Notion Of Entropy Research Articles

Unifying lower bounds for algebraic machines, semantically

We present a new abstract method for proving lower bounds in computational complexity based on the notion of topological and measurable entropy for dynamical systems. It is shown to generalise several previous lower bounds results from the literature in algebraic complexity, thus providing a unifying framework for “topological” proofs of lower bounds. We further use this method to prove that maxflow, a ▪ complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. This improves on a result of Mulmuley since the class of machines considered extends the class “prams without bit operations”, making more precise the relationship between Mulmuley's result and similar lower bounds on real prams.

Information entropy-assisted intuitionistic fuzzy rough feature subset selection

Fuzzy-rough set (FRS) has been effectively implemented as a powerful pre-processing tool to cope with the issues such as vagueness, imprecision, noise and uncertainty in high-dimensional data analysis. However, FRS fails to handle the uncertainty due to identification, which is very much common in heterogeneous data. Moreover, it is always challenging to handle different issues such as vagueness, uncertainty, and noise during elimination of irrelevant and redundant features. This paper investigates an intuitionistic fuzzy rough set (IFRS)-aided feature selection method by using information entropy notion to tackle these issues simultaneously. We establish an intuitionistic fuzzy (IF) similarity relation. Then IFRS model is discussed based on this relation. Next, we present a novel IF granular structure. Based on the granular structure, we present lambda-conditional entropy for IF framework. Further, relevant mathematical theorems are proved to justify the theoretical aspect of the models. Moreover, a positive region preserved attribute selection method is proposed. A comprehensive experimental study is discussed to demonstrate the effectiveness and practical validation of the proposed method. Finally, a methodology is developed based on proposed models, which increases accuracy for the minority RNA silencing suppressors against majority class non-suppressors as evident from better sensitivity and G-means metrics.

Well-posedness results to parabolic problems involving (p(x),q(x))-growth structure with L1-data

We investigate the well-posedness (existence and uniqueness) of solutions to nonlinear parabolic problems with variable exponents and irregular data. Firstly, we establish the existence and uniqueness of weak solutions to the studied model when the data are regular enough. Secondly, we assume that the data belongs only to L1 and we prove the existence and uniqueness of both renormalized and entropy solutions. Finally, we demonstrate the equivalence between the renormalized and entropy notion of solutions to the considered problems. Our approach is based essentially on the use of truncation methods and involves some new technical estimates.

Water and the Origin of Life

This article reviews all the major stages in the origins of life, from the emergence of matter in the initial Big Bang to the modern, civilized human being. On an immaterial level, it is proposed and explained how consciousness necessarily takes precedence over matter. Next, we explain how consciousness, with its ability to process information, selected the water molecule to breathe life into the periodic table of elements. We also explain why the notion of entropy allows us to evolve, “Die Entropie der Welt strebt einem Maximum zu” (second principle), and, therefore, takes precedence over the notion of energy, which, on the contrary, encourages us to preserve what we have, “Die Energie der Welt bleibt konstant” (first principle). This is followed by a discussion of the importance of quantum coherence and the need to rely on a second quantization formalism for a proper understanding of the physical–biochemical properties of water. Moreover, throughout the argument developed on the best and most fundamental things science has to offer, care is taken to link this knowledge to the great philosophies of the West (Greece), the East (China and India), and even to practices of a shamanic nature (Africa and America). Hence, finally, we propose reconsidering all musical practice within the framework of the diapason of water at a frequency of 429.62 Hz, as well as all therapeutic practice on the basis of seven clearly identified and established frameworks of thought.

Coarse entropy of metric spaces

Coarse geometry studies metric spaces on the large scale. The recently introduced notion of coarse entropy is a tool to study dynamics from the coarse point of view. We prove that all isometries of a given metric space have the same coarse entropy and that this value is a coarse invariant. We call this value the coarse entropy of the space and investigate its connections with other properties of the space. We prove that it can only be either zero or infinity, and although for many spaces this dichotomy coincides with the subexponential–exponential growth dichotomy, there is no relation between coarse entropy and volume growth more generally. We completely characterise this dichotomy for spaces with bounded geometry and for quasi-geodesic spaces. As an application, we provide an example where coarse entropy yields an obstruction for a coarse embedding, where such an embedding is not precluded by considerations of volume growth.

Understanding Higher-Order Interactions in Information Space.

Methods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback-Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics.

Information Theory and Coding Techniques for 5G Wireless Communication Systems: Towards Efficient Spectrum Utilization

The introduction of 5G wireless communication networks has initiated a new period of connectivity, offering extremely high data rates, minimal delay, and extensive device compatibility. However, in order to fully use the capabilities of 5G, it is crucial to prioritize the effective exploitation of spectrum. Efficiency in data transmission is achieved by the optimization of data transmission, error minimization, and maximization of throughput, which are facilitated by information theory and coding techniques. This work examines the convergence of information theory and coding methods in the context of 5G wireless communication systems, with an emphasis on improving the efficiency of spectrum usage. The core of information theory is centered around the notion of entropy, which measures the level of uncertainty linked to a random variable. Information theory offers valuable insights into the underlying constraints of communication networks by utilizing concepts like channel capacity and error-correcting codes. When it comes to 5G, it is crucial to comprehend these limitations in order to develop transmission strategies that effectively utilize the spectrum resources at hand. Coding techniques, such as forward error correction (FEC) and channel coding, are essential for reducing the impact of noise, interference, and fading in wireless channels. Forward Error Correction (FEC) is a technique that enhances the reliability of transmitted data by introducing redundancy. This allows receivers to repair errors without requiring retransmissions, thus increasing the efficiency of the available spectrum. Advanced coding techniques such as low-density parity-check (LDPC) codes and polar codes are crucial for delivering high data transfer rates and reliability in 5G networks. By combining various access strategies, such as orthogonal frequency-division multiple access (OFDMA) and non-orthogonal multiple access (NOMA), along with advanced coding schemes, we can achieve higher spectral efficiency and increase the capacity for users. These strategies facilitate the optimal distribution of resources, enabling multiple users to efficiently share the same spectrum while satisfying their quality-of-service demands. Information theory and coding.

Which entropy for general physical theories?

We address the problem of quantifying the information content of a source for an arbitrary information theory, where the information content is defined in terms of the asymptotic achievable compression rate. The functions that solve this problem in classical and quantum theory are Shannon's and von Neumann's entropy, respectively. However, in a general information theory there are three different functions that extend the notion of entropy, and this opens the question as to whether any of them can universally play the role of the quantifier for the information content. Here we answer the question in the negative, by evaluating the information content as well as the various entropic functions in a toy theory called Bilocal Classical Theory.

The Order of the Novel

Abstract: An essay exploring the time organization of novels. The idea of space and time when reading novels and how it can be interpreted is examined thoroughly. It further questions and investigates the notions of probabilities, entropy, superposition, and disorder, in conjunction with the functions and structure of novels. The analyzing of an interpretation of the reader’s view of the novel’s place in time allows for new perspectives and ideas to be formulated.

Dissipation in hydrodynamics from micro- to macroscale: wisdom from Boltzmann and stochastic thermodynamics

We show that macroscopic irreversible thermodynamics for viscous fluids can be derived from exact information-theoretic thermodynamic identities valid at the microscale. Entropy production, in particular, is a measure of the loss of many-particle correlations in the same way in which it measures the loss of system-reservoirs correlations in stochastic thermodynamics (ST). More specifically, we first show that boundary conditions at the macroscopic level define a natural decomposition of the entropy production rate (EPR) in terms of thermodynamic forces multiplying their conjugate currents, as well as a change in suitable nonequilibrium potential that acts as a Lyapunov function in the absence of forces. Moving to the microscale, we identify the exact identities at the origin of these dissipative contributions for isolated Hamiltonian systems. We then show that the molecular chaos hypothesis, which gives rise to the Boltzmann equation at the mesoscale, leads to a positive rate of loss of many-particle correlations, which we identify with the Boltzmann EPR. By generalizing the Boltzmann equation to account for boundaries with nonuniform temperature and nonzero velocity, and resorting to the Chapman–Enskog expansion, we recover the macroscopic theory we started from. Finally, using a linearized Boltzmann equation we derive ST for dilute particles in a weakly out-of-equilibrium fluid and its corresponding macroscopic thermodynamics. Our work unambiguously demonstrates the information-theoretical origin of thermodynamic notions of entropy and dissipation in macroscale irreversible thermodynamics.

Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Ampère mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class $\mathcal{E}^{\frac{n}{n-1}}(X, \theta, \phi)$ (provided $n>1$), while they have the same singularities of $\phi$ when $n=1$.

On preradicals, persistence, and the flow of information

We use Category Theory notions to explore the conceptual axiom that, in a general sense, information flows through a neural network following a path of “least resistance”. To this end, we consider particular endofunctors, called preradicals, whereby we describe persistence in small shape diagrams defined in R-Mod. Specifically, we show that the α preradicals naturally describe persistence in commutative G-modules, for G a directed acyclic graph. Then, we use this results to generalize the notion of persistence to any diagram labeled by a quiver Q. These results, in turn, set the theoretical foundation for our formal framework that explores the notion of “paths of least resistance”. Lastly, we provide a notion of entropy for preradicals in R-Mod, and prove that it respects the order and operations between preradicals.

A new approach to the entropy of a transitive BE-algebra with countable partitions

The concept of entropy and information gain of BE-algebras in scientific disciplines such as information theory, data science, supply chain and machine learning assists us to calculate the uncertanity of the scientific processes of phenomena. In this respect the notion of filter entropy for a transitive BE-algebra is introduced and its properties are investigated. The notion of a dynamical system on a transitive BE-algebra is introduced. The concept of the entropy for a transitive BE-algebra dynamical system is developed and, its characteristics are considered. The notion of equivalent transitive BE-algebra dynamical systems is defined, and it is proved the fact that two equivalent BE-algebra dynamical systems have the same entropy. Theorems to help calculate the entropy are given. Specifically, a new version of Kolmogorov– Sinai Theorem has been proved. The study introduces the concept of information gain of a transitive BE-algebra with respect to its filters and investigates its properties. This study proposes the use of filter entropy to approximate the level of risk introduced by a BE-algebra dynamical system. This aim is reached by defining the information gain with respect to the filters of a BE-algebra. This methodology is well developed for use in engineering, especially in industrial networks. This paper proposes a novel approach to assess the quantity of uncertainty, and the impact of information gain of a BE-algebra dynamical system.

Quantum statistical mechanics from a Bohmian perspective

We develop a general formulation of quantum statistical mechanics in terms of probability currents that satisfy continuity equations in the multi-particle position space, for closed and open systems with a fixed number of particles. The continuity equation for any closed or open system suggests a natural Bohmian interpretation in terms of microscopic particle trajectories, that make the same measurable predictions as standard quantum theory. The microscopic trajectories are not directly observable, but provide a general, simple and intuitive microscopic interpretation of macroscopic phenomena in quantum statistical mechanics. In particular, we discuss how various notions of entropy, proper and improper mixtures, and thermodynamics are understood from the Bohmian perspective.

On the Supposed Mass of Entropy and That of Information.

In the theory of special relativity, energy can be found in two forms: kinetic energy and rest mass. The potential energy of a body is actually stored in the form of rest mass, the interaction energy too, but temperature is not. Information acquired about a dynamical system can be potentially used to extract useful work from it. Hence, the "mass-energy-information equivalence principle" that has been recently proposed. In this paper, it is first recalled that for a thermodynamic system made of non-interacting entities at constant temperature, the internal energy is also constant. So, the energy involved in a variation in entropy (TΔS) differs from a change in the potential energy stored or released and cannot be associated to a corresponding variation in mass of the system, even if it is expressed in terms of the quantity of information. This debate gives us the opportunity to deepen the notion of entropy seen as a quantity of information, to highlight the difference between logical irreversibility (a state-dependent property) and thermodynamical irreversibility (a path-dependent property), and to return to the nature of the link between energy and information that is dynamical.

It Ain't Necessarily So: Ludwig Boltzmann's Darwinian Notion of Entropy.

Ludwig Boltzmann's move in his seminal paper of 1877, introducing a statistical understanding of entropy, was a watershed moment in the history of physics. The work not only introduced quantization and provided a new understanding of entropy, it challenged the understanding of what a law of nature could be. Traditionally, nomological necessity, that is, specifying the way in which a system must develop, was considered an essential element of proposed physical laws. Yet, here was a new understanding of the Second Law of Thermodynamics that no longer possessed this property. While it was a new direction in physics, in other important scientific discourses of that time-specifically Huttonian geology and Darwinian evolution, similar approaches were taken in which a system's development followed principles, but did so in a way that both provided a direction of time and allowed for non-deterministic, though rule-based, time evolution. Boltzmann referred to both of these theories, especially the work of Darwin, frequently. The possibility that Darwin influenced Boltzmann's thought in physics can be seen as being supported by Boltzmann's later writings.

Conservative correction procedures utilizing artificial dissipation operators

The conservative correction procedure of Abgrall [1] is studied from the perspective of filter-based artificial dissipation methods, which motivates the ability to tailor the behavior of the method in both physical and spectral space. Compared to the original formulation, employing diffusion operators biases the correction towards smaller scales and better controls discretization errors when seeking to enforce auxiliary conservation relations. Effective entropy-stable regularization of sharp gradients is furthermore shown to be attainable. Calculations of the Sod shock tube problem as governed by the one-dimensional Euler equations are used to highlight the utility of considering alternate filters within the original correction framework, where the notion of entropy conservation/stability is leveraged for improving non-linear scheme robustness.

Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow

Phase-field models such as the Allen–Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen–Cahn equation with a potential with N\geq 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: as long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen–Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter \varepsilon\searrow 0 . We even establish the rate of convergence O(\varepsilon^{1/2}) . Our approach is based on the gradient-flow structure of the Allen–Cahn equation and its limiting motion: building on the recent concept of “gradient-flow calibrations” for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen–Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen–Cahn operator or additional convergence hypotheses for the energy at positive times.

Improvement in Monte Carlo localization using information theory and statistical approaches

Monte Carlo localization methods deploy a particle filter to resolve a hidden Markov process based on recursive Bayesian estimation, which approximates the internal states of a dynamic system given observation data. When the observed data are corrupted by outliers, the particle filter’s performance may deteriorate, preventing the algorithm from accurately computing dynamic system states such as a robot’s position, which in turn reduces the accuracy of the localization and navigation. In this paper, the notion of information entropy is used to identify outliers. Then, a probability-based approach is used to remove the discovered outliers. In addition, a new mutation process is added to the localization algorithm to exploit the posterior probability density function in order to actively detect the high-likelihood region. The goal of incorporating the mutation operator into this method is to solve the problem of algorithm impoverishment which is due to insufficient representation of the complete probability density function. Simulation experiments are used to confirm the effectiveness of the proposed techniques. They also are employed to predict the remaining viability of a lithium-ion battery. Furthermore, in an experimental study, the modified Monte Carlo localization algorithm was applied to a mobile robot to demonstrate the local planner’s improved accuracy. The test results indicate that developed techniques are capable of effectively capturing the dynamic behavior of a system and accurately tracking its characteristics.

Geometric interpretation of the entropy of so c systems

We consider a geometric approach to the notion of metric entropy. We justify the possibility of this approach for the class of Borel invariant ergodic probability measures on so c systems, which is the rst result of such generality for non-Markovian systems.