Let $f(z)=\sum_{n\ge0}a_n z^n$ be a transcendental entire function and write $M(r,f):=\max_{|z|=r}|f(z)|$ and $μ(r,f):=\max_{n\ge0}|a_n|\,r^n$. A problem of Erdős asks for the value of $$ B:=\sup_f \liminf_{r\to\infty}\frac{μ(r,f)}{M(r,f)}. $$ In 1964, Clunie and Hayman proved that $\frac{4}{7}<B<\frac{2}π$. In this paper we develop a generalization of their construction via a scaling identity and obtain the explicit lower bound $$ B>0.58507, $$ improving the classical constant $\frac{4}{7}$.

Multicategory lattice data arise in a wide variety of disciplines such as image analysis, biology, and forestry. We consider modeling such data with the automultinomial model, which can be viewed as a natural extension of the autologistic model to multicategory responses, or equivalently as an extension of the Potts model that incorporates covariate information into a pure-intercept model. The automultinomial model has the advantage of having a unique parameter that controls the spatial correlation. However, the model's likelihood involves an intractable normalizing function of the model parameters that poses serious computational problems for likelihood-based inference. We address this difficulty by performing Bayesian inference through the Double-Metropolis Hastings algorithm, and implement diagnostics to assess the convergence to the target posterior distribution. Through simulation studies and an application to land cover data, we find that the automultinomial model is flexible across a wide range of spatial correlations while maintaining a relatively simple specification. For large data sets we find it also has advantages over spatial generalized linear mixed models. To make this model practical for scientists, we provide recommendations for its specification and computational implementation.

We introduce and study a novel generalization of the classical Knapsack Problem (KP), called the Colored Knapsack Problem (CKP). In this problem, the items are partitioned into classes of colors and the packed items need to be ordered such that no consecutive items are of the same color. We establish that the problem is weakly NP-hard and propose two exact dynamic programming algorithms with time complexities of $\mathcal{O}(bn^4)$ and $\mathcal{O}(b^2n^3)$, respectively. To enhance practical performance, we derive various dominance and fathoming rules for both approaches. From a theoretical perspective, we analyze the linear programming relaxation of the natural CKP formulation, proving that an optimal solution exists with at most two fractional items. We also show that the relaxation can be solved in $\mathcal{O}(n)$ time, matching the complexity of the classical KP. Finally, we establish a comprehensive benchmark of CKP instances, derived from the Colored Bin Packing Problem. Extensive computational experiments demonstrate that the proposed dynamic programming algorithms significantly outperform state-of-the-art MIP solvers on most of these instances.

We use explicit class field theory of rational function fields to prove a dynamical criterion for a polynomial of the form $x^{p^r}+ax+b$ over a field of characteristic $p$ to have dynamical Galois group as large as possible. When $p=2$ and $r=1$ this yields an analogue in characteristic $2$ of the celebrated criterion of Stoll for quadratic polynomials over fields of characteristic not $2$.

The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the topologies and the differentiable structures of the manifolds. We are concerned with round fold maps, defined as smooth maps locally represented as the product map of a Morse function and the identity map on a smooth manifold, and maps with singular value sets being concentric spheres. A bit different from differential topology, we are concerned with real algebraic geometric aspects of these maps. We discuss real algebraic realization of round fold maps of codimension $-1$ as our new work. Real algebraic realization of these maps is of fundamental and important studies in real algebraic geometry and a new study recently developing mainly due to the author.

We study the energy-momentum tensor of the spherically symmetric non-topological solitons of the $O(3)$ non-linear sigma-model with a standard kinetic term and with a symmetry breaking potential in 3+1 dimensional flat space-time. We evaluate the distributions of the corresponding energy density, shear forces and pressure and study the stability criteria for these solutions. We argue that the presence of domains with negative energy density and violation of the energy conditions most likely do not lead to destabilization of solitons.

We study the computational cost of differential privacy in terms of memory efficiency. While the trade-off between accuracy and differential privacy is well-understood, the inherent cost of privacy regarding memory use remains largely unexplored. This paper establishes for the first time an unconditional space lower bound for user-level differential privacy by introducing a novel proof technique based on a multi-player communication game. Central to our approach, this game formally links the hardness of low-memory private algorithms to the necessity of ``contribution capping'' -- tracking and limiting the users who disproportionately impact the dataset. We demonstrate that winning this communication game requires transmitting information proportional to the number of over-active users, which translates directly to memory lower bounds. We apply this framework, as an example, to the fundamental problem of estimating the number of distinct elements in a stream and we prove that any private algorithm requires almost $\widetildeΩ(T^{1/3})$ space to achieve certain error rates in a promise variant of the problem. This resolves an open problem in the literature (by Jain et al. NeurIPS 2023 and Cummings et al. ICML 2025) and establishes the first exponential separation between the space complexity of private algorithms and their non-private $\widetilde{O}(1)$ counterparts for a natural statistical estimation task. Furthermore, we show that this communication-theoretic technique generalizes to broad classes of problems, yielding lower bounds for private medians, quantiles, and max-select.

Let $p$ be a prime number, $F$ a field of characteristic $p$, and $G$ a cyclic group of order $q =p^a$ for some positive integer $a$. Under these circumstances every indecomposable $F G$-module is cyclic. For indecomposable $F G$-modules $U$ and $W$, we present a new recursive method for identifying a generator for each of the indecomposable components of $U \otimes W$ in terms of a particular $F$-basis of $U \otimes W$.

Holographic entanglement entropy is a key concept linking quantum information theory and gravity. Since the original conjecture of Ryu and Takayanagi, holographic entanglement entropy has been generalized beyond Einstein--Hilbert gravity to include higher-curvature corrections. In most existing generalizations, however, it is implicitly assumed that the bulk spacetime geometry is Riemannian, i.e. torsion-free. Here we propose a prescription for incorporating torsion into holographic entanglement entropy in the boundary theory dual to five-dimensional Chern--Simons gravity. We argue that the entanglement entropy acquires an additional universal divergent term proportional to the logarithm of the UV cutoff, and that this term is generated solely by torsion.

The Event Horizon Telescope (EHT) is providing unprecedented high-resolution images of supermassive black holes. These images are fundamentally related to properties of the luminous accretion disks, since black holes themselves produce no light. We develop a simple prescription to relate observational features of black hole images to a toy model for the intensity profile of the associated accretion disk. We apply our model to the original EHT image of M87*, as well as to the reanalyzed image from the PRIMO algorithm, providing generic, simultaneous constraints on the mass of the black hole and properties of the accretion disk emission. While current images lack the resolution to confidently detect the photon ring, the consideration of multiple image parameters are found to contain enough information to provide constraints on the inner edge of the accretion disk along with the black hole mass. Using observed features of the original EHT image, we constrain the mass of M87* to be $6.6^{+1.2}_{-1.0}\times 10^9 M_\odot$ to 68$\%$ confidence, and find that emission may extend all the way to the black hole horizon. When instead using constraints from the PRIMO algorithm's image along with constraints on the brightness asymmetry provided by the original EHT analysis, we find M87*'s mass to be $ 6.4^{+0.7}_{-0.7}\times 10^9 M_\odot$ to 68$\%$ confidence, with the inner edge of the accretion disk between $3M$ and $5.3M$. Both analyses rule out an inner edge of the accretion disk coinciding with the innermost stable circular orbit for a Schwarzschild black hole. Furthermore, the narrow ring width reported in the PRIMO image also confidently rules out emission increasing all the way down to the black hole horizon. Further assumptions on the mass of M87* and connections between the accretion disk cutoff and physical radii allow for rudimentary black hole spin estimates.

The higher connectivity at infinity for mapping class groups of surfaces with boundary components and punctures is understood with the exceptions of the mapping class groups for the closed surfaces of genus 3 and 4. In this paper we prove a general simply connected at infinity result for finitely presented groups that implies all mapping class groups of closed surfaces of genus $\geq 3$ are simply connected at infinity. As these groups are duality groups the Proper Hurewicz Theorem implies that they are $(n-2)$-connected at infinity where $n$ is the dimension of the group. Combining this result with earlier work we give a complete list of all mapping class groups and their connectivity at infinity.

We analyze increasing propagation of chaos in the high temperature regime of a disordered mean-field model, the Hopfield model. We show that for $β<1$ (the true high temperature region) we have increasing propagation of chaos as long as the size of the marginals $k=k(N)$ and the number of patterns $M=M(N)$ satisfies $Mk/N \to 0$. For $M=o(\sqrt N)$ we show that propagation of chaos breaks down for $k/N \to c>0$. At the ciritcal temperature we show that, for $M$ finite, there is increasing propagation of chaos, for $k=o(\sqrt N)$, while we have breakdown of propagation of chaos for $k=c \sqrt N$, for a $c>0$. All these reulst hold in probability in the disorder.

The field of the mathematical sciences relies on a continuous academic pipeline in which individuals progress from undergraduate study through graduate training and postdoctoral program to long term faculty employment. National statistics report trends in bachelor's, master's, and doctoral degree awards, but these data alone do not explain how individuals move through the academic system or how structural constraints shape downstream career outcomes. Persistent growth in postdoctoral appointments alongside relatively stable faculty employment indicates that degree production alone is insufficient to characterize workforce dynamics. In this study, we develop a discrete time compartmental model of the academic pipeline in the field of the mathematical sciences that links observed degree flows to latent population stocks. Undergraduate and graduate populations are reconstructed directly from nationally reported degree data, allowing postdoctoral and faculty dynamics to be examined under completion, exit, and hiring processes. Advancement to faculty positions is modeled as vacancy limited, with competition for permanent positions depending on downstream population size. Numerical simulations show that increases in degree inflow do not translate into proportional faculty growth when hiring is constrained by limited turnover. Instead, excess supply accumulates primarily at the postdoctoral stage, leading to sustained congestion and elevated competition. Sensitivity analyses indicate that long run workforce outcomes are governed mainly by faculty exit rates and hiring capacity rather than by degree production alone. These results demonstrate the central role of vacancy limited hiring in shaping academic career trajectories within the field of the mathematical sciences.

The charged moments of a reduced density matrix provide a natural starting point for deriving symmetry-resolved Rényi and entanglement entropies, which quantify how entanglement is distributed among symmetry sectors in the presence of a global internal symmetry in a quantum many-body system. In this work, we study charged moments within the framework of Ballistic Fluctuation Theory (BFT). This theory describes large-scale ballistic fluctuations of conserved charges and associated currents and, by exploiting the height-field formulation of twist fields, gives access to the asymptotic behaviour of their two-point correlation functions. In Del Vecchio Del Vecchio et al. $[1]$, this approach was applied to the special case of branch-point twist fields used to compute entanglement entropies within the replica approach. Here, we extend those results by applying BFT to composite branch-point twist fields, obtained by inserting an additional gauge field. Focusing on free fermions, we derive analytic expressions for charged Rényi entropies both at equilibrium, in generalized Gibbs ensembles, and out of equilibrium following a quantum quench from $U(1)$ preserving pair producing integrable initial states. In the latter case, our results agree with the conjecture arising from the quasiparticle picture.

The rapid evolution of cyberattacks continues to drive the emergence of unknown (zero-day) threats, posing significant challenges for network intrusion detection systems in Internet of Things (IoT) networks. Existing machine learning and deep learning approaches typically rely on large labeled datasets, payload inspection, or closed-set classification, limiting their effectiveness under data scarcity, encrypted traffic, and distribution shifts. Consequently, detecting unknown attacks in realistic IoT deployments remains difficult. To address these limitations, we propose SiamXBERT, a robust and data-efficient Siamese meta-learning framework empowered by a transformer-based language model for unknown attack detection. The proposed approach constructs a dual-modality feature representation by integrating flow-level and packet-level information, enabling richer behavioral modeling while remaining compatible with encrypted traffic. Through meta-learning, the model rapidly adapts to new attack types using only a small number of labeled samples and generalizes to previously unseen behaviors. Extensive experiments on representative IoT intrusion datasets demonstrate that SiamXBERT consistently outperforms state-of-the-art baselines under both within-dataset and cross-dataset settings while requiring significantly less training data, achieving up to \num{78.8}\% improvement in unknown F1-score. These results highlight the practicality of SiamXBERT for robust unknown attack detection in real-world IoT environments.

We extend the recent analysis of the rate-reliability tradeoff in deterministic identification (DI) to general linear Gaussian channels, marking the first such analysis for channels with continuous output. Because DI provides a framework that can substantially enhance communication efficiency, and since the linear Gaussian model underlies a broad range of physical communication systems, our results offer both theoretical insights and practical relevance for the performance evaluation of DI in future networks. Moreover, the structural parallels observed between the Gaussian and discrete-output cases suggest that similar rate-reliability behaviour may extend to wider classes of continuous channels.

Tomographic Volumetric Additive Manufacturing(TVAM) is a novel manufacturing method that allows for the fast creation of objects of complex geometry in layerless fashion. The process is based on the solidification of photopolymer that occurs when a sufficient threshold dose of light-energy is absorbed. In order to create complex shapes, an illumination plan must be designed to force solidification in some desired areas while leaving other regions liquid. Determining an illumination plan can be considered as an optimisation problem where a variety of objective functionals (penalties) can be used. This work considers a selection of penalty functions and their impact on selected printing metrics; linking the shape of penalty functions to ranges of light-energy dose levels in in-part regions that should be printed and out-of-part regions that should remain liquid. Further, the threshold parameters that are typically used to demarcate minimum light-energy for in-part regions and maximum light-energy for out-of-part regions are investigated systematically as design parameters on both existing and new methods. This enables the characterisation of their effects on some selected printing metrics as well as informed selection for default values. This work is underpinned by a reproducible and extensible framework, TVAM Adaptive Illumination Design(TVAM AID), which makes use of the open-source Core Imaging Library(CIL) that is designed for tomographic imaging with an emphasis on reconstruction. The foundation of TVAM AID which is presented here can hence be easily enhanced by existing functionality in CIL thus lowering the barrier to entry and encouraging use of strategies that already exist for reconstruction optimisation.

The Joint Replenishment Problem (JRP) is a classical inventory management problem, that aims to model the trade-off between coordinating orders for multiple commodities (and their cost) with holding costs incurred by meeting demand in advance. Moseley, Niaparast and Ravi introduced a natural online generalization of the JRP in which inventory corresponding to demands may be replenished late, for a delay cost, or early, for a holding cost. They established that when the holding and delay costs are monotone and uniform across demands, there is a 30-competitive algorithm that employs a greedy strategy and a dual-fitting based analysis. We develop a 5-competitive algorithm that handles arbitrary monotone demand-specific holding and delay cost functions, thus simultaneously improving upon the competitive ratio and relaxing the uniformity assumption. Our primal-dual algorithm is in the spirit of the work Buchbinder, Kimbrel, Levi, Makarychev, and Sviridenko, which maintains a wavefront dual solution to decide when to place an order and which items to order. The main twist is in deciding which requests to serve early. In contrast to the work of Moseley et al., which ranks early requests in ascending order of desired service time and serves them until their total holding cost matches the ordering cost incurred for that item, we extend to the non-uniform case by instead ranking in ascending order of when the delay cost of a demand would reach its current holding cost. An important special case of the JRP is the single-item lot-sizing problem. Here, Moseley et al. gave a 3-competitive algorithm when the holding and delay costs are uniform across demands. We provide a new algorithm for which the competitive ratio is $ϕ+1 \approx 2.681$, where $ϕ$ is the golden ratio, which again holds for arbitrary monotone holding-delay costs.

This manuscript is concerned with the one-dimensional system \[ \begin{array}{l} τu_{ttt} + αu_{tt} = b \big(γ(Θ) u_{xt}\big)_x + \big( γ(Θ) u_x\big)_x, \\[1mm] Θ_t = D Θ_{xx} + bγ(Θ) u_{xt}^2, \end{array} \] which is connected to the simplified modeling of heat generation in Zener type materials subject to stress from acoustic waves. Under the assumption that the coefficients $τ>0, b>0$ and $α\geq0$ satisfy \begin{align}\tag{$\star$} αb >τ, \end{align} it is shown that for all $Θ_\star>0$ one can find $ν=ν(D,τ,α,b,Θ_\star,γ)>0$ such that an associated Neumann type initial-boundary value problem with Neumann data admits a unique time-global solution in a suitable framework of strong solvability whenever the initial temperature distribution fulfills $$\|Θ_0\|_{L^\infty(Ω)}\leq Θ_\star$$ and the derivatives of the initial data are sufficiently small in the sense of satisfying $$\int_Ωu_{0xx}^2 + \int_Ω(u_{0t})_{xx}^2 + \int_Ω(u_{0tt})_x^2 < ν\quad\text{and}\quad \|Θ_{0x}\|_{L^\infty(Ω)} + \|Θ_{0xx}\|_{L^\infty(Ω)} < ν.$$ The constructed solution moreover features an exponential stabilization property for both components. In particular, the parameter range described by ($\star$) coincides with the full stability regime known for the corresponding Moore--Gibson--Thompson equation despite the fairly strong nonlinear coupling to the temperature variable.

Gamma-hadron discrimination based on shower observables is essential for identifying gamma-ray astrophysical sources at the highest energies. In this work, we introduce $P^{α, T}_{\rm tail}$, a new discrimination variable for ultra-high-energy photon searches within the framework of a water-Cherenkov detector (WCD) array. The observable extends signal-integrated methods by incorporating the time structure of WCD traces, using cumulative signal distributions. Using simulated proton- and gamma-induced air showers at energies around $10^{17}\,\mathrm{eV}$, we evaluate the performance of $P^{α, T}_{\rm tail}$ and compare it with established WCD-based observables such as $S_b$, risetime-based variables, and the SWGO-inspired, $P^α_{\rm tail}$. The new variable attains a background contamination of roughly $2 \times 10^{-2}$ at $50\%$ gamma efficiency, improving upon existing WCD-only methods by nearly a factor of five and approaching the performance of an idealized muon-isolating reference. These results demonstrate the effectiveness of exploiting time-resolved signal tails to enhance ultra-high-energy photon searches in sparse surface arrays.

The amplification of magnetic fields in black hole accretion flows governs key high-energy phenomena such as magnetically arrested disks and relativistic jets. We develop a semi-analytical general relativistic framework that extends classical compressional amplification models by incorporating rotational shear, and apply it to large-scale poloidal magnetic field evolution in accretion flows around a Schwarzschild black hole. By parameterizing the azimuthal velocity as a fraction of the Keplerian value ($ξ\in [0,1]$), from purely radial infall ($ξ=0$) to Keplerian rotation ($ξ=1$), we examine the combined effects of radial compression and shear. Purely radial flows maximize amplification of both $B_r$ and $B_θ$ due to strong compression. In rotating flows, a distinct dichotomy emerges: sub-Keplerian regimes ($ξ<1$) preferentially enhance $B_r$, whereas Keplerian rotation strengthens $B_θ$ via shear. The transition from subsonic outer regions to supersonic relativistic inner regions further accelerates magnetic growth, revealing effects absent in earlier analytical treatments. These results show that rotational support controls both amplification efficiency and magnetic geometry, with sub-Keplerian phases particularly favorable for advecting the radial flux required for MAD formation. This work provides an analytical bridge between classical accretion theory and modern GRMHD simulations, with implications for X-ray binaries, AGNs, and EHT-scale systems.

In this paper, we establish a probabilistic global theory in $H^1$ for the NLS with a Moser-Trudinger nonlinearity posed on compact surfaces. This equation is known to be the two dimensional counterpart to the classical energy-critical Schrödinger equations \cite{CollianderIbrahimMajdoubMasmoudi2009}. The authors of \cite{CollianderIbrahimMajdoubMasmoudi2009} also identified a trichotomy around the criticality of the equation based on the size of the total energy. In particular, for supercritical regimes (large energy), the equation is known to exhibit instabilities : the (uniform) continuity of the flow fails to hold. Large data distributional non unique probabilistic solutions have been obtained in \cite{CasterasMonsaingeon2024}. The setting of \cite{CasterasMonsaingeon2024} does not handle the uniqueness issue for the $H^1$-data and therefore could not define a flow for this regularity. Our main focus here is to build a single probabilistic framework that provides both existence, uniqueness, and continuity with respect to the initial data in $H^1$. Our uniqueness and continuity are based on the so-called Yudowich argument \cite{Judovic1963}, and the probabilistic estimates are derived through the IID limit procedure \cite{Sy2019}. Beyond the difficulties related to the borderline nature of the context, the major challenge resides in the need to satisfy two features that tend to play against each other : obtaining both continuity property of the flow and large data in the support of the reference measure. This made the design of the dissipation operator inherent in the method, as well as the analysis of the resulting quantities, particularly difficult. Regarding the supercritical regime, we show that a modified energy, with regularity similar to the original total energy, admits values as high as desired, suggesting that the constructed set of data contains supercritical ones.

Coherent perfect tunneling in the presence of loss and asymmetry remains a fundamental challenge in wave transport, a universal problem across optics, acoustics, and quantum mechanics. Here we demonstrate coherent perfect tunneling at an exceptional point in a passive one-dimensional waveguide cascade with three coupled interfaces. Using a waveguide-invariant scattering framework, we show that the suppression of a selected output channel originates from a directional scattering degeneracy rather than from resonance or absorption collapse. This exceptional-point condition emerges when interference between boundary-induced feedback loops promotes a simple zero of the scattering response to a second-order degeneracy. As a direct consequence, fixed coherent excitation produces a robust quartic leakage law within a transparency-dominated tunneling window. These results establish directional degeneracy as a general mechanism for loss-tolerant tunneling enabled by exceptional points across a broad class of wave systems.

Deterministic preparation of high-excitation Fock states is a central challenge in bosonic quantum information, with control complexity that generically explodes as the Hilbert space dimension grows. Here we introduce resonant subspace engineering (RSE), a protocol that analytically confines the infinite-dimensional bosonic dynamics to a two-dimensional invariant subspace spanned by an initial coherent state and the target state. State transfer then reduces to a geodesic rotation on a synthetic Bloch sphere, governed by resonance and phase-matching conditions we derive in closed form. For single Fock states, RSE achieves $O(n^{1/4})$ scaling in both evolution time and gate depth, showing a fundamental improvement over existing deterministic schemes. The construction generalizes to $K$-component superpositions via a $(K{+}1)$-dimensional invariant subspace with full $\mathrm{SU}(K{+}1)$ controllability, requiring only 3-5 iterations of operations for superpositions spanning photon numbers 70--100. RSE provides a scalable and analytically transparent framework for large-scale bosonic state engineering and gate synthesis across single- and multimode platforms.

In Universal Extra Dimension (UED) scenarios, the lightest Kaluza-Klein (KK) particle is naturally stable due to a remnant discrete symmetry, KK parity, arising from extra-dimensional compactification. This stability requires no ad hoc symmetry and renders Kaluza-Klein dark matter a well-motivated candidate, provided it reproduces the observed relic abundance. The minimal UED (mUED) framework being highly predictive is strongly constrained by the combined requirements of relic density and collider searches under standard cosmological assumptions. We revisit the dark matter phenomenology of mUED in the presence of a nonstandard cosmological history featuring a low reheating temperature driven by prolonged inflaton decay. Solving the coupled Boltzmann equations for dark matter, radiation, and inflaton energy densities, we show that entropy injection during reheating can dilute the relic abundance by orders of magnitude, reopening large regions of parameter space previously ruled out. We further demonstrate that the revived parameter space is consistent with current collider, direct-detection, and indirect-detection constraints, while remaining testable by upcoming experiments.

Scalable quantum computers and quantum networks require the combination of quantum processing nodes with efficient light-matter interfaces to distribute quantum information in local or long-distance quantum networks. Neutral-atom arrays have both been coupled to Rydberg states to enable high-fidelity quantum gates in universal processing architectures, and to optical cavities to realize interfaces to photons. However, combining these two capabilities and coupling atom arrays to highly excited Rydberg states in the mode of an optical cavity has been an outstanding challenge. Here we present a novel cavity-coupled Rydberg array that achieves this long-standing goal. We prepare, detect, and control individual atoms in a scalable optical tweezer array, couple them strongly to the optical mode of a high-finesse optical cavity and excite them in a controlled way to Rydberg states. We show that strong coupling to an optical cavity - demonstrated via the dispersive shift of the resonance of the cavity in presence of the atoms - and strong Rydberg interactions - demonstrated via the collective enhancement of Rydberg coupling in the atomic array - can be achieved in our setup at the same spatial location. Our presented experimental platform opens the path to several new directions, including the realization of quantum network nodes, quantum simulation of long-range interacting, open quantum systems and photonic-state engineering leveraging high-fidelity Rydberg control.

Serving Large Language Models (LLMs) can benefit immensely from parallelizing both the model and input requests across multiple devices, but incoming workloads exhibit substantial spatial and temporal heterogeneity. Spatially, workloads comprise heterogeneous requests with varying compute and memory demands. Temporally, workload composition varies over time. Nevertheless, existing systems typically assume spatially uniform and temporally stable workloads, employing a homogeneous, static model deployment. This mismatch between the assumption and real-world spatial-temporal heterogeneity results in suboptimal performance. We present OServe, an LLM serving system with heterogeneous and flexible model deployment that addresses both spatial and temporal heterogeneity. First, OServe introduces a novel workload-aware scheduling algorithm that optimizes heterogeneous model deployments according to real-time workload characteristics. Second, OServe proposes an efficient workload-adaptive switching method that migrates model deployments in response to predicted workload changes. Experiments on real-world traces show that OServe improves performance by up to 2$\times$ (average: 1.5$\times$) compared to state-of-the-art serving systems.

In the context of the category $\mathsf{Cap}$ of convergence approach spaces and contractions, we introduce and study approach analogs of the upper and lower Kuratowski convergences, upper-Fell and Fell topologies on the set of closed subsets of the coreflection on the category $\mathsf{Conv}$ of convergence spaces of a convergence approach space. In particular, over a pre-approach space, the $\mathsf{Conv}$-coreflection of the lower Kuratowski convergence approach structure is the lower Kuratowski convergence associated with the $\mathsf{Conv}$-coreflection of the base space, while the $\mathsf{Conv}$-reflection is the lower Kuratowski convergence associated with the $\mathsf{Conv}$-reflection. The $\mathsf{Conv}$-coreflection of the upper Kuratowski convergence approach is is the upper Kuratowski convergence associated with the $\mathsf{Conv}$-reflection of the base space, while the $\mathsf{Conv}$-reflection is the upper Kuratowski convergence associated with the $\mathsf{Conv}$-coreflection of the base space. We show that, over an approach space, the lower Kuratowski convergence approach structure is in fact an approach structure that coincides with the $\vee$-Vietoris approach structure introduced by Lowen and his collaborators, though it may be strictly finer over a general convergence approach space. We show that the upper Fell convergence approach structure is a non-Archimedean approach structure coarser than the upper Kuratowski convergence approach, but finer than the upper Fell approach structure introduced by the first and third author. We also obtain a $\mathsf{Cap}$ abstraction of the classical result that if the upper Kuratowski convergence over a topological space is pretopological, then it is also topological.

The resonating group method (RGM) is widely used to investigate baryon-baryon interactions at the quark level, typically under the assumption that the two baryons involved share an identical harmonic-oscillator frequency. In reality, however, when a specific interaction Hamiltonian is given, different baryons should have unequal oscillator frequencies due to distinct interaction potentials induced by their different quantum numbers. In this work, we develop a new quark-level RGM formalism for baryon-baryon systems with unequal oscillator frequencies, with the aim of describing both single baryons and baryon-baryon interactions in a consistent framework. We present the formalism for solving bound-state and scattering problems, with particular emphasis on constructing the wave functions of two-baryon systems with unequal oscillator frequencies. The proposed formalism is then applied to the $NΔ$ system within a chiral SU(3) quark model, where the quark-quark interaction includes, in addition to the one-gluon exchange (OGE) and a phenomenological confinement potential, the nonet scalar and pseudoscalar meson exchanges arising from the spontaneous breaking of chiral SU(3) symmetry. The distinctive features of the newly developed formalism are elucidated by comparing the results from the new formulation with those from traditional calculations.

Spin networks in Loop Quantum Gravity are traditionally described by unitary holonomies corresponding to noiseless transformations. In this work, we extend this framework to incorporate general quantum channels that model effects of environment, which can become significant at the Planck scale. Specifically, we demonstrate that the transformation properties of Kraus operators, which define completely positive trace-preserving (CPTP) maps, are consistent with the gauge invariance of spin networks. This enables the introduction of generalized spin network states that can be expressed in terms of the Kraus operators. Furthermore, the associated notion of an inner product is proposed, allowing for introduction of the Hilbert space. We illustrate these constructions with examples involving a Wilson loop and a dipole network.