LIPIcs

A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen’s Upper Bound Theorem for polytopes and refine asymptotic bounds due to Clarkson, asserts that for k ⩽ ⌊(n-d-2)/2⌋, the complexity of the (⩽ k)-level in a simple arrangement of n hemispheres in S^d is maximized for arrangements that are polar duals of neighborly d-polytopes. We prove this conjecture in the case n = d+4. By Gale duality, this implies the following result about crossing numbers: In every spherical arc drawing of K_n in S² (given by a set V ⊂ S² of n unit vectors connected by spherical arcs), the number of crossings is at least 1/4 ⌊n/2⌋ ⌊(n-1)/2⌋ ⌊(n-2)/2⌋ ⌊(n-3)/2⌋. This lower bound is attained if every open linear halfspace contains at least ⌊(n-2)/2⌋ of the vectors in V. Moreover, we determine the space of all linear and affine relations that hold between the face numbers of levels in simple arrangements of n hemispheres in S^d. This completes a long line of research on such relations, answers a question posed by Andrzejak and Welzl in 2003, and generalizes the classical fact that the Dehn-Sommerville relations generate all linear relations between the face numbers of simple polytopes (which correspond to the 0-level). To prove these results, we introduce the notion of the g-matrix, which encodes the face numbers of levels in an arrangement and generalizes the classical g-vector of a polytope

Mastitis-related Staphylococcus aureus-derived extracellular vesicles induce a pro-inflammatory response in bovine monocyte-derived macrophages

Staphylococcus aureus (S. aureus) is one of the most common causative agents of mammary gland infection and mastitis, but the specific role of S. aureus-derived extracellular vesicles (SaEVs) in mastitis has been poorly studied to date. Here, we aimed to investigate the response of bovine monocyte-derived macrophages (boMdM) to SaEVs of the genotype B (GTB) mastitis-related strain M5512B. Specifically, we evaluated the effects on the actin cytoskeleton, gene expression, and the SaEV proteomic cargo. Furthermore, we assessed to what extent the cellular and molecular response of boMdM to SaEVs differed from peripheral mononuclear blood cells (PBMCs) used for in vitro derivation of the former. We observed that SaEVs induced morphological changes in boMdM, leading to a pro-inflammatory and pyroptosis-related increased gene expression. Additionally, our study revealed that boMdM and PBMCs exhibited stimulus-specific differing responses. The proteomic analysis of SaEVs identified clusters of proteins related to virulence and antibiotic resistance, supporting the theory that S. aureus might use EVs to evade host defences and colonize the mammary gland. Our results bring new insights into how SaEVs might impact the host during an S. aureus infection, which can be useful for future S. aureus vaccine development

Spontaneous ordering of identical materials into a triboelectric series

When two insulating, neutral materials are contacted and separated, they exchange electrical charge1. Experiments have long suggested that this ‘contact electrification’ is transitive, with different materials ordering into ‘triboelectric series’ based on the sign of charge acquired2. At the same time, the effect is plagued by unpredictability, preventing consensus on the mechanism and casting doubt on the rhyme and reason that series imply3. Here we expose an unanticipated connection between the unpredictability and order in contact electrification: nominally identical materials initially exchange charge randomly and intransitively, but—over repeated experiments—order into triboelectric series. We find that this evolution is driven by the act of contact itself—samples with more contacts in their history charge negatively to ones with fewer contacts. Capturing this ‘contact bias’ in a minimal model, we recreate both the initial randomness and ultimate order in numerical simulations and use it experimentally to force the appearance of a triboelectric series of our choosing. With a set of surface-sensitive techniques to search for the underlying alterations contact creates, we only find evidence of nanoscale morphological changes, pointing to a mechanism strongly coupled with mechanics. Our results highlight the centrality of contact history in contact electrification and suggest that focusing on the unpredictability that has long plagued the effect may hold the key to understanding it

Scales

We introduce the notions of scale for sets and measures on metric space by generalizing the usual notions of dimension. Several versions of scales are introduced such as Hausdorff, packing, box, local and quantization. They are defined for different growth, allowing a refined study of infinite dimensional spaces. We prove general theorems comparing the different versions of scales. They are applied to describe geometries of ergodic decompositions, of the Wiener measure and from functional spaces. The first application solves a problem of Berger on the notions of emergence (2020); the second lies in the geometry of the Wiener measure and extends the work of Dereich–Lifshits (2005); the last refines Kolmogorov–Tikhomirov (1958) study on finitely differentiable functions

Unsupervised extraction of rotational Lagrangian coherent structures

Lagrangian coherent structures (LCSs) are widely recognized as playing a significant role in turbulence dynamics since they can control the transport of mass, momentum or heat. However, the methods used to identify these structures are often based on ambiguous definitions and arbitrary thresholding. While LCSs theory provides precise and frame-indifferent mathematical definitions of coherent structures, some of the commonly used extraction algorithms employed in the literature are still case-specific and involve user-defined parameters. In this study, we present a new, unsupervised extraction algorithm that enables the extraction of rotational LCSs based on Lagrangian average vorticity deviation from an arbitrary 3D velocity field. The algorithm utilizes two alternative methods for the identification of the LCS core (ridge): an unsupervised clustering method and a streamline-based method. In a subsequent step, the ridge curve is parametrized through a pruning procedure of minimum spanning tree graphs. To assess the effectiveness of the algorithm, we test it on two cases: (i) direct numerical simulations of forced homogeneous and isotropic turbulence and (ii) three-dimensional Particle Tracking Velocimetry experiments of a turbulent gravity current

LIPIcs

A quantitative word automaton (QWA) defines a function from infinite words to values. For example, every infinite run of a limit-average QWA obtains a mean payoff, and every word w ∈ Σ^ω is assigned the maximal mean payoff obtained by nondeterministic runs of over w. We introduce quantitative language automata (QLAs) that define functions from language generators (i.e., implementations) to values, where a language generator can be nonprobabilistic, defining a set of infinite words, or probabilistic, defining a probability measure over infinite words. A QLA consists of a QWA and an aggregator function. For example, given a QWA , the infimum aggregator maps each language L ⊆ Σ^ω to the greatest lower bound assigned by to any word in L. For boolean value sets, QWAs define boolean properties of traces, and QLAs define boolean properties of sets of traces, i.e., hyperproperties. For more general value sets, QLAs serve as a specification language for a generalization of hyperproperties, called quantitative hyperproperties. A nonprobabilistic (resp. probabilistic) quantitative hyperproperty assigns a value to each set (resp. distribution) G of traces, e.g., the minimal (resp. expected) average response time exhibited by the traces in G. We give several examples of quantitative hyperproperties and investigate three paradigmatic problems for QLAs: evaluation, nonemptiness, and universality. In the evaluation problem, given a QLA and an implementation G, we ask for the value that assigns to G. In the nonemptiness (resp. universality) problem, given a QLA and a value k, we ask whether assigns at least k to some (resp. every) language. We provide a comprehensive picture of decidability for these problems for QLAs with common aggregators as well as their restrictions to ω-regular languages and trace distributions generated by finite-state Markov chains

High-dimensional analysis of knowledge distillation: Weak-to-Strong generalization and scaling laws

A growing number of machine learning scenarios rely on knowledge distillation where one uses the output of a surrogate model as labels to supervise the training of a target model. In this work, we provide a sharp characterization of this process for ridgeless, high-dimensional regression, under two settings: (i) model shift, where the surrogate model is arbitrary, and (ii) distribution shift, where the surrogate model is the solution of empirical risk minimization with out-of-distribution data. In both cases, we characterize the precise risk of the target model through non-asymptotic bounds in terms of sample size and data distribution under mild conditions. As a consequence, we identify the form of the optimal surrogate model, which reveals the benefits and limitations of discarding weak features in a data-dependent fashion. In the context of weak-to-strong (W2S) generalization, this has the interpretation that (i) W2S training, with the surrogate as the weak model, can provably outperform training with strong labels under the same data budget, but (ii) it is unable to improve the data scaling law. We validate our results on numerical experiments both on ridgeless regression and on neural network architectures

Marginal values of a stochastic game

Zero-sum stochastic games are parameterized by payoffs, transitions, and possibly a discount rate. In this article, we study how the main solution concepts, the discounted and undiscounted values, vary when these parameters are perturbed. We focus on the marginal values, introduced by Mills in 1956 in the context of matrix games—that is, the directional derivatives of the value along any fixed perturbation. We provide a formula for the marginal values of a discounted stochastic game. Further, under mild assumptions on the perturbation, we provide a formula for their limit as the discount rate vanishes and for the marginal values of an undiscounted stochastic game. We also show, via an example, that the two latter differ in general

Prethermalization for deformed Wigner matrices

We prove that a class of weakly perturbed Hamiltonians of the form H_λ= H_0 + λW, with W being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by H_λ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order λ^{-2}. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix H_λ