direct methods of calculus of variation and the use of a Maz'ya type inequality, which is a consequence of the Poincaré inequality. However, when the domain and its boundary are unbounded, such a method is unavailable. In this paper, using the technique of sphericalization developed in the prior paper~[32], we establish the existence of solutions to the Dirichlet boundary value problem for $p$-harmonic functions in unbounded uniform domains with unbounded boundary when $1<p<\infty$. We also explore the issue of whether such solutions are unique by considering $p$-parabolicity and $p$-hyperbolicity properties of the domain.

In 2015 the Open Science Collaboration (OSC) (Nosek et al 2015) published a highly influential paper which claimed that a large fraction of published results in the psychological sciences were not reproducible. In this article we review this claim from several points of view. We first offer an extended analysis of the methods used in that study. We show that the OSC methodology induces a bias that is able by itself to explain the discrepancy between the OSC estimates of reproducibility and other more optimistic estimates made by similar studies. The article also offers a more general literature review and discussion of reproducibility in experimental science. We argue, for both scientific and ethical reasons, that a considered balance of false positive and false negative rates is preferable to a single-minded concentration on false positive rates alone.

We study the properties of a generic object $\mathbb{P}$ in the category of finite graphs. It turns out that this object, being topologically a Cantor set, has the Knaster--Reichbach type property. Namely, every homeomorphism and isomorphism $h\colon K\to L$ where $K$ and $L$ are nowhere dense closed sets in $\mathbb{P}$ and consisting only of isolated vertices in $K$ and $L$ can be extended to the autohomeomorphism and autoisomorphism of the whole graph $\mathbb{P}$.

Quark fragmentation functions describe the hadronization process of a quark where any of the final-state hadrons carries a fraction of its initial momentum. We compute these fragmentation functions for a cascade that includes pions, kaons, and the charmed $D$ and $D_s$ mesons, starting from the elementary quark-to-meson fragmentation process. The latter is obtained from the relevant cut diagram, and employs Poincaré covariant Bethe-Salpeter wave functions and quark propagators. We derive a set of twenty-five coupled jet equations that describe the cascade of emitted mesons in the fragmentation process. Their solutions yield full fragmentation functions that offer a consistent picture of the quark fragmentations across the light and heavy sectors.

We study the generic behavior of Hamiltonian trajectories on a regular level set in the cotangent bundle, after projection to the base. We prove that for a generic submersive level set, projected trajectories have discrete (self-)intersections. Additionally, fixing end-point fibers, we prove that all intersections can be perturbed away if the base has dimension at least three. In particular, this applies to periodic orbits, and both results hold for Reeb flows on fiber-wise star-shaped hypersurfaces, including non-reversible Finsler flows, which answers a question of Rademacher. In the proof we make use of a multi-jet transversality theorem.

Connecting the dynamics of biomolecular networks to experimentally measurable cell phenotypes remains a central challenge in systems biology. Here we introduce a model-based definition of phenotype as a partial steady state that is committed to a certain dynamical outcome while otherwise being minimally constrained. We focus on Boolean models and define \emph{dynamical phenotypes} as complete trap spaces that maximally specify a chosen set of phenotype-determining nodes that correspond to biomarkers while keeping external inputs unconstrained. We show that dynamical phenotypes can be efficiently identified without full attractor enumeration. Using four published models, including a 70-node Boolean model of T cell differentiation, we show that dynamical phenotypes recover known cell types and activation states, and indicate the environmental conditions ensuring their existence. We also propose a method to identify informative phenotype-determining nodes based on the canalization of the Boolean functions. This method reveals biologically relevant cell state information that is complementary to the phenotypes manually defined by model creators and is validated by two attractor-based approaches. Our results demonstrate that dynamical phenotypes provide a scalable framework for linking model structure, external inputs, and phenotypic outcomes, and offer a principled tool for model-guided biomarker selection.

This work explores the potential to detect the nonlinear Integrated Sachs Wolfe effect, namely the Rees-Sciama effect (ISWRS), by cross-correlating current and future Cosmic Microwave Background (CMB) experiments -- Simons Observatory, CMB-S4, CMB-HD, and PICO -- with ongoing Large Scale Structure (LSS) surveys, such as Euclid and the Vera Rubin Observatory (LSST). We model the cross-correlation of the ISWRS effect with gravitational potential tracers like galaxy clustering, cosmic shear, and CMB-lensing potential, to forecast results from these experiments. Our analysis also accounts for the presence of massive neutrinos to assess the feasibility of identifying the $ν$$Λ$CDM model and constraining the neutrino mass sum, M$ν$. Our findings indicate that the CMB-lensing potential reconstructed by CMB-HD is expected to provide the most promising results, achieving $\gtrsim$ 5$σ$ detections even under conservative assumptions for detector noise and foregrounds, thereby allowing differentiation between $ν$$Λ$CDM models. Galaxy clustering can also yield significant detections, whereas cosmic shear can provide valuable results only if non-linearities are accurately modelled, beyond the capabilities of currently available analytical approaches. These latter LSS probes do not provide strong constraining power on M$ν$. While our findings suggest that future CMB experiments and LSS surveys will enable the detection of the ISWRS effect, they do not imply significant prospects for imposing new constraints on neutrino masses in the near future.

Minimizing volatility and adjustment costs is of central importance in many economic environments, yet it is often complicated by evolving feasibility constraints. We study a decision maker who repeatedly selects an action from a stochastically evolving interval of feasible actions in order to minimize either average adjustment costs or variance. We show that for strictly convex adjustment costs (such as quadratic variation), the optimal decision rule is a reference rule in which the decision maker minimizes the distance to a target action. In general, the optimal target depends both on the previous action and the expectation of future constraints; but for the special case where the constraints follow a random walk, the optimal mechanism is to simply target the previous action. If the decision maker minimizes variance, the optimal policy is also a reference rule, but the target is a constant, which is not necessarily equal to the long-term average action. Compared to mid-point heuristics, these optimal rules may substantially reduce quadratic variation and variance, in natural environments by $50\%$ or more. Applied to stock market auctions, our results provide an explanation for the wide-spread use of reference price rules. We also apply our results to bilateral trade in over-the-counter markets, capacity planning in supply chains, and positioning in political agenda setting.

We provide an upper bound for the genus zero logarithmic Gromov-Witten invariants of projective space relative to its toric boundary. The upper bound is polynomial in the contact orders, with degree depending on the number of marked points. The result hinges on the positivity of intersections for projective spaces.

We study the generalization of Correlation Clustering which incorporates fairness constraints via the notion of fairlets. The corresponding Fair Correlation Clustering problem has been studied from several perspectives to date, but has so far lacked a detailed analysis from the parameterized complexity paradigm. We close this gap by providing tractability results for the problem under a variety of structural graph parameterizations, including treewidth, treedepth and the vertex cover number; our results lie at the very edge of tractability given the known NP-hardness of the problem on severely restricted inputs.

Modern digital services have evolved into indispensable tools, driving the present large-scale information systems. Yet, the prevailing platform-centric model, where services are optimized for platform-driven metrics such as engagement and conversion, often fails to align with users' true needs. While platform technologies have advanced significantly-especially with the integration of large language models (LLMs)-we argue that improvements in platform service quality do not necessarily translate to genuine user benefit. Instead, platform-centric services prioritize provider objectives over user welfare, resulting in conflicts against user interests. This paper argues that the future of digital services should shift from a platform-centric to a user-centric agent. These user-centric agents prioritize privacy, align with user-defined goals, and grant users control over their preferences and actions. With advancements in LLMs and on-device intelligence, the realization of this vision is now feasible. This paper explores the opportunities and challenges in transitioning to user-centric intelligence, presents a practical device-cloud pipeline for its implementation, and discusses the necessary governance and ecosystem structures for its adoption.

Numerous studies have explored the SQL query refinement problem, where the objective is to minimally modify an input query so that it satisfies a specified set of constraints. However, these works typically target restricted classes of queries or constraints. We present OmniTune, a general framework for refining arbitrary SQL queries using LLM-based optimization by prompting (OPRO). OmniTune employs a two-step OPRO scheme that explores promising refinement subspaces and samples candidates within them, supported by concise history and skyline summaries for effective feedback. Experiments on a comprehensive benchmark demonstrate that OmniTune handles both previously studied refinement tasks and more complex scenarios beyond the scope of existing solutions.

The required absence of global symmetries in quantum gravity has been used to imply that all non-compact gauge theories are in the swampland. This argument stems from the idea that non-compact gauge symmetries always seem to be accompanied by global symmetries that cannot be broken with a finite number of fields. In this work, we investigate whether these symmetries can be broken by an uncountable infinity of fields. We find that the symmetries can be broken, but as soon as we add these fields the EFT breaks down and in some cases decompactifies to a higher-dimensional theory without the non-compact gauge symmetry, akin to undoing a Kaluza-Klein reduction on a non-compact space. We make various comments on the species scale, free parameters, and the Weak Gravity Conjecture along the way.

Hypothesis tests under order restrictions arise in a wide range of scientific applications. By exploiting inequality constraints, such tests can achieve substantial gains in power and interpretability. However, these gains come at a cost: when the imposed constraints are misspecified, the resulting inferences may be misleading or even invalid, and Type III errors may occur, i.e., the null hypothesis may be rejected when neither the null nor the alternative is true. To address this problem, this paper introduces safe tests. Heuristically, a safe test is a testing procedure that is asymptotically free of Type III errors. The proposed test is accompanied by a certificate of validity, a pre--test that assesses whether the original hypotheses are consistent with the data, thereby ensuring that the null hypothesis is rejected only when warranted, enabling principled inference without risk of systematic error. Although the development in this paper focus on testing problems in order--restricted inference, the underlying ideas are more broadly applicable. The proposed methodology is evaluated through simulation studies and the analysis of well--known illustrative data examples, demonstrating strong protection against Type III errors while maintaining power comparable to standard procedures.

Probabilistic modeling is an effective tool for evaluating team performance and predicting outcomes in sports. However, an important question that hasn't been fully explored is whether these models can reliably reflect actual performance while assigning meaningful probabilities to rare results that differ greatly from expectations. In this study, we create an inference-based probabilistic framework built on expected goals (xG). This framework converts shot-level event data into season-level simulations of points, rankings, and outcome probabilities. Using the English Premier League 2015/16 season as a data, we demonstrate that the framework captures the overall structure of the league table. It correctly identifies the top-four contenders and relegation candidates while explaining a significant portion of the variance in final points and ranks. In a full-season evaluation, the model assigns a low probability to extreme outcomes, particularly Leicester City's historic title win, which stands out as a statistical anomaly. We then look at the ex ante inferential and early-diagnostic role of xG by only using mid-season information. With first-half data, we simulate the rest of the season and show that teams with stronger mid-season xG profiles tend to earn more points in the second half, even after considering their current league position. In this mid-season assessment, Leicester City ranks among the top teams by xG and is given a small but noteworthy chance of winning the league. This suggests that their ultimate success was unlikely but not entirely detached from their actual performance. Our analysis indicates that expected goals models work best as probabilistic baselines for analysis and early-warning diagnostics, rather than as certain predictors of rare season outcomes.

Carbon-ion radiotherapy provides high dose conformity for lung cancer, but its benefit is limited by two sources of uncertainties: interplay between scanned beam delivery and tumor motion, and dose modulation from heterogeneous lung tissue. This study quantifies the separate and combined dosimetric impact of these effects using the GSI TRiP4D treatment planning system. Eighteen lung cancer 4DCT datasets from TCIA were analyzed. A modulation power ($P_{\mathrm{mod}}$) was assigned to lung voxels. Three values were sampled from a Gaussian distribution ($200μ\mathrm{m} \pm 67μ\mathrm{m}$), and an extreme value of $750μ\mathrm{m}$ was tested. Interplay doses were computed by combining scanned-beam delivery with patient-specific respiratory motion. Four scenarios were studied: static, static with modulation, interplay, and interplay with modulation. Metrics included $D95\%$, $V95\%$, homogeneity index (HI), lung $V16\mathrm{Gy}$, and heart $V20\mathrm{Gy}$. Interplay reduced target coverage by $5.2 \pm 1.5$ pp ($D95\%$), $12.1 \pm 5.9$ pp ($V95\%$), and $8.3 \pm 2.4$ pp (HI). Extreme $P_{\mathrm{mod}}$ alone caused small degradations. When combined with interplay, it partially compensated the loss. This effect decreased with 4D optimization. Fractionation mitigated interplay, leaving lung modulation as the main residual effect.

Federated learning security research has predominantly focused on backdoor threats from a minority of malicious clients that intentionally corrupt model updates. This paper challenges this paradigm by investigating a more pervasive and insidious threat: \textit{backdoor vulnerabilities from low-concentration poisoned data distributed across the datasets of benign clients.} This scenario is increasingly common in federated instruction tuning for language models, which often rely on unverified third-party and crowd-sourced data. We analyze two forms of backdoor data through real cases: 1) \textit{natural trigger (inherent features as implicit triggers)}; 2) \textit{adversary-injected trigger}. To analyze this threat, we model the backdoor implantation process from signal aggregation, proposing the Backdoor Signal-to-Noise Ratio to quantify the dynamics of the distributed backdoor signal. Extensive experiments reveal the severity of this threat: With just less than 10\% of training data poisoned and distributed across clients, the attack success rate exceeds 85\%, while the primary task performance remains largely intact. Critically, we demonstrate that state-of-the-art backdoor defenses, designed for attacks from malicious clients, are fundamentally ineffective against this threat. Our findings highlight an urgent need for new defense mechanisms tailored to the realities of modern, decentralized data ecosystems.

We consider the two-dimensional incompressible Navier-Stokes equations with measure initial vorticity. By means of improved Nash inequalities, we establish quantitative estimates for the enstrophy depending on the absolute vorticity decay on balls. The bounds are optimal in several aspects and yield to a conjecturally sharp rate of the dissipation in the Delort's class.

The application of mass-analyzed threshold ionization (MATI) for the state-selective preparation of molecular ions is presented. Based on photoexcitation of long-lived high-$n$ Rydberg states, molecular ions are prepared in a single rovibronic level by pulsed-field ionization. We present a theoretical analysis and a recipe for obtaining an optimal energy ratio between such selected ions and molecular ions in unwanted rovibronic states, created by direct photoionization. It is shown that the second-order chromatic aberration of a dc quadrupole bender can be used to isolate the state-selectively prepared molecular ions. The phase-space properties of ions prepared by MATI are ideally suited for axial injection into a linear radio-frequency trap. A modified approach for carrying out MATI within such an ion trap is also described.