We develop a quantitative mathematical theory that offers new perspectives on nonlinear harmonic generation in plasmonic structures arising from symmetry breaking. Focusing on second harmonic generation--the most fundamental process and the most extensively studied owing to its practical significance--we establish a theoretical framework that can be readily extended to higher-order harmonics. We investigate the plasmonic system in the static regime using a columnar nanowire with \(n\)-fold rotational symmetry (\(n \in \mathbb{N}\)) and construct a phenomenological model in which the second harmonic response originates from nonlinear sources confined to a selvedge region near the surface. By introducing a notion of symmetry degree grounded in group theory, we precisely quantify the second harmonic generation in terms of multipolar contributions. Our theory complements existing physical descriptions of this practically important phenomenon and provides a rigorous account of how nonlinear optical efficiency depends on shape, size, symmetry, and defects in plasmonic structures.

The accurate estimation of observables is a crucial task in quantum computing. Recent advances have highlighted the need for (a) specialized protocols for qudit-based devices, that include (b) error-aware strategies. Here, we present AQUIRE, the first protocol that can (a) accurately estimate both the mean and the error of an observable on qudit-based quantum computers. AQUIRE achieves this by constructing a Bayesian model to accommodate generalized Pauli operators. It is designed to continuously monitor the estimated average and the associated error of the observable, adjusting the subsequent measurements in real-time. Additionally, AQUIRE is (b) device- and experiment-specific error-aware, and accounts for hardware imperfections and experimental noise during the estimation process. We demonstrate AQUIRE's advantage via numerical simulations and showcase its ability to quantify the noise affecting the estimation by implementing it on a trapped-ion qudit quantum processor. By exploiting general commutation relations and overlap grouping measurements, our protocol is state-of-the-art when restricted to qubit-based quantum computers and extends this advantage to the qudit case.

Accurately forecasting the bursty and non-stationary power demand of AI data centers has become increasingly important, as abrupt workload-driven variations at the GPU-node level can affect real-time operational efficiency, power management, and grid-data center coordination. However, high-capacity forecasting models are often difficult to deploy at scale because of their memory and latency requirements, while lightweight predictors may fail to capture short-horizon temporal dynamics. To address this accuracy-deployment tradeoff, this paper proposes a deployment-efficient knowledge distillation framework for short-term load forecasting in AI data centers. The proposed framework first trains a high-capacity sequence teacher model for multi-step load trajectory prediction, where residual learning is used to improve robustness under non-stationary operating conditions. A lightweight point-wise student model is then developed for low-latency rolling inference using a compact neural network architecture. To transfer temporal knowledge from the teacher to the student, a sequence-to-point distillation strategy is introduced by aligning near-term predictive behavior and temporally pooled representations. Case studies on the MIT Supercloud dataset demonstrate that the proposed student model improves forecasting accuracy over recent deep learning baselines while reducing the deployment footprint by over 10x in parameter memory and model size.

We prove the Riemannian Penrose inequality in arbitrary dimension for smooth complete asymptotically flat manifolds with nonnegative scalar curvature and compact outer-minimizing minimal boundary, where the boundary is allowed to have a singular set of Hausdorff dimension at most \(n-8\). Moreover, the equality holds exactly when the manifold is isometric to the Riemannian Schwarzschild exteriors. Our proof extends Bray's conformal-flow method to higher dimensions, where the outer-minimizing enclosures along the flow may be singular.

We study the matter and entropy transport between two ultra-cold neutral Fermi-gas reservoirs linked by a quantum point contact under a chemical-potential gradient. We describe the two leads with a BCS mean-field model and derive the current-bias characteristics for both particle and entropy transport. We compute the out of equilibrium steady-state currents by using the Keldysh formalism. In accordance with previous works in the literature, we confirm the well-known behavior for the particle current and extend the computation to the entropy current in the BCS regime. The entropy current shows an oscillatory behavior at low voltage in the ballistic junction limit. We analyze the results for a wide range of values of the junction's transparency. We also compare our findings with experimental results in cold atomic gases in the unitary regime.

Databases, and datasets more generally, evolve continuously through updates, transformations, versioning, schema changes, streaming operations, and other mechanisms. While prior work has noted connections among some of these areas, they have traditionally been studied in isolation, each with its own abstractions, algorithms, and system implementations. In this paper, we argue for unifying these diverse functionalities under a single abstraction and a common set of computational primitives. We present such an abstraction, powerful enough to encompass existing use cases and to support new ones. Going beyond previous approaches, our framework seamlessly integrates provenance tracking for system-visible operations, conditional propagation of updates, and configurable alerts on change events. It also offers a principled treatment of dependent objects such as views and derived artifacts like machine learning models, by providing declarative mechanisms to control their evolution. Finally, we sketch a prototype implementation in a relational-like database system based on an adaptation of the "Prolly Tree", a Merkle tree-inspired data structure with tunable parameters to meet varying performance requirements, and present some initial experimental results.

We revisit Beukers' modular-form proof of the irrationality of $ζ(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $Γ_0(6)^\star$, we show that Beukers' choice is not isolated: it belongs to a one-parameter affine family. These approximations have the same exponential decay as the classical Apéry approximations and satisfy the same denominator-growth estimate needed in Beukers' irrationality argument. We then apply the same construction to several other genus-zero Fricke groups.

Spectral bifurcation diagrams (SBDs) have recently emerged as an efficient tool for identifying dynamical transitions in nonlinear systems through frequency-domain analysis. Previous studies have been limited to numerical investigations, and the experimental realization of SBDs has remained unexplored. In this work, we develop an automated framework using analog electronic circuits and data acquisition (DAQ) systems to obtain SBDs from real-time measurements. The method enables controlled parameter variation and simultaneous acquisition of time-series data for spectral analysis. Using this approach, we experimentally capture characteristic spectral signatures of dynamical bifurcations, such as period-doubling, quasiperiodicity (two- and three-frequency), and torus length-doubling. The experimental results show strong qualitative agreement with the numerical predictions, despite noise and parameter mismatches. This study establishes SBD as an effective tool for the experimental analysis of nonlinear dynamical systems.

Model counting ($\#\text{SAT}$) is a fundamental yet $\#\text{P}$-complete problem central to probabilistic reasoning. In this work, we address \textit{incremental model counting}, where sequences of structurally similar formulas must be counted. We propose an approach that amortizes computation via a persistent caching mechanism, retaining component data across solver calls to avoid redundant search. Additionally, we investigate branching heuristics adapted for this setting. We focus on the problems of argumentation and soft core, for which incremental model counting is natural. Experiments demonstrate that our method improves performance compared to current model counters, highlighting the capability of structure-aware reuse in dynamic environments.

Multimodal data plays a critical role in web-based recommendation systems, where information from diverse modalities such as vision and text enhances representation learning. However, real-world multimodal datasets often suffer from modality incompleteness due to sensor failures, annotation scarcity, or privacy constraints, which substantially degrade model performance and reliability. One effective solution to address this issue is modality completion, which reconstructs missing features to provide modality-complete graphs for downstream tasks. Given a query node with missing multimodal features, existing modality completion methods typically infer information from the node itself or its neighbors to reconstruct the missing modality. However, these methods may overlook semantically relevant context in the graph, which contains valuable cues that are non-trivial to capture through simple methods like neighborhood aggregation. In this work, we propose GRE-MC, a Graph Retrieval-Enhanced Modality Completion framework, to overcome these limitations. By introducing a modality-aware subgraph retrieval mechanism, GRE-MC selects semantically relevant subgraphs from the entire graph, providing richer contextual information for completing missing modalities. Subsequently, a graph transformer jointly encodes the query node and the retrieved subgraph via global attention to complete the missing features, while a learnable sparse-routing codebook regularizes latent embeddings into compact bases for improved robustness. Extensive experiments on multimodal recommendation benchmarks demonstrate that GRE-MC consistently outperforms state-of-the-art methods, validating the effectiveness of subgraph retrieval and joint-encoding graph transformer for robust modality completion.

Unconventional quantum states defying the ubiquitous Fermi-liquid paradigm can emerge in the presence of strong electronic correlations. Among these, non-Abelian anyons - such as Majorana zero modes and Fibonacci anyons - are of particular interest for topological quantum computing due to their non-integer quantum dimensions d>1, which allows for protected non-local encoding and processing of quantum information. However, despite considerable efforts, the unambiguous characterisation of such anyons via transport measurements has proved challenging. Instead, here we provide experimental evidence for the low-temperature fractional entropy Delta S associated with a single anyon, which directly implies its non-Abelian character through the relation Delta S = kB ln(d). This thermodynamic signature is measured in metal-semiconductor quantum circuits engineered to realize quantum-critical states from frustrated interactions. Using a micrometre-scale metallic island coupled to two or three electronic leads, we tune the system to two-channel and three-channel Kondo critical points. By measuring the island charge and exploiting a thermodynamic Maxwell relation, we estimate the entropy associated with the anyons that emerge in these critical states. Our observations reveal fractional values, exposing non-Abelian anyons. The corresponding scaling dimensions are consistent with theoretical predictions for a Majorana zero mode Delta S = kB ln(sqrt(2)) and a Fibonacci anyon Delta S = kB ln(1 +sqrt(5))/2 for two and three channels. These findings establish entropy measurements as a powerful tool for characterizing exotic quantum states.

Discrete entropy estimation is a classic information theory problem, wherein the average information content of a discrete random variable is estimated from samples alone. Naive approaches, such as the plugin method, fail to account for the probability mass associated with members of the random variable's support that are unobserved in a given sample, known as the "missing mass." The resulting systemic underestimation is particularly problematic when data is time-consuming or costly to gather. We propose SENECA, an entropy estimation scheme based on a novel ``self-consistent'' missing mass calculation. Extensive numerical experiments indicate that our approach outperforms many state-of-the-art alternatives overall in the small-sample setting. We then apply SENECA to two practical use cases, namely biodiversity estimation and the detection of incorrect large language model responses, where our method is competitive with domain-specific approaches. Our work advances SENECA as an effective drop-in replacement for small-sample entropy estimation, with broad utility across several domains.

We present a comprehensive new global QCD analysis of unpolarized parton distribution functions (PDFs) based upon proton, deuteron and $A\!=\!3$ data, including the latest inclusive deep-inelastic scattering (DIS) measurements from Jefferson Lab at high Bjorken-$x$. Using the JAM Bayesian Monte Carlo framework, we systematically explore the stability of the PDFs with respect to variations in the cuts on the invariant mass $W$ of the DIS final state, the implementation of target mass and higher twist corrections, as well as on the nuclear wave functions for the $A\!=\!2$ and 3 data. We find the $u$ and $d$ quark PDFs (and the $d/u$ ratio) are relatively stable up to $x \approx 0.8$, and able to describe DIS data down to $W^2=3.5$ GeV$^2$ and $Q^2=m_c^2$. Within the collinear factorization framework, the fitted higher twist corrections to DIS are found to be positive, and largely isospin independent. The description of the nuclear data also requires nonzero isoscalar and isovector nucleon off-shell PDF contributions, which gives specific predictions for the ratio, $R_D$, of deuteron to isoscalar nucleon structure functions.

Mechanosensitive ion transport plays a central role in enabling living systems to perceive and adapt to their environment through the deformation of soft, embedded ion channels. In this work, we demonstrate that ion transport within a two-dimensional graphite nanoslit can be rationally engineered to achieve a bipolar, pressure-sensitive response without any structural deformation. The mechanosensitivity arises from the selective charging of one channel inlet, which acts as a reversible source of mobile charge carriers. These excess-ions can then be advected in or out of the channel by the pressure-driven water flow, thereby modulating the ionic conductance. This mechanism is captured through a comprehensive electrohydrodynamic model that analytically accounts for coupled diffusion, convection, surface transport, diffusio-osmosis, and interfacial slippage, both inside and outside the nanoslit. The theoretical framework quantitatively reproduces the experimental data, showing that a simple surface charge pattern can give rise to complex, pressure-dependent conductance. These findings reveal how rich nonlinear couplings at the nanoscale can be harnessed to design adaptive, bioinspired nanofluidic systems, exemplified here by ionic pressure sensors.

We prove that the directed seven-dimensional equal-side torus D_7(m) = Cay((Z/mZ)^7, {e_0, e_1, ..., e_6}) admits a directed Hamilton decomposition for every odd integer m >= 3. The proof has two main contributions. First, we introduce the root-flat certificate: a named verification framework in which a Hamilton decomposition of D_n(m) follows from three local conditions on a single root flat -- row Latinness, layer bijectivity, and primitive return maps. This abstraction was used informally in the earlier odd D_5(m) construction; here it appears as a definition and a theorem, providing a common verification interface for prime-dimensional base cases. Second, for every odd m >= 7, we give a uniform prefix-coordinate construction: one-layer prefix maps, a symbol-count criterion, and explicit 7x7 count matrices produce all seven Hamilton factors without a finite search. The remaining moduli m = 3 and m = 5 are exactly the boundary where the prefix-count method provably cannot work; they are handled by finite root-flat certificates whose validity is checked in Lean 4. A Lean 4 formalization verifies the Cayley statement, with the symbolic branch and the finite boundary certificates checked in the same development.

We present the final 1.2mm and 2mm source catalogs and the redshift distribution of the mm-selected population from the NIKA2 Cosmological Legacy Survey (N2CLS) in the COSMOS field. Our aim is to provide a comprehensive dataset for studying the physical properties and evolution of high-redshift DSFGs. N2CLS covers ~1070 arcmin2 with median noise levels of 315$μ$Jy and 91$μ$Jy at 1.2 and 2mm, respectively. Sources are extracted with a S/N threshold of 3.9, ensuring >80% purity. Multi-wavelength counterparts are identified using high-resolution interferometric (sub-)mm data (NOEMA, ALMA) and radio observations (VLA, MeerKAT). Redshifts are compiled from spectroscopic and photometric catalogs (e.g., COSMOSWeb). The N2CLS master catalog includes 323 sources detected at >80% purity, with 104 sources detected in both bands, 197 only at 1.2mm, and 22 only at 2mm. Multi-wavelength identifications are secured for ~89% of the sample. The redshift distribution of 1.2mm sources peaks at 2.8$\pm$0.1, consistent with the epoch of peak cosmic star formation. In the total sample, we lack redshift for ~2% of the identified galaxies, plus 34 sources for which no accurate positional proxy is available, preventing the identification of a multi-wavelength counterpart. We identify 66 galaxies at z>4. The observed redshift distribution agrees well with the SIDES simulations, while four other galaxy evolution models are statistically inconsistent with the data. N2CLS is the largest contiguous deep survey to date with this depth and homogeneous coverage. This homogeneous coverage is important, as 25% of N2CLS sources lack a SCUBA2 850 mic counterpart, likely because the strongly non-uniform noise distribution of the SCUBA2 map results in lower sensitivity in parts of the field.The released data products provide a legacy dataset for studying dust-obscured galaxy evolution.

We study the gambler's ruin problem for a biased random walk on $\{0,1,\dots,a\}$ under multi-site geometric resetting: at each time step, the walker is reset with probability $γ\in(0,1)$ to a random position drawn from a distribution $π$ over $m$ interior sites. Using renewal theory, we derive an exact closed-form expression for the ruin probability $q_z(γ)$, showing that the effect of $π$ is fully encoded in a single scalar quantity, the \emph{coupling constant} $C(π,γ)=\bar{u}_π/\bar{s}_π$. A spectral analysis via Doob symmetrization reveals the structure of this coupling. Our main result is a general criterion -- valid for any absorbed Markov chain admitting a spectral decomposition -- for the existence of a \emph{reset-neutral} distribution $π^*$ such that $C(π^*,γ)$ is independent of $γ$. This occurs under a spectral duality condition: there exists an involution $σ$ on the reset sites and $ν$-independent weights $κ(z)$ such that $B_ν(z) = κ(z)\,A_ν(σ(z))$ for all spectral modes $ν$. When this condition holds, the invariant value is $C^* = q_{a/2}^{(0)}$, the classical ruin probability from the midpoint, independent of the choice of symmetric reset sites or resetting rate. For the biased random walk, the condition reduces to the geometric symmetry $z_i + z_i' = a$. This result holds for any $a$, any number of reset sites $m$, and any bias $p\in(0,1)$. Both analytical and Monte Carlo simulations confirm the theory with high precision, including tests of spectrally neutral sites. Numerical results also reveal a phase-like structure in the space of reset distributions, with $π^*$ acting as a separatrix between monotone regimes.

For continuous-space diffusion processes, there is a strong connection between conservative forces and entropy production. For a given time evolution of the system's state, the entropy production is minimized when the system is driven by a unique conservative force. However, this relation does not extend to jump processes on a discrete state space. In this case, the forces that minimize the entropy production are generally nonconservative, this effect is more pronounced far from equilibrium in the presence of high energy barriers. Here we show that, while conservative forces do not minimize the entropy production for a given time evolution, they are nevertheless uniquely characterized as the minimizer of a quantity we dub the renormalized entropy production. This work explores the properties this quantity shares with entropy production as well as crucial differences between them. We also discuss the conceptual and physical differences between the corresponding optimization problems in finite time. Our theoretical calculations are illustrated with explicit numerical examples.

Erasure enriches type theory with a distinction between runtime relevant and irrelevant data, allowing the compilation step to safely erase the latter. Versions of this feature are implemented by many systems, including Agda, Idris, and Rocq. We present a structural version of type theory with erasure, formulated as a second-order generalised algebraic theory (SOGAT). Erasure is encoded as a phase distinction between runtime and erased terms, in the form of a proposition that can appear in a context. This formulation has several advantages: it has models based on categories with families, is compatible with other structural features such as staging, and provides a better guideline for implementation. Through the model theory of SOGATs, we study the semantics of type theory with erasure in families of sets, which generalises to any Grothendieck topos equipped with a tiny proposition. We establish conservativity over Martin-Löf type theory (MLTT) in both phases. For code extraction, we construct a presheaf model that produces untyped lambda calculus programs and prove its correctness through gluing. Our results are formalised in Agda and we provide a toy elaborator implementation.

The transport and dispersion of multiple species of charged ions are central to many biological and physical processes, including electrokinetic ion separation. However, most theoretical studies of dispersion in channels have focused on neutral solutes, leaving the transport of multiple charged species comparatively unexplored. Differences in ionic diffusivities in a multispecies electrolyte solution generate an self-induced electric fields that drive electromigration. To capture these effects at the macroscopic scale, we combine the lubrication approximation with homogenization theory, under electroneutrality and zero-current constraints, to derive an effective transport equation governing the cross-sectionally averaged concentrations. We apply our model framework to a range of channel geometries and compute the resulting effective dispersion coefficients. Finally, we investigate how channel geometry can be tuned to enhance ionic separation. We observe a geometry-induced electro-diffusive coupling that inhibits solute dispersion in certain channels, leading to a non-monotonic Number of Theoretical Plates (NTP) and making such channels ideal for separation processes.

The Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On missions provide monthly terrestrial water storage anomaly (TWSA) estimates for monitoring large-scale water storage change. The monthly temporal resolution of official products limits the analysis of high-frequency hydrological events, while existing daily GRACE products often have reduced spatial resolution due to sparse groundtrack coverage and required smoothing and regularization. This study introduces D-SHIFT (Daily Spatial High-Resolution Inference via Feature Transformation), a deep learning-based framework for generating daily, high-resolution TWSA fields from daily spherical harmonic coefficient (SHC) solutions. The model is trained in the monthly domain by using low-resolution daily solutions and other auxiliary features as inputs, while targeting on monthly mascon products. The model is then applied to daily SHC inputs to generate products with similar spatial resolution of monthly products. Monthly validation against mascon products gives a global mean root mean square error of about 2.3cm, with good correlation and explained variance agreement. Daily analyses show that D-SHIFT produces spatially coherent day-to-day fields and improves basin-scale trend and seasonality estimates compared with low-resolution SHC. The basin-area double-difference analysis indicates that these gains are most relevant for spatially localized signals affected by smoothing and leakage. In Greenland, D-SHIFT better reproduces coastal mass-loss patterns and gives a basin-mean trend of -10.5cm/yr, close to the CSR Monthly value of -12.0cm/yr.

Small-angle neutron scattering (SANS) data reduction at user facilities follows a largely repetitive workflow. Runs are first classified in the catalog and matched to transmission, background, and empty-beam references within the same instrument configuration. The data are then reduced, placed on an absolute scale using a standard, and stitched across configurations. Although these steps are individually well understood, they remain weakly connected, producing a coordination burden that scales with the number of runs and configurations. This paper describes EQSANS-CLI, a command-line tool that organizes this workflow into a coherent, scriptable, and agent-addressable system. The design rests on four principles: a shared command-handler layer that all input paths converge on; a persistent \emph{working table} that holds every reduction decision as editable rows; two input surfaces (an interactive terminal and a headless JSON mode) that compile to the same handler entry point; and a status-driven re-reduction model that treats parameter changes as first-class events. An \texttt{/autopilot} command chains the full pipeline from the IPTS number to stitched $I(Q)$ curves in one invocation. A Slack bot demonstrates that the headless interface, together with a single skill document loaded into an external agent's system prompt, is sufficient to drive complete reductions by natural language from a mobile device. The architecture is intentionally minimal on the agent side: the CLI is the authoritative executor, and the agent's only job is to translate human intent into commands on a stable contract.

We provide a detailed formulation for calculating the angular power spectrum of the cosmic background photons arising from the dark matter decay or annihilation in a comprehensive manner. We pay particular attention to the case of dark matter decaying or annihilating into line photons. It is pointed out that taking account of the energy resolution of the detector is essential to correctly evaluate the angular power spectrum. We apply our formulation to the observational data from infrared, optical, X-ray and gamma-ray telescopes.

The XTR (conteXtual Token Retrieval) algorithm is a modification to ColBERT retrieval that avoids the costly step of fully gathering and reranking the candidates' embeddings by imputing their missing similarity scores from the initial token retrieval step. The original work proposes a modified training objective as necessary for effective XTR retrieval, arguing that standard ColBERT token scoring is unsuitable for imputation. In this paper, we replicate both the XTR retrieval algorithm and its modified training objective, and extend the evaluation to knowledge-distillation (KD) training and efficient retrieval engines (PLAID and WARP). We confirm the token-level matching characteristics claimed in the original work, but fail to replicate XTR's overall effectiveness advantage over ColBERT under a controlled comparison. We further show that XTR's training modification has a concrete mechanistic consequence for modern retrieval engines: by flattening ColBERT's characteristically peaked token score distribution, XTR training yields more discriminative centroid scores and thus more efficient IVF-based retrieval under PLAID and WARP. The utility of XTR training is therefore not limited to the low-$k'$ regime originally studied, but extends to any deployment setting where IVF-based engines are used. These findings offer practitioners concrete guidance on how and when to use XTR as their multi-vector retriever.

A significant component of the cost of making predictions from lattice QCD stems from the computation of correlation functions on a given ensemble of gauge fields. This cost depends on the observable of interest and the details of its representation, including any approximation needed to estimate it. Moreover, the variance of such estimators may depend strongly on physical and kinematical parameters such as the lattice spacing, volume or separation, which gives an important insight into the costs of reaching the relevant physical limits. In these proceedings, I review some observables involving quark propagators, including both quark-line connected and disconnected Wick contractions, and discuss variance-reduction schemes based on decompositions of the quark propagators. Such strategies have already proven useful for precision physics observables and in future may help reduce the computational cost of reaching large volumes.

We present a number of results concerning infinite-exponent partition relations on linear orders of the form $\langle {}^α2,<_{\text{lex}}\rangle$ for $α$ an ordinal, generalising the setting of the real line, working throughout in ZF without the Axiom of Choice. As a particular consequence of our results, we obtain a full classification of the relation $\langle {}^α2,<_{\text{lex}}\rangle \rightarrow (τ)^τ$ for $τ$ countable.

We establish a general nonlocal approximation principle for the entropy solutions of scalar conservation laws on $\mathbb{R}$. More precisely, we show that the entropy solution to a nonnegative initial datum can be obtained as a weak-star limit of a corresponding scalar nonlocal conservation law. The flux function of the nonlocal conservation law depends on suitable spatial averages of the density. The proof is based on a reformulation on the Hamilton--Jacobi level: working with the primitives, we identify the limit via the stability properties of viscosity solutions; we then recover the entropy solution using the classical relation between Hamilton--Jacobi equations and scalar conservation laws. We further show that the approximation extends, after a suitable shift, to sign-changing initial data, and we prove a quantitative convergence estimate for convex fluxes in terms of the first moments of the nonlocal kernels. This result makes it possible to define entropy solutions for general fluxes using their nonlocal approximations, which satisfy the requirement for a finite speed of mass propagation, a key feature of hyperbolic conservation laws.

Pilgrim's Finite Global Attractor Conjecture has been verified for polynomials [1], but remains open for general rational maps. In this paper, we prove the conjecture for a family of rational maps obtained by gluing two PCF polynomials along the boundaries of their finite superattracting basins. Adapting the idea of [17], we show that a suitably defined intersection number with a finite family of separating arcs eventually decays under pullback, yielding a finite collection of homotopy classes that attracts all non-peripheral curves under iteration.

Detailed numerical results obtained with state-of-the-art Quantum Monte Carlo (QMC) simulations are presented for the superfluid phase of $^4$He at saturated vapor pressure. The aim of this contribution is that of providing reliable, up-to-date estimates for this archetypal superfluid, reflecting the methodological progress that has taken place over the past two decades. We simulate a system comprising 2,048 helium atoms, i.e., an order of magnitude greater in size than those for which results currently regarded as standard references were originally obtained. We offer revised estimates for energetic and structural properties, as well as for the ground state condensate fraction.

Let $G$ be a connected (non-complete) $d$-regular graph with $d\geq3$. Let $c(G-S)$ denote the number of components of $G-S$ for any cut $S$ of $G$. The toughness $t(G)$ of $G$ is defined as $\min\left\{\frac{|S|}{c(G-S)}\right\}$, where the minimum is taken over all proper cuts $S$ of $G$. Let $λ_{2}(G)$ denote the second largest eigenvalue of $G$. In this paper, we prove $$t(G)\geq\min\left\{\frac{d+1}{d}(d-λ_{2}(G)),1\right\}.$$