We identify a sample of 324 red and 273 blue face-on spiral galaxies selected from 115 low-redshift (0.014 < z < 0.18) galaxy clusters imaged with CFHT+MegaCam in u- and r-band, KPNO 0.9-meter 2TKA and MOSAIC 8K camera in B and Rc, and images and catalogs extracted from the WINGS survey. Multi-wavelength photometry and spectroscopy were obtained by cross-matching sources with SDSS, GALEX, and WISE datasets. Primary findings indicate that up to 45% of optically red spiral galaxies exhibit significant dust content compared to blue spiral galaxies, as determined by infrared observations. This dust obscuration can conceal ongoing star formation, which may result in the misclassification of red spirals as passively evolving systems. Conversely, approximately half of the red spirals lack substantial dust abundance and appear optically red due to passive evolutionary processes. Support for the passive nature of these red spirals is provided by SDSS emission line data based on the Dn(4000) spectral index, EW(H-alpha), EW(H-delta), and [O III] 5007 A luminosity, and on a comparison of the star formation rate and the specific star formation rate with cluster blue spirals. Red spirals are an important link in the evolution of galaxies in the high-density cluster environment and play a key role in determining the physical mechanisms that are responsible for transforming blue star-forming galaxies into red spiral systems.

The dynamical Kubo-Toyabe (dKT) function is extended to describe the spin relaxation under the coexisting dynamical and static internal magnetic fields. A detailed re-evaluation of the previous $μ^\pm$SR data in Na$_x$CoO$_2$ using this function disfavors the conventional interpretation based on sodium-ion diffusion and instead supports the $μ^+$ self-diffusion scenario. This also resolves the long-standing inconsistencies in the dKT-function-based $μ$SR studies on ion diffusion from the viewpoint of classical over-barrier-jump mechanism.

Toda lattice or FPUT chain-like dynamics have been regarded as the prerequisite condition to explain the length dependency of high thermal conductivity of low-dimensional systems at the nanoscale. In this paper, a hypothetical condition is introduced that establishes a theoretical connection between the thermal conductivity of a nanoscale low-dimensional system in nonequilibrium steady states(NESS) and the canonical motion of the equation in the Toda lattice in equilibrium. The hypothesis relies on a numerically driven coarse grained molecular dynamics(CGMD) system acquired from the trajectory data of nonequilibrium molecular dynamics(NEMD) simulation. It models the macroscopic motion from longitudinal and flexural modulation observed in NEMD as a separate Hamiltonian in CGMD with a perturbation term governed by an overdamping process, which is assumed to be dominant during heat transfer. The Smoluchowski equation for the perturbation, which is derived from the cross correlated states between two degrees of freedom, suggests that the potential energy function induced from NESS is identical to that of the Toda Lattice under the specific condition in the partition function for coarse grained particles. The restrictions derived from the model are well confirmed by the data from the numerically driven coarse grained model.

This paper is devoted to the study of a convolution structure denoted by $*_α$, which is defined via the Hartley--Bessel transform. This concept was introduced in a recent work by F. Bouzeffour [\emph{J. Pseudo-Differ. Oper. Appl.}, 2024;15, Article 42]. We establish an analog of the Hausdorff--Young inequality for the Hartley--Bessel transform and convolution operator $*_α$. This leads to the convolution $*_α$ being uniformly bounded on the dual space. Moreover, in some special cases, our results yield a better upper bound coefficient for the convolution $*_α$ than those previously obtained by Bouzeffour's result in [Theorem 4.4, \emph{J. Pseudo-Differ. Oper. Appl.}, 2024;15, Article 42]. Finally, we apply the convolution structure $*_α$ to study the solvability of a particular class of integral equations and provide a priori estimates for solutions under appropriate conditions.

We define and study a notion of G-dimension for DG-modules over a non-positively graded commutative noetherian DG-ring $A$. Some criteria for the finiteness of the G-dimension of a DG-module are given by applying a DG-version of projective resolution introduced by Minamoto [Israel J. Math. 245 (2021) 409-454]. Moreover, it is proved that the finiteness of G-dimension characterizes the local Gorenstein property of $A$. Applications go in three directions. The first is to establish the connection between G-dimensions and the little finitistic dimensions of &\mathcal{A}&. The second is to characterize Cohen-Macaulay and Gorenstein DG-rings by the relations between the class of maximal local-Cohen-Macaulay DG-modules and a special G-class of DG-modules. The third is to extend the classical Buchwtweiz-Happel Theorem and its inverse from commutative noetherian local rings to the setting of commutative noetherian local DG-rings.Our method is somewhat different from classical commutative ring.

The prevalence of short form video platforms, combined with the ineffectiveness of age verification mechanisms, raises concerns about the potential harms facing children and teenagers in an algorithm-moderated online environment. We conducted multimodal feature analysis and thematic topic modeling of 4,492 short videos recommended to children and teenagers on Instagram Reels, TikTok, and YouTube Shorts, collected as a part of an algorithm auditing experiment. This feature-level and content-level analysis revealed that unsafe (i.e., problematic, mentally distressing) short videos (a) possess darker visual features and (b) contain explicitly harmful content and implicit harm from anxiety-inducing ordinary content. We introduce a useful framework of online harm (i.e., explicit, implicit, unintended), providing a unique lens for understanding the dynamic, multifaceted online risks facing children and teenagers. The findings highlight the importance of protecting younger audiences in critical developmental stages from both explicit and implicit risks on social media, calling for nuanced content moderation, age verification, and platform regulation.

Two-dimensional superconductors with spin-textured Fermi surfaces can be a platform for realizing unconventional pairing states and are of substantial interest in the context of quantum information science, and superconducting spintronics/orbitronics. We observed an unusual in-plane uniaxial anisotropy in the superconducting 2D electron gas (2DEG) formed at EuOx/KTaO3 (110) interfaces, where the EuOx is magnetic. This anisotropy is not evident in AlOx/KTaO3 (110) where the overlayer is non-magnetic. Our results are consistent with a highly anisotropic 'half-Rashba' spin-textured Fermi surface in 2DEGs formed at the KTaO3 (110) interface that is hidden from external magnetic fields due to a near cancellation between orbital and spin moments but revealed by exchange interactions of the electrons in the 2DEG with Eu moments near the EuOx/KTaO3 (110) interface. The interactions between the uniaxial spin texture and the magnetic overlayer offer new ways to explore the interplay between magnetism and 2D superconductivity.

Pulses from the Crab pulsar are often followed by "echoes", produced by radiation that was deflected by structures in the Crab nebula and thus traveled via longer paths. We describe a simplified but detailed model that treats the structures as cylindrical filaments of dense, neutral material with a thin ionized skin. In this picture, echoes are produced when the line of sight crosses the skin at glancing incidence, which naturally leads to the large electron column density gradients required to get the observed delays even with electron densities comparable to those inferred from optical line emission ratios. We compare the properties of the predicted echoes with those of a relatively isolated observed one identified during daily monitoring with CHIME. We find that the delays of the simulated echoes follow closely the near quadratic evolution known to be a feature of these echoes, and that, unlike in previous models, we match the characteristic observed asymmetry between incoming and outgoing arcs, with the size of the gap in between a consequence of the skin crossing time. However, our model fails to quantitatively reproduce the magnifications of the echoes. We believe this likely is because the structures are not as smooth as envisaged, so that a given echo results from many images. Nevertheless, our results confirm that echoes are produced by sheet-like structures seen edge-on and support the hypothesis that the nebula is filled with small-scale filamentary structures, which may well be substructures of the larger filaments that are seen in optical images.

The Crab Pulsar is known to feature plasma lensing events known as echoes. These events are characterized by additional components in the pulse profile which are produced by additional images formed when the pulsar's radio emission is deflected by ionized nebular material. These components are therefore delayed relative to the primary emission. We observed the Crab pulsar with Canadian Hydrogen Intensity Mapping Experiment (CHIME) during its daily transits, creating an archive of baseband recordings of giant pulses in the 400-800 MHz band. From these, we produced daily stacks of aligned pulses between late October 2021 and March 2024. We find that in these averages, echoes are readily visible throughout the observation period, and we identify clear groups of echoes with distinct behaviour in terms of their evolution with time and frequency. Many echoes exhibit dispersive delays consistent with being observed through excess column densities relative to the unscattered rays, but we also find two events where the dispersive delays indicate column density deficits. For the first time, we also find echoes for which the line of sight never directly intersects the intervening structures, resulting in events with non-zero minimum delays, of around 0.5 ms. The frequency and diversity of the observed echoes make the Crab an excellent target for long-term studies of astrophysical plasma lensing.

Recommender systems often struggle with over-specialization, which severely limits users' exposure to diverse content and creates filter bubbles that reduce serendipitous discovery. To address this fundamental limitation, this paper introduces an adaptive clustering framework with user-controlled exploration that effectively balances personalization and diversity in movie recommendations. Our approach leverages sentence-transformer embeddings to group items into semantically coherent clusters through an online algorithm with dynamic thresholding, thereby creating a structured representation of the content space. Building upon this clustering foundation, we propose a novel exploration mechanism that empowers users to control recommendation diversity by strategically sampling from less-engaged clusters, thus expanding their content horizons while explicitly exposing the relevance-diversity trade-off. Experiments on the MovieLens dataset demonstrate the system's effectiveness, showing that exploration significantly reduces intra-list similarity from 0.34 to 0.26 while simultaneously increasing unexpectedness to 0.73. Furthermore, our Large Language Model-based A/B testing methodology, conducted with 300 simulated users, reveals that 72.7% of long-term users prefer exploratory recommendations over purely exploitative ones. Additional relevance metrics, including NDCG@k, Recall@k, and HitRate@k, reveal the expected relevance-diversity trade-off against CF and MMR baselines, positioning the method as a controllable exploration layer for promoting meaningful content discovery.

A conjecture of Colliot-Thélène predicts that for a smooth projective variety $X$ over a finite extension $k$ of $\mathbb{Q}_p$ the kernel of the Albanese map $\text{CH}_0(X)^{\text{deg}=0}\to Alb_X(k)$ is the direct sum of a divisible group and a finite group. In this article we show that if $π:X\dashrightarrow Y$ is a generically finite rational map between smooth projective surfaces and the conjecture is true for $X\otimes_k L$ for every finite extension $L/k$, then it is true for $Y$. Using work of Raskind and Spiess, this proves the conjecture for surfaces that are geometrically dominated by products of curves, under some assumptions on the reduction type of the Jacobians. The method involves studying similar questions for an open subvariety $U$ of a projective surface $X$ by replacing the Chow group of $0$-cycles with Suslin's singular homology $H_0^{\text{sus}}(U)$.

We characterize when a generalized Baumslag-Solitar group is linear, and extend the result to the fundamental groups of a graph of groups with infinite virtually cyclic vertex and edge groups.

We propose an explicit, oracle-free quantum framework for numerically simulating general linear partial differential equations (PDEs), extending previous work to incorporate (a) Robin boundary conditions - which include Neumann and Dirichlet conditions as special cases - (b) inhomogeneous terms, and (c) variable coefficients in space and time. Our approach begins with a general finite-difference discretization and applies the Schrodingerisation technique to transform the resulting system into one that admits unitary quantum evolution, enabling quantum simulation. For the Schrodinger equation corresponding to the discretized PDE, we construct an efficient block-encoding of the Hamiltonian $H$ that scales polylogarithmically with the number of grid points $N$. This encoding is compatible with quantum signal processing and allows for the implementation of the evolution operator $e^{-iHt}$. The oracle-free nature of our method permits complexity to be measured in fundamental gate units-namely, CNOT gates and single-qubit rotations-bypassing the inefficiencies of oracle queries. Consequently, the overall algorithm scales polynomially with $N$ and linearly with the spatial dimension $d$, achieving a polynomial speedup in $N$ and an exponential advantage in $d$, thereby mitigating the classical curse of dimensionality. The validity and efficiency of the proposed approach are further substantiated by numerical simulations. By explicitly defining the quantum operations and quantifying their resource requirements, our approach offers a practical alternative for numerically solving PDEs, distinct from others that rely on oracle queries and purely asymptotic scaling methods.

A measurement of the angular structure of inclusive jets and those containing a prompt D$^0$ meson in proton-proton collisions at the LHC at a center-of-mass energy of 5.02 TeV is presented. The data corresponding to an integrated luminosity of 301 pb$^{-1}$ were collected by the CMS experiment in 2017. Two jet grooming algorithms, late-$k_\mathrm{T}$ and soft drop, are used to study the intrajet radiation pattern using iterative Cambridge$-$Aachen declustering. The splitting-angle distributions of jets with transverse momentum ($p_\mathrm{T}$) of around 100 GeV, obtained with these two algorithms, show that there is a shift of the distribution for jets containing a prompt D$^0$ meson with respect to inclusive jets. The suppression of emissions at small angles observed in the late-$k_\mathrm{T}$ grooming approach is consistent with the dead-cone effect, whereas the similar suppression for splittings selected with the soft-drop algorithm appears to be induced by gluon splitting to charm quark-antiquark pairs at large angles. The measured distributions are corrected to the particle level and can be used to constrain model predictions for the substructure of high-$p_\mathrm{T}$ charm quark jets.

We investigate qubits coupled to the boundary of a two dimensional photonic lattice that supports dispersionless edge modes, unlike conventional edge modes that sustain propagating photons. As a case study, we consider a honeycomb lattice (photonic graphene) of coupled resonators with a zigzag edge, where the edge modes form a flat band defined only over a restricted region of momentum space. We show that light matter interactions are effectively captured by a dissipative cavity QED model, wherein the emitter coherently couples to a fictitious cavity mode emerging as a superposition of edge modes. This mode has support on only one sublattice and, most notably, displays an unconventional power law localization around the qubit, yet remaining normalizable in the thermodynamic limit, with a spatial range that can be tuned by introducing lattice anisotropy We predict occurrence of vacuum Rabi oscillations and efficient state transfer between distant emitters. An experimental demonstration using superconducting circuits is proposed.

We prove a coarse version of Halin's Grid Theorem: Every one-ended, locally finite graph that contains the disjoint union of infinitely many rays as an asymptotic minor also contains the half-grid as an asymptotic minor. More generally, we show that the same holds for arbitrary (not necessarily one-ended or locally finite) graphs under additional, necessary assumptions on the minor-models of the infinite rays. This resolves a conjecture of Georgakopoulos and Papasoglu. As an application, we show that every one-ended, quasi-transitive, locally finite graph contains the half-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of one-ended finitely generated groups and solves a problem of Georgakopoulos and Papasoglu.

Magic state distillation enables universal fault-tolerant quantum computation by implementing non-Clifford gates via the preparation of high-fidelity magic states. However, it comes at the cost of substantial logical-level overhead in both space and time. In this work, we propose a very low-cost magic state distillation scheme for biased-noise qubits. By leveraging the noise bias, our scheme enables the preparation of a magic state with a logical error rate of $3 \times 10^{-7}$, using only 53 qubits and 5.5 error correction rounds, under a noise bias of $η\gtrsim 5 \times 10^6$ and a phase-flip noise rate of $0.1\%$. This reduces the circuit volume by more than one order of magnitude relative to magic state cultivation for unbiased-noise qubits and by more than two orders of magnitude relative to standard magic state distillation. Moreover, our scheme provides three key advantages over previous proposals for biased-noise qubits. First, it only requires nearest-neighbor two-qubit gates on a 2D lattice. Second, the logical fidelity remains nearly identical even at a more modest noise bias of $η\gtrsim 80$, at the cost of a slightly increased circuit volume. Third, the scheme remains effective even at high physical phase-flip rates, in contrast to previously proposed approaches whose circuit volume grows exponentially with the error rate. Our construction is based on unfolding the $X$ stabilizer group of the Hadamard 3D quantum Reed-Muller code in 2D, enabling distillation at the physical level rather than the logical level, and is therefore referred to as $\textit{unfolded}$ distillation.

In this work, we promote the global $SL(2,\mathbb{R})$ symmetry of the Schwarzian derivative to a local gauge symmetry. To achieve this, we develop a procedure that potentially can be generalized beyond the $SL(2,\mathbb{R})$ case: We first construct a composite field from the fundamental field and its derivative such that it transforms linearly under the group action. Then we write down its gauge-covariant extension and apply standard gauging techniques. Applying this to the fractional linear representation of $SL(2,\mathbb{R})$, we obtain the gauge-invariant analogue of the Schwarzian derivative as a bilinear invariant of covariant derivatives of the composite field. The framework enables a simple construction of Nöther charges associated with the original global symmetry. The gauge-invariant Schwarzian action introduces $SL(2,\mathbb{R})$ gauge potentials, allowing for locally invariant couplings to additional fields, such as fermions. While these potentials can be gauged away on topologically trivial domains, non-trivial topologies (e.g., $S^1$) lead to distinct topological sectors. We mention that in the context of two-dimensional gravity, these sectors could correspond to previously discussed defects in the bulk theory.

We systematically investigate how static symmetry-breaking perturbations and dynamic Floquet terms via a polarized light manipulate the topological phase transitions in the two-dimensional quadratic-band-crossing-point (QBCP) materials. The Berry curvature shows distinct behavior in such two situations. It is linearly and quadratically proportional to the product of microstructural parameters $t_{x,z}$ for the former and the latter, respectively. The static perturbation eliminates the QBCP and opens an energy gap, which leads to the momentum-inversion symmetry of Berry curvature. This yields a nontrivial Chern number determined by the microstructural parameters. In contrast, we demonstrate that either a circularly or an elliptically polarized light breaks the time-reversal symmetry, transforming the QBCP semimetal into a Chern insulator with a quantized anomalous Hall conductivity $σ_{xy} = Ce^2/\hbar$, where the Chern number is governed by the polarization angle. Moreover, the linear polarization preserves the central antisymmetry of the Berry curvature, giving rise to a topological trivial insulator. These results establish a tunable topological phase transition from a QBCP semimetal to Chern insulator in the two-dimensional QBCP materials.

We study helicity correlations of electron-positron pairs created by a homogeneous time-dependent electric field in the Sauter-Schwinger scenario. Our analysis is based on solving the Dirac equation with the Feynman or anti-Feynman boundary conditions, which is equivalent to the scattering matrix approach widely used in high energy physics. Most importantly, both these methods allow to fully account for the helicity (or, more generally, spin) correlations of created particles. The influence of helicity correlations and the carrier-envelope phase of the electric pulse on the properties of topological structures (such as spirals and vortices) in momentum distributions of created particles is investigated. The generation of maximally entangled helicity states is discussed and the possibility of using a short electric pulse as a fast switch between them is indicated.

We propose a detailed study of a canonical bound which relates the numbers of rational points of a curve over a finite field with that over its quadratic extension. Alternative proofs which make a connection with the variance enable to obtain optimal refinements. We focus on the curves reaching the bound, which we call Hallouin-Perret-maximal curves. We provide different characterizations and stress natural links with the curves which attain the Ihara bound. As consequences, we establish the list of such curves with low genus and we outline a maximality result which involves the Suzuki curves. At last we determine which polynomials correspond to the Jacobian of a Hallouin-Perret-maximal curve of genus 2.

How biological networks achieve robust control despite relying on imperfect, local information remains an important open question. Here, we identify thermodynamic constraints that can curtail non-equilibrium steady-state responses so severely that even crude, local feedback rules can achieve globally stable control without requiring precise network design or global information. Specifically, using Markov jump processes as a general framework for biophysical dynamics, we derive general non-equilibrium response constraints showing that for many classes of rate perturbations, steady-state responses have fixed signs across all driving strengths, so that near-equilibrium responses predict far-from-equilibrium behavior regardless of system complexity. These constraints clarify several biological phenomena: monotonicity is thermodynamically guaranteed whenever a perturbation acts on a single transition rate, and non-monotonic responses, as observed for example in transcription factor regulation, arise only when an input simultaneously modulates multiple rates. Even in this case, we identify a graph-theoretic concept termed ``coherence'' that allows for a restoration of monotonicity. We show how coherence naturally and generally emerges in classic biophysical models of adaptation, including E. coli chemotaxis, and transcription factor regulation when biological constraints on network parameterization are included. We next show that, within a control-theoretic framework, these constraints guarantee that simple linear feedback on small subsets of kinetic rates achieves globally stable tracking and adaptation without coordinated manipulation of many variables. For systems with one regulator, local stability implies global stability for arbitrary network topologies without fine tuning, revealing that non-equilibrium thermodynamics fundamentally constrains biochemical network responses.

We settle the problem of K-stability of quasi-smooth Fano 3-fold hypersurfaces with Fano index 1 by providing lower bounds for their delta invariants. We use the method introduced by Abban and Zhuang for computing lower bounds of delta invariants on flags of hypersurfaces in the Fano 3-fold.

Newton's laws of motion pose an apparent problem, sometimes referred to as "the independence problem": the first law seems to be a simple consequence of the second law, raising the question of why it was included as a separate law. Numerous answers to this question have been proposed in the literature. The main contribution of this paper is a novel answer which we call "the formal explanation." Unlike previous accounts it relies on mathematical formalism and argues that the definitions of Euclidean geometry necessitate the inclusion of the first law. We provide evidence in support of this claim. A second contribution is a comprehensive review of previously suggested explanations, which so far have often been treated in a fragmented manner, and a discussion of the plausibility of the various answers.

We examine the joint effects of relativistic accretion and memory-burdened evaporation on the evolution of primordial black holes (PBHs). The memory burden effect, which delays the evaporation by inducing a backreaction and making the evaporation rate scale as an inverse power law of the PBH entropy, opens up a new window that allows PBHs with $M \lesssim 10^{15}~\mathrm{g}$ to survive until the present epoch. Meanwhile, accretion increases the mass of PBHs, thereby enhancing their chances of survival for a given initial mass. We consider two main scenarios: one where PBHs evaporate completely before big bang nucleosynthesis, and another where PBHs persist until today. In the case of evaporation, we analyze the emission of dark matter (DM) and dark radiation (DR) during the process of evaporation. Conversely, in the other case, the surviving PBHs themselves can contribute as DM. We further investigate how relativistic and nonrelativistic accretion, together with memory-burdened evaporation, impact the parameter space of the emitted DM, the abundance of stable PBHs as DM, and the contribution of DR to the effective number of relativistic degrees of freedom, $ΔN_{\mathrm{eff}}$.

In his seminal work on Teichmüller spaces (\cite{Th98}), Thurston introduced the maximal stretch for a pair of hyperbolic metrics on a closed surface of genus $\geq 2$ and showed that the logarithm of this quantity induces an asymmetric metric in the Teichmüller space. He also showed that the subset of the surface on which the maximal stretch is attained is a geodesic lamination. In this paper, we define the maximal stretch analogously for closed manifolds equipped with Riemannian metrics of variable negative curvature and investigate the structure of the related Mather set on the unit tangent bundle. In contrast to the Teichmüller space, the Mather set may not be lifts of geodesic laminations in this broader setting. However, in our paper, we will discuss similar features shared by the Mather set with geodesic laminations. We also connect the study of the Mather set with the theory of best Lipschitz maps.

This study investigates the application of a simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-aided medium-Earth-orbit (MEO) satellite network for providing both global positioning services and communication services in the urban canyons, where the direct satellite-user links are obstructed. Superposition coding (SC) and successive interference cancellation (SIC) techniques are utilized for the integrated navigation and communication (INAC) networks, and the composed navigation and communication signals are reflected or transmitted to ground users or indoor users located in urban canyons. To meet diverse application needs, navigation-oriented (NO)-INAC and communication-oriented (CO)-INAC have been developed, each tailored according to distinct power allocation factors. We then proposed two algorithms, namely navigation-prioritized-algorithm (NPA) and communication-prioritized-algorithm (CPA), to improve the navigation or communication performance by selecting the satellite with the optimized position dilution of precision (PDoP) or with the best channel gain. The effectiveness of the proposed STAR-RIS-aided INAC network is quantified by analyzing the positioning error for navigation services and by evaluating communication performance through achievable ergodic rate metrics. Our satellite selection approach indicates that: the positioning services at the urban canyon users can be completed with the aid of STAR-RIS. 2) Additionally, it is observed that while a single STAR-RIS array can extend the navigational link, it fails to serve users in indoor scenarios, highlighting a limitation in the current system design.

Given a second-order, holomorphic, linear differential equation $Lf=0$ on a punctured Riemann surface, we say that its monodromy group $G\subset\operatorname{GL}(2,\mathbb{C})$ is `unitary' if it preserves a non-degenerate Hermitian form $H$ on $\mathbb{C}^2$ under the action $g\circ H=g^\dagger H g$. In the present work, we give two sets of necessary and sufficient conditions for a monodromy group $G\subset\operatorname{GL}(2,\mathbb{C})$ to be unitary. First, in the case that the natural representation of $G$ on $\mathbb{C}^2$ is irreducible, we show that unitarity is equivalent to a set of easily-verified trace conditions on local monodromy matrices; in the case that the representation is reducible, we show that $G$ is unitary if and only if it is contained in one of two model subgroups of $\operatorname{GL}(2,\mathbb{C})$. Second, we show that unitarity is equivalent to a criterion on the real dimension of the algebra $A$ generated by a rescaled group $G'\subset\operatorname{SL}(2,\mathbb{C})$: that $\dim(A)=1$ if $G\subset S^1$ is scalar, $\dim(A)=2$ if $G$ is abelian, $\dim(A)=3$ if $G$ is non-abelian but its action on $\mathbb{C}^2$ is reducible, and $\dim(A)=4$ otherwise. Our results directly extend the recent work of Adachi (2022, 2024), which treated Fuchsian equations with irreducible monodromy representation on the punctured Riemann sphere. We leverage these results to provide evidence that the spectrum of any Darboux operator should belong to a perturbed, squared lattice in the plane, extending a conjecture of Frits Beukers (2007). Our work makes progress towards characterizing the spectra of second-order operators on Riemann surfaces, and in particular, towards answering the \emph{accessory parameter problem} for Darboux equations.

Over the past 20 years, many new hadron states have been discovered, but understanding their nature remains a key experimental and theoretical challenge. Recent studies have established that hadron-hadron interactions primarily govern the generation of new hadronic states, with their spectroscopy serving as a powerful tool for probing these interactions and determining the corresponding compositeness. In this work, we study four scenarios to determine the $DK$ interaction by reproducing the mass of the $D_{s0}^*(2317)$, i.e., assuming the $D_{s0}^*(2317)$ as a $DK$ molecule, a mixture of a $DK$ molecule and a bare state, a $DK-D_sη$ molecule, and a mixture of a $DK-D_sη$ molecule and a bare state. Using the $D^{0}K^{+}$ interactions derived from these scenarios, we predict the $D^{0}K^{+}$ correlation functions. Our results demonstrate that the lineshape of the $D^{0}K^{+}$ correlation function is sensitive to the admixture effects from the coupled-channel $D^+K^0$ and the bare state. Furthermore, we find that the $D^{0}K^{+}$ correlation function can probe the position of the bare state, if such a QCD bare state exists. Using the shallow-bound state candidate $X(3872)$ as input, we study the $D^0\bar{D}^{*0}$ correlation functions. These functions are highly sensitive to short-range dynamics and bare-state admixtures, resulting in clearly distinguishable correlation-function line shapes across different values of compositeness.

The results of a search for the production of two scalar bosons in final states with two photons and two tau leptons are presented. The search considers both nonresonant production of a Higgs boson pair, HH, and resonant production via a new boson X which decays either to HH or to H and a new scalar Y. The analysis uses up to 138 fb$^{-1}$ of proton-proton collision data, recorded between 2016 and 2018 by the CMS experiment at the LHC at a center-of-mass energy of 13 TeV. No evidence for signal is found in the data. For the nonresonant production, the observed (expected) upper limit at 95% confidence level (CL) on the HH production cross section is set at 930 (740) fb, corresponding to 33 (26) times the standard model prediction. At 95% CL, HH production is observed (expected) to be excluded for values of $κ_λ$ outside the range between $-$12 ($-$9.4) and 17 (15). Observed (expected) upper limits at 95% CL for the X $\to$ HH cross section are found to be within 160 to 2200 (200 to 1800) fb, depending on the mass of X. In the X $\to$ Y($ττ$)H($γγ$) search, the observed (expected) upper limits on the product of the production cross section and decay branching fractions vary between 0.059$-$1.2 fb (0.087$-$0.68 fb). For the X $\to$ Y($γγ$)H($ττ$) search the observed (expected) upper limits on the product of the production cross section and Y $\to$ $γγ$ branching fraction vary between 0.69$-$15 fb (0.73$-$8.3 fb) in the low Y mass search, tightening constraints on the next-to-minimal supersymmetric standard model, and between 0.64$-$10 fb (0.70$-$7.6 fb) in the high Y mass search.