Symbolic regression (SR) aims to discover explicit mathematical expressions that explain observed data and is widely used in domains where interpretability is essential. Because interpretability requires expressions to reflect meaningful regularities, SR is sensitive to observations that deviate from the dominant relationship. Such irregular observations, or outliers, are common in real-world data and can hinder SR from identifying underlying regularities. Robust regression mitigates this by downweighting observations with large residuals. However, deciding which observations should be treated as outliers is often ambiguous and depends on user interpretation and domain knowledge, a perspective largely overlooked in existing SR studies. This motivates approaches that present multiple candidate expressions, allowing users to examine different residual patterns and choose expressions consistent with their expertise. We propose diversified residual symbolic regression (DRSR), which achieves high predictive accuracy while promoting diversity with respect to residual patterns based on the Quality-Diversity paradigm. DRSR collects multiple expressions that fit the data well but differ in how residuals are distributed, enabling post-search selection aligned with domain knowledge. On a synthetic mixture dataset, DRSR produces more diverse expressions than conventional SR while capturing multiple underlying relationships. On a real-world astronomical dataset, DRSR discovers multiple expressions consistent with known physical relationships.

Integrated sensing and communication (ISAC) relies on monostatic sensing (MS) and bistatic positioning (BP) to enable comprehensive environmental awareness and user localization. However, existing frameworks predominantly assume static geometries and optimize these modalities independently, neglecting user mobility and sequential information sharing. In this paper, we propose a velocity-aware sequential beamforming framework that dynamically couples MS and BP in time. We derive the Cramer-Rao bounds (CRBs) in the position domain to formulate a non-convex resource allocation problem. Instead of relying on static weighted-sum tradeoffs, we introduce a sequential Bayesian optimization strategy where MS is executed first to construct a reliable structural prior on the UE and passive targets (PTs). This covariance prior is subsequently passed to the UE to regularize the BP estimation stage. We demonstrate that optimizing a single shared beamformer globally across both phases yields superior synergistic gains compared to a two-stage greedy approach. Simulation results validate that the shared sequential design efficiently balances limited symbol resources, achieving centimeter-level positioning accuracy for both the UE and PTs, robust velocity estimation, and a significantly reduced computational runtime.

Two-dimensional Ruddlesden-Popper lead halide perovskites provide a valuable platform for tailoring charge and spin properties through quantum confinement and reduced symmetry. While the electron and hole Landé $g$-factors in bulk lead halide perovskites exhibit a universal dependence on the band gap energy, their evolution in two-dimensional perovskites has remained largely unexplored. Here, the Zeeman splittings of electrons and holes in (PEA)$_2$MA$_{n-1}$Pb$_n$I$_{3n+1}$ perovskites with the number of inorganic layers ovarying in the range $n=1,...,8$ are measured by means of the spin-flip Raman scattering and time-resolved Kerr rotation magneto-optical techniques. A systematic evolution of the electron and hole $g$-factors with decreasing layer thickness, which deviates from the universal bulk behavior and reveals confinement-driven trends similar to those observed in perovskite nanocrystals, is found. The experimental results are in good qualitative agreement with empirical tight-binding calculations. The exciton $g$-factors are evaluated from the Zeeman splittings of the exciton resonances in reflectivity measured in pulsed magnetic fields up to 55~T. These results provide comprehensive insight into the spin properties of two-dimensional lead halide perovskites and establish them as a tunable platform for engineering spin-dependent phenomena in quantum-confined semiconductors.

In typical single-molecule force spectroscopy experiments the mechanical unfolding of molecular complexes or biomolecules is studied applying a force ramp to one end of the system while the other end is kept fixed in space. The computational counterpart of this type of experiments can routinely be performed using molecular dynamics simulations with atomistic resolution. However, due to the large difference in time scales often coarse graining procedures are applied in the simulations. Most of the applied techniques do not allow to follow the atomistic details of the relevant conformational transitions due to the structural simplifications used to speed up the simulations. Here, we apply an earlier developed dynamic coarse graining technique based on Markov state modeling to a model peptidic system that does not unfold in a simple two-state manner. Using the donor-acceptor distances of the helical hydrogen bonds as collective variables and performing a dimension reduction technique allows us to construct a Markov model of the unfolding process that correctly represents the microscopic behavior of the system. The chosen example shows that the method can be used to mimick the mechanical unfolding process of systems for which the end-to-end distance does not provide a sufficient order parameter and that do not unfold in a simple cooperative manner.

This paper analyzes the security of the Message Queuing Telemetry Transport (MQTT) protocol in the context of the Internet of Things (IoT). The main objective consists of identifying vulnerabilities and proposing security improvements. Adopting a hybrid methodology, a theoretical review was combined with an experimental demonstration in a simulated Smart Home environment. Eavesdropping, Tampering, Denial of Service (DoS), and Brute Force attacks were executed and analyzed. The results evidenced critical risks due to the absence of robust encryption and authentication. Finally, mitigation strategies and best practices are proposed to strengthen MQTT implementations.

In regression models fitted to data from complex survey designs, sampling weights often incorporate non-essential variation, inflating variance estimates. Stabilized weights mitigate this issue by adjusting sampling weights to account for variation explained by covariates. In the context of two-phase sampling, we evaluate the performance of optimal stabilized weights and propose combining the stabilized weight estimator with generalized raking, a class of efficient design-based estimators. This combination improves efficiency by reducing unnecessary weight variation and leveraging information from auxiliary variables. We show this combination can be implemented using the standard statistical package that handles two-phase samples and generalized raking. Simulation studies demonstrate that the proposed estimator enhances precision under realistic two-phase designs, though efficiency gains may be limited in highly informative designs. The developed methods were applied to a large multinational two-phase study of Kaposi sarcoma among people living with HIV.

A code-modulated motion visual evoked potential (c-MVEP) for brain-computer interfacing (BCI) is presented in this study. This paradigm uses pseudo-random sequences to visually stimulate objects using motion as an alternative to flickering. In an offline experiment of this study, EEG data were recorded and compared during sequential stimulation of a single object under four conditions: c-MVEP, code-modulated visual evoked potential (c-VEP), steady-state motion visual evoked potential (SSMVEP), and steady-state visual evoked potential (SSVEP). c-MVEP showed similar time-domain characteristics as c-VEP, and also in the frequency domain c-MVEP evoked a broadband response similar to c-VEP, with a comparable signal-to-noise ratio (SNR), albeit more focused in the lower frequency range. Both SSMVEP and SSVEP showed clear oscillatory responses at the stimulation frequency and harmonics, with a higher SNR for SSVEP than SSMVEP. The spatial distribution of c-MVEP showed the main activation at Oz and spread across multiple electrodes, whereas c-VEP showed less spreading and was more focused at Oz. Similar observations were made for SSMVEP and SSVEP. From subjective ratings, there was no clear preference for the motion-based stimulation of SSMVEP or c-MVEP over flicker-based stimulation of SSVEP or c-VEP. The online experiment of this study, evaluated a 4-class BCI with the same four conditions, testing the practical feasibility of the c-MVEP paradigm. The c-MVEP BCI reached a mean accuracy of 85.67% with an average selection time of 2.61s, which was significantly lower than c-VEP (97.81%; 1.15s) and SSVEP (93.42%; 1.94s), but significantly higher than SSMVEP (64.91%; 4.18s). Overall, this study shows the great potential of the newly proposed c-MVEP paradigm using motion stimulation for BCI applications, providing a valuable alternative to the c-VEP paradigm using flickering stimulation.

The Alliance for Open Media (AOMedia) has developed the AV2 video coding standard to supersede AV1, aiming for substantial compression efficiency gains across diverse media applications. This paper details the quality and performance evaluation methodology defined in the AV2 Common Test Conditions (CTC), which introduces new evaluation methods and content, including convex-hull-based adaptive streaming (AS) configuration, user-generated content (UGC), and extended chroma formats. We present the coding gains of the AV2 (v13.0) against the AV1 baseline. Experimental results show that AV2 achieves significant Bjøntegaard-Delta Rate (BD-rate) reductions of 29.81\% and 33.79\% for PSNR-YUV and VMAF, respectively, under random access configuration, validating the efficiency of AV2 for next-generation streaming applications.

Complex systems display emergent phenomena that vary significantly across spatial and temporal scales. These variations originate from fine-grained system processes, yet arriving at macroscopic dynamics from micro-level data -- particularly when large, high-resolution datasets are available -- remains a persistent challenge. Here we develop event-based spatiotemporal networks, a computational modelling framework that encodes system processes as discrete events anchored in space and time. Event-based spatiotemporal networks offer a unified, flexible and efficient approach to generate emergent behaviour in complex systems across space and time from these events. We demonstrate the effectiveness of event-based spatiotemporal networks through two illustrative real-world applications. First, following a local outbreak of a novel respiratory pathogen in the Netherlands, spatiotemporal networks enable fine-grained tracking of transmission routes and infection patterns through space and time. Second, we use spatiotemporal networks to model propagation of delays in a public transportation system (S-bahn) around Zürich, Switzerland. We also discuss broader uses of event-based spatiotemporal networks in fields like developmental biology and community ecology, where focusing on events rather than static system states can improve data analysis, simulation, and collection strategies.

According to the formulation of the charged large $D$ membrane paradigm, an arbitrary dynamic black hole solution to a theory of gravity with a $U(1)$ gauge field is dual to the dynamics of a membrane in a non-gravitational background. This membrane is endowed with a stress-energy tensor and a charge current, whose conservation equations govern its dynamics. In this work, we demonstrate that the dynamics of these membrane configurations (at the leading nontrivial order in $1/D$) can be mapped to a relativistic charged fluid. Establishing a correspondence for asymptotically flat black holes with a particular class of fluid systems. Unlike the standard AdS/Hydrodynamics correspondence, this dual fluid does not reside on an asymptotic boundary, but is localized strictly on the non-gravitational membrane worldvolume. By evaluating the system in both the Eckart and Landau frames, we systematically extract the out-of-equilibrium transport coefficients. We find that the fluid is governed by a negative effective thermal conductivity and a negative heat capacity, a mechanism that enforces thermodynamic stability in agreement with the quasinormal mode damping in the large $D$ Reissner-Nordström black hole geometry.

We study real-time reconstruction and actuation for two binary Markov sources that share a wireless multi-packet reception (MPR) channel. Each sensor follows a stationary randomized sampling policy, and the receiver maintains source estimates using the most recently decoded updates. We derive closed-form expressions for the steady-state real-time reconstruction error (RTE) and the cost of actuation error (CAE) as functions of the source transition probabilities and the effective update probabilities. We then characterize these update probabilities under randomized sampling, linking the physical-layer MPR model to task-oriented reconstruction and actuation metrics. Using these expressions, we formulate a sampling-constrained optimization problem with a weighted-error objective. The resulting analysis reveals how source dynamics, semantic weights, and MPR coupling affect the allocation of sampling resources. Numerical results show that optimized randomized sampling outperforms random, greedy, and time-sharing baselines.

The Shared Prosperity Internet (SPI) is a network-computing architecture that makes the benefits of automation and Artificial Intelligence (AI) broadly accessible to the society. To ground its design, this paper maps the physical constraints of Shannon, Landauer, Turing, and Einstein to three design principles: trustworthiness, sustainability, and technological sovereignty, and maps them into three technical pillars: i) post-Shannon, goal-oriented communication that transmits only what the task requires; ii) anticipatory decision-making ("negative latency") with confidence-bounded pre-action and correction; and iii) beyond-digital computing that selects energy-optimal substrates under deadline and computability constraints. The SPI is grounded in three societal use cases: remote teaching for pupils, remote teaching of robots and cyber-physical systems, and elder care. Furthermore, this paper defines measurable outcomes for an SPI, including latency decomposition, bits per event, energy and CO2 per task, safety and privacy indicators, and robustness.

Retrieval-Augmented Generation (RAG) improves reliability of large language models by incorporating external knowledge, but the retrieval process can introduce bias that propagates to generated outputs. This issue is particularly challenging in top-k settings, where multiple documents jointly influence generation. We propose a fairness-aware retrieval framework that models and controls this bias. Our approach combines controlled bias injection via reranking, a position-aware model of bias propagation, and an optimization formulation that balances relevance and fairness. We further introduce a scalable solution based on Quadratic Fairness via Dual Hyperplane Approximation (FARO), which enables efficient optimization through problem decomposition. Experimental results show that our method effectively mitigates generation bias while preserving relevance. This work provides a principled approach for fairness-aware retrieval in RAG systems.

We present a new strategic voting model where we use uncertainty representation to model preferences. Specifically, we use probability sets as uncertainty representations, together with lower and upper expected utility gains to take strategic decisions. Focusing on belief functions in particular, we demonstrate that this very expressive model includes in one sweep many existing models based on probabilities, sets or incomplete preferences. Additionally, we generalize several well-known convergence results from the literature to this broader representational setting. Furthermore, we illustrate how this model can capture more realistic scenarios for practical applications but also raises theoretical challenges.

The dynamics of dissipative many-body quantum systems sometimes admit an emergent classical description in terms of a Markov chain even though the corresponding state space contains highly entangled states. In particular, a bad-cavity laser is a paradigm system whose dynamics can be formulated as a Markov chain in a two-dimensional state space spanned by collective angular momentum states. In this article, we investigate the connection between a dissipative phase transition that occurs in the subradiant regime of this system in the large atom number limit, and the properties of the underlying Markov chain. In one of the phases, the Markov chain approaches the detailed-balance condition with increasing atom number $N$ and hence becomes effectively time-reversible. This is caused by a collective atomic emission process that effectively reduces the Markov chain to one dimension. In the other phase, we find the emergence of time-asymmetric probability currents in the two-dimensional state space that signals a breakdown of detailed balance. This is accompanied by a macroscopic internal entropy production rate in the Markov chain that scales extensively with the atom number $N$. An observable consequence of the probability currents is a self-pulsing of the cavity light output in this phase, which can be detected as an anti-correlation dip in the intensity correlation function.

How many measurements are fundamentally required to capture a signal. Shannon's information theory established the bedrock of this question in 1948, the Nyquist Shannon theorem set the first answer, and compressed sensing (CS) rewrote it in 2006 by reducing the required measurement number to M = O(Klog(N/K)) for a K sparse signal. Here, we propose quantum compressed sensing (QCS), a paradigm that reframes signal acquisition as a unitary quantum evolution. By encoding high dimensional signal information into a single quantum probe state, then introducing domain-alignment evolution,a physically realizable unitary transformation that maps the sparse basis directly onto the measurement basis. QCS executes the support-set search at the quantum level without consuming measurement trials. The logarithmic penalty vanishes, compressing the required measurement number from the classical bound to M =O(K) and reducing reconstruction from ill posed optimization to linear estimation. We experimentally validate QCS using frequency and time domain sparse signals, confirming that the measurement number scales linearly with sparsity and decouples entirely from the signal dimension. Our work provides a physical pathway toward ultimate information acquisition efficiency, with broad implications for sensing, imaging, and communication.

We consider multiple and set-indexed sums of random vectors taking values in Euclidean space of growing dimension. It is shown that, when viewed as finite metric spaces, the sets of values of such sums converge in probability. The limit is identified as a generalisation of the Wiener spiral, which appears as the high-dimensional limit of single-index sums.

In this paper, we investigate mean-field backward stochastic differential equation (MFBSDE) with double mean reflections and nonlinear resistance. Specifically, the constraints are formulated in terms of the expectation of the solution, and a compensating term is incorporated into the generator. We establish the existence and uniqueness for both the case of Lipschitz generator and the case where the generator is quadratic and the terminal value is bounded. Finally, when the compensating term is absolutely continuous, we study the well-posedness of a variant type of doubly mean reflected MFBSDE with nonlinear resistance, whose generator depends on the density function of the compensating term.

Multilinear representability extends classical linear representability of matroids by assigning subspaces, rather than vectors, to ground elements. This notion is closely related to almost affine codes. In this paper, we introduce and study a $q$-analogue of multilinear representability for $q$-matroids, motivated by known connections between $q$-matroids, classical matroids, and rank-metric codes. We define $m$-multilinear representability in terms of almost affine matrix rank-metric codes satisfying a natural divisibility condition. We prove that nontrivial uniform $q$-matroids admit no purely multilinear representations, and we derive necessary conditions for multilinear representations of almost uniform $q$-matroids. We further show that the non-Pappus $q$-matroid, if multilinearly representable, must have block size at least $9$. Finally, we prove that no rank-$2$ $q$-matroid on $\mathbb{F}_2^4$ admits a purely $m$-multilinear representation for $1<m<4$, and we classify pure multilinearity for all $q$-matroids on $\mathbb{F}_2^3$ and $\mathbb{F}_2^4$ in the corresponding ranges. At present, no example is known of a purely multilinear $q$-matroid.

We associate to a decorated liability network a liability sheaf on a directed hypergraph whose hyperedges separate the distribution of payments from the collection of receipts. Clearing configurations are precisely the global sections of this sheaf, and the global-section object is canonically the equalizer of the identity and a clearing operator $Φ=A\circ D$ factored into collective distribution $D$ and aggregation $A$; an institution-edge duality identifies it equivalently with the equalizer of the dual operator $D\circ A$ on the edge side. This identifies liability clearing as a finite-limit construction in the ambient data category. The construction is functorial under change of coefficient category: a Clearing Invariance Theorem shows that a finite-limit-preserving functor compatible with constraint subobjects induces a canonical isomorphism on global-section objects, enabling uniform comparison of clearing problems across categories of payment data. Existence, uniqueness, and iterative computation of clearing sections are organized by the structure carried on payment objects: Tarski's theorem yields existence and a complete-lattice structure under complete-lattice global elements; Scott continuity refines this to convergent Kleene iteration; an acyclic underlying graph admits a unique clearing section in finitely many steps with no order or metric hypothesis; and Banach's theorem on global elements yields uniqueness under metric contraction. The Eisenberg--Noe model and lattice liability networks arise as special cases.

This paper presents an analytical framework for downlink pinching antenna systems (PASS) employing waveguide division multiple access (WDMA) and non-orthogonal multiple access (NOMA). A unified channel model is developed to capture antenna deployment, user spatial distribution, and path loss. Closed-form and single-integral expressions for the outage probability and average achievable rate are derived and validated via Monte Carlo simulations. The results show that NOMA achieves higher spectral efficiency at high transmit signal-to-noise ratio (SNR) due to successive interference cancellation (SIC), whereas WDMA offers more reliable performance at low to moderate SNR but suffers from an outage floor and rate saturation at high SNR. Moreover, WDMA performance is more sensitive to the user spatial distribution due to the spatially dependent inter-waveguide interference. These findings provide design insights for access-scheme selection and antenna placement in PASS.

Quantum Gravity remains elusive, largely because its observable effects are suppressed by powers of the Planck scale. Direct detection of single gravitons is widely believed to be impossible. Here we propose a concrete astrophysical mechanism that may overcome this suppression. We show that superradiant axion-like-particle clouds surrounding rotating black holes can generate multimode squeezed states of gravitons containing up to $10^6$ - $10^7$ correlated quanta. Such states exhibit distinctive polarization correlations and quantum-noise signatures that could be detectable in future gravitational-wave interferometers. Observation of these signatures would constitute direct evidence for the quantum nature of gravitational radiation. Conversely, their absence can place constraints on axion-cloud lifetimes. Our approach also provides a test of General Relativity as an effective field theory.

Clean Hoag-like ring galaxies are often characterized by an old compact central component, a depleted gap, and a detached outer ring. We identify a boundary-supported radial-layering mechanism in a shell-deformed Kepler control model. A compact inner boundary supplies the core state, while a localized effective shell deformation, interpreted as the reduced imprint of externally supplied material settled near a finite circularization radius, needs to create only an internal maximum and a subsequent outer minimum. These act as the gap barrier and ring-supporting well. The onset of this structure is organized by a saddle-node threshold of the critical-point equation. In a 10^4-point Monte Carlo scan, shell-localized boundary-supported candidates occupy finite parameter volume under the adopted priors, and none of the localized candidates contains an ordered interior minimum--maximum--minimum subsequence. The same branch gives a scale-free gap-to-ring interval overlapping representative ratios for Hoag's Object, UGC 4599, and PGC 1000714, but not for the environmentally processed comparison object JO171.

Nanoscale quantum light sources are essential building blocks for integrated quantum photonic systems. Here, we report a wavelength-scale entangled-photon source based on van der Waals-engineered NbOBr$_2$, and benchmark its performance for telecom-wavelength quantum light generation. By exploiting the material's second-order nonlinearity, we generate quantum-correlated photon pairs via spontaneous parametric down-conversion. We then use a 90$^{\circ}$ twisted stacking to induce quantum interference in photon-pair generation, yielding polarization-entangled photons. This approach enables tunability of the quantum optical state via control of the excitation laser polarization. We experimentally obtain entanglement fidelities exceeding 95% for Bell states, along with a high coincidence-to-accidental ratio of $\sim$335, and a brightness approximately one order of magnitude higher than recently reported telecom sources based on transition metal dichalcogenide (TMD) 2D materials. These results establish twisted van der Waals engineering as a powerful platform for highly tunable, high-brightness quantum light sources at telecom wavelengths.

A \emph{graceful labeling} of a graph $G$ is an injective function $f : V(G) \to \{0, \ldots, |E(G)|\}$ such that $\{\,|f(u)-f(v)| : uv \in E(G)\,\} = \{1, \ldots, |E(G)|\}$. If such a labeling exists, then we call $G$ \emph{graceful}. Introduced by Rosa in 1967, graceful labeling has been widely studied, and the Graceful Tree Conjecture asserts that every tree is graceful. The conjecture is known to hold for several classes of trees, including caterpillars, trees with at most four leaves, trees of diameter at most five, and certain spiders. An important subclass is that of \emph{$α$-labelings}, where a graceful labeling $f$ admits an integer $α$ such that each edge joins a vertex with label at most $α$ to one with label greater than $α$. A result from 1982 by Huang, Kotzig, and Rosa shows that if $H$ has an $α$-labeling with a vertex $u$ labeled $0$ or $α$, and $G$ has a graceful labeling with a vertex $v$ labeled $0$, then identifying $u$ and $v$ yields a graceful graph, though this requires a $0$-labeled vertex in $G$. We prove a related result that relaxes this condition: if $G$ has a graceful labeling $f$ such that $f(u)+\lfloor n/2 \rfloor + 1 \le n$ and $n \not\equiv 1 \pmod{4}$, where $u\in V(G)$ and $n\ge 2$ is an integer, then joining $u$ to an end vertex of the vertex-disjoint $n$-vertex path $P_n$ yields a graceful graph. As an application, we show that any spider with legs $L_1,\ldots,L_s$ ($s \ge 1$) satisfying $|E(L_{2})| \ge 2|E(L_1)|+ 4$ and $|E(L_{i+1})| \ge 2|E(L_i)|+ 2$ for $i \in \{2,\ldots, s-1\}$ is graceful. Furthermore, we give an explicit graceful labeling for spiders with one leg of arbitrary length and all others of length at most two such that the center is labeled by $0$. This labeling enables the construction of larger graceful spiders by attaching paths at the center.

The security of networked control systems (NCS) is receiving increasing attention from both cyber-security and system-theoretic perspectives. The former focuses on classical IT security goals such as confidentiality, integrity, and availability of process data, while the latter investigates tailored attacks (and detection schemes), including covert and zero-dynamics attacks. Confidentiality in control systems can, for instance, be achieved by securely outsourcing the evaluation of the controller to third-party platforms, such as cloud services. The underlying technology enabling such secure computation often is homomorphic encryption (HE). Recent works in encrypted control have proposed modifications to underlying HE schemes to achieve not only confidentiality but also resilience to certain types of integrity attacks. While extensions in this direction are desirable in principle, we show that the integrity problem in encrypted control cannot be solved by public-key HE schemes alone due to their inherent malleability. In other words, the same homomorphisms that enable encrypted control in the first place can be leveraged not only constructively but also destructively. More precisely, we demonstrate that NCS are vulnerable to covert attacks, even when encrypted control is employed. Remarkably, this remains possible without knowledge of an unencrypted model. Yet, resilience to such attacks can still be achieved through complementary techniques. We present an approach based on verifiable computation that integrates with modern homomorphic cryptosystems and is asymptotically secure while incurring no communication overhead.

Upcoming experiments need improved simulations of neutrino scattering. This work uses the popular GENIE event generator to test a variety of neutrino interaction models against recent MicroBooNE measurements of pionless charged-current interactions. The GENIE code can easily interchange model components, including nucleon form factor parameterizations, quasielastic cross-section calculations, treatments of the nuclear ground state and hadronic final-state interactions. Leveraging this software capability in comparisons with MicroBooNE data, the performance of some of GENIE's most theoretically sophisticated model components is evaluated and contrasted with more empirically-driven alternatives.

In this paper, we presented an approximate analytical treatment of the Coulomb plus logarithmic potential using perturbation theory to investigate the mass spectra of bottomonium and charmonium mesons for the low-order quantum states. The derived energy equation, to first-order corrections, was employed to model the free potential parameters through fitting to experimental data of the Particle Data Group. The proposed potential successfully reproduces asymptotic freedom at short distances through one-gluon exchange interactions and quark confinement at large distances, which are the essential features of the strong interactions in Quantum chromodynamics theory. The calculated bottomonium masses exhibited excellent agreement with experimental values, yielding an absolute percentage average deviation (APAD) of 0.24%, which improves upon several previously reported theoretical results. Similarly, the vector and pseudoscalar charmonium masses were obtained with an APAD of 1.65%, demonstrating improved and comparable accuracy relative to existing competing theoretical calculations. Although our results were limited to first-order corrections to the energy spectra within the perturbation theory, the reliability of the approximation was validated by comparison with exact numerical solutions obtained using the matrix Numerov method. The small percentage errors obtained confirm the effectiveness of the phenomenological potential and perturbation approximation in describing quarkonia systems. The results suggest that the approach can be reliably extended to higher excited states.

The replication dynamics (differential equation system) is the foundation of evolutionary game theory. When n=2, there are four possible types of replication dynamics. When n=3, there are 49 possible types of replication dynamics. However, when n>3, the classification of replication dynamics has not been solved. In this article, the sufficient and necessary conditions of the replication dynamics equation with a unique fixed point in the interior of simplex $S_n$(Int$S_n$) for $n\geq 2$ are presented. Furthermore, the different types of replication dynamics equations with a unique fixed point in IntSn is discussed.

In quantum many-body dynamics, locally encoded information typically scrambles across the entire system, becoming inaccessible to local probes. The upper bound of accessible information of local probes can be characterized by the Holevo information via optimal measurement. In this work, we investigate the information dynamics of quantum scrambling utilizing local randomized probes, quantified by the averaged accessible information (AAI). We derive an analytical expression for the AAI under Haar-random measurements and demonstrate that it is a function of purity of local reduced density matrix. Operationally, we employ the classical shadow protocol, using only single-qubit randomized Pauli measurements, to efficiently extract the AAI across extended subsystems. Through numerical simulations across diverse many-body paradigms, we show that the AAI can reveal distinct scrambling behaviors, resolving phenomena that range from dynamical confinement and ballistic transport to persistent scar revivals and many-body localization. This work highlights a pragmatic paradigm shift--from relying on optimal measurements to utilizing randomized local probes--for the characterization of complex quantum information dynamics.