Entropy最新文献

Kolmogorov complexity asks whether a string can be outputted by a Turing Machine (TM) whose description is shorter. Analogously, a real number is considered computable if a Turing machine can generate its decimal expansion. The modern ϵ-approximation definition of computability, widely used in practical computation, ensures that computable reals are constructively closed under addition. However, Turing's original 1936 digit-by-digit notion, which demands the direct output of the n-th digit, presents a stark divergence. Though the set of Turing-computable reals is not constructively closed under addition, we prove that a Turing machine capable of computing x+y non-constructively exists. The core constructive computational barrier arises from determining the ones digit of a sum like 0.333¯+0.666¯=0.999¯=1.000¯. This particular example is ambiguous because both 0.999¯ and 1.000¯ are legitimate decimal representations of the same number. However, if any of the infinite number of 3s in the first term is changed to a 2 (e.g., 0.33…32…+0.666¯), the sum's leading digit is definitely zero. Conversely, if it is changed to a 4 (e.g., 0.33…34…+0.666¯), the leading digit is definitely one. This implies an inherent undecidability in determining these digits. Recent papers and our work address this issue. Hamkins provides an informal argument, while Berthelette et al. present more complicated formal proof, and our contribution offers a simple reduction to the Halting Problem. We demonstrate that determining when carry propagation stops can be resolved with a single query to an oracle that tells if and when a given TM halts. Because a concrete answer to this query exists, so does a TM computing the digits of x+y, though the proof is non-constructive. As far as we know, the analogous question for multiplication remains open. This, we feel, is an interesting addition to the story. This reveals a subtle but significant difference between the modern ϵ-approximation definition and Turing's original 1936 digit-by-digit notion of a computable number, as well as between constructive and non-constructive proof. This issue of computability and numerical precision ties into algorithmic information and Kolmogorov complexity.

We numerically investigate vibrational resonance (VR) and vibrational energy harvesting (VEH) in a mechanical system driven by a low-frequency periodic force, using time-periodic phase modulation of the potential function. We focus on how the characteristics of high-frequency excitations influence frequency response, power output, and harvesting efficiency. We uncover two modulation-induced phenomena-resonant induction and resonant amplification-that together produce a double VR effect. We demonstrate that in the weak low-frequency regime (ω≤0.3), the power output can exceed that of the moderate regime (ω≈1). Among the modulating waveforms, square waveform (SQW) demonstrated superior efficiency over other waveforms, which corresponds to higher response amplitude. In addition, the frequency ratio K=6.7 yielded optimal performance compared to other frequency ratios, thereby providing both maximum power output and efficiency. These findings suggest a new design strategy for energy harvesters, leveraging both primary and induced VR to enhance performance.

Percolation describes the formation of a giant cluster once the average degree of a network exceeds a critical value. A hybrid percolation transition (HPT) denotes a phenomenon in which a discontinuous jump of the order parameter and the critical behavior, a basic pattern of a continuous transition, appear together at the same threshold. Such HPTs have been reported in many different systems. In this review, we present several representative examples of HPTs and classify them into two categories: global suppression-induced HPTs and cascading failure-induced HPTs. In the former class, critical behavior manifests itself in the distribution of cluster sizes, whereas in the latter it emerges in the distribution of avalanche sizes. We further outline the universal scaling relations shared by both types.

Accurately and efficiently predicting redox potentials and Hammett constants using simple density-based functions derived from information-theoretic approach (ITA) quantities remains an unresolved challenge. In this work, we employ two recently proposed protocols, DL(ITA) (deep learning) and QML(ITA) (quantum machine learning), to a broad range of quinone derivatives with available experimental data. The molecular electrostatic potential (MEP) at the nucleus of the acidic atom and the sum of valence natural atomic orbital (NAO) energies are used within a linear regression (LR) framework to assess the first redox potentials and Hammett parameters of these quinone derivatives. The DL(ITA) protocol enables the construction of a transferable model trained on quinone derivatives that can be applied to both quinone and non-quinone systems. Interestingly, the QML(ITA) model exhibits superior performance compared to the DL(ITA) approach. Moreover, the structure of the QML(ITA) method suggests that it may be readily implemented on real quantum hardware in the near future.

A description of the Kibble-Zurek mechanism with linear response theory has been done previously, but ad hoc hypotheses were used, such as the rate-dependent impulse window via the Zurek equation in the context of no driving in the relaxation time. In this work, I present a new framework where such hypotheses are unnecessary while preserving all the characteristics of the phenomenon. The Kibble-Zurek scaling obtained for the excess work is close to 2/5, a result that holds for open and thermally isolated systems with relaxation time that diverges at the critical point and the first zero of the relaxation function is finite. I exemplify the results using four different but significant types of scaling functions.

Relation extraction serves as an essential task for knowledge acquisition and management, defined as determining the relation between two annotated entities from a piece of text. Over recent years, zero-shot learning has been introduced to train relation extraction models due to the expensive cost of incessantly annotating emerging relations. Current methods endeavor to transfer knowledge of seen relations into predictions of unseen relations by conducting relation extraction through different tasks. Nonetheless, the divergence in task formulations prevents relation extraction models from acquiring informative semantic representations, resulting in inferior performance. In this paper, we strive to exploit the relational knowledge contained in pre-trained language models, which may generate enlightening information for the representation of unseen relations from seen relations. To this end, we investigate a Prompt-Contrastive learning perspective for Relation Extraction under a zero-shot setting, namely PCRE. To be specific, based on leveraging semantic knowledge from pre-trained language models with prompt tuning, we augment each instance with different prompt templates to construct two views for an instance-level contrastive objective. Additionally, we devise an instance-description contrastive objective to elicit relational knowledge from relation descriptions. With joint optimization, the relation extraction model can learn how to separate relations. The experimental results show our PCRE method outperforms state-of-the-art baselines in zero-shot relation extraction. The further extensive analysis verifies that our proposal is robust in different datasets, the number of seen relations, and the number of training instances.

We explore a rigorous formulation of agent-based SIR epidemic dynamics as a discrete-state Markov process, capturing the stochastic propagation of infection or an invading agent on networks. Using indicator functions and corresponding marginal probabilities, we derive a hierarchy of evolution equations that resembles the classical BBGKY hierarchy in statistical mechanics. The structure of these equations clarifies the challenges of closure and highlights the principal problem of systemic complexity arising from stochastic but generally not fully chaotic interactions. Monte Carlo simulations are used to validate simplified closures and approximations, offering a unified perspective on the interplay between network topology, stochasticity, and infection dynamics. We also explore the impact of lockdown measures within a networked agent framework, illustrating how SIR dynamics and structural complexity of the network shape epidemic with propagation of the COVID-19 pandemic in Northern Italy taken as an example.

Accurate prediction of sensor network data in critical domains such as electric power systems and traffic planning is a core task for ensuring grid stability and enhancing urban operational efficiency. Although deep learning models have achieved significant architectural advancements, their training strategy implicitly assumes that all future events are equally predictable, ignoring that the future evolution of sensor signals intertwines deterministic patterns with stochastic events and that prediction difficulty increases with temporal distance. Forcing a model to fit inherently unpredictable events with a uniform supervision may impair its ability to learn generalizable patterns. To address this, we introduce an Unpredictability Perception loss that dynamically computes a supervision weight. The computation of this weight unifies two assessment dimensions of the intrinsic unpredictability of the forecasting task. The first originates from a posterior analysis of the signal content's randomness, while the second stems from an a priori consideration of temporal distance. The first dimension, through a complexity-aware weight derived from local spectral entropy, reduces supervision on random segments of the signal. The second dimension, through a temporal decay weight based on exponential decay, lessens supervision for distant future points. Applied to the advanced TimeMixer model, experimental results show that our approach achieves performance improvements across multiple public benchmark datasets. By matching the supervision strength to the intrinsic predictability of the signals, our proposed Unpredictability Perception loss function enhances the forecasting accuracy for sensor network data, providing a more reliable technical foundation for ensuring the stability of critical infrastructures like power grids and optimizing urban traffic systems.

While deep learning models have demonstrated superior performance in cryptocurrency forecasting, their deployment is often hindered by a lack of interpretability and trustworthiness. To bridge this gap, this paper proposes the Cryptocurrency Counterfactual Explanation (CryptoForecastCF) model. Recognizing the inherent volatility and complex non-linear dynamics of cryptocurrency markets, we argue that understanding the sensitivity of model outputs to slight variations in historical conditions is fundamental to robust risk management. CryptoForecastCF employs a gradient-based optimization strategy to generate meaningful counterfactual explanations. Specifically, it identifies minimal modifications, defined as the optimal perturbations to historical market features such as price constrained by ℓ1 or ℓ2 norms, that are sufficient to steer the model's future predictions into user-specified target intervals. This approach not only elucidates the key driving factors and decision boundaries of opaque models but also equips traders and risk managers with actionable insights, enabling them to identify the specific market shifts required to navigate high-stakes scenarios and mitigate unfavorable predictive outcomes.

In the spacetime of a charged rotating accelerated black hole, the dynamics equations of fermions and bosons are modified by Lorentz invariance violation (LIV). The correction effects of LIV on the quantum tunneling radiation of this black hole are investigated. New expressions for the quantum tunneling rate, Hawking temperature, and Bekenstein-Hawking entropy of this black hole, which depend on the charge parameter and acceleration parameter, are derived, incorporating LIV correction terms. The physical implications of these results are discussed in depth.