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Entropy, Volume 26, Issue 11 (November 2024) – 110 articles

Cover Story (view full-size image): Quantum information scrambling describes how initially localized information spreads across many degrees of freedom in quantum systems, a process crucial to understanding thermalization. While typically studied in sudden quenches, this work explores scrambling during adiabatic evolution in critical systems. The cover illustrates an adiabatic cycle where information, initially encoded in a symmetry-breaking state, becomes scrambled as the system transitions between quantum phases. Although the populations of the initial state remain intact, relative phases among eigenstates change, rendering the encoded information inaccessible and factually lost. This research connects information scrambling with quantum phase transitions, offering insights for experimental studies of quantum many-body systems. View this paper
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23 pages, 553 KiB  
Article
Inferring About the Average Value of Audit Errors from Sequential Ratio Tests
by Grzegorz Sitek and Mariusz Pleszczyński
Entropy 2024, 26(11), 998; https://doi.org/10.3390/e26110998 - 20 Nov 2024
Viewed by 350
Abstract
The book amounts are modeled as values of a random variable, represented by a mixture of distributions of both the correct and error-contaminated amounts. The mixing coefficient represents the proportion of items with non-zero error amounts. This study addresses the problem of determining [...] Read more.
The book amounts are modeled as values of a random variable, represented by a mixture of distributions of both the correct and error-contaminated amounts. The mixing coefficient represents the proportion of items with non-zero error amounts. This study addresses the problem of determining the sample size needed for testing statistical hypotheses regarding mean accounting errors. The average sample size is estimated using the Sequential Probability Ratio Test (SPRT), applying the Monte Carlo method. Estimating average audit errors is a common challenge in economic research. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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21 pages, 6191 KiB  
Article
How Do Transformers Model Physics? Investigating the Simple Harmonic Oscillator
by Subhash Kantamneni, Ziming Liu and Max Tegmark
Entropy 2024, 26(11), 997; https://doi.org/10.3390/e26110997 - 19 Nov 2024
Viewed by 650
Abstract
How do transformers model physics? Do transformers model systems with interpretable analytical solutions or do they create an “alien physics” that is difficult for humans to decipher? We have taken a step towards demystifying this larger puzzle by investigating the simple harmonic oscillator [...] Read more.
How do transformers model physics? Do transformers model systems with interpretable analytical solutions or do they create an “alien physics” that is difficult for humans to decipher? We have taken a step towards demystifying this larger puzzle by investigating the simple harmonic oscillator (SHO), x¨+2γx˙+ω02x=0, one of the most fundamental systems in physics. Our goal was to identify the methods transformers use to model the SHO, and to do so we hypothesized and evaluated possible methods by analyzing the encoding of these methods’ intermediates. We developed four criteria for the use of a method within the simple test bed of linear regression, where our method was y=wx and our intermediate was w: (1) Can the intermediate be predicted from hidden states? (2) Is the intermediate’s encoding quality correlated with the model performance? (3) Can the majority of variance in hidden states be explained by the intermediate? (4) Can we intervene on hidden states to produce predictable outcomes? Armed with these two correlational (1,2), weak causal (3), and strong causal (4) criteria, we determined that transformers use known numerical methods to model the trajectories of the simple harmonic oscillator, specifically, the matrix exponential method. Our analysis framework can conveniently extend to high-dimensional linear systems and nonlinear systems, which we hope will help reveal the “world model” hidden in transformers. Full article
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16 pages, 971 KiB  
Article
Derangetropy in Probability Distributions and Information Dynamics
by Masoud Ataei and Xiaogang Wang
Entropy 2024, 26(11), 996; https://doi.org/10.3390/e26110996 - 18 Nov 2024
Viewed by 343
Abstract
We introduce derangetropy, which is a novel functional measure designed to characterize the dynamics of information within probability distributions. Unlike scalar measures such as Shannon entropy, derangetropy offers a functional representation that captures the dispersion of information across the entire support of a [...] Read more.
We introduce derangetropy, which is a novel functional measure designed to characterize the dynamics of information within probability distributions. Unlike scalar measures such as Shannon entropy, derangetropy offers a functional representation that captures the dispersion of information across the entire support of a distribution. By incorporating self-referential and periodic properties, it provides insights into information dynamics governed by differential equations and equilibrium states. Through combinatorial justifications and empirical analysis, we demonstrate the utility of derangetropy in depicting distribution behavior and evolution, providing a new tool for analyzing complex and hierarchical systems in information theory. Full article
(This article belongs to the Special Issue Mathematics in Information Theory and Modern Applications)
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18 pages, 317 KiB  
Article
Information-Theoretic Generalization Bounds for Batch Reinforcement Learning
by Xingtu Liu
Entropy 2024, 26(11), 995; https://doi.org/10.3390/e26110995 - 18 Nov 2024
Viewed by 360
Abstract
We analyze the generalization properties of batch reinforcement learning (batch RL) with value function approximation from an information-theoretic perspective. We derive generalization bounds for batch RL using (conditional) mutual information. In addition, we demonstrate how to establish a connection between certain structural assumptions [...] Read more.
We analyze the generalization properties of batch reinforcement learning (batch RL) with value function approximation from an information-theoretic perspective. We derive generalization bounds for batch RL using (conditional) mutual information. In addition, we demonstrate how to establish a connection between certain structural assumptions on the value function space and conditional mutual information. As a by-product, we derive a high-probability generalization bound via conditional mutual information, which was left open and may be of independent interest. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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19 pages, 1472 KiB  
Article
Generalized Filter Bank Orthogonal Frequency Division Multiplexing: Low-Complexity Waveform for Ultra-Wide Bandwidth and Flexible Services
by Yu Xin, Jian Hua, Tong Bao, Yaxing Hao, Ziheng Xiao, Xin Nie and Fanggang Wang
Entropy 2024, 26(11), 994; https://doi.org/10.3390/e26110994 - 18 Nov 2024
Viewed by 436
Abstract
Terahertz (THz) communication is a crucial technique in sixth generation (6G) mobile networks, which allow for ultra-wide bandwidths to enable ultra-high data rate wireless communication. However, the current subcarrier spacing and the size of fast Fourier transform (FFT) of the orthogonal frequency division [...] Read more.
Terahertz (THz) communication is a crucial technique in sixth generation (6G) mobile networks, which allow for ultra-wide bandwidths to enable ultra-high data rate wireless communication. However, the current subcarrier spacing and the size of fast Fourier transform (FFT) of the orthogonal frequency division multiplexing (OFDM) in 5G NR are insufficient regarding the bandwidth requirements of terahertz scenarios. In this paper, a novel waveform is proposed to address the ultra-wideband issue, namely the generalized filter bank orthogonal frequency division multiplexing (GFB-OFDM) waveform. The main advantages are summarized as follows: (1) The K-point IFFT is implemented by two levels of IFFTs in smaller sizes, i.e, performing M-point IFFT in N times and performing N-point IFFT in M times, where K=N×M. (2) The proposed waveform can accommodate flexible subcarrier spacings and different numbers of the subbands to provide various services in a single GFB-OFDM symbol. (3) Different bandwidths can be supported using a fixed filter since the filtering is performed on each subband. In contrast, the cyclic prefix orthogonal frequency division multiplexing (CP-OFDM) in 4G/5G requires various filters. (4) The existing detection for CP-OFDM can be directly employed as the detector of the proposed waveform. Lastly, the comprehensive simulation results demonstrate that GFB-OFDM outperforms CP-OFDM in terms of the out-of-band leakage, complexity and error performance. Full article
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26 pages, 766 KiB  
Article
Still No Evidence for an Effect of the Proportion of Non-Native Speakers on Natural Language Complexity
by Alexander Koplenig
Entropy 2024, 26(11), 993; https://doi.org/10.3390/e26110993 - 18 Nov 2024
Viewed by 439
Abstract
In a recent study, I demonstrated that large numbers of L2 (second language) speakers do not appear to influence the morphological or information-theoretic complexity of natural languages. This paper has three primary aims: First, I address recent criticisms of my analyses, showing that [...] Read more.
In a recent study, I demonstrated that large numbers of L2 (second language) speakers do not appear to influence the morphological or information-theoretic complexity of natural languages. This paper has three primary aims: First, I address recent criticisms of my analyses, showing that the points raised by my critics were already explicitly considered and analysed in my original work. Furthermore, I show that the proposed alternative analyses fail to withstand detailed examination. Second, I introduce new data on the information-theoretic complexity of natural languages, with the estimates derived from various language models—ranging from simple statistical models to advanced neural networks—based on a database of 40 multilingual text collections that represent a wide range of text types. Third, I re-analyse the information-theoretic and morphological complexity data using novel methods that better account for model uncertainty in parameter estimation, as well as the genealogical relatedness and geographic proximity of languages. In line with my earlier findings, the results show no evidence that large numbers of L2 speakers have an effect on natural language complexity. Full article
(This article belongs to the Special Issue Complexity Characteristics of Natural Language)
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24 pages, 1736 KiB  
Article
Multi-Label Feature Selection with Feature–Label Subgraph Association and Graph Representation Learning
by Jinghou Ruan, Mingwei Wang, Deqing Liu, Maolin Chen and Xianjun Gao
Entropy 2024, 26(11), 992; https://doi.org/10.3390/e26110992 - 18 Nov 2024
Viewed by 419
Abstract
In multi-label data, a sample is associated with multiple labels at the same time, and the computational complexity is manifested in the high-dimensional feature space as well as the interdependence and unbalanced distribution of labels, which leads to challenges regarding feature selection. As [...] Read more.
In multi-label data, a sample is associated with multiple labels at the same time, and the computational complexity is manifested in the high-dimensional feature space as well as the interdependence and unbalanced distribution of labels, which leads to challenges regarding feature selection. As a result, a multi-label feature selection method based on feature–label subgraph association with graph representation learning (SAGRL) is proposed to represent the complex correlations of features and labels, especially the relationships between features and labels. Specifically, features and labels are mapped to nodes in the graph structure, and the connections between nodes are established to form feature and label sets, respectively, which increase intra-class correlation and decrease inter-class correlation. Further, feature–label subgraphs are constructed by feature and label sets to provide abundant feature combinations. The relationship between each subgraph is adjusted by graph representation learning, the crucial features in different label sets are selected, and the optimal feature subset is obtained by ranking. Experimental studies on 11 datasets show the superior performance of the proposed method with six evaluation metrics over some state-of-the-art multi-label feature selection methods. Full article
(This article belongs to the Section Multidisciplinary Applications)
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46 pages, 15416 KiB  
Review
Mathematical Modeling of Physical Reality: From Numbers to Fractals, Quantum Mechanics and the Standard Model
by Marian Kupczynski
Entropy 2024, 26(11), 991; https://doi.org/10.3390/e26110991 - 18 Nov 2024
Viewed by 605
Abstract
In physics, we construct idealized mathematical models in order to explain various phenomena which we observe or create in our laboratories. In this article, I recall how sophisticated mathematical models evolved from the concept of a number created thousands of years ago, and [...] Read more.
In physics, we construct idealized mathematical models in order to explain various phenomena which we observe or create in our laboratories. In this article, I recall how sophisticated mathematical models evolved from the concept of a number created thousands of years ago, and I discuss some challenges and open questions in quantum foundations and in the Standard Model. We liberated nuclear energy, landed on the Moon and built ‘quantum computers’. Encouraged by these successes, many believe that when we reconcile general relativity with quantum theory we will have the correct theory of everything. Perhaps we should be much humbler. Our perceptions of reality are biased by our senses and by our brain, bending them to meet our priors and expectations. Our abstract mathematical models describe only in an approximate way different layers of physical reality. To describe the motion of a meteorite, we can use a concept of a material point, but the point-like approximation breaks completely when the meteorite hits the Earth. Similarly, thermodynamic, chemical, molecular, atomic, nuclear and elementary particle layers of physical reality are described using specific abstract mathematical models and approximations. In my opinion, the theory of everything does not exist. Full article
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20 pages, 9405 KiB  
Article
Integration of Sense and Control for Uncertain Systems Based on Delayed Feedback Active Inference
by Mingyue Ji, Kunpeng Pan, Xiaoxuan Zhang, Quan Pan, Xiangcheng Dai and Yang Lyu
Entropy 2024, 26(11), 990; https://doi.org/10.3390/e26110990 - 18 Nov 2024
Viewed by 431
Abstract
Asa result of the time lag in transmission, the data obtained by the sensor is delayed and does not reflect the state at the current moment. The effects of input delay are often overlooked in active inference (AIF), which may lead to significant [...] Read more.
Asa result of the time lag in transmission, the data obtained by the sensor is delayed and does not reflect the state at the current moment. The effects of input delay are often overlooked in active inference (AIF), which may lead to significant deviations in state estimation and increased prediction errors, particularly when the system is subjected to a sudden external stimulus. In this paper, a theoretical framework of delayed feedback active inference (DAIF) is proposed to enhance the applicability of AIF to real systems. The probability model of DAIF is defined by incorporating a control distribution into that of AIF. The free energy of DAIF is defined as the sum of the quadratic state, sense, and control prediction error. A predicted state derived from previous states is defined and introduced as the expectation of the prior distribution of the real-time state. A proportional-integral (PI)-like control based on the predicted state is taken to be the expectation of DAIF preference control, whose gain coefficient is inversely proportional to the measurement accuracy variance. To adaptively compensate for external disturbances, a second-order inverse variance accuracy replaces the fixed sensory accuracy of preference control. The simulation results of the trajectory tracking control of a quadrotor unmanned aerial vehicle (UAV) show that DAIF performs better than AIF in state estimation and disturbance resistance. Full article
(This article belongs to the Section Multidisciplinary Applications)
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39 pages, 416 KiB  
Article
“In Mathematical Language”: On Mathematical Foundations of Quantum Foundations
by Arkady Plotnitsky
Entropy 2024, 26(11), 989; https://doi.org/10.3390/e26110989 - 18 Nov 2024
Viewed by 450
Abstract
The argument of this article is threefold. First, the article argues that from its rise in the sixteenth century to our own time, the advancement of modern physics as mathematical-experimental science has been defined by the invention of new mathematical structures. Second, the [...] Read more.
The argument of this article is threefold. First, the article argues that from its rise in the sixteenth century to our own time, the advancement of modern physics as mathematical-experimental science has been defined by the invention of new mathematical structures. Second, the article argues that quantum theory, especially following quantum mechanics, gives this thesis a radically new meaning by virtue of the following two features: on the one hand, quantum phenomena are defined as essentially different from those found in all previous physics by purely physical features; and on the other, quantum mechanics and quantum field theory are defined by purely mathematical postulates, which connect them to quantum phenomena strictly in terms of probabilities, without, as in all previous physics, representing or otherwise relating to how these phenomena physically come about. While these two features may appear discordant, if not inconsistent, I argue that they are in accord with each other, at least in certain interpretations (including the one adopted here), designated as “reality without realism”, RWR, interpretations. This argument also allows this article to offer a new perspective on a thorny problem of the relationships between continuity and discontinuity in quantum physics. In particular, rather than being concerned only with the discreteness and continuity of quantum objects or phenomena, quantum mechanics and quantum field theory relate their continuous mathematics to the irreducibly discrete quantum phenomena in terms of probabilistic predictions while, at least in RWR interpretations, precluding a representation or even conception of how these phenomena come about. This subject is rarely, if ever, discussed apart from previous work by the present author. It is, however, given a new dimension in this article which introduces, as one of its main contributions, a new principle: the mathematical complexity principle. Full article
24 pages, 1445 KiB  
Article
A Novel Information Complexity Approach to Score Receiver Operating Characteristic (ROC) Curve Modeling
by Aylin Gocoglu, Neslihan Demirel and Hamparsum Bozdogan
Entropy 2024, 26(11), 988; https://doi.org/10.3390/e26110988 - 17 Nov 2024
Viewed by 560
Abstract
Performance metrics are measures of success or performance that can be used to evaluate how well a model makes accurate predictions or classifications. However, there is no single measure since each performance metric emphasizes a different classification aspect. Model selection procedures based on [...] Read more.
Performance metrics are measures of success or performance that can be used to evaluate how well a model makes accurate predictions or classifications. However, there is no single measure since each performance metric emphasizes a different classification aspect. Model selection procedures based on information criteria offer a quantitative measure that balances model complexity with goodness of fit, providing a better alternative to classical approaches. In this paper, we introduce and develop a novel Information Complexity–Receiver Operating Characteristic, abbreviated as ICOMP-ROC, criterion approach to fit and study the performance of ROC curve models. We construct and derive the Universal ROC (UROC) for a combination of sixteen Bi-distributional ROC models to choose the best Bi-distributional ROC by minimizing the ICOMP-ROC criterion. We conduct large-scale Monte Carlo simulations using the sixteen Bi-distributional ROC models with the Normal–Normal and Weibull–Gamma pairs as the pseudo-true ROC models. We report the frequency of hits of the ICOMP-ROC criterion, showing its remarkable recovery rate. In addition to Bi-distributional fitting, we consider a high-dimensional real Magnetic Resonance Imaging (MRI) of the Brain dataset and Wisconsin Breast Cancer (WBC) dataset to study the performance of the common performance metrics and the ICOMP-ROC criterion using several machine learning (ML) classification algorithms. We use the genetic algorithm (GA) to reduce the dimensions of these two datasets to choose the best subset of the features to study and compare the performance of the newly proposed ICOMP-ROC criterion along with the traditional performance metrics. The choice of a suitable metric is not just contingent upon the ML model used, but it also depends upon the complexity and high dimensionality of the input datasets, since the traditional performance metrics give different results and have inherent limitations. Our numerical results show the consistency and reliability of the ICOMP-ROC criterion over the traditional performance metrics as a clever model selection criterion to choose the best fitting Bi-distributional ROC model and the best classification algorithm among the ones considered. This shows the utility and the versatility of our newly proposed approach in ROC curve modeling that integrates and robustifies currently used procedures. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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15 pages, 3776 KiB  
Article
Toward Transparent and Controllable Quantum Generative Models
by Jinkai Tian and Wenjing Yang
Entropy 2024, 26(11), 987; https://doi.org/10.3390/e26110987 - 17 Nov 2024
Viewed by 292
Abstract
Quantum generative models have shown promise in fields such as quantum chemistry, materials science, and optimization. However, their practical utility is hindered by a significant challenge: the lack of interpretability. In this work, we introduce model inversion to enhance both the interpretability and [...] Read more.
Quantum generative models have shown promise in fields such as quantum chemistry, materials science, and optimization. However, their practical utility is hindered by a significant challenge: the lack of interpretability. In this work, we introduce model inversion to enhance both the interpretability and controllability of quantum generative models. Model inversion allows for tracing generated quantum states back to their latent variables, revealing the relationship between input parameters and generated outputs. We apply this method to models generating ground states for Hamiltonians, such as the transverse-field Ising model (TFIM) and generalized cluster Hamiltonians, achieving interpretability control without retraining the model. Experimental results demonstrate that our approach can accurately guide the generated quantum states across different quantum phases. This framework bridges the gap between theoretical models and practical applications by providing transparency and fine-tuning capabilities, particularly in high-stakes environments like drug discovery and material design. Full article
(This article belongs to the Section Quantum Information)
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9 pages, 553 KiB  
Article
Gravitational Hamiltonian Systems and the Retarded Gravity Inequality
by Asher Yahalom
Entropy 2024, 26(11), 986; https://doi.org/10.3390/e26110986 - 17 Nov 2024
Viewed by 371
Abstract
Gravity and electromagnetic interactions are the only fundamental physical interactions (outside the nuclear domain). In this work, we shall concentrate on Hamiltonians containing gravitational interaction, which according to general relativity must be retarded. In recent years, retarded gravity has explained many of the [...] Read more.
Gravity and electromagnetic interactions are the only fundamental physical interactions (outside the nuclear domain). In this work, we shall concentrate on Hamiltonians containing gravitational interaction, which according to general relativity must be retarded. In recent years, retarded gravity has explained many of the mysteries surrounding the “missing mass” related to galactic rotation curves, the Tully–Fisher relations, and gravitational lensing phenomena. Indeed, a recent paper analyzing 143 galaxies has demonstrated that retarded gravity will suffice to explain galaxies’ rotation curves without the need to postulate dark matter for multiple types of galaxies. Moreover, it also demystified the “missing mass” related to galactic clusters and elliptic galaxies in which excess matter was derived through the virial theorem. Here, we give a mathematical criterion that specifies the cases in which retardation is important for gravity (and when it is not). The criterion takes the form of an inequality. Full article
(This article belongs to the Special Issue Unstable Hamiltonian Systems and Scattering Theory)
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34 pages, 459 KiB  
Article
Dynamic Bayesian Networks, Elicitation, and Data Embedding for Secure Environments
by Kieran Drury and Jim Q. Smith
Entropy 2024, 26(11), 985; https://doi.org/10.3390/e26110985 - 17 Nov 2024
Viewed by 400
Abstract
Serious crime modelling typically needs to be undertaken securely behind a firewall where police knowledge and capabilities remain undisclosed. Data informing an ongoing incident are often sparse; a large proportion of relevant data only come to light after the incident culminates or after [...] Read more.
Serious crime modelling typically needs to be undertaken securely behind a firewall where police knowledge and capabilities remain undisclosed. Data informing an ongoing incident are often sparse; a large proportion of relevant data only come to light after the incident culminates or after police intervene—by which point it is too late to make use of the data to aid real-time decision-making for the incident in question. Much of the data that are available to the police to support real-time decision-making are highly confidential and cannot be shared with academics, and are therefore missing to them. In this paper, we describe the development of a formal protocol where a graphical model is used as a framework for securely translating a base model designed by an academic team to a fully embellished model for use by a police team. We then show, for the first time, how libraries of these models can be built and used for real-time decision support to circumvent the challenges of data missingness seen in such a secure environment through the ability to match ongoing plots to existing models within the library.The parallel development described by this protocol ensures that any sensitive information collected by police and missing to academics remains secured behind a firewall. The protocol nevertheless guides police so that they are able to combine the typically incomplete data streams that are open source with their more sensitive information in a formal and justifiable way. We illustrate the application of this protocol by describing how a new entry—a suspected vehicle attack—can be embedded into such a police library of criminal plots. Full article
(This article belongs to the Special Issue Bayesian Network Modelling in Data Sparse Environments)
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17 pages, 757 KiB  
Article
Bayesian Mechanics of Synaptic Learning Under the Free-Energy Principle
by Chang Sub Kim
Entropy 2024, 26(11), 984; https://doi.org/10.3390/e26110984 - 16 Nov 2024
Viewed by 378
Abstract
The brain is a biological system comprising nerve cells and orchestrates its embodied agent’s perception, behavior, and learning in dynamic environments. The free-energy principle (FEP) advocated by Karl Friston explicates the local, recurrent, and self-supervised cognitive dynamics of the brain’s higher-order functions. In [...] Read more.
The brain is a biological system comprising nerve cells and orchestrates its embodied agent’s perception, behavior, and learning in dynamic environments. The free-energy principle (FEP) advocated by Karl Friston explicates the local, recurrent, and self-supervised cognitive dynamics of the brain’s higher-order functions. In this study, we continue to refine the FEP through a physics-guided formulation; specifically, we apply our theory to synaptic learning by considering it an inference problem under the FEP and derive the governing equations, called Bayesian mechanics. Our study uncovers how the brain infers weight changes and postsynaptic activity, conditioned on the presynaptic input, by deploying generative models of the likelihood and prior belief. Consequently, we exemplify the synaptic efficacy in the brain with a simple model; in particular, we illustrate that the brain organizes an optimal trajectory in neural phase space during synaptic learning in continuous time, which variationally minimizes synaptic surprisal. Full article
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29 pages, 2051 KiB  
Review
Quantum Stream Cipher Based on Holevo–Yuen Theory: Part II
by Osamu Hirota and Masaki Sohma
Entropy 2024, 26(11), 983; https://doi.org/10.3390/e26110983 - 15 Nov 2024
Viewed by 365
Abstract
This paper discusses the foundation of security theory for the Quantum stream cipher based on the Holevo–Yuen theory, which allows the use of “optical amplifiers”. This type of cipher is a technology that provides information-theoretic security (ITS) to optical data transmission by randomizing [...] Read more.
This paper discusses the foundation of security theory for the Quantum stream cipher based on the Holevo–Yuen theory, which allows the use of “optical amplifiers”. This type of cipher is a technology that provides information-theoretic security (ITS) to optical data transmission by randomizing ultrafast optical communication signals with quantum noise. In general, the quantitative security of ITS is evaluated in terms of the unicity distance in Shannon theory. However, the quantum version requires modeling beyond the Shannon model of a random cipher to utilize the characteristics of the physical layer. Therefore, as the first step, one has to develop a generalized unicity distance theory and apply it to the evaluation of security. Although a complete theoretical formulation has not yet been established, this paper explains a primitive structure of a generalization of the Shannon random cipher and shows that the realization of this is the generalized quantum stream cipher. In addition, we present several implementation methods of generalized quantum stream ciphers and their security. Full article
(This article belongs to the Special Issue Quantum Communication, Quantum Radar, and Quantum Cipher, 2nd Edition)
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13 pages, 498 KiB  
Article
New Variable-Weight Optical Orthogonal Codes with Weights 3 to 5
by Si-Yeon Pak, Hyo-Won Kim, DaeHan Ahn and Jin-Ho Chung
Entropy 2024, 26(11), 982; https://doi.org/10.3390/e26110982 - 15 Nov 2024
Viewed by 334
Abstract
In optical networks, designing optical orthogonal codes (OOCs) with appropriate parameters is essential for enhancing the overall system performance. They are divided into two categories, constant-weight OOCs (CW-OOCs) and variable-weight OOCs (VW-OOCs), based on the number of distinct Hamming weights present in their [...] Read more.
In optical networks, designing optical orthogonal codes (OOCs) with appropriate parameters is essential for enhancing the overall system performance. They are divided into two categories, constant-weight OOCs (CW-OOCs) and variable-weight OOCs (VW-OOCs), based on the number of distinct Hamming weights present in their codewords. This paper introduces a method for constructing VW-OOCs of length kp by using the structure of an integer ring and the Chinese Remainder Theorem. In particular, we present some specific VW-OOCs with weights of 3, 4, or 5. The results demonstrate that certain optimal VW-OOCs can be obtained with parameters that are not covered in the existing literature. Full article
(This article belongs to the Special Issue New Advances in Error-Correcting Codes)
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50 pages, 751 KiB  
Article
Non-Equilibrium Quantum Brain Dynamics: Water Coupled with Phonons and Photons
by Akihiro Nishiyama, Shigenori Tanaka and Jack Adam Tuszynski
Entropy 2024, 26(11), 981; https://doi.org/10.3390/e26110981 - 15 Nov 2024
Viewed by 517
Abstract
We investigate Quantum Electrodynamics (QED) of water coupled with sound and light, namely Quantum Brain Dynamics (QBD) of water, phonons and photons. We provide phonon degrees of freedom as additional quanta in the framework of QBD in this paper. We begin with the [...] Read more.
We investigate Quantum Electrodynamics (QED) of water coupled with sound and light, namely Quantum Brain Dynamics (QBD) of water, phonons and photons. We provide phonon degrees of freedom as additional quanta in the framework of QBD in this paper. We begin with the Lagrangian density QED with non-relativistic charged bosons, photons and phonons, and derive time-evolution equations of coherent fields and Kadanoff–Baym (KB) equations for incoherent particles. We next show an acoustic super-radiance solution in our model. We also introduce a kinetic entropy current in KB equations in 1st order approximation in the gradient expansion and show the H-theorem for self-energy in Hartree–Fock approximation. We finally derive conserved number density of charged bosons and conserved energy density in spatially homogeneous system. Full article
(This article belongs to the Section Quantum Information)
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15 pages, 27235 KiB  
Article
Dynamics of Aggregation in Systems of Self-Propelled Rods
by Richard J. G. Löffler and Jerzy Gorecki
Entropy 2024, 26(11), 980; https://doi.org/10.3390/e26110980 - 15 Nov 2024
Viewed by 485
Abstract
We highlight camphene–camphor–polypropylene plastic as a useful material for self-propelled objects that show aggregation while floating on a water surface. We consider self-propelled rods as an example of aggregation of objects characterized by non-trivial individual shapes with low-symmetry interactions between them. The motion [...] Read more.
We highlight camphene–camphor–polypropylene plastic as a useful material for self-propelled objects that show aggregation while floating on a water surface. We consider self-propelled rods as an example of aggregation of objects characterized by non-trivial individual shapes with low-symmetry interactions between them. The motion of rods made of the camphene–camphor–polypropylene plastic is supported by dissipation of the surface-active molecules. The physical processes leading to aggregation and the mathematical model of the process are discussed. We analyze experimental data of aggregate formation dynamics and relate them to the system’s properties. We speculate that the aggregate structure can be represented as a string of symbols, which opens the potential applicability of the phenomenon for information processing if objects floating on a water surface are regarded as reservoir computers. Full article
(This article belongs to the Special Issue Matter-Aggregating Systems at a Classical vs. Quantum Interface)
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23 pages, 324 KiB  
Article
Bowen’s Formula for a Dynamical Solenoid
by Andrzej Biś, Wojciech Kozłowski and Agnieszka Marczuk
Entropy 2024, 26(11), 979; https://doi.org/10.3390/e26110979 - 15 Nov 2024
Viewed by 306
Abstract
More than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen’s formula, that relates the Hausdorff dimension of a conformal repeller to the [...] Read more.
More than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen’s formula, that relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function defined by a single conformal map. In this paper, we extend the result of Bowen to a sequence of conformal maps. We present a dynamical solenoid, i.e., a generalized dynamical system obtained by backward compositions of a sequence of continuous surjections (fn:XX)nN defined on a compact metric space (X,d). Under mild assumptions, we provide a self-contained proof that Bowen’s formula holds for dynamical conformal solenoids. As a corollary, we obtain that the Bowen’s formula holds for a conformal surjection f:XX of a compact Full article
(This article belongs to the Section Statistical Physics)
13 pages, 284 KiB  
Article
Quantum Control Design by Lyapunov Trajectory Tracking and Optimal Control
by Hongli Yang, Guohui Yu and Ivan Ganchev Ivanov
Entropy 2024, 26(11), 978; https://doi.org/10.3390/e26110978 - 15 Nov 2024
Viewed by 471
Abstract
In this paper, we investigate a Lyapunov trajectory tracking design method that incorporates a Schrödinger equation with a dipole subterm and polarizability. Our findings suggest that the proposed control law can overcome the limitations of certain existing control laws that do not converge. [...] Read more.
In this paper, we investigate a Lyapunov trajectory tracking design method that incorporates a Schrödinger equation with a dipole subterm and polarizability. Our findings suggest that the proposed control law can overcome the limitations of certain existing control laws that do not converge. By integrating a quadratic performance index, we introduce an optimal control law, which we subsequently analyze for stability and optimality. We also simulate the spin-1/2 particle system to illustrate our results. These findings are further validated through numerical illustrations involving a 3D, 5D system, and a spin-1/2 particle system. Full article
(This article belongs to the Special Issue Information Theory in Control Systems, 2nd Edition)
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12 pages, 359 KiB  
Article
Statistical Properties of Superpositions of Coherent Phase States with Opposite Arguments
by Miguel Citeli de Freitas and Viktor V. Dodonov
Entropy 2024, 26(11), 977; https://doi.org/10.3390/e26110977 - 15 Nov 2024
Viewed by 377
Abstract
We calculate the second-order moments, the Robertson–Schrödinger uncertainty product, and the Mandel factor for various superpositions of coherent phase states with opposite arguments, comparing the results with similar superpositions of the usual (Klauder–Glauber–Sudarshan) coherent states. We discover that the coordinate variance in the [...] Read more.
We calculate the second-order moments, the Robertson–Schrödinger uncertainty product, and the Mandel factor for various superpositions of coherent phase states with opposite arguments, comparing the results with similar superpositions of the usual (Klauder–Glauber–Sudarshan) coherent states. We discover that the coordinate variance in the analog of even coherent states can show the most strong squeezing effect, close to the maximal possible squeezing for the given mean photon number. On the other hand, the Robertson–Schrödinger (RS) uncertainty product in superpositions of coherent phase states increases much slower (as function of the mean photon number) than in superpositions of the usual coherent states. A nontrivial behavior of the Mandel factor for small mean photon numbers is discovered in superpositions with unequal weights of two components. An exceptional nature of the even and odd superpositions is demonstrated. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness V)
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31 pages, 1865 KiB  
Article
Robustness Analysis of Multilayer Infrastructure Networks Based on Incomplete Information Stackelberg Game: Considering Cascading Failures
by Haitao Li, Lixin Ji, Yingle Li and Shuxin Liu
Entropy 2024, 26(11), 976; https://doi.org/10.3390/e26110976 - 14 Nov 2024
Viewed by 464
Abstract
The growing importance of critical infrastructure systems (CIS) makes maintaining their normal operation against deliberate attacks such as terrorism a significant challenge. Combining game theory and complex network theory provides a framework for analyzing CIS robustness in adversarial scenarios. Most existing studies focus [...] Read more.
The growing importance of critical infrastructure systems (CIS) makes maintaining their normal operation against deliberate attacks such as terrorism a significant challenge. Combining game theory and complex network theory provides a framework for analyzing CIS robustness in adversarial scenarios. Most existing studies focus on single-layer networks, while CIS are better modeled as multilayer networks. Research on multilayer network games is limited, lacking methods for constructing incomplete information through link hiding and neglecting the impact of cascading failures. We propose a multilayer network Stackelberg game model with incomplete information considering cascading failures (MSGM-IICF). First, we describe the multilayer network model and define the multilayer node-weighted degree. Then, we present link hiding rules and a cascading failure model. Finally, we construct MSGM-IICF, providing methods for calculating payoff functions from the different perspectives of attackers and defenders. Experiments on synthetic and real-world networks demonstrate that link hiding improves network robustness without considering cascading failures. However, when cascading failures are considered, they become the primary factor determining network robustness. Dynamic capacity allocation enhances network robustness, while changes in dynamic costs make the network more vulnerable. The proposed method provides a new way of analyzing the robustness of diverse CIS, supporting resilient CIS design. Full article
(This article belongs to the Special Issue Robustness and Resilience of Complex Networks)
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13 pages, 668 KiB  
Article
Sensitivity of Bayesian Networks to Errors in Their Structure
by Agnieszka Onisko and Marek J. Druzdzel
Entropy 2024, 26(11), 975; https://doi.org/10.3390/e26110975 - 14 Nov 2024
Viewed by 389
Abstract
There is a widespread belief in the Bayesian network (BN) community that while the overall accuracy of the results of BN inference is not sensitive to the precision of parameters, it is sensitive to the structure. We report on the results of a [...] Read more.
There is a widespread belief in the Bayesian network (BN) community that while the overall accuracy of the results of BN inference is not sensitive to the precision of parameters, it is sensitive to the structure. We report on the results of a study focusing on the parameters in a companion paper, while this paper focuses on the BN graphical structure. We present the results of several experiments in which we test the impact of errors in the BN structure on its accuracy in the context of medical diagnostic models. We study the deterioration in model accuracy under structural changes that systematically modify the original gold standard model, notably the node and edge removal and edge reversal. Our results confirm the popular belief that the BN structure is important, and we show that structural errors may lead to a serious deterioration in the diagnostic accuracy. At the same time, most BN models are forgiving to single errors. In light of these results and the results of the companion paper, we recommend that knowledge engineers focus their efforts on obtaining a correct model structure and worry less about the overall precision of parameters. Full article
(This article belongs to the Special Issue Bayesian Network Modelling in Data Sparse Environments)
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27 pages, 12110 KiB  
Article
Exploring the Impact of Additive Shortcuts in Neural Networks via Information Bottleneck-like Dynamics: From ResNet to Transformer
by Zhaoyan Lyu and Miguel R. D. Rodrigues
Entropy 2024, 26(11), 974; https://doi.org/10.3390/e26110974 - 14 Nov 2024
Viewed by 519
Abstract
Deep learning has made significant strides, driving advances in areas like computer vision, natural language processing, and autonomous systems. In this paper, we further investigate the implications of the role of additive shortcut connections, focusing on models such as ResNet, Vision Transformers (ViTs), [...] Read more.
Deep learning has made significant strides, driving advances in areas like computer vision, natural language processing, and autonomous systems. In this paper, we further investigate the implications of the role of additive shortcut connections, focusing on models such as ResNet, Vision Transformers (ViTs), and MLP-Mixers, given that they are essential in enabling efficient information flow and mitigating optimization challenges such as vanishing gradients. In particular, capitalizing on our recent information bottleneck approach, we analyze how additive shortcuts influence the fitting and compression phases of training, crucial for generalization. We leverage Z-X and Z-Y measures as practical alternatives to mutual information for observing these dynamics in high-dimensional spaces. Our empirical results demonstrate that models with identity shortcuts (ISs) often skip the initial fitting phase and move directly into the compression phase, while non-identity shortcut (NIS) models follow the conventional two-phase process. Furthermore, we explore how IS models are still able to compress effectively, maintaining their generalization capacity despite bypassing the early fitting stages. These findings offer new insights into the dynamics of shortcut connections in neural networks, contributing to the optimization of modern deep learning architectures. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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20 pages, 752 KiB  
Article
DUS Topp–Leone-G Family of Distributions: Baseline Extension, Properties, Estimation, Simulation and Useful Applications
by Divine-Favour N. Ekemezie, Kizito E. Anyiam, Mohammed Kayid, Oluwafemi Samson Balogun and Okechukwu J. Obulezi
Entropy 2024, 26(11), 973; https://doi.org/10.3390/e26110973 - 13 Nov 2024
Viewed by 575
Abstract
This study introduces the DUS Topp–Leone family of distributions, a novel extension of the Topp–Leone distribution enhanced by the DUS transformer. We derive the cumulative distribution function (CDF) and probability density function (PDF), demonstrating the distribution’s flexibility in modeling various lifetime phenomena. The [...] Read more.
This study introduces the DUS Topp–Leone family of distributions, a novel extension of the Topp–Leone distribution enhanced by the DUS transformer. We derive the cumulative distribution function (CDF) and probability density function (PDF), demonstrating the distribution’s flexibility in modeling various lifetime phenomena. The DUS-TL exponential distribution was studied as a sub-model, with analytical and graphical evidence revealing that it exhibits a unique unimodal shape, along with fat-tail characteristics, making it suitable for time-to-event data analysis. We evaluate parameter estimation methods, revealing that non-Bayesian approaches, particularly Maximum Likelihood and Least Squares, outperform Bayesian techniques in terms of bias and root mean square error. Additionally, the distribution effectively models datasets with varying skewness and kurtosis values, as illustrated by its application to total factor productivity data across African countries and the mortality rate of people who injected drugs. Overall, the DUS Topp–Leone family represents a significant advancement in statistical modeling, offering robust tools for researchers in diverse fields. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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24 pages, 916 KiB  
Article
An Instructive CO2 Adsorption Model for DAC: Wave Solutions and Optimal Processes
by Emily Kay-Leighton and Henning Struchtrup
Entropy 2024, 26(11), 972; https://doi.org/10.3390/e26110972 - 13 Nov 2024
Viewed by 534
Abstract
We present and investigate a simple yet instructive model for the adsorption of CO2 from air in porous media as used in direct air capture (DAC) processes. Mathematical analysis and non-dimensionalization reveal that the sorbent is characterized by the sorption timescale and [...] Read more.
We present and investigate a simple yet instructive model for the adsorption of CO2 from air in porous media as used in direct air capture (DAC) processes. Mathematical analysis and non-dimensionalization reveal that the sorbent is characterized by the sorption timescale and capacity, while the adsorption process is effectively wavelike. The systematic evaluation shows that the overall adsorption rate and the recommended charging duration depend only on the wave parameter that is found as the ratio of capacity and dimensionless air flow velocity. Specifically, smaller wave parameters yield a larger overall charging rate, while larger wave parameters reduce the work required to move air through the sorbent. Thus, optimal process conditions must compromise between a large overall adsorption rate and low work requirements. Full article
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9 pages, 238 KiB  
Article
Dirac Equation and Fisher Information
by Asher Yahalom
Entropy 2024, 26(11), 971; https://doi.org/10.3390/e26110971 - 12 Nov 2024
Viewed by 532
Abstract
Previously, it was shown that Schrödinger’s theory can be derived from a potential flow Lagrangian provided a Fisher information term is added. This approach was later expanded to Pauli’s theory of an electron with spin, which required a Clebsch flow Lagrangian with non-zero [...] Read more.
Previously, it was shown that Schrödinger’s theory can be derived from a potential flow Lagrangian provided a Fisher information term is added. This approach was later expanded to Pauli’s theory of an electron with spin, which required a Clebsch flow Lagrangian with non-zero vorticity. Here, we use the recent relativistic flow Lagrangian to represent Dirac’s theory with the addition of a Lorentz invariant Fisher information term as is required by quantum mechanics. Full article
(This article belongs to the Special Issue Applications of Fisher Information in Sciences II)
28 pages, 13144 KiB  
Article
Complexity and Variation in Infectious Disease Birth Cohorts: Findings from HIV+ Medicare and Medicaid Beneficiaries, 1999–2020
by Nick Williams
Entropy 2024, 26(11), 970; https://doi.org/10.3390/e26110970 - 12 Nov 2024
Viewed by 528
Abstract
The impact of uncertainty in information systems is difficult to assess, especially when drawing conclusions from human observation records. In this study, we investigate survival variation in a population experiencing infectious disease as a proxy to investigate uncertainty problems. Using Centers for Medicare [...] Read more.
The impact of uncertainty in information systems is difficult to assess, especially when drawing conclusions from human observation records. In this study, we investigate survival variation in a population experiencing infectious disease as a proxy to investigate uncertainty problems. Using Centers for Medicare and Medicaid Services claims, we discovered 1,543,041 HIV+ persons, 363,425 of whom were observed dying from all-cause mortality. Once aggregated by HIV status, year of birth and year of death, Age-Period-Cohort disambiguation and regression models were constructed to produce explanations of variance in survival. We used Age-Period-Cohort as an alternative method to work around under-observed features of uncertainty like infection transmission, receiver host dynamics or comorbidity noise impacting survival variation. We detected ages that have a consistent, disproportionate share of deaths independent of study year or year of birth. Variation in seasonality of mortality appeared stable in regression models; in turn, HIV cases in the United States do not have a survival gain when uncertainty is uncontrolled for. Given the information complexity issues under observed exposure and transmission, studies of infectious diseases should either include robust decedent cases, observe transmission physics or avoid drawing conclusions about survival from human observation records. Full article
(This article belongs to the Special Issue Stability and Flexibility in Dynamic Systems: Novel Research Pathways)
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13 pages, 4232 KiB  
Article
Universality of Dynamical Symmetries in Chaotic Maps
by Marcos Acero, Sean Lyons, Andrés Aragoneses and Arjendu K. Pattanayak
Entropy 2024, 26(11), 969; https://doi.org/10.3390/e26110969 - 12 Nov 2024
Viewed by 539
Abstract
Identifying signs of regularity and uncovering dynamical symmetries in complex and chaotic systems is crucial both for practical applications and for enhancing our understanding of complex dynamics. Recent approaches have quantified temporal correlations in time series, revealing hidden, approximate dynamical symmetries that provide [...] Read more.
Identifying signs of regularity and uncovering dynamical symmetries in complex and chaotic systems is crucial both for practical applications and for enhancing our understanding of complex dynamics. Recent approaches have quantified temporal correlations in time series, revealing hidden, approximate dynamical symmetries that provide insight into the systems under study. In this paper, we explore universality patterns in the dynamics of chaotic maps using combinations of complexity quantifiers. We also apply a recently introduced technique that projects dynamical symmetries into a “symmetry space”, providing an intuitive and visual depiction of these symmetries. Our approach unifies and extends previous results and, more importantly, offers a meaningful interpretation of universality by linking it with dynamical symmetries and their transitions. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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