Characterising hadronisation across the phase space
academic
Banfi, El-Menoufi

Characterising hadronisation across the phase space

Leading hadronisation corrections to two-jet global event shapes amount to a shift in the corresponding perturbative distributions. It has been recently established that this shift depends significantly on the value of the considered event shape. These analyses consider perturbative configurations with only three partons emitting an ultra-soft non-perturbative gluon. These are dominant in the three-jet region. However, multiple soft and/or collinear emissions need to be considered to accurately describe event shape distributions near their Sudakov peak. In this region, non-perturbative shifts are usually computed by considering only two hard emitters. In this paper, we find that this approximation misses an important correction to the shift due to the emission of an additional soft wide-angle gluon. We then compute its contribution, and embed it in a general treatment for the shift that is valid in both the two-jet and three-jet regions.

#arxiv #academic-paper #hep-ph +2
On approximation of convex functionals with a convexity constraint and general Lagrangians
academic
Kim

On approximation of convex functionals with a convexity constraint and general Lagrangians

In this note, we prove that minimizers of convex functionals with a convexity constraint and a general class of Lagrangians can be approximated by solutions to fourth-order equations of Abreu type. Our result generalizes that of Le (Twisted Harnack inequality and approximation of variational problems with a convexity constraint by singular Abreu equations. Adv. Math. 434 (2023)) where the case of quadratically growing Lagrangians was treated.

#arxiv #academic-paper #math.AP +1
On the Extreme Value Behavior of $\vartheta$-Expansions
academic
Sebe, Lascu, Selmi

On the Extreme Value Behavior of $\vartheta$-Expansions

The main objective of this paper is to develop extreme value theory for $\vartheta$-expansions. We establish the limit distribution of the maximum value in a $\vartheta$-continued fraction mixing stationary stochastic process, along with some related results. These findings are analogous to the theorems of J. Galambos and W. Philipp for regular continued fractions. Additionally, we emphasize that a Borel-Bernstein type theorem plays a crucial role.

#arxiv #academic-paper #math.PR +2
Class Prototypes based Contrastive Learning for Classifying Multi-Label and Fine-Grained Educational Videos
academic
Gupta, Roy, Christensen et al.

Class Prototypes based Contrastive Learning for Classifying Multi-Label and Fine-Grained Educational Videos

The recent growth in the consumption of online media by children during early childhood necessitates data-driven tools enabling educators to filter out appropriate educational content for young learners. This paper presents an approach for detecting educational content in online videos. We focus on two widely used educational content classes: literacy and math. For each class, we choose prominent codes (sub-classes) based on the Common Core Standards. For example, literacy codes include `letter names', `letter sounds', and math codes include `counting', `sorting'. We pose this as a fine-grained multilabel classification problem as videos can contain multiple types of educational content and the content classes can get visually similar (e.g., `letter names' vs `letter sounds'). We propose a novel class prototypes based supervised contrastive learning approach that can handle fine-grained samples associated with multiple labels. We learn a class prototype for each class and a loss function is employed to minimize the distances between a class prototype and the samples from the class. Similarly, distances between a class prototype and the samples from other classes are maximized. As the alignment between visual and audio cues are crucial for effective comprehension, we consider a multimodal transformer network to capture the interaction between visual and audio cues in videos while learning the embedding for videos. For evaluation, we present a dataset, APPROVE, employing educational videos from YouTube labeled with fine-grained education classes by education researchers. APPROVE consists of 193 hours of expert-annotated videos with 19 classes. The proposed approach outperforms strong baselines on APPROVE and other benchmarks such as Youtube-8M, and COIN. The dataset is available at https://github.com/rohit-gupta/MMContrast/tree/main/APPROVE

#arxiv #academic-paper #cs.CV +1
Algebraic billiards in the Fermat hyperbola
academic
Weinreich

Algebraic billiards in the Fermat hyperbola

We prove two results on the algebraic dynamics of billiards in generic algebraic curves of degree $d \geq 2$. First, the dynamical degree grows quadratically in $d$; second, the set of complex periodic points has measure 0, implying the Ivrii Conjecture for the classical billiard map in generic algebraic domains. To prove these results, we specialize to a new billiard table, the Fermat hyperbola, on which the indeterminacy points satisfy an exceptionality property. Over $\mathbb{C}$, we construct an algebraically stable model for this billiard via an iterated blowup. Over more general fields, we prove essential stability, i.e. algebraic stability for a particular big and nef divisor.

#arxiv #academic-paper #math.DS +1
Non-commutative linear logic fragments with sub-context-free complexity
academic
Nishimiya, Taniguchi

Non-commutative linear logic fragments with sub-context-free complexity

We present new descriptive complexity characterisations of classes REG (regular languages), LCFL (linear context-free languages) and CFL (context-free languages) as restrictions on inference rules, size of formulae and permitted connectives in the Lambek calculus; fragments of the intuitionistic non-commutative linear logic with direction-sensitive implication connectives. Our identification of the Lambek calculus fragments with proof complexity REG and LCFL is the first result of its kind. We further show the CFL complexity of one of the strictly `weakest' possible variants of the logic, admitting only a single inference rule. The proof thereof, moreover, is based on a direct translation between type-logical and formal grammar and structural induction on provable sequents; a simpler and more intuitive method than those employed in prior works. We thereby establish a clear conceptual utility of the Cut-elimination theorem for comparing formal grammar and sequent calculus, and identify the exact analogue of the Greibach Normal Form in Lambek grammar. We believe the result presented herein constitutes a first step toward a more extensive and richer characterisation of the interaction between computation and logic, as well as a finer-grained complexity separation of various sequent calculi.

#arxiv #academic-paper #cs.LO +4
SSPO: Subsentence-level Policy Optimization
academic
Yang, chen, Wang et al.

SSPO: Subsentence-level Policy Optimization

As a significant part of post-training of the Large Language Models (LLMs), Reinforcement Learning from Verifiable Reward (RLVR) has greatly improved LLMs' reasoning skills. However, some RLVR algorithms, such as GRPO (Group Relative Policy Optimization) and GSPO (Group Sequence Policy Optimization), are observed to suffer from unstable policy updates and low usage of sampling data, respectively. The importance ratio of GRPO is calculated at the token level, which focuses more on optimizing a single token. This will be easily affected by outliers, leading to model training collapse. GSPO proposed the calculation of the response level importance ratio, which solves the problem of high variance and training noise accumulation in the calculation of the GRPO importance ratio. However, since all the response tokens share a common importance ratio, extreme values can easily raise or lower the overall mean, leading to the entire response being mistakenly discarded, resulting in a decrease in the utilization of sampled data. This paper introduces SSPO, which applies sentence-level importance ratio, taking the balance between GRPO and GSPO. SSPO not only avoids training collapse and high variance, but also prevents the whole response tokens from being abandoned by the clipping mechanism. Furthermore, we apply sentence entropy to PPO-CLIP to steadily adjust the clipping bounds, encouraging high-entropy tokens to explore and narrow the clipping range of low-entropy tokens. In particular, SSPO achieves an average score of 46.57 across five datasets, surpassing GRPO (43.01) and GSPO (44.42), and wins state-of-the-art performance on three datasets. These results highlight SSPO's effectiveness in leveraging generated data by taking the essence of GSPO but rejecting its shortcomings.

#arxiv #academic-paper #cs.CL +1
Variation of the disk thickness across ice bands: A method to determine ice abundances in highly inclined protoplanetary disks
academic
Martinien, Duchêne, Ménard et al.

Variation of the disk thickness across ice bands: A method to determine ice abundances in highly inclined protoplanetary disks

The James Webb Space Telescope provides unprecedented information to study ices in protoplanetary disks. However, the saturation of ice bands in highly inclined disks hinders the measurement of ice abundances using classical spectroscopy. This is unfortunate as the presence and more importantly abundance of ices plays a key role in, e.g., the evolution of dust (because it modifies the sticking properties) and the composition of planetesimals and exoplanetary atmospheres. To overcome this issue and quantify the ice abundance within disks, we introduce a new method based on measuring the changes in the apparent disk thickness as a function of wavelength, which is directly and quantitatively related to the grain opacity. Specifically, we expect i) that the increased opacity within ice bands should result in a thicker disk than in the adjacent continuum, and ii) the thickness variations to be proportional to the abundance of ice. We extracted the disk thickness in model images of edge-on disks containing different abundances of water ice, as well as in James Webb Space Telescope spectral imaging of four edge-on disks. For both models and observations, the disk thickness decreases toward longer wavelengths except across the positions of ice absorption features where the thickness is enhanced across the band. In the model images, we demonstrate that this effect increases with ice abundance without any hint of saturation. This definitely demonstrates the presence of the ice species within each disk and confirms our expectation that this method can be applied to estimate ice abundances. Thanks to this method, it will thus be possible to constrain the ice abundance in highly inclined disks with disks model fitting. Unlike spectroscopic analysis, this method is not subject to saturation and should therefore be more robust and applicable to all disks for which the two surfaces can be resolved.

#arxiv #academic-paper #astro-ph.SR +2
Extensions of Operator-Valued Kernels on $\mathbb{F}^{+}_{d}$
academic
Tian

Extensions of Operator-Valued Kernels on $\mathbb{F}^{+}_{d}$

We study the problem of extending a positive-definite operator-valued kernel, defined on words of a fixed finite length from a free semigroup, to a global kernel defined on all words. We show that if the initial kernel satisfies a natural one-step dominance inequality on its interior, a global extension that preserves this interior data and the dominance property is always possible. This extension is constructed explicitly via a Cuntz-Toeplitz model. For the problem of matching the kernel on the boundary, we introduce an intrinsic shift-consistency condition. We prove this condition is sufficient to guarantee the existence of a global extension that agrees with the original kernel on its entire domain.

#arxiv #academic-paper #math.FA +1
Optimised neural networks for online processing of ATLAS calorimeter data on FPGAs
academic
Aad, Bertrand, Laatu et al.

Optimised neural networks for online processing of ATLAS calorimeter data on FPGAs

A study of neural network architectures for the reconstruction of the energy deposited in the cells of the ATLAS liquid-argon calorimeters under high pile-up conditions expected at the HL-LHC is presented. These networks are designed to run on the FPGA-based readout hardware of the calorimeters under strict size and latency constraints. Several architectures, including Dense, Recurrent (RNN), and Convolutional (CNN) neural networks, are optimised using a Bayesian procedure that balances energy resolution against network size. The optimised Dense, CNN, and combined Dense+RNN architectures achieve a transverse energy resolution of approximately 80 MeV, outperforming both the optimal filtering (OF) method currently in use and RNNs of similar complexity. A detailed comparison across the full dynamic range shows that Dense, CNN, and Dense+RNN accurately reproduce the energy scale, while OF and RNNs underestimate the energy. Deep Evidential Regression is implemented within the Dense architecture to address the need for reliable per-event energy uncertainties. This approach provides predictive uncertainty estimates with minimal increase in network size. The predicted uncertainty is found to be consistent, on average, with the difference between the true deposited energy and the predicted energy.

#arxiv #academic-paper #physics.ins-det +1
Investigating Solid-Fluid Phase Coexistence in DC Plasma Bilayer Crystals: The Role of Particle Pairing and Mode Coupling
academic
Mangamuri, Jaiswal, Couëdel

Investigating Solid-Fluid Phase Coexistence in DC Plasma Bilayer Crystals: The Role of Particle Pairing and Mode Coupling

This article presents a detailed investigation of solid-fluid phase coexistence in a bilayer dusty plasma crystal subjected to varying confinement ring bias voltages in a DC glow discharge argon plasma. Melamine formaldehyde particles were employed to form a stable, hexagonally ordered bilayer crystal within a confinement ring electrically isolated from the grounded cathode. By systematically adjusting the confinement ring bias, a distinct phase coexistence emerged: it is characterized by a fluid-like melted core surrounded by a solid crystalline periphery. Crucially, analysis of the phonon spectra revealed frequency shifts that deviate significantly from the predictions of classical monolayer Mode-Coupling Instability (MCI) theory. Stability analysis further demonstrated that dynamic interlayer particle pairing and non-reciprocal interactions play a pivotal role in destabilizing the bilayer structure. These findings highlight previously underappreciated mechanisms driving the melting transition in bilayer dusty plasmas, offering a more comprehensive understanding of phase behavior in complex plasma systems. The results underscore the importance of interlayer coupling and confinement effects in tuning structural transitions.

#arxiv #academic-paper #physics.plasm-ph +1
Harmonic Analysis associated with a discrete Laplacian
academic
Ciaurri, Gillespie, Roncal et al.

Harmonic Analysis associated with a discrete Laplacian

It is well-known that the fundamental solution of $$ u_t(n,t)= u(n+1,t)-2u(n,t)+u(n-1,t), \quad n\in\mathbb{Z}, $$ with $u(n,0) =δ_{nm}$ for every fixed $m \in\mathbb{Z}$, is given by $u(n,t) = e^{-2t}I_{n-m}(2t)$, where $I_k(t)$ is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series $$ W_tf(n) = \sum_{m\in\mathbb{Z}} e^{-2t} I_{n-m}(2t) f(m). $$ By using semigroup theory, this formula allows us to analyze some operators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted $\ell^p(\mathbb{Z})$-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essentially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.

#arxiv #academic-paper #math.CA +1
On quantifying the spin angular momentum density of light
academic
Zheng, Palffy-Muhoray

On quantifying the spin angular momentum density of light

In addition to energy, light carries linear and angular momentum. These are key quantities in rapidly developing optics research and in technologies focusing on light induced forces and torques on materials. Spin angular momentum (SAM) density is of particular interest, since unlike orbital angular momentum, it is uncoupled from linear momentum. The SAM density of light was first estimated in 1909 by Poynting, using a mechanical analogy. Exact expressions, based on results from quantum mechanics and field theory were subsequently developed, and are in common use today. In this paper, we show that the SAM density of light can be obtained directly from the Coulomb force and Maxwell's equations, without reliance on quantum mechanics or field theories; it could have been calculated by Maxwell and his contemporaries. Besides its historical significance, the simple derivation of our result makes it readily accessible to non-experts in the field.

#arxiv #academic-paper #physics.optics +1
Transitions of bifurcation diagrams of a forced heteroclinic cycle
academic
Labouriau, Rodrigues

Transitions of bifurcation diagrams of a forced heteroclinic cycle

A family of periodic perturbations of an attracting robust heteroclinic cycle defined on the two-sphere is studied by reducing the analysis to that of a one-parameter family of maps on a circle. The set of zeros of the family forms a bifurcation diagram on the cylinder. The different bifurcation diagrams and the transitions between them are obtained as the strength of attraction of the cycle and the amplitude of the periodic perturbation vary. We determine a threshold in the cycle's attraction strength above which frequency locked periodic solutions with arbitrarily long periods bifurcate from the cycle as the period of the perturbation decreases. Below this threshold further transitions are found giving rise to a frequency locked invariant torus and to a frequency locked suspended horseshoe, arising from heteroclinic tangencies in the family of maps.

#arxiv #academic-paper #math.DS +1
Some arithmetic aspects of ortho-integral surfaces
academic
Doan, Le

Some arithmetic aspects of ortho-integral surfaces

We investigate ortho-integral (OI) hyperbolic surfaces with totally geodesic boundaries, defined by the property that every orthogeodesic (i.e. a geodesic arc meeting the boundary perpendicularly at both endpoints) has an integer cosh-length. We prove that while only finitely many OI surfaces exist for any fixed topology, infinitely many commensurability classes arise as the topology varies. Moreover, we completely classify OI pants and OI one-holed tori, and show that their doubles are arithmetic surfaces of genus 2 derived from quaternion algebras over $\mathbb{Q}$.

#arxiv #academic-paper #math.GT +2
KLAP: KYP lemma based low-rank approximation for $\mathcal{H}_2$-optimal passivation
academic
Nicodemus, Voigt, Gugercin et al.

KLAP: KYP lemma based low-rank approximation for $\mathcal{H}_2$-optimal passivation

We present a novel passivity enforcement (passivation) method, called KLAP, for linear time-invariant systems based on the Kalman-Yakubovich-Popov (KYP) lemma and the closely related Lur'e equations. The passivation problem in our framework corresponds to finding a perturbation to a given non-passive system that renders the system passive while minimizing the $\mathcal{H}_2$ or frequency-weighted $\mathcal{H}_2$ distance between the original non-passive and the resulting passive system. We show that this problem can be formulated as an unconstrained optimization problem whose objective function can be differentiated efficiently even in large-scale settings. We show that any minimizer of the unconstrained problem yields the same passive system. Furthermore, we prove that, in the absence of a feedthrough term, every local minimizer is also a global minimizer. For cases involving a non-trivial feedthrough term, we analyze global minimizers in relation to the extremal solutions of the Lur'e equations, which can serve as tools for identifying local minima. To solve the resulting numerical optimization problem efficiently, we propose an initialization strategy based on modifying the feedthrough term and a restart strategy when it is likely that the optimization has converged to a non-global local minimum. Numerical examples illustrate the effectiveness of the proposed method.

#arxiv #academic-paper #math.OC +1
Hardy spaces for the Lamé equation
academic
Barceló, Peréz-Esteva, Marmolejo-Olea et al.

Hardy spaces for the Lamé equation

We study, for $1 \leq p \leq \infty$, the Hardy space $\bm{h}_e^p(\B)$, the elastic analogue of the classical Hardy spaces of harmonic functions in the unit ball of $\mathbb{R}^3$. The space consists of vector-field solutions of the Lamé system satisfying the standard integrability condition on concentric spheres centered at the origin. Using the elastic Poisson kernel, we establish a Fatou-type theorem and show that $\bm{h}_e^p(\B)$ is isomorphic to the $\mathbb{R}^3$-valued Lebesgue space $L^p$ on the unit sphere for $1 < p \leq \infty$, while $\bm{h}_e^1(\B)$ corresponds to the space of $\mathbb{R}^3$-valued Borel measures on the unit sphere. For $1 < p < \infty$, we prove that $\bm{h}_e^p(\B)$ decomposes as the direct sum of three subspaces. The main contribution of this paper is to describe each of these subspaces along with the corresponding spaces of boundary values. In particular, two of these spaces consist of solutions of the Lamé equation for all eligible choices of the Lamé constants: one of them is the space of Riesz fields (solutions of the generalized Cauchy--Riemann equations) in $\bm{h}_e^p(\B)$; the second is the space of fields given by the cross product of $x$ with such Riesz fields. The results rely on the classical decomposition of $L^2$ vector fields on the sphere into the direct sum of three spaces of vector spherical harmonics, which we extend to $L^p$.

#arxiv #academic-paper #math.FA +1
Stable degeneration of families of klt singularities with constant local volume
academic
Chen

Stable degeneration of families of klt singularities with constant local volume

We prove that for a locally stable family of klt singularities with constant local volume, the ideal sequences of the minimizing valuations for the normalized volume function form a family of ideals with flat cosupport, which induces a degeneration to a locally stable family of K-semistable log Fano cone singularities. Our proof is a family version of the method of C. Xu and Z. Zhuang proving finite generation by Kollár models and multiple degenerations.

#arxiv #academic-paper #math.AG +1
Equivalence criteria for the two--term functional equations for Herglotz--Zagier functions
academic
Sathyanarayana, Sharan

Equivalence criteria for the two--term functional equations for Herglotz--Zagier functions

We establish Kronecker limit type formula for the generalized Mordell-Tornheim zeta function $Θ(r,r,t,x)$ as a function of the third argument around $t=1-r$. We then show that the above Kronecker limit type formula is equivalent to the two-term functional equation for the higher Herglotz function obtained by Vlasenko and Zagier. We also show the equivalence between a previously known Kronecker limit type formula for $Θ(1,1,t,x)$ around $t=0$ and the two-term functional equation for the Herglotz-Zagier function obtained by Zagier. Using the theory of the Mordell-Tornheim zeta function, we obtain results of Ramanujan, Guinand, Zagier, and Vlasenko-Zagier as consequences, to further show that the Mordell-Tornheim zeta function lies centrally between many modular relations in the literature, thus providing the means to view them under one umbrella.

#arxiv #academic-paper #math.NT +1
Maximal orders optimal embedding of central simple algebras over number fields
academic
Yang

Maximal orders optimal embedding of central simple algebras over number fields

Given a number field $F$ and $R$ be the ring of integers of $F$, the problem of embedding a field extension $K/F$ into a central simple algebra $B$ is classical. This paper proves that when the central simple algebra has degree $p$, the $R$-order $S\subset K$ can be optimal embedded into all maximal $R$-orders $O\subset B$, unless satisfies the optimal selectivity condition.

#arxiv #academic-paper #math.NT +2
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