In modern correlation imaging systems, also known as ghost imaging (GI), particularly under low-light or noisy conditions, preserving high image fidelity presents a significant challenge. This paper introduces an innovative approach by integrating Bose-Chaudhuri-Hocquenghem (BCH) error control coding (ECC) into CGI systems to assist imaging. By encoding target image with BCH codes and using order-statistic decoding (OSD) for error correction during reconstruction, this approach significantly improves image quality across various signal-to-noise ratio (SNR) conditions. Simulation and experiment results validate that BCH coding assisted imaging achieves significantly enhanced robustness against additive white Gaussian noise (AWGN) and improved image reconstruction quality. In addition, the imaging performance of different BCH codes varies, with each code exhibiting distinct advantages based on factors such as code length and coding efficiency.

Existing scientific document retrieval (SDR) methods primarily rely on document-centric representations learned from inter-document relationships for document-document (doc-doc) retrieval. However, the rise of LLMs and RAG has shifted SDR toward question-driven retrieval, where documents are retrieved in response to natural-language questions (q-doc). This change has led to systematic mismatches between document-centric models and question-driven retrieval, including (1) input granularity (long documents vs. short questions), (2) semantic focus (scientific discourse structure vs. specific question intent), and (3) training signals (citation-based similarity vs. question-oriented relevance). To this end, we propose UniFAR, a Unified Facet-Aware Retrieval framework to jointly support doc-doc and q-doc SDR within a single architecture. UniFAR reconciles granularity differences through adaptive multi-granularity aggregation, aligns document structure with question intent via learnable facet anchors, and unifies doc-doc and q-doc supervision through joint training. Experimental results show that UniFAR consistently outperforms prior methods across multiple retrieval tasks and base models, confirming its effectiveness and generality.

In this work, an extensive class of coherent states is introduced by taking the Fox Wright function as the normalization function. It is demonstrated that these states satisfy the key requirements of continuity, normalizability and resolution of unity. Furthermore, coherent states associated with the continuous spectrum are obtained through a discrete to continuous limiting procedure. Moreover, FW generalized multi parameter nu function is introduced and shown to act as the normalization function for the Fox Wright coherent states in the continuous spectrum. Later the Fox Wright function with bicomplex arguments has been introduced and its existence has been investigated. Bicomplex Fox Wright coherent states are also developed for the discrete spectrum based on this new function and their properties are analyzed. Subsequently, the results regarding Fox Wright coherent states are generalized to the bicomplex setting. In addition, a bicomplex FW generalized multi-parameter nu function is defined to demonstrate that it provides the normalization for these states in the continuous spectrum.

We document the first systematic evidence of negative spillover effects in crypto asset returns across blockchains. Using on-chain data from Ethereum, Solana, Binance Smart Chain, Arbitrum, and Avalanche (2022-2025), we show that surges on one chain often coincide with declines on others, in contrast to the positive co-movements typical of equity markets. These spillovers intensify during attention shocks, proxied by chain activity and extreme return events, and persist after controlling for global equity returns, interest rates, and Bitcoin. Nonlinear factor models reveal that attention-driven capital reallocation, rather than common information, underlies these dynamics. Our findings introduce a new form of cross-market linkage, attention-induced substitution, that shapes risk transmission in crypto markets. The results carry implications for portfolio diversification, systemic risk measurement, and regulation of token launches that may trigger cross-chain capital flight.

Drone Remote Identification (RID) plays a critical role in low-altitude airspace supervision, yet its broadcast nature and lack of cryptographic protection make it vulnerable to spoofing and replay attacks. In this paper, we propose a consistency verification-based physical-layer authentication (PLA) algorithm for drone RID frames. A RID-aware sensing and decoding module is first developed to extract communication-derived sensing parameters, including angle-of-arrival, Doppler shift, average channel gain, and the number of transmit antennas, together with the identity and motion-related information decoded from previously authenticated RID frames. Rather than fusing all heterogeneous information into a single representation, different types of information are selectively utilized according to their physical relevance and reliability. Specifically, real-time wireless sensing parameter constraints and previously authenticated motion states are incorporated in a yaw-augmented constant-acceleration extended Kalman filter (CA-EKF) to estimate the three-dimensional position and motion states of the drone. To further enhance authentication reliability under highly maneuverable and non-stationary flight scenarios, a data-driven long short-term memory-based motion estimator is employed, and its predictions are adaptively combined with the CA-EKF via an error-aware fusion strategy. Finally, RID frames are authenticated by verifying consistency in the number of transmit antennas, motion estimates, and no-fly-zone constraints. Simulation results demonstrate that the proposed algorithm significantly improves authentication reliability and robustness under realistic wireless impairments and complex drone maneuvers, outperforming existing RF feature-based and motion model-based PLA schemes.

Modern cloud servers routinely co-locate multiple latency-sensitive microservice instances to improve resource efficiency. However, the diversity of microservice behaviors, coupled with mutual performance interference under simultaneous multithreading (SMT), makes large-scale placement increasingly complex. Existing interference aware schedulers and isolation techniques rely on coarse core-level profiling or static resource partitioning, leaving asymmetric hyperthread-level heterogeneity and SMT contention dynamics largely unmodeled. We present Hestia, a hyperthread-level, interference-aware scheduling framework powered by self-attention. Through an extensive analysis of production traces encompassing 32,408 instances across 3,132 servers, we identify two dominant contention patterns -- sharing-core (SC) and sharing-socket (SS) -- and reveal strong asymmetry in their impact. Guided by these insights, Hestia incorporates (1) a self-attention-based CPU usage predictor that models SC/SS contention and hardware heterogeneity, and (2) an interference scoring model that estimates pairwise contention risks to guide scheduling decisions. We evaluate Hestia through large-scale simulation and a real production deployment. Hestia reduces the 95th-percentile service latency by up to 80\%, lowers overall CPU consumption by 2.3\% under the same workload, and surpasses five state-of-the-art schedulers by up to 30.65\% across diverse contention scenarios.

We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$ \begin{equation*} Δ^{2}u+V(\varepsilon x)u=λu+μ|u|^{q-2}u+|u|^{2^{**}-2}u \mbox{ in }\ \mathbb{R}^{N}, \quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2}, \end{equation*} where $Δ^{2}$ is the biharmonic operator, $N\geq5$, $μ,c>0$, $\varepsilon>0,$ $λ\in\mathbb{R}$, $q\in(2,2+\frac{8}{N}),$ and $2^{**}=\frac{2N}{N-4}$ is the Sobolev critical exponent. The potential $V$ is a bounded and continuous nonnegative function, satisfying some suitable global conditions. Using minimization techniques and a truncation argument, we show that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $\varepsilon$ is sufficiently small. To overcome the loss of compactness of the energy functional due to the critical growth, we apply the concentration-compactness principle. To the best of our knowledge, this study is the first contribution regarding the concentration and multiplicity properties of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$. To some extent, the main results included in this paper complement several recent contributions to the study of biharmonic equations with combined nonlinearities.

A physics-based analytical model describing the phase transition from crystalline to conformationally disordered (condis) crystalline phase is developed. In the model, the free energy is written as a function of temperature and the lattice parameter (mean distance between neighboring chains). It consists of two contributions: elastic and conformational. The elastic contribution describes the interaction between neighboring chains, while the conformational part takes into account the conformation of one chain inside the potential tube, formed by the neighboring chains. To verify this approach, polyethylene - the simplest polymer possessing the condis phase - was chosen as a modeling object. Previous experiments and molecular dynamics simulations show that the typical conformation of a polymer chain in a crystalline phase consists mainly of trans dihedrals and a small fraction of gauche dihedrals, which can be considered as defects of the crystalline lattice. These defects displace the chain inside the tube thus increasing the potential energy. The energy required to form such a defect decreases rapidly with increasing distance between neighboring chains. This leads to a first-order phase transition at a certain temperature to the condis phase, in which distance between neighboring chains is large and a fraction of gauche dihedrals is high. This physical picture of the phase transition is described by the proposed analytical model, the parameters of which were calibrated against the results of molecular dynamics simulations for atmospheric pressure. The model predictions for the pressure of 500 atm and 1000 atm are in perfect agreement with the results of molecular dynamics simulations.

In this paper, we study an acoustic black hole in Hayward spacetime from the relativistic Gross-Pitaevskii theory. By examining the critical null geodesics, the shadow of the acoustic horizon is sketched. Then the quasinormal mode (QNM) frequencies of the acoustic Hayward black hole are computed numerically using the WKB method, which are shown to be more stable than those of the Hayward black hole, and the variations in the QNM frequencies are shown to correlate with the behavior of the effective potential. Moreover, the WKB method is also employed to calculate the grey-body factor and energy emission rate of the analogue Hawking radiation. It is shown that, as the tuning parameter increases, both the grey-body factor and the energy emission rate are enhanced, which can likewise be attributed to changes in the effective potential. Besides, the radius of acoustic shadow increases with the tuning parameter as well. Our results not only construct an acoustic black hole in regular black hole spacetime, but may also provide potential applications in future observations of astrophysical black holes.

We use a quantum formalism for the partition function of the classical $XY$ model to identify a resilience phase transition in a noisy toric-rotor code. Specifically, we consider the toric-rotor code under phase-shift noise described by a von Mises probability distribution and show that the fidelity between the final state after noise and the initial state is proportional to the partition function of the $XY$ model. We map the temperature of the $XY$ model to the width of the noise in the toric-rotor code, such that a Kosterlitz--Thouless phase transition at a critical temperature $T_{c}$ corresponds to a mixed-state phase transition at a critical width $σ_c$. To characterize this phase transition, we develop a quantum formalism for the spin stiffness in the $XY$ model and show that it is mapped to the gate fidelity in the logical subspace of the toric-rotor code. In particular, we introduce a topological order parameter that characterizes the resilience of the toric-rotor code to decoherence within the logical subspace. We show that the logical subspace does not exhibit complete resilience to noise, which is a necessary condition for correctability. However, it exhibits partial resilience to noise for widths less than $σ_c\approx 0.89$, where the resilience order parameter takes values near $1$ and then drops to zero at $σ_c$. We also use our results to shed light on the correctability of toric-rotor codes in higher dimensions $d > 2$. Our work shows that the quantum formalism for partition functions provides a mathematically rigorous framework for studying correctability in continuous-variable quantum codes.

Strong coupling between molecular excitations and quantized electromagnetic fields in optical cavities provides a powerful means to control the physical and chemical properties of molecular systems. Here, we study electron transfer (ET) dynamics in cavity-coupled molecules using the numerically exact hierarchical equations of motion (HEOM) method, which captures nonperturbative and non-Markovian effects beyond standard perturbative theories. We identify distinct resonance and collective effects associated with polariton formation and show that the ET rate saturates in the strong-coupling regime, a feature not captured by perturbative approaches. We further extend the cavity-modified ET model by incorporating the nuclear-coordinate dependence of molecular electric dipole moments, which gives rise to a three-body interaction involving molecular electronic and vibrational degrees of freedom and cavity photons. This vibronic polariton formation leads to non-monotonic, oscillatory dependencies of the ET rate on the light-matter coupling strength and cavity frequency, which we attribute to quantum interference among multiple transfer pathways. These findings establish cavity-modified electron transfer as a multichannel quantum process governed by the interplay of electronic, vibrational, and photonic degrees of freedom.

Hand-tracking enables controller-free XR interaction but does not have the tactile feedback controllers provide. Rather than treating this solely as a missing-sensation problem, we explore whether pseudo-haptic cues on an embodied virtual hand act as tactile or as affect substitutes that shape how interactions feel. We used a mixed reality prototype that keeps the contacted surface visually neutral, rendering cues on the hand with motion modulation for texture, color glow, and movement-coupled sound. In a within-subjects study (n=12), participants experienced 12 conditions (4 effects x 3 modalities: audio, visual, both) and reported subjective affect and cognitive demand. Participants rarely reported sustained tactile, thermal sensations, yet affect shifted systematically: rough-hot lowered valence increasing arousal, while smooth-cold produced calmer pleasant states. These findings suggest that pseudo-haptics in XR may be better understood as an affective feedback channel rather than a direct replacement for physical touch in controller-free systems.

For a graph $G$, let $c_1(G)$ be the largest distortion necessary to embed any shortest-path metric on $G$ into $\ell_1$, and for any natural number $n,m\in\mathbb{N}$, denote $K_{n,m}$ as the complete bipartite graph. In this note, we caculate the value of $c_1(K_{2,n})$, more precisely we prove $c_1(K_{2,n})=\frac{3k-2}{2k-1}$ where $k=\lceil\frac{n}{2}\rceil$.

We experimentally investigate the Lagrangian dynamics of finite-sized, neutrally buoyant droplets in homogeneous isotropic turbulence. The droplet size follows a log-normal distribution whose average value decreases with increasing Reynolds number, reflecting enhanced turbulent breakup. While size-conditioned velocity and acceleration statistics show only weak finite-size dependence, temporal measures reveal clear size-dependent dynamics: larger droplets exhibit longer Lagrangian velocity integral times and an extended ballistic regime in their mean squared displacement. These findings indicate that though droplets exhibit mild deformation and internal circulation, they behave similarly to finite-size rigid particles in terms of Lagrangian dynamics. Our study opens the way to study droplet-laden turbulence and droplet-flow interactions.

Pulse-profile modeling of rotation-powered millisecond pulsars targeted by NICER has enabled mass--radius constraints of several neutron star sources, with implications for the dense-matter equation of state. For the bright isolated pulsar PSR J0030+0451, the inferred mass--radius was previously found to depend strongly on the assumed hot spot model. These hot-spot models yielded different mass--radius constraints, with the statistically preferred model exhibiting some mild tension with results inferred for PSR J0437$-$4715, PSR~J0614$-$3329, and GW170817. We present an updated pulse-profile analysis of PSR J0030+0451 using new NICER observations obtained between 2017 July to 2023 January, increasing the number of X-ray counts by about 50% compared to previous analyses. We jointly analyze the NICER data with archival XMM-Newton observations to better constrain the source spectrum and background. The new analysis significantly reduces the discrepancy between the hot spot models. The inferred mass and radius are $M = 1.43^{+0.20}_{-0.17}\,M_\odot$ and $R_{\rm eq} = 12.68^{+1.31}_{-1.04}$ km (68% credible intervals), reducing the tension with the results from other sources. In addition, the inferred hot spot configurations suggest the presence of intra-spot temperature gradients.

The source localization problem, fundamental to applications like GPS, is typically approached as a minimization problem in the presence of various types of noise. Ensuring the uniqueness of solutions in GPS technology is vital for the reliability and accuracy of applications, from everyday navigation to critical military operations. In this paper, we examine two key minimization problems: one focused on distance error and the other on squared distance error. We explore these problems in both three-dimensional space, the standard scenario, and in two-dimensional space as a simplified case. Furthermore, we discuss the number of possible source solutions when the number of measurements is fewer than three.

Surface electromyography (sEMG) signals exhibit substantial inter-subject variability and are highly susceptible to noise, posing challenges for robust and interpretable decoding. To address these limitations, we propose a discrete representation of sEMG signals based on a physiology-informed tokenization framework. The method employs a sliding window aligned with the minimal muscle contraction cycle to isolate individual muscle activation events. From each window, ten time-frequency features, including root mean square (RMS) and median frequency (MDF), are extracted, and K-means clustering is applied to group segments into representative muscle-state tokens. We also introduce a large-scale benchmark dataset, ActionEMG-43, comprising 43 diverse actions and sEMG recordings from 16 major muscle groups across the body. Based on this dataset, we conduct extensive evaluations to assess the inter-subject consistency, representation capacity, and interpretability of the proposed sEMG tokens. Our results show that the token representation exhibits high inter-subject consistency (Cohen's Kappa = 0.82+-0.09), indicating that the learned tokens capture consistent and subject-independent muscle activation patterns. In action recognition tasks, models using sEMG tokens achieve Top-1 accuracies of 75.5% with ViT and 67.9% with SVM, outperforming raw-signal baselines (72.8% and 64.4%, respectively), despite a 96% reduction in input dimensionality. In movement quality assessment, the tokens intuitively reveal patterns of muscle underactivation and compensatory activation, offering interpretable insights into neuromuscular control. Together, these findings highlight the effectiveness of tokenized sEMG representations as a compact, generalizable, and physiologically meaningful feature space for applications in rehabilitation, human-machine interaction, and motor function analysis.

PoCo is a technique that aims to enhance modern coverage-based seed selection (CSS) techniques (such as afl-cmin) by gradually removing obstacle conditional statements and conducting deeper seed selection.

We present numerical results obtained in a finite-temperature study of the Sp(4) Yang-Mills theory on the lattice. We study its first-order confinement/deconfinement phase transition, by reconstructing the density of states via the Logarithmic Linear Relaxation (LLR) algorithm. We perform our measurements on lattices with different extents of space and time (and aspect ratios). We estimate the size of discretisation and finite-volume artefacts. We find clear signatures of a first-order transition. We determine the critical coupling, the specific heat, and the surface tension, for finite extents of the thermal circle, and use the results to set bounds for the continuum theory.

This paper investigates channel-aware decision fusion empowered by massive MIMO systems and reconfigurable intelligent surfaces (RIS). By integrating both, we aim to improve goal-oriented (fusion) performance despite the unique propagation challenges introduced. Specifically, we investigate traditional favorable propagation properties in the context of RIS-aided Massive MIMO decision fusion. The above analysis is then leveraged (i) to design three sub-optimal simple fusion rules suited for the large-array regime and (ii) to devise an optimization criterion for RIS reflection coefficients based on long-term channel statistics. Simulation results confirm the appeal of the presented design.

We introduce kALDo2.0, an open-source Python package for computing vibrational, elastic, and thermal transport properties of solids from first principles and machine-learned interatomic potentials. Building on the anharmonic lattice dynamics (ALD) framework, kALDo2.0 provides efficient CPU and GPU-accelerated implementations of the Boltzmann transport equation (BTE) for crystals and the quasi-harmonic Green-Kubo (QHGK) method. QHGK extends thermal transport predictions beyond crystals to disordered materials, including glasses, alloys, and complex nanostructures. kALDo2.0 introduces native integration with modern machine-learned potentials (MLPs), enabling thermal transport workflows that combine the accuracy of first-principles methods with the scalability of classical force fields. It also features comprehensive support for temperature-dependent effective potentials workflows, flexible storage backends for large-scale calculations, and advanced quantification of anharmonicity. The software seamlessly interfaces with electronic structure codes (Quantum ESPRESSO, VASP), molecular dynamics packages (LAMMPS), and MLPs (ACE, NEP, MACE, MatterSim, Orb), enabling thermal transport studies from 0 K to finite temperatures. kALDo2.0 implements multiple BTE solution strategies and essential physical corrections, including isotopic scattering and non-analytical terms for polar materials. A modular Python architecture with lazy evaluation and multiple storage formats (ASCII, NumPy, HDF5) enables simulations of systems containing up to tens of thousands of atoms. This paper describes the theoretical framework, implementation details, software architecture, and validation examples demonstrating kALDo2.0's capabilities for studying complex materials, including halide perovskites with strong anharmonicity and polar oxides requiring long-range electrostatic corrections.

We study convex-concave saddle point problems with bilinear coupling, covering linearly constrained convex optimization and more general nonsmooth or constrained models via a proximable term in the dual objective. In linearly convergent regimes, we characterize how spectral properties of the coupling matrix and objective conditioning jointly determine the attainable linear rates. We propose direct spectral acceleration for first-order primal--dual methods for a class of bilinear-coupled saddle point problems, including affinely constrained smooth strongly convex optimization and extensions with proximable dual terms. The resulting algorithms distinguish objective-dominated and coupling matrix-dominated regimes and attain optimal linear convergence without Chebyshev inner loops or double-loop designs. We further develop stochastic block-coordinate extensions in the affinely constrained case with separable objectives; we also establish optimal linear rates matching the block-coordinate lower bound. For both deterministic and stochastic methods, we provide matching worst-case lower bounds via explicit finite-dimensional hard instances.

Despite of the low transition temperature, the recently identified superconductor CeRh$_2$As$_2$ has garnered significant interest due to its unique symmetry and magnetic characteristics, particularly the existence of two superconducting (SC) phases under a magnetic field, one of which exceeds the Pauli-Clogston limit. The field-induced transition from a low-field even-parity state to a high-field odd-parity state is usually described as a singlet-triplet transition. However, it is uncommon for a single compound to exhibit both triplet and singlet SC scenarios. The aim of this paper is to investigate the possibilities of symmetry changes in the SC state without a change of spin multiplicity. To this end, we construct the SC order parameter based on Anderson pair functions, considering the phase winding within the symmetry of the point group $D_{4h}$ and the magnetic group $4/mm^{\prime }m^{\prime}$. It was found that two triplets with opposite-spin and equal-spin pairing states of symmetry $E_{1u}^{\prime +}$, are nodeless but exhibit distinct internal structures and may be associated with low-and high-field phases. Additionally, nontrivial Cooper pairing resulting from the non-symmorphic structure of the space group was examined, particularly in the case where the Fermi surface intersects with the boundaries of a Brillouin zone (BZ). It was determined that at the X point, triplet pairs are even, while singlet pairs can be either even or odd. Furthermore, at the X point, pair density waves that alter phase by $π$ at the atomic centers linked by lattice translations are also feasible. To explore the possibility of such scenarios, precise DFT calculations of the band structure were performed, revealing the contribution of Ce $4f$ electrons to the states at the Fermi level. Thus, the even-odd transition can take place in a triplet scenario at symmetry points of a BZ.

We investigate the impact of a Coronal Mass Ejection (CME) on the transport and acceleration of relativistic protons in the solar wind using a coupled 3D Magnetohydrodynamics (MHD) simulation and a test-particle approach. The CME is driven by a spheromak injected into a Parker solar wind at a heliocentric distance of 0.139 AU. The trajectories of 5 GeV protons, injected towards the CME from 3 AU, are integrated in the guiding-centre approximation limit and scattered in velocity space with a mean free path $λ_{\|}$. Our results show that the CME can increment the protons energy by several GeV. The acceleration occurs during the time particles stream along the portion of a magnetic field line subject to compression downstream of the quasi-perpendicular portion of the CME-driven shock. In our configuration, the maximum energy gain, which is of the order of a few percent per shock crossing, occurs when the shock approaches 0.3 AU. Large energy gains require multiple passes through the acceleration region, which is made possible by the combined action of the mirror force and pitch angle scattering. The efficiency of the acceleration on time scales of the order of hours scales as $λ_{\|}^{-3/2}$. Energy spectra harden for decreasing parallel mean free path $λ_{\|}$.

Airbnb, a two-sided online marketplace connecting guests and hosts, offers a diverse and unique inventory of accommodations, experiences, and services. Search filters play an important role in helping guests navigate this variety by refining search results to align with their needs. Yet, while search filters are designed to facilitate conversions in online marketplaces, their direct impact on driving conversions remains underexplored in the existing literature. This paper bridges this gap by presenting a novel application of machine learning techniques to recommend search filters aimed at improving booking conversions. We introduce a modeling framework that directly targets lower-funnel conversions (bookings) by recommending intermediate tools, i.e. search filters. Leveraging the framework, we designed and built the filter recommendation system at Airbnb from the ground up, addressing challenges like cold start and stringent serving requirements. The filter recommendation system we developed has been successfully deployed at Airbnb, powering multiple user interfaces and driving incremental booking conversion lifts, as validated through online A/B testing. An ablation study further validates the effectiveness of our approach and key design choices. By focusing on conversion-oriented filter recommendations, our work ensures that search filters serve their ultimate purpose at Airbnb - helping guests find and book their ideal accommodations.

In this paper, we study the structure of the global attractor for weak and regular solutions of a problem governed by a scalar semilinear reaction-diffusion equation with a non-regular nonlinearity, such that uniquness of solutions can fail to happen. First, using the Moser--Alikakos iterations we obtain the estimates of the weak solutions in the space $L^{\infty }(Ω)$. After that, using these estimates we improve the existing results on the structure of the attractor. Finally, estimates of the Hausdorff and fractal dimension of the attractor are obtained.

The eccentricity matrix of a graph is obtained from the distance matrix by keeping the largest entries in their row or column, and the remaining entries are replaced by zeros. The eccentricity energy of a graph is the sum of the absolute values of the eigenvalues of its eccentricity matrix. In this paper, we investigate the effect of edge deletion on the eccentricity energy of graphs of the form $$G=K_{2n}\circ_{n} K_{2n}\circ_{n}\cdots \circ_{n} K_{2n}, (\text{\textit{l} copies of } K_{2n}),$$ where $n\geq 3,$ $l\geq 2,$ and $\circ_{n}$ denotes the $n-$coalescence of graphs, and prove that the eccentricity energy increases whenever an edge is removed. This result identifies a class of graphs whose eccentricity energy exhibits monotonic growth under edge deletion.

A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, every edge-length preserving continuous motion of the vertices preserves the distances between all pairs of non-adjacent vertices as well. In this paper, we present several new results on the rigidity of random graphs. In particular, we show that there exists $c>0$ such that, for $p\ge 2 \log{n}/n$, the binomial random graph $G(n,p)$ is with high probability (whp) $\lfloor c n p\rfloor$-rigid. This is sharp up to the constant $c$, and complements recent results of Peled and Peleg (in the regime $p= o(n^{-1/2})$), and of Jordán, Liu, and Villányi (in the constant $p$ regime). Moreover, we show that for every fixed $d\ge 2$ and $r\ge 501d$, a random $r$-regular graph is whp $d$-rigid, and that for $100/n\le p\le 2\log{n}/n$, the binomial random graph $G(n,p)$ contains whp an $\lfloor np/251\rfloor$-rigid subgraph with at least $(1-e^{-np/2})n$ vertices. Both results are sharp up to the multiplicative constant. In addition, we present a new sufficient condition for rigidity in terms of the minimum codegree of the graph (the minimum number of common neighbours of a pair of vertices in the graph). A main tool in our arguments is a new combinatorial sufficient condition for rigidity, which provides a common generalization to Whiteley's vertex-splitting lemmas, and to the "rigid partitions" method, developed in works by Crapo, Lindemann, Lew, Nevo, Peled and Raz, and by the present authors.

Nitrogen-vacancy centers in diamond have been shown to be capable of detecting AC magnetic fields with high sensitivity, spectral resolution, and spatial resolution. However, most studies so far have focused on the regime of time-averaged or time-correlated measurements, while little attention has been paid to the single-shot regime. Here we show that the amplitude and phase of an AC field can be retrieved from a single pair of two consecutive measurements. We demonstrate this concept by measuring a 4 MHz AC field with a per-shot amplitude and phase sensitivity of 78 nT and 63 mrad, respectively, at a temporal resolution of 320 us. We also investigate the effects and quantify the errors resulting from probe frequency detunings, as well as operating in the strong field regime. Moreover, we showcase the ability of the measurement protocol to dynamically change the probe frequency in real-time. This work advances the use of NV centers for real-time measurements of AC magnetic fields.

We propose that the symmetry-breaking bifurcation of coupled topological edge states (CTESs) can be used as a general principle for achieving spontaneous symmetry breaking (SSB) in a nonlinear topological lattice. Using an optical resonator array composed of two Su-Schrieffer-Heeger (SSH) chains as an example, we find that as the nonlinearity strength increases, the symmetric CTESs undergo a supercritical bifurcation. Beyond the critical threshold, the originally stable symmetric state becomes unstable, leading to the formation of a pair of stable asymmetric states. Both sides of the symmetric CTESs exhibit sublattice polarization, while the side of the asymmetric CTESs that is predominantly occupied demonstrates stronger sublattice polarization. We further find that as interchain coupling increases, the frequency range for stable CTESs expands, while the frequency range for stable asymmetric CTESs decreases. Our work provides a universal mechanism for realizing SSB in nonlinear topological lattices.