We establish a general min-max type theorem that produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature. An important intermediate step is to show that, in a generic metric with positive Ricci curvature, any family of smooth embedded surfaces, possibly with finitely many singularities, can be deformed into a certain topologically optimal family. Results in this paper will be crucial to our program on the construction of multiple minimal surfaces with prescribed genus in 3-spheres via topological methods.

A short proof of convergence for the discretization of the Hodge-Dirac operator in the framework of discrete exterior calculus (DEC) is provided using the techniques established in [Johnny Guzmán and Pratyush Potu, A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms, arXiv:2505.08934, 2025]

The Laurent series expansions of zeta-functions play an important role in understanding their behavior near singularities, and their coefficients often encode significant arithmetic information. In the case of the Riemann and Hurwitz zeta-functions, these coefficients are given by the Euler-Stieltjes constants and their generalizations. In this paper, we investigate the Laurent series expansions of the Barnes double zeta-function $ζ_2(s,α;v,w)$ at the singular points $s=1$ and $s=2$. We derive explicit limit expressions for the Laurent coefficients, providing analogues of the Euler-Stieltjes constants in this setting. In particular, we obtain representations of the coefficients in terms of finite double sums together with logarithmic correction terms. Furthermore, we study the asymptotic behavior of the Laurent coefficients and show that, in contrast to the Hurwitz zeta-function, they exhibit a much simpler structure. These results provide a clearer understanding of the analytic structure of the Barnes double zeta-function and highlight notable differences from the classical theory.

As a special case of Bass' theory of perfect rings, one obtains the assertion that, over a finite-dimensional associative algebra over a field, all flat modules are projective. In this paper we prove the following relative version of this result. Let $R\rightarrow A$ be a homomorphism of associative rings such that $A$ is a finitely generated projective right $R$-module. Then every flat left $A$-module is a direct summand of an $A$-module filtered by $A$-modules $A\otimes_RF$ induced from flat left $R$-modules $F$. In other words, a left $A$-module is cotorsion if and only if its underlying left $R$-module is cotorsion. The proof is based on the cotorsion periodicity theorem.

It is known that swift heavy ion (SHI) irradiation at temperatures below $\sim$1000 K does not cause structural damage in the bulk SiC. However, the effect of the SiC film thickness on the formation and structure of SHI tracks over a wide range of irradiation temperatures remains unexplored. To address this gap, we used a model sensitive to irradiation temperature that describes all stages of ion track formation: from material excitation, considering the emission of excited electrons from the film surface (MC code TREKIS-3), to the reaction of the material's atomic system to the excitation (classical molecular dynamics). We observed the appearance of two different types of nanostructures on the surface of SiC films with thicknesses ranging from 10 nm to 100 nm after impacts of 710 MeV Bi ions: craters and hillocks. The transition from hillocks to craters occurs with the irradiation temperature. The transition temperature increases with the film thickness, tending to $\approx 1534 \pm 100$ K for the surface of bulk SiC.

In this paper, we introduce and study multiple $\wp$-functions, which generalize the classical Weierstrass $\wp$-function to iterated sums over lattice points, and we establish explicit formulas expressing them in terms of single $\wp$-functions with coefficients given by multiple Eisenstein series. As an application, we derive some relations among multiple Eisenstein series and multiple zeta values by exploiting the double periodicity of the multiple $\wp$-functions.

Over the past two centuries, the frequency of word usage in major Western languages has exhibited small amplitude regular cycles, superimposed on larger background trends. We show that these cycles of word usage organize into semantically coherent clusters that track, interact with, or precede macroeconomic rhythms. To explore the nature of this interaction, we build on a minimal mathematical model that captures usage dynamics through parameters linked to real world processes. Parameter fitting reveals that word usage operates near a Hopf bifurcation, a critical regime associated with heightened sensitivity, uncovering a robust and specific coupling between language usage and economic activity. In linguistic datasets that allow long term quantitative analysis, rooted in capitalist contexts, this coupling appears as an oscillatory dynamic, reflecting a feedback process in which language both shapes material reality and manages the changes it produces.

Lightweight, high-strength structural materials are component enablers in transportation and aerospace, reducing carbon footprints and enhancing fuel efficiency. Cast aluminium alloys, mainly based on eutectic compositions, make up 85% of these materials but often fail catastrophically due to inefficient load transfer across the interfaces between the brittle eutectic phase and the ductile matrix. Here, we discovered that promoting a superlattice nano-layer (SNL) around the eutectic fibres, achieved by adding Zr to an Al-Gd near-eutectic alloy, enables excellent load transfer capabilities, resulting in a 400% increase in tensile ductility. The primary Al matrix also contains a high number density of superlattice core-shell nano-particles. This exceptional increase in formability is attributed to the ability of the SNL to prevent dislocations from accumulating at the weak and brittle eutectic fibre/matrix interfaces, thereby avoiding stress concentrations that would otherwise initiate fibre breakage and debonding. The core-shell nano-particles in Al cause a large number of dislocation cross/multiple-slips on {111} planes, forming ultra-fine (12 nm) dislocation networks that leverage substantial plastic strain accumulation. This atomic interface design overcomes the ductility limitations of cast-eutectic alloys, enabling them for structural applications.

We show that, under suitable conditions, finite-dimensional systems describing invariant solutions of partial differential equations (PDEs) inherit local Hamiltonian operators through the mechanism of invariant reduction, which applies uniformly to point, contact, and higher symmetries. The inherited operators endow the reduced systems with Poisson bivectors that relate constants of motion to symmetries. Applying the same mechanism to invariant conservation laws, we further show that the induced Poisson brackets agree with those of the original systems, up to sign. The results are illustrated by two examples in which the inherited Poisson brackets and inherited constants of motion yield integrability of the reduced systems. The construction is independent of the choice of an $\ell$-normal inclusion of a PDE system into jet spaces.

This paper concerns supersonic flows with nonzero vorticity governed by the steady Euler-Poisson system, under the coupled effects of the electric potential and the geometry of a convergent nozzle. By the coordinate rotation, the existence of radially symmetric supersonic flows is proved. We then establish the structural stability of these background supersonic flows under multi-dimensional perturbations of the boundary conditions. One of the crucial ingredients of the analysis is the reformulation of the steady Euler-Poisson system into a deformation-curl-Poisson system and several transport equations via the deformation-curl-Poisson decomposition. Another one is to obtain the well-posedness of the boundary value problem for the associated linearized hyperbolic-elliptic coupled system, which is achieved through a delicate choice of multiplier to derive a priori estimates. The result indicates that the electric field force in compressible flows can counteract the geometric effects of the convergent nozzle and thereby stabilize key physical features of the flow.

We investigate how structural ordering, i.e. crystallization, affects the flow of bidisperse granular materials in a quasi-two-dimensional silo. By systematically varying the mass fraction of two particle sizes, we finely tune the degree of local order. Using high-speed imaging and kinematic modeling, we show that crystallization significantly enhances the diffusion length $b$, a key parameter controlling the velocity profiles within the flowing medium. We reveal a strong correlation between $b$ and the hexatic order parameter $\left<|ψ_6|\right>_t$, highlighting the role of local structural organization in governing macroscopic flow behavior. Furthermore, we demonstrate that pressure gradients within the silo promote the stabilization of orientational order even in the absence of crystallization, thus intrinsically increasing $b$ with height. These findings establish a direct link between microstructural order, pressure, and transport properties in granular silo flows.

We present new and updated methods for reducing transport caused by electrostatic ion temperature gradient (ITG) driven turbulence in quasi-isodynamic (QI) configurations. We first show an updated model for the threshold (critical) gradient of localized toroidal ITG modes. It is then argued that it is desirable for ITG modes to "split" and localize in separate curvature drift wells, which is leveraged to produce a six-field-period QI configuration with a high critical gradient. We show that the destabilizing effect of kinetic electrons (Costello & Plunk 2025a) on localized ITG modes can be minimized in a magnetic field structure, which we dub "inverse mirror". Applying a general optimization target that improves ITG stability above the critical gradient yields an inverse mirror configuration, which produces heat fluxes below or equal to the W7-X high mirror configuration for a range of applied density gradients.

3C403 is a well-known FRII radio galaxy with jets extending up to kiloparsec scales. We report its identification as the second most significant candidate among more than 150 sources examined using the 15-year neutrino dataset from the ANTARES Collaboration, making it one of the most promising radio-galaxy candidates for high-energy neutrino emission. Motivated by previous associations between blazars and neutrino events, we investigated the jet properties of 3C403 and their possible role in neutrino production. Multi-scale radio observations, from parsec to kiloparsec scales, reveal a stable, two-sided jet lying close to the plane of the sky, with no evidence of strong Doppler boosting, while X-ray data indicate a dominant, heavily absorbed accretion-related component. We also examined the recently proposed correlation between neutrino and hard X-ray fluxes - originally identified in blazars and Seyfert galaxies - and find that 3C403 occupies an intermediate location in the $L_ν$--$L_{\rm hX}$ plane between jet-dominated and corona-dominated systems. However, the current upper limit on its neutrino flux prevents a firm assessment of whether it follows the proposed relation. With radiatively efficient accretion ($λ_{\rm Edd}\sim10^{-2}$), strong hard X-ray emission, and a powerful but misaligned jet, 3C403 provides a physically motivated laboratory for exploring the interplay between coronal activity and jet environments in multimessenger scenarios of neutrino production in active galaxies.

Efficient operation of quantum batteries requires not only fast energy transfer but also the ability to halt the charging process to prevent reverse flow. Existing approaches typically rely on direct control of the charger-battery interaction, which can be experimentally demanding. Here we propose a modulator-assisted quantum battery protocol that enables indirect control of energy transfer while keeping the interaction always on. By applying repeated local unitary operations to an auxiliary modulator qubit, we exploit a Zeno-like mechanism to dynamically reshape the effective Hamiltonian and switch the charger-battery coupling on and off. We demonstrate this mechanism in a minimal three-body model and show that it remains effective beyond the ideal fast-control limit. We further extend the protocol to a collective many-body architecture, where it preserves the characteristic enhancement of charging power, scaling as $N^{3/2}$ with the number of battery units. Our results establish modulator-assisted Zeno control as a scalable route to regulating energy transfer in quantum batteries, and we further discuss a possible implementation in an NV-$\Cs$ spin platform.

Adaptive control in biological systems, such as intestinal immunity, remains poorly understood despite detailed knowledge of underlying regulatory networks. We propose an alternative framework based on stochastic martingale turnover, in which cells proliferate through mutual competition and decay without cell-type-specific regulation. Through stochastic simulations and mathematical analysis, we show that this process autonomously generates balanced population compositions associated with low decay probabilities. The compositional dynamics can be described as a random walk whose step lengths decrease in low-decay regions. Reduced decay leads to larger total population sizes and an increase in the number of compatible microscopic states, which in turn shapes the distribution of compositions under fluctuating conditions. More generally, the dynamics follow a modified Langevin equation, in which constant mass is replaced by a fitness-dependent effective mass proportional to the total population size. Thus, biological systems regulate resistance to change, not merely direction, in shaping their macroscopic behavior.

The Carroll group arises in the vanishing speed of light limit of the Poincaré group and was initially discarded as just a mathematical curiosity. However, recent developments have proved otherwise. Carroll and conformal Carroll symmetries are now ubiquitous, appearing in diverse physical phenomena starting from condensed matter physics to quantum gravity. This review aims to provide the reader a gateway into this fast-developing field. After an introduction and setting the stage with basics of the symmetry in question, we detail the construction of Carrollian and Carrollian Conformal field theories (CCFT). We then focus on applications. By far the most popular of these applications is in the context of the construction of holography in asymptotically flat spacetimes (AFS) in terms of a co-dimension one dual CCFT. We review the early work on AFS$_3$ /CCFT$_2$ before delving into an in-depth analysis for the construction of the dual to 4D AFS. Two other important sets of applications are in hydrodynamics and in condensed matter physics, which we discuss in detail. Carroll hydrodynamics is introduced as the $c\to 0$ limit of relativistic hydrodynamics first and then reconstructed from a symmetry based approach. Relations to ultrarelativistic flows and connections to the quark-gluon plasma are discussed with concrete examples of the Bjorken and Gubser flow models. In condensed matter applications, we cover connections to fractons, flat bands, and phase separation in Luttinger liquid models. To conclude, we give very brief outlines of other topics of interest including string theory and black hole horizons.

Accurate volumetric velocity estimation is crucial in ultrasound imaging for both diagnostic and therapeutic applications. Traditional ultrasound systems, though effective for two-dimensional imaging, face major limitations in 3D imaging due to hardware and computational demands. Row-column addressed (RCA) ultrasound probes offer a promising alternative by reducing hardware complexity, thereby reducing the gap between research prototypes and clinical systems. However, this typically comes at the expense of stronger sidelobes compared with fully populated matrix arrays, leading to reduced image contrast. Several approaches have been proposed to improve the contrast of power Doppler imaging, yet the accuracy and performance of velocity Doppler estimation have received comparatively little attention. In this study, we present a method that exploits the phase information from RCA row and column signals to derive a novel velocity estimator based on cross-correlation of orthogonal apertures. This extends the XDoppler scheme, initially developed for power Doppler imaging, to velocity estimation. The XDoppler estimator is shown to provide accurate measurements of axial velocities and to outperform the traditional phase-shift autocorrelator, while offering a theoretical Nyquist velocity twice as high. In vitro experiments further demonstrate enhanced sensitivity to slow flows and reduced bias in flow rate estimation. In vivo data from a carotid artery confirm the reduced sensitivity to aliasing and reveal the ability to track dynamic blood flow velocity changes associated with arterial pulsatility. These findings suggest that the XDoppler velocity estimator could improve volumetric velocity imaging in clinical contexts.

This survey synthesizes the principal descriptive set-theoretic perspectives on deterministic Cantor sets on the real line and charts directions for future study. After recounting their historical genesis and compiling an up-to-date taxonomy, we review the Borel hierarchy and four hierarchically ordered representations-general, nested, iterated-function-system (IFS), and q-ary expansion-presented from the most general to the most specific set-theoretic description of deterministic Cantor sets. We then present explicit and recursive descriptions for two thin families of measure-zero Cantor sets and an augmented "tick" family of positive measure, respectively, showing that the classical middle-third set lies in the intersection of all three families of after-mentioned Cantor sets. The survey closes by isolating several open problems in four directions, aiming to provide mathematicians with a coherent platform for further descriptive set-theoretic investigations into Cantor-type sets on the real line.

Elemental abundances hold important information about the star formation history in the Galactic Center. The thermal X-ray spectra of certain stars can provide a robust probe of elemental abundances, mainly through the presence of K-shell emission lines. In this work, based on deep archival {\it Chandra} observations, we obtain X-ray measurements of five heavy elements (Si, S, Ar, Ca and Fe) for three sources in the Arches cluster, one source in the Quintuplet cluster, as well as a field source known as Edd 1, which are all probable WR stars exhibiting a high quality X-ray spectrum. A two-temperature, non-equilibrium ionization plasma model is employed for the spectral fit, taking into account light element compositions characteristic of WR star winds, which is substantially depleted in hydrogen but enriched in nitrogen and/or carbon. It is found that the Arches and Quintuplet WR stars share similar abundances of Si, S, and Ar, while exhibiting distinct Ca and Fe abundances, which may be understood as due to dust depletion of the latter two elements in Quintuplet. The observed near-solar or sub-solar metallicity of the WR star winds can be naturally understood as the result of nucleosynthesis and internal mixing of the parent star, which have a supersolar initial metallicity as expected for the Galactic center in general. Implications of our findings on the origin of the young star clusters and isolated massive stars in the Galactic center, as well as the elemental composition of the accretion flow onto Sgr A*, are addressed.

Modern cybersecurity requires systematic ways to evaluate how detection systems respond to evolving and previously unseen attack behaviors. Existing malware repositories largely capture known patterns and provide limited support for stress-testing defenses against novel threats. To address this, we present MalGEN, a modular testbed that models adversarial workflows and generates executable artifacts in a controlled environment. The framework decomposes high-level attack objectives into structured stages, enabling the synthesis of diverse and multi-stage behaviors. We evaluate MalGEN across 1,920 benchmark settings covering multiple platforms and behavioral objectives, resulting in 977 executable samples. Analysis shows that the generated artifacts exhibit a wide range of malicious techniques and multi-stage attack patterns. However, 45.71% of these samples remain undetected by existing detection engines, which reveals notable gaps in current defenses. These findings provide practical insights into the limitations of widely used detection approaches and support the development of more robust security evaluation and testing practices.

This paper develops a synthetic framework for the geometric and analytic study of null (lightlike) hypersurfaces in non-smooth spacetimes. Drawing from optimal transport and recent advances in Lorentzian geometry and causality theory, we define a synthetic null hypersurface as a triple $(H, G, \mathfrak{m})$: $H$ is a closed achronal set in a topological causal space, $G$ is a gauge function encoding affine parametrizations along null generators, and $\mathfrak{m}$ is a Radon measure serving as a synthetic analog of the rigged measure. This generalizes classical differential geometric structures to potentially singular spacetimes. The central object is the synthetic null energy condition ($\mathsf{NC}^e(N)$), defined via the concavity of an entropy power functional along optimal transport, with parametrization given by the gauge $G$. This condition is invariant under changes of gauge and measure within natural equivalence classes. It agrees with the classical Null Energy Condition in the smooth setting and it applies to low-regularity spacetimes. A key property of the $\mathsf{NC}^e(N)$ condition is the stability under convergence of synthetic null hypersurfaces. The $\mathsf{NC}^e(N)$ condition is also remarkably fruitful for applications. First, it provides a framework for a synthetic version of Hawking's area theorem. Second, the celebrated Penrose's singularity theorem is proved for continuous spacetimes, and the existence of trapped regions is settled in the general setting of topological causal spaces satisfying the $\mathsf{NC}^e(N)$.

We construct explicit integrators of arbitrary even orders of accuracy for massive point vortex dynamics in binary mixture of Bose--Einstein condensates proposed by Richaud et al. The integrators are symplectic and preserve the angular momentum of the system exactly. Our main focus is the small-mass regime in which the minor component of the binary mixture comprises a very small fraction of the total mass. The solution behaviors in this regime change significantly depending on the initial momenta: they are highly oscillatory unless the momenta satisfy certain conditions. The standard Runge--Kutta method performs very poorly in preserving the Hamiltonian showing a significant drift in the long run, especially for highly oscillatory solutions. On the other hand, our integrators nearly preserve the Hamiltonian without drifts. We also give an estimate of the error in the Hamiltonian by finding an asymptotic expansion of the modified Hamiltonian for our second-order integrator.

The rapidly evolving cryptocurrency market presents unique challenges for investment due to its inherent volatility and evolving regulatory environment. Collective price movements can be exploited to construct diversified portfolios with improved risk-return profiles. This paper introduces an integrated framework that combines network analysis, price forecasting, and portfolio theory to identify stable groups of highly correlated cryptocurrencies for profitable portfolio construction. We employ the Louvain community detection algorithm together with consensus clustering to extract temporally persistent correlation clusters, and incorporate ARIMA-based price forecasts to enhance forward-looking cluster formation. Using 5 years of daily closing prices, we evaluate portfolio performance across multiple strategies and holding horizons, assessing both profitability and downside risk with return-based and tail-risk metrics. Our empirical results show that predictive consensus-clustering portfolios maintain consistently positive and stable performance up to a 14-day horizon, exhibit favourable gain-loss asymmetry, and achieve tighter tail-risk control. These findings demonstrate that stable interdependencies in cryptocurrency markets can be leveraged to construct profitable and risk-aware portfolios across short-term holding horizons.

Multi-agent reinforcement learning (MARL) has achieved significant progress in large-scale traffic control, autonomous vehicles, and robotics. Drawing inspiration from biological systems where roles naturally emerge to enable coordination, role-based MARL methods have been proposed to enhance cooperation learning for complex tasks. However, existing methods exclusively derive roles from an agent's past experience during training, neglecting their influence on its future trajectories. This paper introduces a key insight: an agent's role should shape its future behavior to enable effective coordination. Hence, we propose Role Discovery and Diversity through Dynamics Models (R3DM), a novel role-based MARL framework that learns emergent roles by maximizing the mutual information between agents' roles, observed trajectories, and expected future behaviors. R3DM optimizes the proposed objective through contrastive learning on past trajectories to first derive intermediate roles that shape intrinsic rewards to promote diversity in future behaviors across different roles through a learned dynamics model. Benchmarking on SMAC and SMACv2 environments demonstrates that R3DM outperforms state-of-the-art MARL approaches, improving multi-agent coordination to increase win rates by up to 20%. The code is available at https://github.com/UTAustin-SwarmLab/R3DM.

Alzheimer's Disease Neuroimaging Initiative (ADNI) diagnostic groups present strong heterogeneous associations among demographic, imaging, and cognitive data. We propose a novel PArtially-shared Imaging Regression (PAIR) model to represent imaging coefficients as weighted combinations of smooth spatial components. A Total Variation penalty is applied to enforce spatial smoothness, and a Selective Integration penalty is introduced to adaptively learn partial-sharing structures across groups. Theoretically, we establish minimax-optimal error bounds that dynamically adapt to varying sharing paradigms. Numerically, PAIR achieves predictive accuracy comparable to advanced deep learning models while providing superior interpretability. Applied to ADNI data, PAIR reveals substantial heterogeneity in brain-cognition pathways between cognitively normal (CN) and cognitively impaired (CI) groups, with hippocampal imaging contributing minimally in the CN group but substantially in the CI group, particularly in the CA1, CA3, and presubiculum subfields.

The photon statistics of Spontaneous Parametric Down Conversion (SPDC) exhibit dependencies on wavelength, pump power, and coincidence time. Notably, the average photon numbers were found to asymmetrically increase with increasing pump power around the degenerate wavelength of emission. By the coupling of the detection scheme to a spectrometer, studying different bandwidths within the emission revealed that shorter wavelengths increased nonlinearly with pump power, while longer wavelengths showed more linear behavior, indicating a wavelength dependent efficiency in the generation of the SPDC. We employ the use of a four detector Hanbury Brown and Twiss Interferometer to study the photon statistics of the signal beam, where the idler serves as the herald. The measured statistics were found to be best described by a Negative Binomial Distribution, which is a characteristic of thermal light sources. The detection and characterization of complex light sources has wide ranging applications in the fields of quantum metrology, quantum communications, and quantum computing, more specifically, a system that is sensitive to wavelength and photon number distribution.

The hydroxyl radical, hereafter referred to as the OH molecule (OHM), carries both electric and magnetic dipole moments and, as a diatomic molecule, admits a comparatively simple and accurate model. This makes it a natural quantum probe for the joint estimation of electric and magnetic fields. Here we study simultaneous estimation of both fields using the tools of multiparameter quantum estimation theory, explicitly accounting for the performance loss caused by measurement incompatibility. We analyze and optimize both stationary and dynamical estimation strategies. In the stationary regime we consider ground and thermal states of the Stark-Zeeman Hamiltonian and identify optimal operating points. For thermal probes we find a nontrivial multiparameter effect: increasing the temperature can reduce the overall estimation error by weakening parameter correlations. In the dynamical regime we study both pure and thermal initial states, illustrating nontrivial manifestations of incompatibility for mixed probes. Finally, we show that an optimal sequential control protocol can overcome limitations due to noncommutativity, and we assess its robustness in the multiparameter setting.

Microbial populations exhibit a broad spectrum of nutrient utilization strategies, ranging from strategies utilizing diverse nutrients, called "generalists," to those being highly adapted to specific nutrients, called "specialists." The mathematical conditions for the diversification of nutrient utilization strategies are central questions in theoretical ecology. Previous studies have shown that trade-offs among different resource utilization functions that cells cannot utilize broad types of substrates at near-maximum speed are crucial for the emergence of diverse strategies. However, in natural settings, nutrient availability often fluctuates over time, imposing additional trade-offs on cells. Cells that grow rapidly under nutrient-rich conditions will suffer a higher death rate under nutrient-poor conditions, creating a growth-death trade-off that intersects with the classical resource-use trade-off. Here, we introduce a unified mathematical model that simultaneously incorporates the resource-use trade-off and the growth-death trade-off. The nutrient supply was modeled as discrete stochastic events, capturing realistic temporal fluctuations. We show that the relative balance between growth and death rates critically influences the dominance of either generalist or specialist strategies. Specifically, under conditions of high average growth rates among different environments and a weak trade-off between growth and death rates, generalists prevail. In contrast, when the growth-death trade-off is intense, specialists emerge as the dominant strategy. Our findings reveal that accounting for the growth-death trade-off is crucial for understanding how microbial communities adapt and evolve in temporally varying environments.

We consider processing networks where multiple dispatchers are connected to single-server queues by a bipartite compatibility graph, modeling constraints that are common in data centers and cloud networks due to geographic reasons or data locality issues. We prove lower bounds for the steady-state occupancy, i.e., the complementary cumulative distribution function of the empirical queue length distribution. The lower bounds are geometric with ratios given by two flexibility metrics: the average degree of the dispatchers and a novel metric that averages the minimum degree over the compatible dispatchers across the servers. Using these lower bounds, we establish that the asymptotic performance of a growing processing network cannot match that of the classic Power-of-$d$ or JSQ policies unless the flexibility metrics approach infinity in the large-scale limit.

The Higgs inflation model with nonminimal coupling, while disfavored by the 1$σ$ region of the latest Atacama Cosmology Telescope Data Release 6 (ACT DR6) observational data, can be reconciled with the ACT DR6 data by incorporating the effects of reheating. In this paper, we consider reheating with a constant equation of state $w_{re}$. For the strong coupling case $ξ=100$, we find that reconciling the model with both the ACT DR6 constraints and the minimum reheating temperature required for successful Big Bang Nucleosynthesis (BBN) demands $w_{re} \geq 0.92$. Specifically, the reheating $e$-folding number must be $ N_{ re}= 28.3$ for $w_{ re} = 0.92$, and within $24.9 \leq N_{ re} \leq 27.2$ for $w_{ re} = 1$. In the more general case without assuming the strong coupling limit, consistency with both ACT and BBN requires the nonminimal coupling to satisfy $ξ\geq 0.05$. Our findings suggest that by considering reheating, a wide range of inflationary models, such as $R^2$ inflation, hilltop inflation, E-model inflation, and T-model inflation, can also be made consistent with the ACT DR6 observational data.