Let $\Bbbk$ be an algebraically closed field of characteristic $0$ and let $H$ be a finite-dimensional Hopf algebra over $\Bbbk$ with the dual Chevalley property. In this paper, we give a description of the link quiver of $H$ for different corepresentation types. Moreover, we show that $\operatorname{gr}^c(H)$ is of tame corepresentation type if and only if $\operatorname{gr}^c(H)\cong (\k\langle x,y\rangle/I)^* \times H_0$ for some special ideals $I$. Using the methods of link quivers and bosonization, we then discuss which of the above ideals occur when $(\Bbbk\langle x,y\rangle/I)^* \times H_0$ is a Hopf algebra of tame corepresentation type under certain assumptions.

In 1939 Pál Turán and János Erőd initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a matter of normalization they considered the family $\mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given convex, compact subset $K\Subset {\mathbb C}$. While Turán obtained the first results for the interval $I:=[-1,1]$ and the disk $D:=\{ z\in {\mathbb C}~:~ |z|\le 1\}$, Erőd extended investigations to other compact convex domains, too. The order of the optimal constant was found to be $\sqrt{n}$ for $I$ and $n$ for $D$. It took until 2006 to clarify that all compact convex \emph{domains} (with nonempty interior), follow the pattern of the disk, and admit an order $n$ inequality. For $L^q(\partial K)$ norms with any $1\le q <\infty$ we obtained order $n$ results for various classes of domains. Further, in the generality of all convex, compact domains we could show a $c n/\log n$ lower bound together with an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all compact convex domains admit an order $n$ Turán type inequality. Here we prove this for all \emph{polygonal} convex domains and any $0< q <\infty$.

Constrained maximization of submodular functions is a central problem in combinatorial optimization. In many realistic scenarios, multiple agents each need to maximize their own submodular objective over a common ground set, subject to individual constraints, with the requirement that their solutions be disjoint. We study this setting through the lens of algorithmic fairness and constrained fair division. Inspired by the fair division literature, we propose and analyze a simple Round-Robin protocol in which agents take turns building their solutions one item at a time; each agent is free to use any internal algorithm, and the protocol itself performs no computation. We show that agents following simple greedy policies enjoy solid guarantees for both monotone and non-monotone objectives subject to constraints as general as $p$-systems. For monotone objectives, a greedy agent $i$ with a $p_i$-system constraint achieves a $1/(n+p_i)$ fraction of the best value available when they first get to choose. On instances that are robust to competition -- where no agent's optimal value is greatly affected by losing some items to others -- these guarantees improve to a $1/Θ(p_i)$ approximation of the unconstrained optimum, which is asymptotically best-possible in polynomial time. We further establish novel fairness guarantees: greedy agents produce approximately feasible-envy-free-up-to-one-item (FEF1) and approximately feasible-envy-free-towards-unallocated-items (FEFu) allocations for monotone and non-monotone objectives. Via a simple augmented protocol and a self-contained polynomial-time proxy algorithm, we also obtain the first $Θ(1/p_i)$-approximate feasible maximin share (FMMS) guarantees for submodular agents with combinatorial constraints. Finally, although greedy policies may not be individually optimal, consistently improving upon them is NP-hard even in the simplest settings.

In this paper we define an efficient implementation of Runge-Kutta methods of Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems. The proposed implementation relies on an alternative low-rank formulation of the methods, for which a splitting procedure is easily defined. The linear convergence analysis of this splitting procedure exhibits excellent properties, which are confirmed by its performance on a few numerical tests.

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g. the Lagrangian Hofer metric or the spectral metric). We exhibit many metric and symplectic properties of these spaces, such that they have compact completions and that they contain only finitely many Hamiltonian isotopy classes. We then use this to exclude many unusual phenomena from happening in these bounded spaces. Taking limits in the bounds, we also conclude that there are at most countably many Hamiltonian isotopy classes of exact Lagrangian submanifolds in a Liouville manifold. Under some mild topological assumptions, we get analogous results for monotone Lagrangian submanifolds with a fixed monotonicity constant. Finally, in the process of showing these results, we get new results on the Riemannian geometry of cotangent bundles and surfaces which might be of independent interest.

We introduce and document a class of probability distributions, called bilateral generalized inverse Gaussian (BGIG) distributions, that are obtained by convolution of two generalized inverse Gaussian distributions supported by the positive and negative semi-axis. We prove several results regarding their analyticity, shapes and asymptotics, and we introduce the associated Lévy processes as well as their main properties. We study the behaviour of these processes under change of measure, their simulations and the structure of their sample paths, and we introduce a stock market model constructed by means of exponential BGIG processes. Based on real market data, we show that this model is easy to calibrate thanks notably to idiosyncratic properties of BGIG distributions, and that it is well suited to Monte Carlo and Fourier option pricing.

$ler$ is a Python package for simulating compact-binary gravitational-wave populations and estimating detectable event rates for current and future LIGO-Virgo-KAGRA detector networks. The package provides a unified framework for unlensed and strongly lensed binary black hole, binary neutron star, and neutron star black hole mergers. It samples source and lens populations, evaluates detector selection effects, solves lens equations for strongly lensed systems, and computes image properties such as magnifications and time delays. The framework supports multiple source-population and lens models, including SIS, SIE, and EPL plus external shear, and allows users to replace default distributions and detection criteria through modular interfaces. Computational efficiency is obtained through vectorized sampling, inverse-transform and importance-sampling strategies, multiprocessing, and just-in-time compiled routines. $ler$ is designed for large-scale Monte Carlo studies in which rates and selected populations must be evaluated repeatedly, including forecasts for observing runs, studies of lensing candidate validation, and selection-function calculations for population inference. The package is distributed with documentation, validation examples, and reproducible workflows.

Broadcast is a ubiquitous distributed computing problem that underpins many other system tasks. In static, connected networks, it was recently shown that broadcast is solvable without any node memory and only constant-size messages in worst-case asymptotically optimal time (Hussak and Trehan, PODC'19/STACS'20/DC'23). In the dynamic setting of adversarial topology changes, however, existing algorithms rely on identifiers, port labels, or polynomial memory to solve broadcast and compute functions over node inputs. We investigate space-efficient, terminating broadcast algorithms for anonymous, synchronous, 1-interval connected dynamic networks and introduce the first memory lower bounds in this setting. Specifically, we prove that broadcast with termination detection is impossible for idle-start algorithms (where only the broadcaster can initially send messages) and otherwise requires $Ω(\log n)$ memory per node, where $n$ is the number of nodes in the network. Even if the termination condition is relaxed to stabilizing termination (eventually no additional messages are sent), we show that any idle-start algorithm must use $ω(1)$ memory per node, separating the static and dynamic settings for anonymous broadcast. This lower bound is not far from optimal, as we present an algorithm that solves broadcast with stabilizing termination using $\mathcal{O}(\log n)$ memory per node in worst-case asymptotically optimal time. In sum, these results reveal the necessity of non-constant memory for nontrivial terminating computation in anonymous dynamic networks.

Training deep neural networks often requires large-scale datasets, necessitating storage and processing on cloud servers due to computational constraints. The procedures must follow strict privacy regulations in domains like healthcare. Split Learning (SL), a framework that divides model layers between client(s) and server(s), is widely adopted for distributed model training. While Split Learning reduces privacy risks by limiting server access to the full parameter set, previous research has identified that intermediate outputs exchanged between server and client can compromise client's data privacy. Homomorphic encryption (HE)-based solutions exist for this scenario but often impose prohibitive computational burdens. To address these challenges, we propose CURE, a novel system based on HE, that encrypts only the server side of the model and optionally the data. CURE enables secure SL while substantially improving communication and parallelization through advanced packing techniques. We propose two packing schemes that consume one HE level for one-layer networks and generalize our solutions to n-layer neural networks. We demonstrate that CURE can achieve similar accuracy to plaintext SL while being 16x more efficient in terms of the runtime compared to the state-of-the-art privacy-preserving alternatives.

We investigate numerical methods for wave equations in $n+2$ spacetime dimensions, written in spherical coordinates, decomposed in spherical harmonics on $S^n$, and finite-differenced in the remaining coordinates $r$ and $t$. Such an approach is useful when the full physical problem has spherical symmetry, for perturbation theory about a spherical background, or in the presence of boundaries with spherical topology. The key numerical difficulty arises from lower-order $1/r$ terms at the origin $r=0$. As a toy model for this, we consider the flat space linear wave equation in the form $\dotπ=ψ'+pψ/r$, $\dotψ=π'$, where $p=2l+n$, and $l$ is the leading spherical harmonic index. We propose a class of summation by parts (SBP) finite differencing methods that conserve a discrete energy up to boundary terms, thus guaranteeing stability and convergence in the energy norm. We explicitly construct SBP schemes that are second and fourth-order accurate at interior points and the symmetry boundary $r=0$, and first and second-order accurate at the outer boundary $r=R$.

The Hamiltonian governing the gravitational interaction of N relativistic particles in a four-dimensional anti-de Sitter background is derived to leading order in Newton's constant. The resulting pairwise interactions, combined with the confining nature of motion in anti-de Sitter spacetime, are expected to lead to classical chaos. In the context of the AdS/CFT correspondence, the emergence of a chaotic classical limit on the gravity side has important implications for the dual three-dimensional conformal field theory, including that the spectrum of conformal primary operators at strong coupling should exhibit level repulsion in line with the Wigner surmise.

We present a multi-frequency, multi-epoch catalog of extragalactic sources. The catalog is based on 150, 220, and 280 GHz observations carried out in 2008, 2009, and 2010 using the Millimeter Bolometric Array Camera on the Atacama Cosmology Telescope. We also present and release 280 GHz maps from 2008 and 2010. The catalog contains 483 sources found in a sky area of ${\sim}600$ square degrees. It was obtained by cross-matching sources found in 11 sub-catalogs, one for each season and frequency band. We also include co-added data from ${\sim}150$ and ${\sim}160$ square degrees using two and three years of overlapping observations. We divide the sources into two populations, synchrotron and dusty emitters, based on their spectral behavior in the 150-280 GHz frequency range. We find 284 synchrotron sources and 183 dusty source candidates. Our cross-matching with catalogs from radio to X-ray results in 251 synchrotron sources (88%) and 92 dusty sources (51%) with counterparts and suggests that 91 dusty candidates are not in existing catalogs. We study the variability and number counts of each population. In the case of synchrotron sources, we find year-to-year variability, with a mean value around 35%. As expected, we find no evidence of dusty source variability. Our number counts generally agree with previous measurements and models, except for dusty sources at 280 GHz, where some models overestimate our results.

Local interactions drive emergent collective behavior, which pervades biological and social complex systems. But uncovering the interactions that produce a desired behavior remains a core challenge. In this paper, we present EvoSOPS, an evolutionary framework that searches landscapes of stochastic distributed algorithms for those that achieve a mathematically specified target behavior. These algorithms govern self-organizing particle systems (SOPS) comprising individuals with no persistent memory and strictly local sensing and movement. For aggregation, phototaxing, and separation behaviors, EvoSOPS discovers algorithms that achieve 4.2-15.3% higher fitness than those from the existing "stochastic approach to SOPS" based on mathematical theory from statistical physics. EvoSOPS is also flexibly applied to new behaviors such as object coating where the stochastic approach would require bespoke, extensive analysis. Finally, we distill insights from the diverse, best-fitness genomes produced for aggregation across repeated EvoSOPS runs to demonstrate how EvoSOPS can bootstrap future theoretical investigations into SOPS algorithms for new behaviors.

We report data analysis results about the outburst evolution and spectral properties during the hard state of the recently discovered X-ray transient Swift J1727.8--163 as observed by \emph{Insight}-HXMT. The broadband X-ray spectrum of Swift J1727.8--163 is more complex than the most typical spectral patterns of black hole X-ray binary systems, with not only a comparatively weaker reflection component but also an additional high-energy component, manifesting itself as a hard X-ray tail beyond the thermal Comptonization description detectable below 100 keV. Using reflection models combining thermal ({\tt reflkerr}) and hybrid ({\tt reflkerr\_bb}) Comptonization, we find that the emission arises from a plasma containing both thermal and non-thermal electrons, with the non-thermal tail forming a smooth extension of the thermal continuum. This supports a compact hybrid corona. We find that the inner disk radius remains at the ISCO and the reflection fraction stays low, with no abrupt spectral evolution across epochs. These results highlight the role of hybrid corona in shaping hard-state emission in black hole X-ray binaries.

In this paper, we begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 = c^2$, these closed forms become even more intriguing. This approach allows us to provide new integral representations for the error function. Additionally, by leveraging an identity we derived for the inverse Laplace transform and applying a result based on Srivastava and Yürekli's identity, we provide a closed form for a nontrivial generalized integral.

We consider the problem of minimizing a polynomial $f$ over the (binary) hypercube. We show that, for a specific set of polynomials, their binary non-negativity (i.e. on the hypercube) can be checked in polynomial time via minimum cut algorithms, from which we construct a linear programming representation for this set of polynomials. We categorize binary polynomials according to their signed support patterns and develop parameterized linear programming representations for binary non-negative polynomials. This allows the construction of signed certificates of binary non-negativity with adjustable signed support patterns and representation complexities; and we propose a method for minimizing $f$ by decomposing it as a sum of signed certificates. This method yields new hierarchies of linear programming relaxations for binary polynomial optimization. Moreover, since our decomposition depends only on the support of $f$, the new hierarchies are sparsity-preserving.

We investigate the limits of epistemic models in reproducing the empirical predictions of general quantum preparations. This involves comparing the common quantum overlap determined by the anti-distinguishability of a set of preparations with the common epistemic overlap of the probability distribution over the ontic states describing these preparations. We show that no epistemic model with non-zero common overlap can reproduce the predictions of quantum theory for mixed preparations, even for qubit systems. We explicitly provide sets of distinct mixed preparations, starting from qutrit systems, that give rise to identical quantum mixed states for which the common epistemic overlap must be zero. Finally, we demonstrate the strongest form of preparation contextuality by presenting pairs of mixed preparations that yield identical quantum mixed states while their epistemic overlap must vanish.

The patch dynamics scheme in equation-free multiscale modelling has the potential to efficiently predict the macroscopic behaviours by simulating the microscale problem in a fraction of the space-time domain. The patch dynamics schemes developed so far are mainly on rectangular domains with uniform grids and uniform rectangular patches. In real-life problems, the geometry of the domain is not regular or simple, where rectangular and uniform grids or patches may not be useful. To address this kind of complexity, for the first time, a generalised orthogonal curvilinear coordinate system is employed in the patch dynamics scheme, applicable to both rectangular domains with non-uniform grids and non-rectangular domains; while applying this, the concept of non-uniform and non-rectangular patch configurations in the physical domain is also adopted for the first time. An explicit representation of a patch dynamics scheme on a generalised curvilinear coordinate system in a two-dimensional domain is proposed for unsteady, linear, heterogeneous convection-diffusion-reaction (CDR) problems. The proposed scheme is validated through heterogeneous convection-diffusion-reaction and non-axisymmetric diffusion problems on generalised curvilinear coordinate systems. The results demonstrate excellent accuracy and show that the method significantly outperforms full-domain simulations in terms of computational efficiency, memory usage and overall performance.

For a polarized toric variety, we provide a sufficient criterion ensuring that a uniformly K-stable polarized toric variety $(X,L)$ is asymptotically Chow polystable, under the assumption that the obstruction to asymptotic Chow semistability (the Futaki-Ono invariant) vanishes. Our approach is based on a detailed study of triangulations of neighborhoods of the vertices of the associated moment polytope $Δ$. As an application, we prove that every uniformly K-stable polarized smooth toric variety $(X,L)$ with vanishing Futaki-Ono invariant is asymptotically Chow polystable.

In this work, we consider the $H_2$ optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive gradients of the squared $H_2$ system error with respect to the reduced model quantities and, from the stationary points of these gradients, introduce Gramian-based first-order necessary conditions for the $H_2$ optimal approximation of a linear quadratic output (LQO) system. The resulting $H_2$ optimality framework neatly generalizes the analogous Gramian-based optimality framework for purely linear systems. Computationally, we show how to enforce the necessary optimality conditions using Petrov-Galerkin projection; the corresponding projection matrices are obtained from a pair of Sylvester equations. Based on this result, we propose an iteratively corrected algorithm for the $H_2$ model reduction of LQO systems, which we refer to as LQO-TSIA (linear quadratic output two-sided iteration algorithm). Numerical examples are included to illustrate the effectiveness of the proposed computational method against other existing approaches.

The paper provides an introductory discussion about two fundamental models of oscillator synchronization: the (continuous-time) diffusive model, that dominates the mathematical literature on synchronization, and the (hybrid) kick model, that accounts for most popular examples of synchronization, but for which only few theoretical results exist. The paper stresses fundamental differences between the two models, such as the different contraction measures underlying the analysis, as well as important analogies that can be drawn in the limit of weak coupling.

We establish a central limit theorem for tensor product random variables $c_k:=a_k \otimes a_k$, where $(a_k)_{k \in \mathbb{N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables $a_k$ and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.

Let $R$ be a commutative noetherian local ring, and let $M$ be a finitely generated $R$-module. Inspired by works of Vasconcelos and Briggs on characterization of complete intersection local rings through the homological properties of the conormal module, in this paper, we define the characteristic module T$_M$ and the cocharacteristic module E$_M$ of $M$, and investigate their properties. Our main results include characterizations of Cohen--Macaulay and Gorenstein local rings. Also, we show that if the injective dimension of the conormal module over an almost complete intersection ring is finite, then $R$ is a complete intersection.

We theoretically study the radiance properties of a pair of qubits inside a single-mode cavity driven by a two-photon drive. Our results show that, when the two qubits are strongly coupled to the cavity field, the collective radiation emitted from the qubits exhibits hyperradiance which can be detected as a signature of two qubit entanglement in the weak-driving regime. We quantify the entanglement in terms of concurrence. Additionally, we study the radiance behaviour in the presence of an intracavity Kerr-nonlinear medium that leads to two-photon blockade. Our results suggest that this system with nonlinearity may act as a quadrature-squeezed and hyperradiant two-photon source.

We report on the absolute frequency determination of the 5s5p $^1P_1$ - 5s5d $^1D_2$ transition in atomic strontium, achieved through frequency comb-referenced laser-induced-fluorescence (LIF) spectroscopy. We excite the 5s$^2$ $^1S_0$ - 5s5p $^1P_1$ transition using an on-resonance laser at $\approx$461 nm, and then measure the variation in the LIF signal while scanning the laser at $\approx$767 nm across the 5s5p $^1P_1$ - 5s5d $^1D_2$ transition. We determine the absolute frequency of $390599571.7\pm0.4$ MHz, with an accuracy that surpasses the previous most accurate measurement by two orders of magnitude. This measurement technique can be readily applied for precision spectroscopy of high-lying states not only in strontium, but also in other atomic species.

The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems and to some Higman-Thompson groups.

String attractors are a combinatorial tool coming from the field of data compression. It is a set of positions within a word which captures an occurrence of every factor. While one-sided infinite words admitting a finite string attractor are eventually periodic, the situation is different for two-sided infinite words. In this article, we characterise the bi-infinite words admitting a finite string attractor as the characteristic Sturmian words and their morphic images. For words that do not admit finite string attractors, we study the structure and properties of their infinite string attractors.

We study the equilibrium properties and the wetting behavior of a simple liquid on a polymer brush, with and without presence of lubricant by multibody Dissipative Particle Dynamics simulations. The lubricant is modelled as a polymeric liquid consisting of short chains that are chemically identical to the brush polymers. We investigate the behavior of the brush in terms of the grafting density and the amount of lubricant present. Regarding the wetting behavior, we study a sessile droplet on top of the brush. The droplet consists of non-bonded particles that form a dense phase. Our model and choice of parameters result in the formation of a wetting ridge and in the cloaking of the droplet by the lubricant, i.e. the lubricant chains creep up onto the droplet and eventually cover its surface completely. Cloaking is a phenomenon that is observed experimentally and is of integral importance to the dynamics of sliding droplets. We quantify the cloaking in terms of its thickness, which increases with the amount of lubricant present. The analysis reveals a well-defined transition point where the cloaking sets in. We propose a thermodynamic theory to explain this behavior. In addition we investigate the dependence of the contact angles on the size of the droplet and the possible effect of line tension. We quantify the variation of the contact angle with the curvature of the contact line on a lubricant free brush and find a negative value for the line tension. Finally we investigate the effect of cloaking/lubrication on the contact angles and the wetting ridge. We find that lubrication and cloaking reduce the contact angles by a couple of degrees. The effect on the wetting ridge is a reduction in the extension of the brush chains near the three phase contact line, an effect that was also observed in experiments of droplets on crosslinked gels.

The paper discusses an applicability criterion for a cutoff regularization in the coordinate representation in the Euclidean space with a dimension larger than two. It is shown that the set of functions satisfying the criterion is not empty. As an example, an explicit function is presented. It is proved by explicit construction that there are functions satisfying the criterion in a stronger formulation.

Understanding the dynamics of drops on polymer-coated surfaces is crucial for optimizing applications such as self-cleaning materials or microfluidic devices. While the static and dynamic properties of deposited drops have been well characterised, a microscopic understanding of the underlying dynamics is missing. In particular, it is unclear how drop dynamics depends on the amount of uncrosslinked chains in the brush, because experimental techniques fail to quantify those. Here we use coarse-grained simulations to study droplets moving on a lubricated polymer brush substrate under the influence of an external body force. The simulation model is based on the many body dissipative particle dynamics (mDPD) method and designed to mimic a system of water droplets on polydimethylsiloxane (PDMS) brushes with chemically identical PDMS lubricant. In agreement with experiments, we find a sublinear power law dependence between the external force $F$ and the droplet velocity $v$, $F \propto v^α$ with $α<1$; however, the exponents differ ($α\sim 0.6-0.7$ in simulations versus $α\sim 0.25$ in experiments). With increasing velocity, the droplets elongate and the receding contact angle decreases, whereas the advancing contact angle remains roughly constant. Analyzing the flow profiles inside the droplet reveals that the droplets do not slide, but roll, with vanishing slip at the substrate surface. Surprisingly, adding lubricant has very little effect on the effective friction force between the droplet and the substrate, even though it has a pronounced effect on the size and structure of the wetting ridge, especially above the cloaking transition.