It was recently shown that under mild assumptions second-order conformally superintegrable systems can be encoded in a $(0,3)$-tensor, called structure tensor. For abundant systems, this approach led to algebraic integrability conditions that essentially allow one to restore a system from the knowledge of its structure tensor in a point on the manifold. Here we study the geometric structure formalising such systems, which we call an abundant manifold. The underlying Riemannian manifold is necessarily conformally flat. We establish a correspondence between these superintegrable systems and the geometry of affine hypersurfaces. More precisely, we show that abundant manifolds correspond to certain non-degenerate relative affine hypersurfaces normalisations in $\mathbb R^{n+1}$ ($n\ge 2$). We also formulate the necessary and sufficient conditions non-degenerate relative affine hypersurface normalisations in $\mathbb R^{n+1}$ need to satisfy, if they arise from abundant manifolds. These relative affine hypersurface normalisations are called abundant hypersurface normalisations. Both for abundant manifolds and for relative affine hypersurface normalisations a natural concept of conformal equivalence can be defined. We prove that they are compatible, permitting us to identify conformal classes of abundant manifolds with abundant hypersurface immersions (without specified normalisation).

We prove the uniqueness of general Bessel models for $\mathrm{GSpin}$ groups over a local field of characteristic zero. The proof is to reduce it to the spherical case, which has been proved by Emory and Takeda in the non-archimedean case and by Emory, Kim, and Maiti in the archimedean case.

We first prove Bosch-Lütkebohmert-Raynaud's conjectures on existence of global Néron models of not necessarily semi-abelian algebraic groups in the perfect residue fields case. We then give a counterexample to the existence in the imperfect residue fields case. Finally, as a complement to the conjectures, we classify unirational wound unipotent groups "up to relative perfection", again in the perfect residue fields case. The key ingredient for all these is the duality for relatively perfect unipotent groups.

We prove the existence of global-in-time weak solutions to volume-preserving mean curvature flow with in the presence of obstacles by the phase field method in all dimensions. Namely, we prove the convergence of solutions to the Allen-Cahn equation with a multiplier to a weak solution to the flow. The choice of the multiplier is motivated from [Mugnai-Seis-Spadaro '16], [Kim-Kwon '20], and [Takasao '23], which enables us to complete the comparison between the multiplier and the forcing that stops the intrusion into the obstacle. We also prove the vanishing of the discrepancy measure by dealing with the forcing term that is now spatially dependent due to the obstacles.

We provide a new, elementary, algebraic proof of the fundamental theorem of algebra. Furthermore, the method recovers a more general version of the theorem recently obtained by Joseph Shipman. The key idea is to understand extension fields in which a polynomial gains a factor of a given degree.

The COVID-19 pandemic highlighted the urgent need for effective vaccine policies, but traditional clinical trials often lack sufficient data to capture the diverse population characteristics necessary for comprehensive public health strategies. Ethical concerns around randomized trials during a pandemic further complicate policy development for public health. Reinforcement Learning (RL) offers a promising alternative for vaccine policy development. However, direct online RL exploration in real-world scenarios can result in suboptimal and potentially harmful decisions. This study proposes a novel framework combining tabular Q-learning with microsimulation, where a Recurrent Neural Network (RNN) serves as a digital twin environment simulator of the target population. This digital twin captures temporal associations between infection and patient characteristics to generate realistic individual disease trajectories, enabling safe and efficient policy learning without real-world interaction. Our tabular Q-learning model produces an interpretable policy table that balances the risks of severe infection against vaccination side effects. Applied to COVID-19 booster policies, the learned Q-learning-based policy outperforms current practices, offering a path toward more effective vaccination strategies. A project webpage introducing our work, including links to the software, a brief introductory video, and a step-by-step tutorial video, is available at https://public.websites.umich.edu/~jiankang/software/dtpl_website_umich/index.html.

Preference heterogeneity massively increases the social cost of carbon. We call this the Weitzman premium. Uncertainty about an exponential discount rate implies a hyperbolic discount rate, which in the near term is equal to the average discount rate but in the long term falls to the minimum discount rate. We generalise Weitzman's (2001, AER) gamma discounting to zero-inflation and two dimensions but find that the analytical solution is a poor approximation of the non-parametric heterogeneity. We calibrate the pure rate of time preference and the inverse of the elasticity of intertemporal substitution of 79,273 individuals from 76 countries and compute the corresponding social cost of carbon. Compared to the social cost of carbon for average time preferences, the average social cost of carbon is 6 times as large in the base calibration, and up to 200 times as large in sensitivity analyses.

Augmenting randomized controlled trials (RCTs) with external real-world data (RWD) has the potential to improve the finite sample efficiency of treatment effect estimators. We describe using adaptive targeted maximum likelihood estimation (A-TMLE) for estimating the average treatment effect (ATE) by decomposing the ATE estimand into two components: a pooled-ATE estimand that combines data from both the RCT and external sources, and a bias estimand that captures the conditional effect of RCT enrollment on the outcome. This approach views the RCT data as the reference and corrects for inconsistencies of any kind between the RCT and the external data source. Given the growing abundance of external RWD from modern electronic health records, determining the optimal strategy to select candidate external patients for data integration remains an open yet critical problem. In this work, we begin by studying the robustness property of the A-TMLE estimator and then propose a matching-based sampling strategy that attempts to improve the robustness of the estimator with respect to the target estimand. Our proposed strategy is outcome-blind and involves matching based on two one-dimensional scores: the trial enrollment score and the propensity score in the external data. We demonstrate in simulations that our sampling strategy improves the coverage and narrows the widths of confidence intervals produced by A-TMLE. We illustrate our method with a case study of augmenting the DEVOTE cardiovascular safety trial by using the Optum Clinformatics claims database.

The main goals of the present paper are to determine the structure of the $C^\ast$-algebras associated to a finitely presented system and to develop the basic theory of the Ruelle algebras associated to a general synchronizing system. The later is related to the former in the sense that we show that Ruelle algebras associated to a finitely presented system are explicitly related to the Smale space case. The relevant $C^\ast$-algebras are the synchronizing heteroclinic algebras that were introduced in our previous work on synchronizing systems. They are very much related to previous work of Thomsen, who in turn was building on work of Ruelle, Putnam, and Spielberg.

The Fréchet mean (or barycenter) generalizes the expectation of a random variable to metric spaces by minimizing the expected squared distance to the random variable. Similarly, the median can be generalized by its property of minimizing the expected absolute distance. We consider the class of transformed Fréchet means with nondecreasing, convex transformations that have a concave derivative. This class includes the Fréchet median, the Fréchet mean, the Huber loss-induced Fréchet mean, and other statistics related to robust statistics in metric spaces. We study variance inequalities for these transformed Fréchet means. These inequalities describe how the expected transformed distance grows when moving away from a minimizer, i.e., from a transformed Fréchet mean. Variance inequalities are useful in the theory of estimation and numerical approximation of transformed Fréchet means. Our focus is on variance inequalities in Hadamard spaces - metric spaces with globally nonpositive curvature. Notably, some results are new also for Euclidean spaces. Additionally, we are able to characterize uniqueness of transformed Fréchet means, in particular of the Fréchet median.

We study homotopic-to-the-identity torus homeomorphisms, whose rotation set has nonempty interior. We prove that any such map is monotonically semiconjugate to a homeomorphism that preserves the Lebesgue measure, and that has the same rotation set. Furthermore, the dynamics of the quotient map has several interesting chaotic traits: for instance, it is topologically mixing, has a dense set of periodic points and is continuum-wise expansive. In particular, this shows that a convex compact set of $\mathbb{R}^2$ with nonempty interior is the rotation set of the lift of a homeomorphism of $\mathbb{T}^2$ if and only if it is the rotation set of the lift of a conservative homeomorphism.

The sixth Painlevé equation (PVI) admits dual isomonodromy representations of type $2$-dimensional Fuchsian and $3$-dimensional Birkhoff. Taking the multiplicative middle convolution of a Teichmüller $\mathcal{X}$-coordinatization for the Fuchsian monodromy group, we give Okamoto's symmetry $w_2$ of PVI a monodromic realization in the language of cluster $\mathcal{X}$-mutations. The explicit mutation formula is encoded in dual geometric terms of colored equilateral triangulations and star-shaped fat graphs. Moreover, this realization has a known additive analogue through the middle convolution for Fuchsian systems, and dual formulations for both the Birkhoff representation and its Stokes data exist. We give this quadruple of maps, each one realizing $w_2$, a unified diagrammatic description in purely convolutional terms.

Ekedhal Sieve can be used to count squarefree bounded values of a polynomial closely related to the discriminant polynomial and one can also get a weighted lower bounds for the number of number fields with bounded discriminant which show a limsup much larger than current known bounds.

Superconducting nanowire single-photon detectors (SNSPDs) were integrated with plasmonic nanostructures to enhance the absorption efficiency of superconducting BSCCO stripes. A numerical investigation of optimized nanocavity array (NCAI) and nanocavity-trench-array (NCTAI) SNSPDs has revealed that more than one order of magnitude larger absorptance can be achieved at perpendicular incidence, when compared to the corresponding meandered BSCCO pattern in a resonant optical cavity. The SNSPDs were considerably improved either via first and third quarter cavity resonances, as evidenced by the near-field maps and validated by the standard retrieval method. Although, NCAI-SNSPD exhibits slightly larger absorptance, NCTAI-SNSPD remains competitive due to its larger period and significantly smaller filling factor, thereby allowing for quicker electric response.

The aim of change-point detection is to identify behavioral shifts within time series data. This article focuses on scenarios where the data is derived from an inhomogeneous Poisson process or a marked Poisson process. We present a methodology for detecting multiple offline change-points using a minimum contrast estimator. Specifically, we address how to manage the continuous nature of the process given the available discrete observations. Additionally, we select the appropriate number of changes via a cross-validation procedure which is particularly effective given the characteristics of the Poisson process. Lastly, we show how to use this methodology for self-exciting processes with changes in the intensity. Through experiments, with both simulated and real datasets, we showcase the advantages of the proposed method, which has been implemented in the R package.

A statistically significant detection of the diffuse supernova neutrino background (DSNB) is around the corner. To this purpose, we assess the contribution to the DSNB of magnetorotational collapses of massive stars, relying on a suite of state-of-the-art three-dimensional neutrino-magnetohydrodynamic simulations. We find that neutrinos from magnetorotational core collapses boost the high-energy tail of the DSNB spectrum, similar to what is expected from neutrino-driven black hole-forming collapses. The latest data from the Super-Kamiokande Collaboration can already exclude that more than $9\%$ of all collapsing massive stars undergo magnetorotational collapses under optimistic assumptions. A DSNB detection at $3 σ$ could take place up to $4$ yr earlier at Super-Kamiokande-Gadolinium or JUNO if the fraction of magnetorotational collapses should be larger than $10\%$. Fascinatingly, if the fraction of magnetorotational stellar collapses should be larger than $19\%$ ($13\%$), Hyper-Kamiokande could measure such a fraction at $3σ$ after ($10$ yr) $20$ yr of DSNB data taking. The combination of DSNB and electromagnetic data has the potential to resolve the degenerate contributions from magnetorotational and neutrino-driven black hole-forming collapses, providing crucial insight on the properties of the population of collapsing massive stars.

This paper proposes the second-order Esscher transform as a tractable extension of the classical Esscher framework for option pricing and risk management in Lévy-driven markets. For a general Lévy process, we derive the associated densities and equivalent pricing measures, characterize the martingale condition in closed form, and obtain FFT-based valuation formulas for European call options. For jump-diffusion models, we establish explicit pricing formulas under the second-order Esscher measure and show that the resulting option prices lie in an interval bounded below by the Black--Scholes price and above by the underlying asset value. For the constant jump-diffusion model, we further prove monotonicity of option prices with respect to the second-order Esscher parameter. An empirical analysis based on market data shows that this additional parameter provides a tractable tool for stress testing, delta-hedging evaluation, and the construction of interval-valued risk measures in incomplete markets. We further document a strong association between the estimated second-order Esscher parameter and standard indicators of market stress, including the VIX, news sentiment, and crisis regimes. The proposed framework preserves analytical tractability while enlarging the class of admissible pricing measures, thereby supporting pricing, hedging, and stress-based risk assessment in incomplete markets with jump and general Lévy dynamics.

Locally classical behavior is often interpreted as evidence that a global classical description should exist. In particular, when individual processes admit classical (measure-and-prepare) realizations, it is natural to expect at least one compatible joint implementation that remains classical. We show that this intuition fails in a simple broadcast setting with one input system and two outputs. Specifically, we construct pairs of compatible channels that are entanglement-breaking individually and thus admit classical descriptions, yet every joint broadcast realization is necessarily Bell-nonlocal between the outputs, even for a maximally mixed input. This establishes a form of compatibility-induced activation of nonlocality: Bell nonlocality arises not from entanglement in the inputs or marginal channels, but solely from the requirement that these marginals admit a common global implementation. In this sense, compatibility is not merely a consistency condition on local descriptions, but a mechanism that can enforce nonclassical correlations at the global level.

In this work, we study nonsingular bouncing cosmology in quadratic modified teleparallel gravity $f(T)=T+βT^{2}$ within a flat FLRW background. Using dimensionless variables, we derive modified Friedmann equations and reformulate the dynamics as a 2D autonomous system. Phase-space analysis identifies saddle, unstable, and stable critical points, with a scalar field-dominated late-time attractor under specific potential conditions. A genuine nonsingular bounce is realized via a regular scale factor, finite and positive throughout, with Hubble parameter satisfying $H(t_b)=0$, $\dot{H}(t_b)>0$, ensuring a smooth transition from contraction to expansion without singularity. The torsion scalar, effective energy density, and geometry remain finite. The deceleration and effective equation of state parameters show a temporary phantom-like regime near the bounce, violating the null energy condition as required. The reconstructed bounce aligns with the dynamical system's qualitative structure, asymptotically approaching a matter-dominated saddle while avoiding unstable stiff-matter solutions. Our results confirm that quadratic $f(T)$ gravity offers a self-consistent and viable framework for regular bouncing cosmology.

As quantum computing moves towards practical deployment, ensuring the reliability of quantum software becomes increasingly important. Mutation testing is a promising technique in this context; however, existing exhaustive mutation generators have primarily been developed for legacy versions of Qiskit (0.x), limiting their applicability to current stable releases. This work presents QCRMut, a mutation testing tool for quantum circuits compatible with stable Qiskit versions, which supports controlled preservation of circuit structure through immutable positions and enables efficient, representative random mutant generation as an alternative to exhaustive mutation. We develop QCRMut according to four design principles: unicity, similarity, representativity, and coverability. We evaluate the tool empirically by comparing mutation scores obtained from randomly sampled mutant subsets against those produced by exhaustive mutation over a benchmark suite of quantum circuits. We further analyse sensitivity to random seeds and apply statistical tests to assess the robustness of the observed differences. Finally, we compare QCRMut with existing mutation testing tools. Across the benchmark suite, randomly generated mutant subsets produced by QCRMut achieve mutation scores that are comparable to those obtained via exhaustive mutation. The results are stable across different random seeds and highlight practical limitations in existing approaches that are addressed by our method. Overall, QCRMut provides a practical and extensible solution for mutation testing of quantum circuits by combining compatibility with stable Qiskit releases, controlled structure preservation, statistically sound evaluation, and efficient mutant generation. In addition, it enables the mutation and analysis of circuits that cannot be handled by previous tools.

The dynamics of energetic particles (EPs) interacting with Alfvén eigenmodes (AEs) for the ITER 15MA baseline scenario was described using a reduced 1D model in [Carlevaro et al. PPCF 64, 035010 (2022)], and successfully tested against nonlinear wave-particle simulations. In this paper, we introduce a detailed phase-space and statistical analysis of this case to characterize the emerging EP transport regimes. Deviations from pure diffusive dynamics are quantitatively addressed, indicating the limitations of standard quasi-linear descriptions. The phase space diagnostics introduced allows to describe the emergence of a very complex dynamics of overlapping resonances and substructure formation, reinforcing the evidence of non-diffusive domino-like AEs.

In this article, we prove that Ramsey's theorem for pairs and two colors is $Π^1_1$-conservative over~$\mathsf{RCA}_0 + \mathsf{B}Σ^0_2 + \mathsf{WF}(ε_0)$ and over~$\mathsf{RCA}_0 + \mathsf{B}Σ^0_2 + \bigcup_n \mathsf{WF}(ω^ω_n)$. These results improve theorems from Chong, Slaman and Yang and Kołodziejczyk and Yokoyama and belong to a long line of research towards the characterization of the first-order part of Ramsey's theorem for pairs.

By using optimal transport theory, we prove a sharp dimension-free isoperimetric inequality involving the volume entropy, in metric measure spaces with non-negative Ricci curvature in the sense of Lott--Sturm--Villani. We show that this isoperimetric inequality is attained by a non-trivial open set, if and only if the space satisfies a certain foliation property. For metric measure spaces with non-negative Riemannian Ricci curvature, we show that the sharp Cheeger constant is achieved by a non-trivial measurable set, if and only if a one-dimensional space is split off. Our isoperimetric inequality and the rigidity theorems are proved in non-smooth framework, but new even in the smooth setting. In particular, our results provide some new understanding of logarithmically concave measures.

The recently discovered altermagnets exhibit collinear magnetic order with zero net magnetization but with unconventional spin-polarized d/g/i-wave band structures, expanding the known paradigms of ferromagnets and antiferromagnets. In addition to novel current-driven electronic transport effects, the unconventional time-reversal symmetry breaking in these systems also makes it possible to obtain a spin response to \emph{linearly polarized} fields in the optical frequency domain. We show through ab-initio calculations of the prototypical d-wave altermagnet RuO$_2$, with a symmetry combining twofold spin rotation with fourfold lattice rotation, $[C_2\|C_{4z}]$, that there is an optical analogue of a spin splitter effect, as the coupling to a linearly polarized exciting laser field makes the d-wave character of the altermagnet directly visible. By magneto-optical measurements on RuO$_2$ films of different thicknesses ranging from $2$ to $8\,$nanometers, we demonstrate the predicted connection of the linear polarization of an ultrashort pump pulse to the sign and magnitude of the optically excited electronic spin polarization in the ultrathin RuO$_2$ films. The possibility of exciting and controlling an electronic spin polarization by linearly polarized optical pulses in a compensated system is a unique consequence of the altermagnetic material properties. Our experimental results therefore establish an optical pump-probe based protocol for detection of altermagnetic characteristics in ultrathin RuO$_2$ films, but our all-optical approach should apply more generally to materials in this altermagnetic symmetry class.

We construct a new family $\left( η_α^{\left( q\right) }\right) _{α\in\operatorname*{Comp}}$ of quasisymmetric functions for each element $q$ of the base ring. We call them the "enriched $q$-monomial quasisymmetric functions". When $r:=q+1$ is invertible, this family is a basis of $\operatorname{QSym}$. It generalizes Hoffman's "essential quasi-symmetric functions" (obtained for $q=0$) and Hsiao's "monomial peak functions" (obtained for $q=1$), but also includes the monomial quasisymmetric functions as a limiting case. We describe these functions $η_α^{\left( q\right) }$ by several formulas, and compute their products, coproducts and antipodes. The product expansion is given by an exotic variant of the shuffle product which we call the "stufufuffle product" due to its ability to pick several consecutive entries from each composition. This "stufufuffle product" has previously appeared in recent work by Bouillot, Novelli and Thibon, generalizing the "block shuffle product" from the theory of multizeta values.

In recent years, there has been an effort to extend the classical notion of phylogenetic balance, originally defined in the context of trees, to networks. One of the most natural ways to do this is with the so-called $B_2$ index. In this paper, we study the $B_2$ index for a prominent class of phylogenetic networks: galled trees. We show that the $B_2$ index of a uniform leaf-labeled galled tree converges in distribution as the network becomes large. We characterize the corresponding limiting distribution, and provide a way to compute its moments. This is the first time that a balance index has been studied to this level of detail for a random phylogenetic network. One specificity of this work is that we use two different and independent approaches, each with its advantages: analytic combinatorics, and local limits. The analytic combinatorics approach is more direct, as it relies on standard tools; but it involves slightly more complex calculations. Because it has not previously been used to study such questions, the local limit approach requires developing an extensive framework beforehand; however, this framework is interesting in itself and can be used to tackle other similar problems.

We introduce and study several random combinatorial billiard trajectories. Such a system, which depends on a fixed parameter $p\in(0,1)$, models a beam of light that travels in a Euclidean space, occasionally randomly reflecting off of a hyperplane in the Coxeter arrangement of an affine Weyl group with some probability that depends on the side of the hyperplane that it hits. In one case, we recover Lam's reduced random walk in the limit as $p$ tends to $0$. The investigation of our random billiard trajectories relies on an analysis of new finite Markov chains that we call stoned exclusion processes. These processes have remarkable stationary distributions determined by well-studied polynomials such as ASEP polynomials, inhomogeneous TASEP polynomials, and open boundary ASEP polynomials; in many cases, it was previously not known how to construct Markov chains with these stationary distributions. Using multiline queues, we analyze correlations in the stoned multispecies TASEP, allowing us to determine limit directions for reduced random billiard trajectories and limit shapes for new random growth processes for $n$-core partitions.

We demonstrate that the observed parameters, related to the flavor oscillation of the neutrinos, are consistent with a system with three degenerate ground states. These ground states must have the same energy and they are connected by a symmetry with its corresponding generator. This symmetry is not spontaneously broken because the neutrino never selects a specific vacuum state. Instead, the neutrino keeps oscillating between the three ground states as it moves through spacetime. The order parameter in this case is the neutrino flavor and then the superposition of all the possible vacuum flavor states for one neutrino gives a trivial (zero) result. From this trivial condition, we can find constraints over the observed mixing angles $θ_{ij}$. By using these constraints we make the predictions of the mixing angles which are in agreement with the observations. Subsequently, complementing these arguments with a triangular formulation of the neutrino oscillation, we find relations between the the mass eigenvalues and the mixing angles, making again predictions consistent with the observations and consistent with a normal order in hierarchy with $m_3>>m_2\approx m_1$. Finally, we analyze the symmetry related to the flavor oscillation.

Neural embedding models have become a fundamental component of modern information retrieval (IR) pipelines. These models produce a single embedding $x \in \mathbb{R}^d$ per data-point, allowing for fast retrieval via highly optimized maximum inner product search (MIPS) algorithms. Recently, beginning with the landmark ColBERT paper, multi-vector models, which produce a set of embedding per data point, have achieved markedly superior performance for IR tasks. Unfortunately, using these models for IR is computationally expensive due to the increased complexity of multi-vector retrieval and scoring. In this paper, we introduce MUVERA (MUlti-VEctor Retrieval Algorithm), a retrieval mechanism which reduces multi-vector similarity search to single-vector similarity search. This enables the usage of off-the-shelf MIPS solvers for multi-vector retrieval. MUVERA asymmetrically generates Fixed Dimensional Encodings (FDEs) of queries and documents, which are vectors whose inner product approximates multi-vector similarity. We prove that FDEs give high-quality $ε$-approximations, thus providing the first single-vector proxy for multi-vector similarity with theoretical guarantees. Empirically, we find that FDEs achieve the same recall as prior state-of-the-art heuristics while retrieving 2-5$\times$ fewer candidates. Compared to prior state of the art implementations, MUVERA achieves consistently good end-to-end recall and latency across a diverse set of the BEIR retrieval datasets, achieving an average of 10$\%$ improved recall with $90\%$ lower latency.

We consider the problem of recovering an unknown matching between a set of $n$ randomly placed points in $\mathbb{R}^d$ and random perturbations of these points. This can be seen as a model for particle tracking and more generally, entity resolution. We use matchings in random geometric graphs to derive minimax lower bounds for this problem that hold under great generality. Using these results we show that for a fixed $d$, as long as the noise distribution has finite $d$-th moment, and both initial positions and noise have bounded continuous densities, the minimax rate for the problem scales as $Θ(n^2σ^d \wedge n)$. Under the stronger assumptions that the tail of the noise is sub-Gaussian, we show that the order of the number of mistakes made by an estimator that minimizes the sum of squared Euclidean distances is minimax optimal when $d$ is fixed and is optimal up to $n^{o(1)}$ factors when $d = o(\log n)$. In the high-dimensional regime we consider a setup where both initial positions and perturbations have independent sub-Gaussian coordinates. In this setup we give sufficient conditions under which the same estimator makes no mistakes with high probability. We prove an analogous result for an adapted version of this estimator that incorporates information on the covariance matrix of the perturbations.