The Wasserstein barycenter extends the Euclidean mean to the space of probability measures by minimizing the weighted sum of squared 2-Wasserstein distances. We develop a free-support algorithm for computing Wasserstein barycenters that avoids entropic regularization and instead follows the formal Riemannian geometry of Wasserstein space. In our approach, barycenter atoms evolve as particles advected by averaged optimal-transport displacements, with barycentric projections of optimal transport plans used in place of Monge maps when the latter do not exist. This yields a geometry-aware particle-flow update that preserves sharp features of the Wasserstein barycenter while remaining computationally tractable. We establish theoretical guarantees, including consistency of barycentric projections, monotone descent and convergence to stationary points, stability with respect to perturbations of the inputs, and resolution consistency as the number of atoms increases. Empirical studies on averaging probability distributions, Bayesian posterior aggregation, image prototypes and classification, and large-scale clustering demonstrate accuracy and scalability of the proposed particle-flow approach, positioning it as a principled alternative to both linear programming and regularized solvers.

Large language models (LLMs) promise to democratize financial analysis by reducing information-processing costs. Yet equal access does not ensure equal outcomes, as the locus of friction may shift from processing information to evaluating model outputs. We study GPT's earnings forecasts following corporate earnings releases and document two patterns. First, GPT's narrative attention is consistent and human-like but not always associated with higher forecast accuracy. Second, its quantitative reasoning varies substantially across contexts, challenging the view that LLMs are uniformly weak at numerical tasks. Building on these insights, we propose a diagnostic framework that links forecast accuracy to observable processing features (i.e., narrative focus, numerical reasoning, and self-assessed confidence). These indicators serve as proxies for this new form of information friction and alert investors when to exercise caution. Our study has implications for information frictions, regulatory oversight, and the economics of AI-mediated financial markets.

We present a rigorously formulated, novel operator-based framework that merges ideas from condensed matter physics, continuum mechanics, and quantum-inspired theory to analyze collective virion behavior in complex environments. By modeling metabolically inert virions, whose main interactions are Coulombic and Lennard-Jones, as nodes in a viral lattice linked by effective springs, we obtain collective vibrational modes akin to phonons in solids. A coupled, complex-valued displacement field PDE underpins our approach, enabling wave mechanics and functional analysis to define key observables, ranging from stress fields and thermodynamic responses to effective Hamiltonians. We interpret the apparent stochasticity of virion motion as deterministic chaos arising from rapid transitions among 'viral phonon' modes that store and redistribute energy from the host. Short-lived lattice states (picoseconds) may thus help sustain infectivity in aerosolized droplets of respiratory virions. Extending our operator framework, we derive an uncertainty-like relation coupling a virion's self-stiffness operator with its phononic frequency operator, underscoring fundamental limits on specifying capsid rigidity and vibrational response. We also explore transient coherence effects, akin to entanglement witness operators, which may expose critical correlations affecting capsid resilience or genome release. Our approach invites experimental validation via advanced imaging, high-resolution spectroscopy, and mechanical perturbations. By integrating aspects of soft matter physics, crystallography, statistical mechanics, and wave-based modeling, this methodology suggests new antiviral strategies targeting specific vibrational modes and guides the rational design of virus-based vectors in therapeutic or diagnostic applications.

We propose a novel evolutionary algorithm for optimizing real-valued objective functions defined on the Grassmann manifold Gr}(k,n), the space of all k-dimensional linear subspaces of R^n. While existing optimization techniques on Gr}(k,n) predominantly rely on first- or second-order Riemannian methods, these inherently local methods often struggle with nonconvex or multimodal landscapes. To address this limitation, we adapt the Differential Evolution algorithm - a global, population based optimization method - to operate effectively on the Grassmannian. Our approach incorporates adaptive control parameter schemes, and introduces a projection mechanism that maps trial vectors onto the manifold via QR decomposition. The resulting algorithm maintains feasibility with respect to the manifold structure while enabling exploration beyond local neighborhoods. This framework provides a flexible and geometry-aware alternative to classical Riemannian optimization methods and is well-suited to applications in machine learning, signal processing, and low-rank matrix recovery where subspace representations play a central role. We test the methodology on a number of examples of optimization problems on Grassmann manifolds.

The density conjecture for activated random walk on the interval was recently resolved using a new tool called layer percolation. As a step towards understanding how layer percolation extends to activated random walk on more complex graphs, we develop its analog for the comb graph and use it to prove bounds on the critical density. Additionally, we provide simulation evidence suggesting that the comb has different critical densities on its spine and teeth, both of which are smaller than the critical density for the interval.

We develop a comprehensive theoretical biophysics model grounded in a path integral perspective and an m-sectorial open-system framework, to describe complex, damped viral phonon dynamics in resource limited and noise driven environments. By unifying wave mechanics (via PDEs with multiplicative noise), Markov jumps for occupant or arrangement transitions, and second quantized (Fock-space) expansions, our construction accommodates an unbounded number of viral lattices in a single global wavefunction. In doing so, we capture how an entire population potentially numbering in the millions may be represented by a single operator theoretic state, or orbit, whose evolution is governed by non-unitary semigroups with potential equilibrium or non equilibrium steady states. This approach admits action functionals over the space of system trajectories, enabling large deviation and flux analyses whenever detailed balance is broken by sustained resource inputs, as often happens in real infections. Such a global wavefunction thus synthesizes PDE wavefront modes, occupant transitions, and stochastically induced rearrangements into a single evolution equation, capturing how local capsid vibrations might catalyze large scale replication bursts, and vice versa. Our proofs show that, despite unbounded occupant expansions or morphological continuums, solutions remain finite norm over finite times. The well posedness extends to non-self-adjoint operators with complex damping, irreversibility, and operator-valued-noise, thus mirroring host constraints and immune factors that restrict virus proliferation. Our formalism invites direct experimental cross validation, from single virion tracking to population assays, and offers predictive insights for resource limited replication, capsid reorganizations, and potential intervention strategies in modern virology.

Present knowledge of QCD n-point functions of Wilson lines at high energies is rather limited. In practical applications, it is therefore customary to factorize higher n-point functions into products of two-point functions (dipoles) which satisfy the BK evolution equation. We employ the JIMWLK formalism to derive explicit evolution equations for the 4- and 6-point functions of fundamental Wilson lines and show that if the Gaussian approximation is carried out before the rapidity evolution step is taken, then many leading order N_c contributions are missed. Our evolution equations could specifically be used to improve calculations of forward dijet angular correlations, recently measured by the STAR collaboration in deuteron-gold collisions at the RHIC collider. Forward dijets in proton-proton collisions at the LHC probe QCD evolution at even smaller light-cone momentum fractions. Such correlations may provide insight into genuine differences between the JIMWLK and BK approaches.

We introduce a novel, geometry-aware distance metric for the family of von Mises-Fisher (vMF) distributions, which are fundamental models for directional data on the unit hypersphere. Although the vMF distribution is widely employed in a variety of probabilistic learning tasks involving spherical data, principled tools for comparing vMF distributions remain limited, primarily due to the intractability of normalization constants and the absence of suitable geometric metrics. Motivated by the theory of optimal transport, we propose a Wasserstein-like distance that decomposes the discrepancy between two vMF distributions into two interpretable components: a geodesic term capturing the angular separation between mean directions, and a variance-like term quantifying differences in concentration parameters. The derivation leverages a Gaussian approximation in the high-concentration regime to yield a tractable, closed-form expression that respects the intrinsic spherical geometry. We show that the proposed distance exhibits desirable theoretical properties and induces a latent geometric structure on the space of non-degenerate vMF distributions. As a primary application, we develop the efficient algorithms for vMF mixture reduction, enabling structure-preserving compression of mixture models in high-dimensional settings. Empirical results on synthetic datasets and real-world high-dimensional embeddings, including biomedical sentence representations and deep visual features, demonstrate the effectiveness of the proposed geometry in distinguishing distributions and supporting interpretable inference. This work expands the statistical toolbox for directional data analysis by introducing a tractable, transport-inspired distance tailored to the geometry of the hypersphere.

We propose a highly efficient and accurate methodology for generating synthetic financial market data using a diffusion model approach. The synthetic data produced by our methodology align closely with observed market data in several key aspects: (i) they pass the two-sample Cramer - von Mises test for portfolios of assets, and (ii) Q - Q plots demonstrate consistency across quantiles, including in the tails, between observed and generated market data. Moreover, the covariance matrices derived from a large set of synthetic market data exhibit significantly lower condition numbers compared to the estimated covariance matrices of the observed data. This property makes them suitable for use as regularized versions of the latter. For model training, we develop an efficient and fast algorithm based on numerical integration rather than Monte Carlo simulations. The methodology is tested on a large set of equity data.

The Wasserstein barycenter extends the Euclidean mean to the space of probability measures by minimizing the weighted sum of squared 2-Wasserstein distances. We develop a free-support algorithm for computing Wasserstein barycenters that avoids entropic regularization and instead follows the formal Riemannian geometry of Wasserstein space. In our approach, barycenter atoms evolve as particles advected by averaged optimal-transport displacements, with barycentric projections of optimal transport plans used in place of Monge maps when the latter do not exist. This yields a geometry-aware particle-flow update that preserves sharp features of the Wasserstein barycenter while remaining computationally tractable. We establish theoretical guarantees, including consistency of barycentric projections, monotone descent and convergence to stationary points, stability with respect to perturbations of the inputs, and resolution consistency as the number of atoms increases. Empirical studies on averaging probability distributions, Bayesian posterior aggregation, image prototypes and classification, and large-scale clustering demonstrate accuracy and scalability of the proposed particle-flow approach, positioning it as a principled alternative to both linear programming and regularized solvers.

We study the problem of the optimal execution of a large trade in the presence of nonlinear transient impact. We propose an approach based on homotopy analysis, whereby a well behaved initial strategy is continuously deformed to lower the expected execution cost. We find that the optimal solution is front loaded for concave impact and that its expected cost is significantly lower than that of conventional strategies. We then consider brute force numerical optimization of the cost functional; we find that the optimal solution for a buy program typically features a few short intense buying periods separated by long periods of weak selling. Indeed, in some cases we find negative expected cost. We show that this undesirable characteristic of the nonlinear transient impact model may be mitigated either by introducing a bid-ask spread cost or by imposing convexity of the instantaneous market impact function for large trading rates.

The goal of this whitepaper is to give a comprehensive overview of the rich field of forward physics. We discuss the occurrences of BFKL resummation effects in special final states, such as Mueller-Navelet jets, jet gap jets, and heavy quarkonium production. It further addresses TMD factorization at low x and the manifestation of a semi-hard saturation scale in (generalized) TMD PDFs. More theoretical aspects of low x physics, probes of the quark gluon plasma, as well as the possibility to use photon-hadron collisions at the LHC to constrain hadronic structure at low x, and the resulting complementarity between LHC and the EIC are also presented. We also briefly discuss diffraction at colliders as well as the possibility to explore further the electroweak theory in central exclusive events using the LHC as a photon-photon collider.

We generalize Pach and de Zeeuw's bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including exponential and logarithmic functions, rational and irrational powers, trigonometric functions and their inverses, integration, and more. The bound remains $\Omega(\min\{m^{3/4}n^{3/4},m^2,n^2\})$, as obtained from the proximity technique of Solymosi and Zahl.

We study configurations of $n$ points and $n$ lines that form $\Theta(n^{4/3})$ incidences, when the point set is a Cartesian product. We prove structural properties of such configurations, such that there exist many families of parallel lines or many families of concurrent lines. We show that the line slopes have multiplicative structure or that many sets of $y$-intercepts have additive structure. We introduce the first infinite family of configurations with $\Theta(n^{4/3})$ incidences. We also derive a new variant of a different structural point-line result of Elekes. Our techniques are based on the concept of line energy. Recently, Rudnev and Shkredov introduced this energy and showed how it is connected to point-line incidences. We also prove that their bound is tight up to sub-polynomial factors.

This article introduces a novel approach to the classification of categorical time series under the supervised learning paradigm. To construct meaningful features for categorical time series classification, we consider two relevant quantities: the spectral envelope and its corresponding set of optimal scalings. These quantities characterize oscillatory patterns in a categorical time series as the largest possible power at each frequency, or spectral envelope, obtained by assigning numerical values, or scalings, to categories that optimally emphasize oscillations at each frequency. Our procedure combines these two quantities to produce an interpretable and parsimonious feature-based classifier that can be used to accurately determine group membership for categorical time series. Classification consistency of the proposed method is investigated, and simulation studies are used to demonstrate accuracy in classifying categorical time series with various underlying group structures. Finally, we use the proposed method to explore key differences in oscillatory patterns of sleep stage time series for patients with different sleep disorders and accurately classify patients accordingly.

We examine how large language models (LLMs) interpret historical stock returns and compare their forecasts with estimates from a crowd-sourced platform for ranking stocks. While stock returns exhibit short-term reversals, LLM forecasts over-extrapolate, placing excessive weight on recent performance similar to humans. LLM forecasts appear optimistic relative to historical and future realized returns. When prompted for 80% confidence interval predictions, LLM responses are better calibrated than survey evidence but are pessimistic about outliers, leading to skewed forecast distributions. The findings suggest LLMs manifest common behavioral biases when forecasting expected returns but are better at gauging risks than humans.

Product manifolds arise when heterogeneous geometric variables are recorded jointly. While the Fr\'{e}chet mean on Riemannian manifolds separates cleanly across factors, the canonical geometric median couples them, and its behavior in product spaces has remained largely unexplored. In this paper, we give the first systematic treatment of this problem. After formulating the coupled objective, we establish general existence and uniqueness results: the median is unique on any Hadamard product, and remains locally unique under sharp conditions on curvature and injectivity radius even when one or more factors have positive curvature. We then prove that the estimator enjoys Lipschitz stability to perturbations and the optimal breakdown point, extending classical robustness guarantees to the product-manifold setting. Two practical solvers are proposed, including a Riemannian subgradient method with global sublinear convergence and a product-aware Weiszfeld iteration that achieves local linear convergence when safely away from data singularities. Both algorithms update the factors independently while respecting the latent coupling term, enabling implementation with standard manifold primitives. Simulations on parameter spaces of univariate and multivariate Gaussian distributions endowed with the Bures-Wasserstein geometry show that the median is more resilient to contamination than the Fr\'{e}chet mean. The results provide both theoretical foundations and computational tools for robust location inference with heterogeneous manifold-valued data.

This review examines data from hadron-nucleus collisions, primarily focusing on hard processes like Drell-Yan, heavy flavor and quarkonium production. It highlights observed modifications of particle yields as functions of momentum and rapidity, aiming to clarify the underlying QCD effects in cold nuclear matter (CNM). The paper outlines strategies for future experiments, including the Electron-Ion Collider (EIC), to distinguish between these effects. Key questions address the universality of suppression mechanisms and the role of non-perturbative physics, providing a road map for upcoming nuclear data.

We propose an Economic Lexicon (EL) specifically designed for textual applications in economics. We construct the dictionary with two important characteristics: 1) to have a wide coverage of terms used in documents discussing economic concepts, and 2) to provide a human-annotated sentiment score in the range [-1,1]. We illustrate the use of the EL in the context of a simple sentiment measure and consider several applications in economics. The comparison to other lexicons shows that the EL is superior due to its wider coverage of domain relevant terms and its more accurate categorization of the word sentiment.

We show that the key features of the CMS result on the ridge correlation seen for high multiplicity events in sqrt(s)=7TeV proton-proton collisions at the LHC can be understood in the Color Glass Condensate framework of high energy QCD. The same formalism underlies the explanation of the ridge events seen in A+A collisions at RHIC, albeit it is likely that flow effects may enhance the magnitude of the signal in the latter.