We address the question of the large-time behavior of solutions to reaction-diffusion equations in periodic media. We start with the description of the asymptotic shape of the invasion set, which is characterized by the Freidlin-Gärtner formula. We outline a proof of the formula that holds true for general types of reaction terms. We then present some recent results, obtained in collaboration with H. Guo and F. Hamel, for (weakly) bistable equations. They include a regular version of the Freidlin-Gärtner formula and the convergence in profile towards pulsating traveling fronts for solutions with either bounded or unbounded initial support.

{\it Jacob's ladder} -- a classic children's toy -- is a simple mechanical frame comprising rigid blocks connected by strings that shows curious unidirectional flipping waves. Nonetheless, its physical origin remains elusive. By combining experiment, numeral simulation, and theory, we show that understanding the underlying design principle of this toy requires diverse physical ideas. First, we conduct a water-tank experiment that excludes the domino-like mechanism, thus defying widespread expectations. Subsequently, we analytically demonstrate that the toy is bistable under gravity, thus implying its kink wave as a class of topological solitons. The waves are surprisingly reminiscent -- both experimentally and theoretically -- to those in the Kane--Lubensky topological chain, owing to the stiffening of zero modes by the pretension under gravity. However, a close examination based on the index theorem reveals that the similarity remains superficial and that the floppiness of the toy underlies the kink and antikink coexistence -- a forbidden mode in the topological chain. By analyzing a generalized asymmetric toy, we reveal that its symmetric connection renders it topologically singular, thus resulting in amusing motions. We demonstrate these ideas by experimentally observing a dramatic pair annihilation of kink and antikink waves.

We consider a class of nonlinear parabolic equations \[ \dfrac{\partial}{\partial t} b(u)-\nabla \cdot (A(x,t,u,\nabla u))+H(x,t,\nabla u)=f , \] where $H$ is a nonlinear lower order term satisfied the Carath$\acute{e}$odory condition and \[ \left\lvert H(x,t,\nabla u)\right\rvert\leqslant g(x,t)\left\lvert \nabla u\right\rvert^{δ(x)} \] with \[ δ(x)=\frac{p(x)(N+1)-N}{(N+2)(p(x)-1)}(p^--1) \quad \text{and} \quad p^-=\underset{x\in\barΩ}{min}\,p(x). \] By virtue of truncation metheod,the monotone operator theory and a gradient estimate we prove existence of renormalized solutions without coercivity condition on lower order term in the framework of variable exponents.

Optoelectronic and nonlinear transport experiments probe the quantum geometric tensor of Bloch states, whose real and imaginary components -- the quantum metric and the Berry curvature -- are typically constrained by symmetry. Here, we show that geometric bending provides a route to engineer such responses in centrosymmetric ferromagnets. Curvature-induced strain gradients across the shell thickness break inversion symmetry and activate an orbital Rashba coupling. In the presence of in-plane magnetization and spin-orbit coupling, this generates spin textures with a nontrivial quantum geometry, leading to an intrinsic nonlinear anomalous Hall effect (NAHE) governed by the quantum metric and maximized when the magnetization aligns with the applied electric field. When geometric deformations further break twofold rotational symmetry around the out-of-plane axis, an additional NAHE emerges, maximal for magnetization perpendicular to the driving electric field and governed by the Berry curvature dipole, thus giving access to the imaginary component of the quantum geometric tensor. These results establish curved ferromagnetic shells as a platform for engineering anisotropic nonlinear transport and for selectively probing both components of the quantum geometric tensor.

I briefly discuss the search for low mass scalars at Higgs factories as well as available models that render such scalars feasible, where I focus on new developments since the review presented in arXiv:2205.09687 (see also arXiv:2504.11969 for a recent update).

Body-fitted arbitrary Lagrangian-Eulerian (ALE) methods provide a sharp representation of the fluid-structure interface but rely on mesh-update strategies that incrementally deform a reference configuration. To address this issue, we reformulate the ALE mesh-motion problem in the isogeometric setting as a sequence of independent domain parameterization problems. At each time step, a multi-patch spline parameterization of the fluid domain is constructed from the current interface geometry. Three technical components realize this framework: (i) a barrier-function-based spline parameterization that enforces a strictly positive Jacobian at every time step; (ii) a tangential-slip reparameterization that handles unbounded cumulative rotations of closed domains, where no fixed boundary-to-parameter correspondence is admissible; and (iii) a constant-preserving quasi-interpolation operator for solution transfer between consecutive parameterizations, ensuring that the discrete geometric conservation law holds algebraically. We validate the method on three two-dimensional FSI benchmarks, covering standard and large-rotation regimes, and on a three-dimensional rotor problem. On a rotating-square benchmark, the tangential-slip strategy enables simulations under sustained rotation far beyond the range accessible to classical mesh-update schemes--a regime that is fundamentally inaccessible to any mesh-deformation formulation, not merely numerically difficult. A three-dimensional rotor example further demonstrates that the framework extends naturally to volumetric spline parameterizations. Finally, we show that the per-step spline parameterizations can be used directly within a standard finite element solver.

Let $ n \in \mathbb{N} $ with $ n \geq 3 $, and let $\mathcal{G} = \{G_i:i\in [n]\} $ be a family of $ n $-vertex graphs on a common vertex set $V$, where the graphs in the family do not need to be distinct. A graph $H$ with vertex set $V$ is \emph{rainbow} in $\mathcal{G}$ if there exists an injection $ ϕ: E(H) \to [n] $ such that $e \in E(G_{ϕ(e)})$ for every edge $e \in E(H)$, where $|E(H)|\leq n$. In 2020, Joos and Kim proved that $\mathcal{G}$ contains a rainbow Hamiltonian cycle under the Dirac-type condition. Recently, Liu, Chen, and Ma generalized this result by replacing the Dirac-type condition with a more general Ore-type condition involving degree sums of non-adjacent vertices: If $σ(\mathcal{G}) \geq n$, then $\mathcal{G}$ contains a rainbow Hamiltonian cycle, where the Ore-type condition $σ(\mathcal{G})$ is defined as follows: $ σ(\mathcal{G}) = \min\{d_p(u) + d_q(v) \mid uv \notin E(G_i) \text{ for some } i \in [n] \text{ and for all } p, q \in [n]\}. $ In this paper, under the Ore-type condition, we show that either each vertex of $V$ is contained in a rainbow cycle of length $\ell$ for every $\ell\in[4,n]$, or $G_1=\cdots=G_n=K_{\frac{n}{2},\frac{n}{2}}$. As a corollary, we deduce the rainbow pancyclicity of $\mathcal{G}$, which supports the famous meta-conjecture posed by Bondy. Furthermore, we prove rainbow vertex-pancyclicity of $\mathcal{G}$ under the Ore-type condition and provide an extremal graph family to show that the result is sharp.

News consumption behavior is shaped by the coupling between temporal dynamics and content selection. This study proposes a multi-scale temporal-content framework and validates it on two large real-world news datasets, MIND and Adressa. Results reveal hierarchical temporal patterns. At the macroscale, Fourier modeling identifies clear circadian rhythms; at the mesoscale, session intervals follow a power-law distribution with $α\approx 1$; and at the microscale, within-session action counts and inter-action intervals follow exponential distributions with $λ\approx 0.3$ and $λ\approx 0.02$, respectively. Content analysis shows that clicks are mainly driven by historical interests, while this dependence weakens as content diversity increases. Temporal-content coupling further indicates that users' historical interests dominate active time periods in shaping behavior. Preference groups also differ: timeliness and entertainment-oriented users click more frequently and rely more on historical interests, whereas diversified users click less and are more sensitive to content diversity.

A self-contained introduction to infinite dimensional representations over a tame hereditary algebra is provided, assuming a basic knowledge of the category of finite dimensional representations. This includes a complete description of all pure-injective modules. Of particular interest are the torsionfree divisible modules, which are precisely the direct sums of copies of the unique generic module.

We define a torus $U \subset T = (\mathbb{C}^\times)^K$ which acts on the $Δ$-Springer varieties $Y_{n,λ,s}$ defined by Griffin-Levinson-Woo and give a Borel-style presentation for the equivariant cohomology ring $H^*_U(Y_{n,λ,s})$. Our presentation arises from the orbit harmonics deformation technique, and uses methods of Chou-Matsumura-Rhoades and Abe-Horiguchi.

We give a unified proof of property $R_\infty$ for the Higman groups $H_n$ ($n\ge 4$) and for their generalizations studied by Martin and Horbez--Huang. As a key step, we prove that the automorphism groups of these groups are acylindrically hyperbolic. As a byproduct, we obtain acylindrical hyperbolicity of the groups themselves. In addition, we give an independent proof, based on Delzant's lemma, of the criterion of Fournier-Facio and collaborators stating that if $\operatorname{Aut}(G)$ is acylindrically hyperbolic and $\operatorname{Inn}(G)$ is infinite, then $G$ has property $R_\infty$.

We prove that the Lorentz--FitzGerald contraction is the unique deformation of a resonant cavity moving through a mechanical wave medium that preserves spherical-harmonic phase closure. For a cavity moving at speed $v = βc$ through a medium supporting nondispersive wave propagation at speed $c$, the round-trip phase of an internal ray at angle $θ$ to the motion depends on the boundary radius $r(θ)$ according to $Φ(θ) = 2k\,r(θ)\sqrt{1-β^2\sin^2θ}/(1-β^2)$. Requiring $Φ(θ)$ to be independent of $θ$ -- the necessary condition for retaining a spherical-harmonic eigenstructure -- uniquely fixes the Lorentzian aspect ratio \[ \frac{a_\parallel}{a_\perp} = \frac{1}γ = \sqrt{1-β^2}. \] Substituting this unique boundary into the round-trip time yields the resonant period dilation $T = γT_0$, without additional assumptions. Both results -- contraction and dilation -- follow from a single mechanical constraint: preservation of eigenstructure under motion. This is the missing uniqueness theorem of the constructive relativity program initiated by FitzGerald, Lorentz, and Heaviside: the proof that Lorentzian kinematics are not merely consistent with, but uniquely required by, phase closure in a mechanical wave medium.

Piezoelectricity is a technologically important property of certain insulators in which mechanical strain induces an electrical polarization. However, the rate at which a piezoelectric response can be established over a macroscopic volume is limited by the sound velocity, constraining applications in high-bit-rate transduction and sensing. Furthermore, the strength of the piezoelectric effect is not readily tunable, as it depends on intrinsic anharmonic coupling between strain and intra-unit-cell distortions in a given material. Lastly, the maximum amplitude of the effect is bounded by material fracture, which sets in already at percent level strain values. Here we overcome these limitations by realizing a strain-free, piezoelectric-like response driven solely by photo-excited optical phonon distortions. We demonstrate such optical phonoelectricity in the weak piezoelectric BPO$_4$, in which we induce electrical polarization through phonon rectification. This effect is established over macroscopic volumes with four orders of magnitude higher speed than piezoelectric responses, ultimately limited by the speed of light. The maximum induced polarization is estimated to be far in excess of that attainable through strain at the fracture limit. Ultrafast phonoelectricity opens up new opportunities for optical control in quantum materials, but also for device applications.

In this paper we introduce the energy efficiency as a new metric for evaluating both hardware platforms based on Graphic Processor Units (GPU), and algorithm optimisations at High Energy Physics (HEP) experiments. We develop a method to compute the energy efficiency for the case of the first high level trigger (HLT1) of the LHCb experiment, relating the throughput with GPU specifications such as the number of cores, clock frequency, memory bandwidth and thermal design power. The model can be extended to other HEP experiments to make decisions and reach sustainable computing ecosystems.

This paper provides a diffeomorphism classification of smooth manifolds homeomorphic to the complex projective space $\mathbb{C}P^m$ for $m \in \{5, 6, 7, 8\}$. The classification is obtained by computing the group of concordance classes of smooth structures on $\mathbb{C}P^m$ and determining the orbit space under the action induced by the group of self-homeomorphisms. Using these computations in conjunction with the tangential surgery exact sequence and techniques from stable homotopy theory, we determine the diffeomorphism classes of smooth manifolds within the tangential homotopy type of $\mathbb{C}P^m$ for $4 \le m \le 8$. We also investigate the relationship between these two classification problems by studying the natural map from the homeomorphism type to the tangential homotopy type. As a consequence, we prove that for $m = 4$, there exists a unique smooth manifold, up to diffeomorphism, that is tangentially homotopy equivalent to $\mathbb{C}P^4$ but not homeomorphic to it. Furthermore, for $m = 8$, there exist exactly two pairwise non-diffeomorphic smooth manifolds that are tangentially homotopy equivalent to $\mathbb{C}P^8$ but not homeomorphic to it.

The tragedy of the commons has traditionally been framed as a problem of resource overuse driven by self-interested exploitation. In contrast, growing empirical evidence shows that insufficient use or abandonment of natural resources, known as underuse, can also lead to ecological degradation and loss of ecosystem services. Despite its relevance, underuse has rarely been examined within evolutionary theories of resource use. Here, we develop a simple eco-evolutionary model that integrates both provisioning and non-provisioning ecosystem services to analyze the evolution of resource-use strategies. Using adaptive dynamics, we investigate how individual resource use evolves while altering resource abundance. The model shows that overuse and underuse arise naturally as alternative evolutionary outcomes of the same underlying process, alongside intermediate use and evolutionary branching. We derive analytical conditions for the existence, number, and stability of evolutionarily singular strategies, and show that the qualitative evolutionary fate is primarily determined by the shape of provisioning benefits. Only when provisioning benefits increase in a concave manner does evolutionary dynamics converge to a unique intermediate strategy that is continuously stable. In contrast, convex increasing benefits generate a broader range of outcomes: overuse, underuse, bi-stability, and evolutionary branching. By explicitly comparing the continuously stable strategy with the socially optimal strategy, we further quantify how their deviations depend on the valuation of non-provisioning services. Our results provide a theoretical framework for viewing the common-pool resource dilemmas as intrinsically two-sided evolutionary problems, and offer a baseline for future studies exploring interventions to address overuse and underuse simultaneously.

Motivated by an optimal-matching problem (Leighton-Shor) and the random-field Ising model (Aizenman-Wehr, Ding-Wirth), we consider a variational problem for graphs in $1+1$ dimension maximizing an action that is the difference of a field term given by integrating white noise over the subgraph on the one hand, and the Dirichlet integral of the (continuum) height function $h=h(x)$ on the other hand. This problem is scale-invariant in law, and requires a small-scale cut-off which we implement by restricting to $h$ that are piecewise linear on intervals of size $1$ and vanish at $x=0,L$. We show that with overwhelming probability, the maximal action $A$ satisfies $A=a\ln L+O(1)$ for a deterministic constant $a\in(0,\infty)$. This can be considered as a homogenization result that is quantitative in an optimal way. The present result sharpens a recent qualitative homogenization result by the authors with C. Wagner; it does so by finding bounds for the action that are locally uniform in the boundary conditions. Like the earlier result, the present one relies on pointwise bounds on the optimizer, as provided by Dembin-Elboim-Hadas-Peled in a more general setting. In the earlier work, the small-scale cut-off also involved an explicit discretization of the field term, yielding a Brownian potential that is i.i.d. in $x\in\mathbb{Z}$; this had the benefit of allowing for a comparison argument, but is inconvenient for the coarse-graining used here.

This paper presents lpviz, a browser-based visualization tool for linear programming. lpviz is deeply interactive, offering an intuitive interface where users can directly draw and edit the feasible region and objective vector, without requiring cumbersome manipulation of raw numerical coefficients. lpviz lets users compare the behavior of several classes of linear programming algorithms, namely Simplex, Interior-Point, Primal-Dual Hybrid Gradient, and Central Path. In the 3D mode, lpviz places iterates at heights corresponding to important solver metadata such as complementarity gap or KKT residual, helping users gain further insight into algorithm behavior beyond the primal iterates alone. lpviz has been used in both research and classroom settings, to help develop intuition for the strengths and weaknesses of different solvers and the impact of solver settings on convergence behavior. lpviz is open-source, permissively licensed, and freely available on any device with a web browser at https://lpviz.net .

Digital journaling creates an authenticity gap: users consciously translate raw emotions into text, often sanitizing narratives even in private writing. We formalize this as Cross-Modal Affective Dissonance Detection (CADD), a directional three-way classification distinguishing Masking (positive text, negative acoustics), Coping (negative text, positive acoustics), and Congruent utterances, grounded in Gross's process model of emotion regulation. We present three further contributions: (i) CADD-Journal, a 1,800-sample TTS dataset with a shared-sentence-pool design that provably isolates acoustic signal from textual content; (ii) DACM, a dual-encoder model with asymmetric cross-modal attention that re-solves a gradient degeneracy in pooled fusion, achieving macro-F1 0.711 - with a four-step ablation demonstrating that asymmetric attention is the dominant driver (+ 0.242) while the DIM is effective only on cross-modal features (+0.033); and (iii) a domain gap quantification: zero-shot evaluation across three naturalistic corpora reveals a substantial gap between TTS-trained models and real speech, and we identify two concrete requirements for future in-the-wild corpus construction. ReflectJournal, a proof-of-concept iOS application, operationalizes the framework and provides a deployment platform for naturalistic data collection.

In this paper we study when the origin $(0,0)$ is a regular (or irregular) boundary point for the so-called soda can domains of the type \[ Θ_{l,θ}:= \{(x,t) \in \mathbf{R}^{n+1}: 0<-t < θ|x|^l <θ\}, \quad \text{with $l,θ>0$,} \] for the $p$-parabolic equation $\partial_t u- Δ_p u=0$, where $1<p<\infty$. For $p<2n/(n+1)$ and for the heat equation (i.e.\ $p=2$) we completely determine when the origin is regular for soda can domains. The domains $Θ_{l,θ}$ have nonconvex time sections with power dependence on time. For domains with rotationally symmetric convex time sections with power dependence on time, the regularity of the origin as the last point was characterized by Petrovskii (in 1935) for the heat equation, and almost completely in the nonlinear case ($p \ne 2$) in our earlier paper (joint with Gianazza, Math. Ann. 368 (2017), 885--904).

Let $P\subset\mathbb R^n$ be a convex polytope and let $\ell$ be a linear functional which is nonconstant on every edge of $P$. The induced acyclic orientation determines positive and negative Białynicki-Birula type partitions of $P$ into unions of relative interiors of faces. Our first result establishes a duality: the positive partition is a stratification if and only if the negative one is a stratification. Our second result connects poset invariants with monotone path polytopes. Assuming the induced vertex relation admits the structure of a graded poset, we prove that the Chow polynomial of the resulting vertex poset agrees with the $h$-polynomial of a (dual) monotone path polytope.

We define the topological complexity sequence of a group as the sequence of topological complexities of its Milnor constructions. This sequence may be regarded as an intrinsic refinement of the topological complexity of a group and, unlike topological complexity itself, is meaningful for groups of infinite cohomological dimension. We show that the topological complexity sequence of every group of infinite cohomological dimension is weakly increasing and unbounded. We then estimate its growth and determine its asymptotic behavior for a finite group of even order.

This paper develops the numerical inverse scattering transform (NIST) framework for the coupled modified Korteweg-de Vries (mKdV) equation based on its associated Riemann-Hilbert problem. The coupled system gives rise to a $3\times3$ matrix-valued Riemann-Hilbert problem, whose jump matrix and scattering data have a more involved structure than in the scalar case. This matrix setting makes the extension of NIST to the coupled system nontrivial, both in the direct scattering computation and in the numerical solution of the inverse problem. Within this framework, the scattering data are first computed by solving the matrix direct scattering problem using a Chebyshev collocation method with suitable mappings. The Deift-Zhou nonlinear steepest descent method is then used to analyze and deform the oscillatory Riemann-Hilbert problem. In particular, the phase function admits two stationary points symmetric about the origin, and the analysis leads to a division of the $(x,t)$-plane into three regions with corresponding contour deformations. Compared with traditional numerical methods, the NIST computes the solution directly at prescribed spatial and temporal points without relying on time-stepping. Numerical experiments illustrate the performance of the proposed NIST in long-time simulations and indicate that it captures the main asymptotic features of the coupled mKdV solutions.

Based on a discontinuous Galerkin method in the spatial directions and an improved implicit-explicit pressure-correction scheme in the temporal direction, this paper discusses a fully discrete scheme for the Poisson-Nernst-Planck-Navier-Stokes equations. Optimal error estimates are derived in $L^2$ and in the energy norms for the concentrations of positive and negative ions, the electrostatic potential, the fluid velocity, and the $L^2$ norm of the fluid pressure. The discrete mass conservation properties of both ions are established. Finally, numerical simulations are performed, whose results confirm our theoretical findings.

We present a model-independent, photometry-only framework for identifying strongly lensed supernovae when multiple images are unresolved and blended into a single point source. Building on the simulation-based methodology of Bag et al. (2021), we apply this approach to real Zwicky Transient Facility (ZTF) data using a validation sample of spectroscopically confirmed Type Ia supernovae. The method models the observed flux as a superposition of two time-shifted components, and Bayesian inference is used to estimate the relative scaling and time delay. Applying this framework to 445 well-converged supernovae, we find that only a single object satisfies the selection criteria when adopting a conservative threshold of $Δt \ge 12$ days, corresponding to a false positive fraction of $1/445 \approx 0.22\%$. A laxer threshold of $Δt \ge 10$ days yields fourteen objects, for a false positive fraction of $3.15\%$. The method provides a scalable and model-independent first-stage filter for identifying lens-like candidates in large time-domain surveys such as the Rubin Observatory's Legacy Survey of Space and Time (LSST).

We demonstrate a magnetic quantum critical point embedded within the superconducting state of Zn-doped CeCoIn$_5$, revealed by a pronounced peak in the magnetic penetration depth at zero temperature $λ(0)$. Using scanning SQUID microscopy, we determine the local superconducting transition temperature $T_{\mathrm c}$ and $λ(0)$. By parameterizing $λ(0)$ in terms of the local $T_{\mathrm c}$ rather than nominal Zn substitution, we circumvent the ambiguity caused by doping inhomogeneity and enable a more precise extraction of the critical exponent. The extracted exponent exceeds the clean spin-density-wave value, indicating a disorder-modified quantum critical regime. The enhancement of $λ(0)$ reflects the suppression of the superfluid stiffness and is consistent with critical scaling. Our approach provides a route to uncover intrinsic quantum critical behavior hidden by inhomogeneity in unconventional superconductors.

This study examined university students' discontinuance intention towards AI-mediated informal digital learning of English (AI-IDLE). Drawing on the cognition-affect-conation framework, the study investigated how three cognitive factors, namely disconfirmation, perceived complexity, and perceived risk, influence two affective responses, namely dissatisfaction and frustration, and how these affective responses predict discontinuance intention. A cross-sectional survey was conducted with 746 Chinese university students who had experience using AI tools for informal English learning. Data were analysed using structural equation modelling (SEM) and fuzzy-set qualitative comparative analysis (fsQCA). The SEM results showed that dissatisfaction and frustration positively predicted discontinuance intention, with frustration showing the stronger effect. Disconfirmation, perceived complexity, and perceived risk also positively influenced dissatisfaction and frustration. The fsQCA results further identified multiple sufficient configurations leading to high AI-IDLE discontinuance intention, indicating that discontinuance is shaped by causal complexity and equifinality rather than by a single necessary condition. These findings extend AI-IDLE research from adoption and engagement to post-adoption disengagement and provide implications for reducing learners' dissatisfaction, frustration, perceived complexity, and risk in AI-supported informal English learning.

We analyzed 19 years of $R$-band data of the blazar 3C 454.3 from the Whole Earth Blazar Telescope (WEBT) archive, along with new data from its members and from public archives such as those provided by the Small and Moderate Aperture Research Telescope System (SMARTS) and the Steward Observatory projects to search for quasi-periodic oscillations (QPOs). We detected a QPO of $\sim$ 433 days using Lomb-Scargle periodogram, which lasted from MJD 54980--58450 as detected by the weighted wavelet Z-transform technique, making it one of the most persistent QPOs ever detected in the optical regime. The phase dispersion minimization technique was also performed to further validate this QPO claim. We detected this signal at a global significance of $2.53σ$ across all methodologies. To explain the observed QPO, we have considered both models focused on the accretion disk around the super-massive black hole (SMBH), and those based purely on jet emissions. Plausible jet-based models involve a shock moving down the jet in a helical magnetic field, whereas the SMBH models could involve Lense-Thirring effect-induced jet precession or dual jets in a binary SMBH system. We introduce a novel approach to distinguish genuine QPOs from spurious signals arising from annual seasonal gaps, a common limitation of ground-based observations.

We present an extensive X-ray spectral variability study of the TeV photon-emitting high-energy-peaked BL Lacertae object PG 1553+113, using the data from EPIC-PN camera of XMM-Newton, which observed the source during its operational period from Sep 2001 to Nov 2024. X-ray spectra in this energy range, $0.6-7.0$ keV, were fitted with absorbed Power-law (PL) and absorbed Log-Parabola (LP) models. We found with 99$\%$ confidence that 14 of them were fit well by LP models having parameters in the range $α\simeq2.13-2.80$, and $β\simeq0.04-0.18$, one spectrum favours a LP model with $β<0$, while simple PL models with $Γ\simeq2.53-2.69$ were sufficient to describe the X-ray spectra of the remaining 15. Two of these 30 observations showed strong signatures of an additional inverse Compton component, while one showed weaker indications. On fitting joint Optical Monitor and EPIC-PN data with LP models, we found synchrotron peaks in the energy range of $ν_s\simeq4.59-48.61$ eV. This indicates that the spectral evolution is probably caused by variations in particle acceleration or cooling conditions within the jet.

Let $F$ be a finite field of odd characteristic. We prove that any set $A\subset F$ with $|A|\geq C|F|^{5/6}$ contains a nontrivial quadratic progression $(x, x+y, x+y^2), y\neq 0.$ For prime fields, this improves the previous best-known exponent of $7/8$, due to Kavrut and Wu. Unlike some of the previous papers, which rely on Katz's deep multivariate exponential-sum estimates, our argument uses only one-variable Weil-type estimates. We also construct, over certain non-prime finite fields, progression-free sets of size $c|F|^{2/3}$. A key idea in the proof was suggested to the author by ChatGPT 5.5.