The theory of Shannon entropy was applied to the Choi-Williams time-frequency distribution (CWD) of time series in order to extract entropy information in both time and frequency domains. In this way, four novel indexes were defined: (1) partial instantaneous entropy, calculated as the entropy of the CWD with respect to time by using the probability mass function at each time instant taken independently; (2) partial spectral information entropy, calculated as the entropy of the CWD with respect to frequency by using the probability mass function of each frequency value taken independently; (3) complete instantaneous entropy, calculated as the entropy of the CWD with respect to time by using the probability mass function of the entire CWD; (4) complete spectral information entropy, calculated as the entropy of the CWD with respect to frequency by using the probability mass function of the entire CWD. These indexes were tested on synthetic time series with different behavior (periodic, chaotic and random)
We review the relative entropy method in the context of hyperbolic and diffusive relaxation limits of entropy solutions for various hyperbolic models. The main example consists of the convergence from multidimensional compressible Euler equations with friction to the porous medium equation \cite{LT12}. With small modifications, the arguments used in that case can be adapted to the study of the diffusive limit from the Euler-Poisson system with friction to the Keller-Segel system \cite{LT13}. In addition, the $p$--system with friction and the system of viscoelasticity with memory are then reviewed, again in the case of diffusive limits \cite{LT12}. Finally, the method of relative entropy is described for the multidimensional stress relaxation model converging to elastodynamics \cite[Section 3.2]{LT06}, one of the first examples of application of the method to hyperbolic relaxation limits.. ...
Alexis Vasseur speaking at BIRS workshop, Model reduction in continuum thermodynamics: Modeling, analysis and computation, on Wednesday, September 19, 2012 on the topic: Relative entropy method applied to model reduction in fluid mechanics.
ARTICLE SYNOPSIS...Cycle Analysis: A comparison of the Fourier and Maximum Entropy methods by John F. Ehler The motivation for reducing price history to a mathematical expression is clear. If we can describe the prices mathematically, we have the means to extend the equat. ...
TY - GEN. T1 - Renyi entropy in identification of cardiac autonomic neuropathy in diabetes. AU - Jelinek, Herbert. AU - Tarvainen, M.P.. AU - Cornforth, David. N1 - Imported on 03 May 2017 - DigiTool details were: 086 FoR could not be migrated (80201 - ). publisher = United States: Institute of Electrical and Electronics Engineers, 2012. Event dates (773o) = 9-12 September, 2012; Parent title (773t) = Computing in Cardiology Conference. ISSNs: 0276-6574; PY - 2012. Y1 - 2012. N2 - Heart rate variability (HRV) has been conventionally analyzed with time- and frequency-domain methods. More recent nonlinear analysis has shown an increased sensitivity for identifying risk of future morbidity and mortality in diverse patient groups. Included in the domain of nonlinear analysis are the multiscale entropy measures. The Renyi entropy is such a measure. It is calculated by considering the probability of sequences of values occurring in the HRV data. An exponent a of the probability can be varied to ...
The problem of analyzing data from a multisphere neutron spectrometer to infer the energy spectrum of the incident neutrons is discussed. The main features of the code MAXED, a computer program developed to apply the maximum entropy principle to the deconvolution (unfolding) of multisphere neutron spectrometer data, are described, and the use of the code is illustrated with an example. A user`s guide for the code MAXED is included in an appendix. The code is available from the authors upon request.
The aim of this book is to provide an overview of current works addressing the topics of research that explore the geometric structures of information and entropy. The papers in this book includes the extended versions of a selection of the paper published in Proceedings of the 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2014), Amboise, France, 21-26 September 2014. Chapter 1 of the book is a historical review of the origins of thermodynamics and information theory. Chapter 2 discusses the mathematical and physical foundations of geometric structures related to information and entropy. Lastly, Chapter 3 is dedicated to applications with numerical schemes for geometric structures of information and entropy. ...
Why do you think the entropy of 0 is incorrect? It intuitively makes sense, as there is no uncertainty about the base at position 3, and thus there is no entropy.. However, what is plotted in a sequence logo isnt the entropy, but rather a measure of the decrease in uncertainty as the sequence is aligned. This is calculated by taking the entropy at this position if we randomly aligned sequences ($H_g(L)$), and subtracting from it the entropy of the alignment ($H_g(s)$): $$R_{sequence}=H_g(L) - H_s(L)$$. The entropy at position 3 based on a random alignment is calculated by assuming there are 4 equally likely events (one for each base), and thus: \begin{align*} H_g(L) & = -((1/4 \times -2) + (1/4 \times -2) + (1/4 \times -2) + (1/4 \times -2)) \\ H_g(L) & = 2 \end{align*} Notably, $H_g(L)$ will always be 2 when dealing with nucleotide sequences.. So in your example, we have:  \begin{align*} R_{sequence}&=H_g(L) - H_s(L) \\ R_{sequence}&=2 - 0 \\ R_{sequence}&=2 \text{ bits of entropy} ...
TY - JOUR. T1 - The Effect of Threshold Values and Weighting Factors on the Association between Entropy Measures and Mortality after Myocardial Infarction in the Cardiac Arrhythmia Suppression Trial (CAST). AU - Mayer, Christopher. AU - Bachler, Martin. AU - Holzinger, Andreas. AU - Stein, Phyllis K.. AU - Wassertheurer, Siegfried. PY - 2016/4/20. Y1 - 2016/4/20. N2 - Heart rate variability (HRV) is a non-invasive measurement based on the intervals between normal heart beats that characterize cardiac autonomic function. Decreased HRV is associated with increased risk of cardiovascular events. Characterizing HRV using only moment statistics fails to capture abnormalities in regulatory function that are important aspects of disease risk. Thus, entropy measures are a promising approach to quantify HRV for risk stratification. The purpose of this study was to investigate this potential for approximate, corrected approximate, sample, fuzzy, and fuzzy measure entropy and its dependency on the ...
Depending on the nature of the specimen to be reconstructed, the commonly used reconstruction techniques can create a problem, since in order to obtain creditable reconstructions, filters, which are obtained in an ad-hoc manner, have to be applied. These filters which effectively exclude high resolution data (e.g. noise), also remove any non-noise data thereby leading to artifacts in the reconstruction. The generation of these artifacts however can be overcome if the object lends itself to some sort of symmetry or if many copies of the object being reconstructed have the same structure. However, chromatin fibres do not lend itself towards this type of analysis and hence in order to get reconstructions which are devoid of artifacts, all the information present in the sample has to be preserved. Thus an alternate method namely the maximum entropy method, which does not involve the use of filters, was developed.. The maximum entropy method is used extensively in astronomy and medical tomography and ...
Employs a multi-stage algorithm that makes use of spatial contextual information in a hierarchical clustering procedure for unsupervised image segmentation
Title of Book: Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering ...
Electric power is a basic industry in the national economy in China. It influences the economy growth in China seriously owing to the surplus and shortage
In the continuation of [6], we study reversible reaction-diffusion systems via entropy methods (based on the free energy functional). We show for a particular model problem with two reacting species in which one of the diffusion constants vanishes that the solutions decay exponentially with explicit rate and constant towards the unique constant equilibrium state.
Heres a paper for the proceedings of a workshop on Information and Entropy in Biological System this spring: • John Baez and Blake Pollard, Relative entropy in biological systems, with Blake S. Pollard, Entropy 18 (2016), 46. Wed love any comments or questions you might have. Im not happy with the title. In the paper…
The study compares permutation-based and coarse-grained entropy approaches for the assessment of complexity of short heart period (HP) variability recordings. Shannon permutation entropy (SPE) and conditional permutation entropy (CPE) are computed as examples of permutation-based entropies, while the k-nearest neighbor conditional entropy (KNNCE) is calculated as an example of coarse-grained conditional entropy. SPE, CPE and KNNCE were applied to ad-hoc simulated autoregressive processes corrupted by increasing amounts of broad band noise and to real HP variability series recorded after complete vagal blockade obtained via administration of a high dose of atropine (AT) in nine healthy volunteers and during orthostatic challenge induced by 90° head-up tilt (T90) in 15 healthy individuals. Over the simulated series the performances of SPE and CPE degraded more rapidly with the amplitude of the superimposed broad band noise than those of KNNCE. Over real data KNNCE identified the expected decrease ...
The support vector machine is used as a data mining technique to extract informative hydrologic data on the basis of a strong relationship between error tolerance and the number of support vectors. Hydrologic data of flash flood events in the Lan-Yang River basin in Taiwan were used for the case study. Various percentages (from 50% to 10%) of hydrologic data, including those for flood stage and rainfall data, were mined and used as informative data to characterize a flood hydrograph. Information on these mined hydrologic data sets was quantified using entropy indices, namely marginal entropy, joint entropy, transinformation, and conditional entropy. Analytical results obtained using the entropy indices proved that the mined informative data could be hydrologically interpreted and have a meaningful explanation based on information entropy. Estimates of marginal and joint entropies showed that, in view of flood forecasting, the flood stage was a more informative variable than rainfall. In addition,
Conformational entropy is a potentially important thermodynamic parameter contributing to protein function. Quantitative measures of conformational entropy are necessary for an understanding of its role but have been difficult to obtain experimentally. We have recently introduced an empirical calibration method that utilizes the changes in conformational dynamics as a proxy for changes in conformational entropy. This approach raises several questions with regards to the microscopic origins of the measured conformational entropy as well as its general applicability. One of the goals in this work was to probe the microscopic origins of the link between conformational dynamics and conformational entropy. Using MD simulations, we find that the motions of methyl-bearing side chains are sufficiently coupled to that of other side chains and serve as excellently reporters of the overall side chain conformational entropy. We also propose a modified weighting scheme to project the change in NMR-measured methyl
Extracted from text ... Southern African Journal of Anaesthesia & Analgesia ? March 2006 21 Spectral entropy and haemodynamic response to surgery during sevoflurane anaesthesia Introduction Apart from somatic responses, surgery also evokes autonomic responses, including haemodynamic responses. Spectral entropy has been validated as a means to monitor the hypnotic state during sevoflurane anaesthesia. Aim To investigate the relationship between spectral entropy, heart rate, and blood pressure during sevoflurane anaesthesia. FJ Smith, E Dannhauser Patients and methods The sample consisted of 43 patients scheduled for elective abdominal surgery. Patients were premedicated with oral midazolam. Induction of anaesthesia was achieved with alfentanil 15 mg/kg, ..
The two entropy expressions, log B and −B logB (where B is the local brightness of the object or its spatial spectral power) used in maximum entropy (ME) image restoration, are derived as limiting cases of a general entropy formula. The brightness B is represented by the n photons emitted from a small unit area of the object and imaged in the receiver. These n photons can be distributed over z degrees of freedom in q(n,z) different ways calculated by the Bose-Einstein statistics. The entropy to be maximized is interpreted, as in the original definition of entropy by Boltzmann and Planck, as logq(n,z). This entropy expression reduces to log B and −B logB in the limits of ...
Measuring complexity in terms of the predictability of time series is a major area of research in science and engineering, and its applications are spreading throughout many scientific disciplines, where the analysis of physiological signals is perhaps the most widely reported in literature. Sample entropy is a popular measure for quantifying signal irregularity. However, the sample entropy does not take sequential information, which is inherently useful, into its calculation of sample similarity. Here, we develop a method that is based on the mathematical principle of the sample entropy and enables the capture of sequential information of a time series in the context of spatial dependence provided by the binary-level co-occurrence matrix of a recurrence plot. Experimental results on time-series data of the Lorenz system, physiological signals of gait maturation in healthy children, and gait dynamics in Huntingtons disease show the potential of the proposed method.. ...
So the reason John is surprised to see his cards while Dave is not surprised is that Johns hand is in a very rare macrostate, while Daves hand is in a very common macrostate. We expect to see high entropy hands like Daves junk (with many microstates) and not low entropy hands like Johns straight flush. High entropy hands are more common than low entropy hands, simply because there are more of them. If we buy a new deck of cards, we can immediately deal two straight flushes right off the top of the deck. (The cards in a new deck are all in order.) The entropy of the cards is low. When we shuffle the deck, the entropy increases. We start dealing lousy, high entropy, hands. As you learn more about entropy, youll see that increasing entropy is the way of the world. In the meantime, next time you play poker, take the time to savor those lousy hands. Each one is just as rare as a great one ...
Abstract: We consider words as a network of interacting letters, and approximate the probability distribution of states taken on by this network. Despite the intuition that the rules of English spelling are highly combinatorial (and arbitrary), we find that maximum entropy models consistent with pairwise correlations among letters provide a surprisingly good approximation to the full statistics of four letter words, capturing ~92% of the multi-information among letters and even discovering real words that were not represented in the data from which the pairwise correlations were estimated. The maximum entropy model defines an energy landscape on the space of possible words, and local minima in this landscape account for nearly two-thirds of words used in written English ...
We study the problem of privacy amplification: key agreement between two parties who both know a weak secret w, such as a password. (Such a setting is ubiquitous on the internet, where passwords are the most commonly used security device.) We assume that the key agreement protocol is taking place in the presence of an active computationally unbounded adversary Eve. The adversary may have partial knowledge about w, so we assume only that w has some entropy from Eves point of view. Thus, the goal of the protocol is to convert this non-uniform secret w into a uniformly distributed string R R that is fully secret from Eve. R may then be used as a key for running symmetric cryptographic protocols (such as encryption, authentication, etc.).. Because we make no computational assumptions, the entropy in R can come only from w. Thus such a protocol must minimize the entropy loss during its execution, so that R is as long as possible. The best previous results have entropy loss of Θ(κ 2 ) Θ(κ2) , ...
print(Information Gain: %.3f bits % gain). 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. # calculate the information gain. from math import log2. # calculate the entropy for the split in the dataset. def entropy(class0, class1):. return -(class0 * log2(class0) + class1 * log2(class1)). # split of the main dataset. class0 = 13 / 20. class1 = 7 / 20. # calculate entropy before the change. s_entropy = entropy(class0, class1). print(Dataset Entropy: %.3f bits % s_entropy). # split 1 (split via value1). s1_class0 = 7 / 8. s1_class1 = 1 / 8. # calculate the entropy of the first group. s1_entropy = entropy(s1_class0, s1_class1). print(Group1 Entropy: %.3f bits % s1_entropy). # split 2 (split via value2). s2_class0 = 6 / 12. s2_class1 = 6 / 12. # calculate the entropy of the second group. s2_entropy = entropy(s2_class0, s2_class1). print(Group2 Entropy: %.3f bits % s2_entropy). # calculate the information gain. gain = s_entropy ...
Jain, R., Radhakrishnan, J., Sen, P. (2009). A property of quantum relative entropy with an application to privacy in quantum communication. Journal of the ACM 56 (6). [email protected] Repository. https://doi.org/10.1145/ ...
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These processes are accompanied by increase of randomness and hence increase of entropy i.e. fro these processes, entropy change is positive.. Consider the following processes:. 1) Cooling down of a cup of tea. 2)Reaction taking place between a piece of marble or sodium hydroxide and Hydrochloric acid in an open vessel.. These processes involve exchange of matter and energy with the surrounding.Hence they are not isolated systems.. For these processes, we have to consider the total entropy change of the system and the surrounding.. ΔStotal = ΔSsystem + ΔSsurrounding. For the process to be spontaneous, ΔStotal must be positive.. For all spontaneous processes, the total entropy change ( ΔStotal ) must be positive.. ΔStotal = ΔSsystem + ΔSsurrounding . ,0. The randomness and hence the entropy keeps on increasing till ultimately an equilibrium is reached.. The entropy of the system at equilibrium is maximum and there is no further change in entropy i.e. ΔS =0.. If Stotal is negative, the ...
This issue starts with a basic introduction to the far from equilibrium thermodynamics of the Earth system by Kleidon (2010). This paper reviews the basics of thermodynamics to demonstrate that thermodynamics provides the means of describing practically all Earth system processes in purely thermodynamic terms. Entropy production is not just defined for heat fluxes and temperature gradients, but rather for a very broad range of conjugated variables, demonstrating that MEP has potentially wide-ranging applications within Earth and environmental systems.. This introduction is followed by a critical outside view by Volk & Pauluis (2010) on the contributions of this special issue. Rather than discussing every contribution of this issue in detail, they focus on a few key questions that summarize the challenges of applying MEP and that should guide future developments. Using the example of dry versus moist convection, they show that systems can produce the same amount of entropy but by very different ...
Entropy rate estimates how much information is present in the action potential signal, without considering how much of the action potential timing represents encoded mechanical signal. In contrast, maximum entropy rate is the amount of information that could be encoded in an action potential train at a given firing rate (Rieke et al., 1997). A perfectly regular action potential train would have a very low entropy rate, regardless of firing rate or maximum entropy rate.. The data compression method of estimating entropy has the advantage of being independent of any assumptions about the mechanism of information coding. However, it often gives higher values than the maximum entropy rate. Possible reasons for this include a failure to achieve maximal possible compression and the use of Stirlings approximation in the derivation of maximum entropy (French et al., 2003).. Entropy rate and maximum entropy rate had similar absolute values to the coherence-based information capacity, suggesting that ...
Constructing ultrasound statistical parametric images by using a sliding window is a widely adopted strategy for characterizing tissues. Deficiency in spatial resolution, the appearance of boundary artifacts, and the prerequisite data distribution limit the practicability of statistical parametric imaging. In this study, small-window entropy parametric imaging was proposed to overcome the above problems. Simulations and measurements of phantoms were executed to acquire backscattered radiofrequency (RF) signals, which were processed to explore the feasibility of small-window entropy imaging in detecting scatterer properties. To validate the ability of entropy imaging in tissue characterization, measurements of benign and malignant breast tumors were conducted (n = 63) to compare performances of conventional statistical parametric (based on Nakagami distribution) and entropy imaging by the receiver operating characteristic (ROC) curve analysis. The simulation and phantom results revealed that entropy
Parent Directory - entropy_decoder_model_kernel_abstract.h.html 2017-12-19 19:15 6.3K entropy_decoder_model_kernel_6.h.html 2017-12-19 19:15 9.2K entropy_decoder_model_kernel_5.h.html 2017-12-19 19:15 60K entropy_decoder_model_kernel_4.h.html 2017-12-19 19:15 45K entropy_decoder_model_kernel_3.h.html 2017-12-19 19:15 26K entropy_decoder_model_kernel_2.h.html 2017-12-19 19:15 18K entropy_decoder_model_kernel_1.h.html 2017-12-19 19:15 12K ...
Entropy is that thermodynamic quantity that is associated with representing non-availability of thermal energy in a system for conversion into mechanical energy that is termed as a disorder in the system. The entropy changes that are there in a specific system is specifically driven by heat flow as well as signs are determined by that.. When exothermic reactions are taken into consideration, there is a constant temperature within the system that would cause heat to flow into the surrounding area and thereby ensure that the surrounding is rendered positive.. When any such system and its surroundings come across each other in irreversible process, there is an increase in entropy. When entropy of one increases, entropy of another decreases. So, it is taken that this change in entropy in an irreversible process is greater than zero, while in reversible process it is less than zero.. ...
In the maximum entropy method, among all the solutions compatible with the underdetermined equation system, the solution with the smallest information content, or the maximum entropy, is selected. s results seem to indicate, the maximum entropy reconstruction is distinguished favorably from the ART Three-Dimensional Reconstruction of Nonperiodic Macromolecular Assemblies 25 reconstruction by a smaller amount of artifacts due to angular limitation. However, the great computational expense has thus far prevented a widespread use of this method. 1974; FRANK et ai. , 1986 b); the lateral views of the 40 S eukaryotic ribosomal subunit (BOUBLIK and HELLMAN 1978; FRANK et ai. 1981 b, 1982); and the top view of glutamine synthetase (VALENTINE et ai. 1968; FRANK et ai. 1978; see Figs. 4 and 5). , 1986). The azimuthal angles of the particles can be determined from an additional micrograph of the same specimen area at 0°, by employing rotational correlation techniques originally developed for the ...
The role of side-chain entropy (SCE) in protein folding has long been speculated about but is still not fully understood. Utilizing a newly developed Monte Carlo method, we conducted a systematic investigation of how the SCE relates to the size of the protein and how it differs among a proteins X-ray, NMR, and decoy structures. We estimated the SCE for a set of 675 nonhomologous proteins, and observed that there is a significant SCE for both exposed and buried residues for all these proteins-the contribution of buried residues approaches ∼40% of the overall SCE. Furthermore, the SCE can be quite different for structures with similar compactness or even similar conformations. As a striking example, we found that proteins X-ray structures appear to pack more
Calculates entropy of the reads coverage over each grl group. The entropy value per group is a real number in the interval (0:1), where 0 indicates no variance in reads over group. For example c(0,0,0,0) has 0 entropy, since no reads overlap.
Entropy sources are designed to provide unpredictable random numbers for cryptographic systems. As an assessment of the sources, Shannon entropy is usually adopted to quantitatively measure the unpredictability of the outputs. In several related works about the entropy evaluation of ring oscillator-based (RO-based) entropy sources, authors evaluated the unpredictability with the average conditional Shannon entropy (ACE) of the source, moreover provided a lower bound of the ACE (LBoACE). However, in this paper, we have demonstrated that when the adversaries have access to the history outputs of the entropy source, for example, by some intrusive attacks, the LBoACE may overestimate the actual unpredictability of the next output for the adversaries. In this situation, we suggest to adopt the specific conditional Shannon entropy (SCE) which exactly measures the unpredictability of the future output with the knowledge of previous output sequences and so is more consistent with the reality than the ...
1] J.-P. Eckmann, S. Oliffson Kamphorst & D. Ruelle, Recurrence Plots of Dynamical Systems, Europhysics Letters, 4, 973-977, 1987.. [2] L. L. Trulla, A. Giuliani, J. P. Zbilut & C. L. Webber Jr., Recurrence quantification analysis of the logistic equation with transients, Physics Letters A, 223 (4), 255-260, 1996.. [3] This latter detail was missing in the definition provided in C. Letellier, Estimating the Shannon entropy : recurrence plots versus symbolic dynamics, Physical Review Letters, 96, 254102, 2006. ...
TY - CONF. T1 - Shannon Entropy: From Probabilities to Particles. AU - Wallek, Thomas. AU - Pfleger, Martin. AU - Pfennig, Andreas. PY - 2012/2/13. Y1 - 2012/2/13. M3 - (Altdaten) Vortrag oder Präsentation. ER - ...
2] M. M. Kaur, P. Nath: On some characterizations of the Shannon entropy using extreme symmetry and block symmetry. Inform, and Control 53 (1982), 9-20. MR 0715518 , Zbl 0511.94012 ...
Entropy is a measurement of uncertainty. It is a way to assign a score of uncertainty to a stochastic variable. Entropy is a key component of information theory, a branch of mathematics designed to quantify information. It first came of age in 1948 with the publication of Claude Shannons paper A Mathematical Theory of Communication.. Before we can adequately define entropy, we need to first define information. Information, for our purposes, is what you get when the uncertainty about something is diminished or erased. Take a fair coin. Before we flip the coin, we are uncertain about what will happen when we flip it. After we flip the coin, our uncertainty about that coin flip is effectively 0 since we know the outcome (heads or tails). Weve gained information. But how do we measure the uncertainty about the coin flip before it happens? How do we quantify that uncertainty? The answer is entropy.. Consider an information source X that can produce, with equal probability, 6 symbols (also known ...
Many research problems are extremely complex, making interdisciplinary knowledge a necessity; consequently cooperative work in mixed teams is a common and increasing research procedure. In this paper, we evaluated information-theoretic network measures on publication networks. For the experiments described in this paper we used the network of excellence from the RWTH Aachen University, described in [1]. Those measures can be understood as graph complexity measures, which evaluate the structural complexity based on the corresponding concept. We see that it is challenging to generalize such results towards different measures as every measure captures structural information differently and, hence, leads to a different entropy value. This calls for exploring the structural interpretation of a graph measure [2] which has been a challenging problem.
in British Journal of Anaesthesia (2006), 97(6), 842-847. Background. The spectral entropy of the electroencephalogram has been proposed to monitor the depth of anaesthesia. State Entropy (SE) reflects the level of hypnosis. Response Entropy (RE), computed from ... [more ▼]. Background. The spectral entropy of the electroencephalogram has been proposed to monitor the depth of anaesthesia. State Entropy (SE) reflects the level of hypnosis. Response Entropy (RE), computed from electroencephalogram and facial muscle activity, reflects the response to nociceptive stimulation. We evaluated the effect of rocuronium on Bispectral Index (TM) (BIS) and entropy responses to laryngoscopy. Methods. A total of 25 patients were anaesthetized with propofol using a target-controlled infusion. At steady state, they randomly received 0.6 mg kg(-1) rocuronium (R) or saline (S). After 3 min, a 20 s laryngoscopy was applied. BIS, RE and SE were recorded continuously and averaged over 1 min during baseline, at ...
There is great significance in lie detection for the criminal investigations and lawtrials. In this study,according to the nonlinear characteristics of electroencephalography( EEG),it is the first time to use the sample entropy( SE),a nonlinear dynamical parameter of EEG,to see if someone is lying. The sample entropy values of 30 subjects EEG signals in lying or honesty states were calculated and analyzed. The study found that the fluctuating range of SE values in honesty was obviously less than that in lying. It is more important that the SE values in lying was significantly higher than the honesty,which indicated that SE could be used to distinguish EEG signals between two different states of honesty and lying. This research provides a newway for EEG-based lie detection.
For information, address State University of New York Press, 90 State Street, Suite 700, Albany, NY 12207 Production by Michael Haggett Marketing by Anne M. We have passed the tipping point: While information policy is among the. Information Theory: A tutorial Introduction is a highly readable first account of Shannon&39;s mathematical theory of communication, now an introduction to information theory pdf an introduction to information theory pdf known as information theory.. information, hidden actions or incomplete contracts are present) and in nonmar-ket interactions (such as between a regulator and a an introduction to information theory pdf firm, a boss and a worker, and so on). Basic properties of the classical Shannon entropy and the quantum an introduction to information theory pdf von Neumann entropy are described, along with related concepts such as classical and quantum relative entropy, conditional entropy, and mutual information. A simple physical example (gases) 36. Introduction ...
Bayesian probability theory and maximum entropy methods are at the core of a new view of scientific inference. These new ideas, along with the revolution in computational methods afforded by modern computers, allow astronomers, electrical engineers, image processors of any type, NMR chemists and physicists, and anyone at all who has to deal with incomplete and noisy data, to take advantage of methods that, in the past, have been applied only in some areas of theoretical physics. This volume records the Proceedings of Eleventh Annual Maximum Entropy Workshop, held at Seattle University in June, 1991. These workshops have been the focus of a group of researchers from many different fields, and this diversity is evident in this volume. There are tutorial papers, theoretical papers, and applications in a very wide variety of fields. Almost any instance of dealing with incomplete and noisy data can be usefully treated by these methods, and many areas of theoretical research are being
This dissertation takes a step toward a general framework for solving network information theory problems by studying the capacity region of networks through the entropy region. We first show that the capacity of a large class of acyclic memoryless multiuser information theory problems can be formulated as convex optimization over the region of entropy vectors of the network random variables. This capacity characterization is universal, and is advantageous over previous formulations in that it is single letter. Besides, it is significant as it reveals the fundamental role of the entropy region in determining the capacity of network information theory problems. With this viewpoint, the rest of the thesis is dedicated to the study of the entropy region, and its consequences for networks. A full characterization of the entropy region has proven to be a very challenging problem, and thus, we mostly consider inner bound constructions. For discrete random variables, our approaches include ...
A polycrystalline Ni 41Co 9Mn 40Sn 10 (at. %) magnetic shape memory alloy was prepared by arc melting and characterized mainly by magnetic measurements, in-situ high-energy X-ray diffraction (HEXRD), and mechanical testing. A large magnetoresistance of 53.8% (under 5 T) and a large magnetic entropy change of 31.9 J/(kg K) (under 5 T) were simultaneously achieved. Both of these values are among the highest values reported so far in Ni-Mn-Sn-based Heusler alloys. The large magnetic entropy change, closely related to the structural entropy change, is attributed to the large unit cell volume change across martensitic transformation as revealed by our in-situ HEXRD experiment. Furthermore, good compressive properties were also obtained. Lastly, the combination of large magnetoresistance, large magnetic entropy change, and good compressive properties, as well as low cost makes this alloy a promising candidate for multifunctional applications. ...
You might better understand the macroscopic definition of entropy (uakron, 9min) if you consider isothermal reversible expansion of an ideal gas. Note the word isothermal is different from adiabatic. If the expansion was an adiabatic and reversible expansion of an ideal gas, then we know from Chapter 2 that the temperature would go down, ie. T2/T1=(P2/P1)^(R/Cp)=(V1/V2)^(R/Cv). Therefore, holding the temperature constant must require the addition of heat. We can calculate the change in entropy for this isothermal process from the microscopic balance, then show that the amount of heat added is exactly equal to the change in entropy (of this reversible process) times the (isothermal) temperature. Studying the energy and entropy balance for the irreversible process helps us to appreciate how entropy is a state function. As suggested by the hint at the end of this video, you can turn this perspective around and infer the relation of entropy to volume by starting with the macroscopic definition ...
The channel capacity of a deterministic system with confidential data is an upper bound on the amount of bits of data an attacker can learn from the system. We encode all possible attacks to a system using a probabilistic specification, an Interval Markov Chain. Then the channel capacity computation reduces to finding a model of a specification with highest entropy. Entropy maximization for probabilistic process specifications has not been studied before, even though it is well known in Bayesian inference for discrete distributions. We give a characterization of global entropy of a process as a reward function, a polynomial algorithm to verify the existence of a system maximizing entropy among those respecting a specification, a procedure for the maximization of reward functions over Interval Markov Chains and its application to synthesize an implementation maximizing entropy. We show how to use Interval Markov Chains to model abstractions of deterministic systems with confidential data, and use the
We investigate the amplitude (Higgs) mode of a diluted quantum rotor model in two dimensions close to the superfluid-Mott glass quantum phase transition. After mapping the Hamiltonian onto a classical (2$+$1)d XY model, scalar susceptibility is calculated in imaginary time by means of large-scale Monte Carlo simulations. Analytic continuation of the imaginary time data is performed via maximum entropy methods and yields the real-frequency spectral function. The spectral peak associated with the Higgs mode is identified and its fate upon approaching the disordered quantum phase transition is determined ...
TY - GEN. T1 - Expertise classification using functional brain networks and normalized transfer entropy of EEG in design applications. AU - Baig, Muhammad Zeeshan. AU - Kavakli, Manolya. PY - 2019/2/23. Y1 - 2019/2/23. N2 - Expertise prediction is a challenging tasks for the development of state-of-the-art next generation computer-aided design (CAD) system. To develop an adaptive system that can accommodate the lack of expertise, the system needs to classify the expertise level. In this paper, we have presented a method to estimate the cognitive activity of the novice and expert user in the 3D modelling environment. The method has the capability of predicting novice and expert users. Normalized Transfer Entropy (NTE) of Electroencephalography (EEG) was used as a connectivity measure to calculate the information flow between the EEG electrodes. Functional brain networks (FBNs) were created from the NTE matrix and graph theory was used to analyze the complex network. The results from graph ...
This is the lowest point on the Kelvin scale. When a system goes from an ordered state to a disordered state the entropy is increased. The second law of thermodynamics states that the total entropy of the universe or an isolated system never decreases. V= 1 V. The third law defines absolute zero and helps to explain that the entropy, or disorder, of the universe is heading towards a constant, nonzero value. Keywords: Nernst postulate, thermodynamics, entropy, quantum laws The third law of thermodynamics states that as the temperature approaches absolute zero in a system, the absolute entropy of the system approaches a constant value. The law of conservation of mass is also an equally fundamental concept in the theory of thermodynamics, but it is not generally included as a law of thermodynamics. This makes sense because the third law suggests a limit to the entropy value for different systems, which they approach as the temperature drops. It has had great influence on thermodynamics. THIRD LAW ...
Pulse oximetry is routinely used for monitoring patients oxygen saturation levels with little regard to the variability of this physiological variable. There are few published studies on oxygen saturation variability (OSV), with none describing the variability and its pattern in a healthy adult population. The aim of this study was to characterise the pattern of OSV using several parameters: the regularity (sample entropy analysis), the self-similarity (detrended fluctuation analysis (DFA)), and the complexity (multiscale entropy (MSE) analysis). Secondly, to determine if there were any changes that occur with age.. The study population consisted of 36 individuals. The young population consisted of 20 individuals [Mean age = 21.0 (SD = 1.36 years)] and the old population consisted of 16 individuals [Mean age = 50.0 (SD = 10.4 years)]. Through DFA analysis, OSV was shown to exhibit fractal-like patterns. The sample entropy revealed the variability to be more regular than heart rate ...
Pulse oximetry is routinely used for monitoring patients oxygen saturation levels with little regard to the variability of this physiological variable. There are few published studies on oxygen saturation variability (OSV), with none describing the variability and its pattern in a healthy adult population. The aim of this study was to characterise the pattern of OSV using several parameters: the regularity (sample entropy analysis), the self-similarity (detrended fluctuation analysis (DFA)), and the complexity (multiscale entropy (MSE) analysis). Secondly, to determine if there were any changes that occur with age.. The study population consisted of 36 individuals. The young population consisted of 20 individuals [Mean age = 21.0 (SD = 1.36 years)] and the old population consisted of 16 individuals [Mean age = 50.0 (SD = 10.4 years)]. Through DFA analysis, OSV was shown to exhibit fractal-like patterns. The sample entropy revealed the variability to be more regular than heart rate ...
This dissertation attempts to motivate, derive and imply potential uses for a generalized perceptual theory of musical harmony called psychoacoustic entropy theory. This theory treats the human auditory system as a physical system which takes acoustic measurements. As a result, the human auditory system is subject to all the appropriate uncertainties and limitations of other physical measurement systems. This is the theoretic basis for defining psychoacoustic entropy. Psychoacoustic entropy is a numerical quantity which indexes the degree to which the human auditory system perceives instantaneous disorder within a sound pressure wave. Chapter one explains the importance of harmonic analysis as a tool for performance practice. It also outlines the critical limitations for many of the most influential historical approaches to modeling harmonic stability, particularly when compared to available scientific research in psychoacoustics. Rather than analyze a musical excerpt, psychoacoustic entropy is ...
Principal Solar, Inc., a solar power developer and operator, has agreed to terms to co-develop its first major solar asset with affiliates of Entropy Investment Management, LLC. The 100MW facility, located in Cumberland County, North Carolina, will produce enough electricity to power approximately 20,000 average American homes. Construction began the week of August 17, 2015, and the project is expected to begin generating power before the end of 2015. In the transaction, PSI will continue to play an important role in completing the projects development phase and sold its interest in the project to affiliates of Entropy.. [Native Advertisement] This agreement with affiliates of Entropy represents a significant milestone for the Company, says Michael Gorton, CEO, PSI. We have established a relationship with a high-caliber team of solar and finance professionals at Entropy, and have positioned PSI to build additional utility scale projects with them. This is the next major step toward our goal ...
In prokaryotes, RNA thermometers regulate a number of heat shock and virulence genes. These temperature sensitive RNA elements are usually located in the 5-untranslated regions of the regulated genes. They repress translation initiation by base pairing to the Shine-Dalgarno sequence at low temperatures. We investigated the thermodynamic stability of the temperature labile hairpin 2 of the Salmonella fourU RNA thermometer over a broad temperature range and determined free energy, enthalpy and entropy values for the base-pair opening of individual nucleobases by measuring the temperature dependence of the imino proton exchange rates via NMR spectroscopy. Exchange rates were analyzed for the wild-type (wt) RNA and the A8C mutant. The wt RNA was found to be stabilized by the extraordinarily stable G14-C25 base pair. The mismatch base pair in the wt RNA thermometer (A8-G31) is responsible for the smaller cooperativity of the unfolding transition in the wt RNA. Enthalpy and entropy values for the ...
The Second Law of Thermodynamics is stateed as follows in wikipedia: The entropy of an isolated system consisting of two regions of space, isolated from one another, each in thermodynamic equilibrium in itself, but not in equilibrium with each other, will, when the isolation that separates the two regions is broken, so that the two regions become able to exchange matter or energy, tend to increase over time, approaching a maximum value when the jointly communicating system reaches thermodynamic equilibrium. In more practical terms, I have defined the second law of thermodynamics with the two corresponding ideas: 1) In a closed systems, entropy will increase. Conversely, in open systems, entropy will tend to increase. 2 ) If energy is inputted into a system and a state change occurs, the result of the change will be simpler/less ordered than the original participants. Conversely, if energy dissipates from a system and state-change occurs, the result is more complex/more ordered than the original
Downloadable! In this paper, we consider a number of algorithms for estimating the parameters of ME models, including iterative scaling, gradient ascent, conjugate gradient, and variable metric methods. Surprisingly, the standardly used iterative scaling algorithms perform quite poorly in comparison to the others, and for all of the test problems, a limitedmemory variable metric algorithm outperformed the other choices. Maximum entropy (ME) models, variously known as log-linear, Gibbs, exponential, and multinomial logit models, provide a general purpose machine learning technique for classification and prediction which has been successfully applied to fields as diverse as computer vision and econometrics.
Entropy Optimization of Social Networks Using an Evolutionary Algorithm - : Recent work on social networks has tackled the measurement and optimization of these networks robustness and resilience to both failures and attacks. Different metrics have been used to quantitatively measure the robustness of a social network. In this work, we design and apply a Genetic Algorithm that maximizes the cyclic entropy of a social network model, hence optimizing its robustness to failures. Our social network model is a scale-free network created using Barabási and Alberts generative model, since it has been demonstrated recently that many large complex networks display a scale-free structure. We compare the cycles distribution of the optimally robust network generated by our algorithm to that belonging to a fully connected network. Moreover, we optimize the robustness of a scale-free network based on the links-degree entropy, and compare the outcomes to that which is based on cyclesentropy. We show that both
A solution to this problem was found almost 30 years later, by Jorma Rissanen, under the name of Arithmetic coder. Explaining how it works is outside of the scope of this blog post, since its complex and would require a few chapters; I invite you to read the Wikipedia page if you want to learn more about it. For the purpose of this presentation, its enough to say that Arithmetic encoding, and its little brother Range encoding, solved the fractional bit issue of Huffman, and with only some minimal losses to complain about due to rounding, get closer to Shannon limit. So close in fact that entropy encoding is, since then, considered a solved problem ...
The STP Entropy Einstein Solid program calculates the entropy of two Einstein solids that can exchange energy. The purpose of this calculation is to illustrate that the entropy is a maximum at thermal equilibrium. The default system is two…
Entropies of fusion and vaporisation of a variety of elements and compounds have been derived from literature data. Fusion entropies range from low values for metals and certain cyclic hydrocarbons (e.g. cyclopentane) through modest values for salts to high values for materials undergoing drastic rearrangement or disentanglement such as aluminium chloride and n-alkanes. Entropies of vaporisation for most substances are close to the Troutons Law value of ∼100 J deg.-1 mol.-1, with low values for species which associate on boiling (e.g. acetic acid) and higher values signifying simple dissociation (e.g. nitrogen tetroxide) or total decomposition (e.g. some ionic liquids). The nature of inorganic and semi-organic salts in all 3 phases is discussed.. ...
Functional segregation requires convergence and divergence of neuroanatomical connections. Furthermore, the nature of functional segregation suggests that (1) signals in convergent afferents are correlated and (2) signals in divergent efferents are uncorrelated. The aim of this article is to show that this arrangement can be predicted mathematically, using information theory and an idealized model of cortical processing. In theory, the existence of bifurcating axons limits the number of independent output channels from any small cortical region, relative to the number of inputs. An information theoretic analysis of this special (high input:output ratio) constraint indicates that the maximal transfer of information between inputs, to a cortical region, and its outputs will occur when (1) extrinsic connectivity to the area is organized such that the entropy of neural activity in afferents is optimally low and (2) connectivity intrinsic to the region is arranged to maximize the entropy measured at the
Claude Shannon developed a model of information transmission in terms of information entropy. It was developed to describe the transfer of information through a noisy channel. Digitized information consists of bits with quantized amounts. Computers typically use a binary system, with 0 or 1 as allowed values. Genetic information can be thought of as digitized, with A, C, G, and T as allowed values. If each position has one specific possible value, it can be said to have low information content, or in more colloquial terms, no news. As more values are possible at each point in the signal, it becomes less predictable, and hence the information content of any particular message, or instance of the signal, increases. Shannon developed his theory to provide a rigorous model of the transmission of information. Importantly, information entropy provides an operational and mathematical way to describe the amount of information that is transmitted, as well as the amount of redundancy required to get a ...
Suppose you receive an e-mail filled with Chinese characters and one picture of a pile of blue-ish powder. Its quite easy to measure the information contained in that e-mail, including the image, and express it in bits and bytes. Thats what information is, a measurable quantity used in information science and physics. Information and physical systems are linked through the concept of entropy. Shannon Entropy may be loosely defined as the amount of bits needed to describe the unique features of an information structure. Analogously, entropy in physics could be described as the amount of information that is needed to describe the unique modes of behaviour of a physical system. A completely random system has many such modes, is disordered, anything can happenand and thus a high entropy. A deterministic system has low entropy, is highly ordered, only a few things described by a deterministic rule can happen. The information of a physical system are therefore its degrees of freedom and the ...
The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant additivity properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the Markov order). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally most random distributions ...
The Third Law of Thermodynamics . ln This is because a system at zero temperature exists in its ground state, so that its entropy is determined only by the degeneracy of the ground state. Q No, seriously, how cold is it? [9] If there were an entropy difference at absolute zero, T = 0 could be reached in a finite number of steps. {\displaystyle \Delta S=S-S_{0}=k_{\text{B}}\ln(\Omega )={\frac {\delta Q}{T}}}, S [citation needed], The third law is equivalent to the statement that. Third law of thermodynamics 1. The Nernst-Simon statement of the third law of thermodynamics concerns thermodynamic processes at a fixed, low temperature: The entropy change associated with any condensed system undergoing a reversible isothermal process approaches zero as the temperature at which it is performed approaches 0 K. Here a condensed system refers to liquids and solids. The Third Law of Thermodynamics. B Following thermodynamics
For evaluating performance of nonlinear features and iterative and non-iterative classification algorithms (i.e. kernel support vector machine (KSVM), random forest (RaF), least squares SVM (LS-SVM) and multi-surface proximal SVM based oblique RaF (ORaF) for ECG quality assessment we compared the four algorithms on 7 feature schemes yielded from 27 linear and nonlinear features including four features derived from a new encoding Lempel-Ziv complexity (ELZC) and the other 26 features. Seven feature schemes include the first scheme consisting of 7 waveform features, the second consisting of 15 waveform and frequency features, the third consisting of 19 waveform, frequency and approximate entropy (ApEn) features, the fourth consisting of 19 waveform, frequency and permutation entropy (PE) features, the fifth consisting of 19 waveform, frequency and ELZC features, the sixth consisting of 23 waveform, frequency, PE and ELZC features, and the last consisting of all 27 features. Up to 1500 mobile ECG ...
In physics entropy is typically referred to as a measure of disorder in a physical system however I have seen it referred to as a measure of statistical uncertainty for a set of data. I also recall the function defined the statistical uncertainty took the form of an integral and is used in information theory. Could anybody shed any light on this thought of entropy representing a statistical measure of uncertainty ...
Thats enough background on information theory. It is a theory which has long held a fascination for me, and I have used it in several of my research papers over the years. Lets now think how we might use it to ask whether the information content of genomes increases in evolution. First, recall the three way distinction between total information capacity, the capacity that is actually used, and the true information content when stored in the most economical way possible. The total information capacity of the human genome is measured in gigabits. That of the common gut bacterium Escherichia coli is measured in megabits. We, like all other animals, are descended from an ancestor which, were it available for our study today, wed classify as a bacterium. So perhaps, during the billions of years of evolution since that ancestor lived, the information capacity of our genome has gone up about three orders of magnitude (powers of ten) - about a thousandfold. This is satisfyingly plausible and ...
Thats enough background on information theory. It is a theory which has long held a fascination for me, and I have used it in several of my research papers over the years. Lets now think how we might use it to ask whether the information content of genomes increases in evolution. First, recall the three way distinction between total information capacity, the capacity that is actually used, and the true information content when stored in the most economical way possible. The total information capacity of the human genome is measured in gigabits. That of the common gut bacterium Escherichia coli is measured in megabits. We, like all other animals, are descended from an ancestor which, were it available for our study today, wed classify as a bacterium. So perhaps, during the billions of years of evolution since that ancestor lived, the information capacity of our genome has gone up about three orders of magnitude (powers of ten) - about a thousandfold. This is satisfyingly plausible and ...
Common methods of causal inference generate directed acyclic graphs (DAGs) that formalize causal relations between n variables. Given the joint distribution of all these variables, the DAG contains all information about how intervening on one variable would change the distribution of the other n-1 variables. It remains, however, a non-trivial question how to quantify the causal influence of one variable on another one.Here we propose a measure for causal strength that refers to direct effects and measure the strength of an arrow or a set of arrows. It is based on a hypothetical intervention that modifies the joint distribution by cutting the corresponding edge. The causal strength is then the relative entropy distance between the old and the new distribution.We discuss other measures of causal strength like the average causal effect, transfer entropy and information flow and describe their limitations. We argue that our measure is also more appropriate for time series than the known ones.
Whether we can perform this type of analysis or not is constrained by the data available to us. Performing this type of analysis would be very similar to performing phylogenetic analysis using SNP data. Information content of genes can be calculated if we have SNP data representing the probabilities of individual nucleotides at a certain position. Genes with less SNPs have less entropy, and less information content. Vice versa for genes with more SNPs.. If we were able to get our hands on SNP data for ancient species, we would be able to compare the information content of individual genes and perhaps make a statement relating a change in entropy to a change in selective pressure. Perhaps a function that is lost gradually throughout evolution will be represented by a gradual increase in entropy of the group of genes responsible for the function.. Instead of looking at entropy of genes at the species level with SNP data, Dr. Adami recently looked at the entropy of gene families across taxons (C. ...
Entropy, in thermodynamics, is a state function originally introduced to explain why part of a thermodynamic systems total energy is unavailable to do useful work. Entropy may also refer to: .mw-parser-output .tocright{float:right;clear:right;width:auto;background:none;padding:.5em 0 .8em 1.4em;margin-bottom:.5em}.mw-parser-output .tocright-clear-left{clear:left}.mw-parser-output .tocright-clear-both{clear:both}.mw-parser-output .tocright-clear-none{clear:none} ...
9. Conclusions. We have seen that by making three assumptions about the thermal diffusion of a free particle, we are able to show that entropy is generated at a rate equal to twice the particles temperature (when expressed in the correct units).. This result will be applicable to all studies on free particles and other environments that are governed by similar equations. Also a myriad of applications exist in computer modeling, including but not limited to the following: finite difference time domain methods, Blocks equations for nuclear magnetic resonance imaging, and plasma and semiconductor physics.. To check the primary result, one would perform a quantum non-demolition measurement on the quantum state of an ensemble of free particles. The minimum bit rate needed to describe the resulting string of numbers that describe the trajectory would be the entropy rate and should be equal to twice the temperature.. However, even before an experiment can be conducted, this result is useful by ...
Physical symptoms which cannot be adequately explained by organic disease are a common problem in all fields of medicine. Reduced complexity, shown using nonlinear dynamic analysis, has been found to be associated with a wide range of illnesses. These methods have been applied to short time series of mood but not to self-rated physical symptoms. We tested the hypothesis that self-reported medically unexplained physical symptoms display reduced complexity by measuring the approximate entropy of self-reported emotions and physical symptoms collected twice daily over 12 weeks and comparing the results with series-specific surrogate data. We found that approximate entropy (ApEn) was lower for actual data series than for surrogate data. There was no significant difference in entropy between different types of symptoms and no significant correlation between entropy and the diurnal variation of the data series. Future studies should concentrate on specific symptoms and conditions, and evaluate the effect of
Vijay P. Singh, Ph.D., D.Sc., D.Eng. (Hon.), Ph.D. (Hon.), P.E., P.H., Hon.D.WRE is a Distinguished Professor in the Department of Biological and Agricultural Engineering at Texas A&M University. He specializes in surface-water hydrology, groundwater hydrology, hydraulics, irrigation engineering, environmental quality, and water resources. Dr. Singh has published more than 20 textbooks. Description: A thorough introduction to entropy theory and its applications in hydrologic science and engineering. This comprehensive volume addresses basic concepts of entropy theory from a hydrologic engineering perspective. The application of these concepts to a wide range of hydrologic engineering problems is discussed in detail. The book is divided into sections-preliminaries, rainfall and evapotranspiration, subsurface flow, surface flow, and environmental conditions. Helpful equations, solutions, tables, and diagrams are included throughout this practical resource. ...
At the end of the nineties, Jordan, Kinderlehrer, and Otto discovered a new interpretation of the heat equation in R^n, as the gradient flow of the entropy in the Wasserstein space of probability measures. In this talk, I will present a discrete counterpart to this result: given a reversible Markov kernel on a finite set, there exists a Riemannian metric on the space of probability densities, for which the law of the continuous time Markov chain evolves as the gradient flow of the entropy ...
Pierottis theory for the solubility of gases in liquids is tested by means of a large amount of literature data on solubility and entropy of solution. The involved solutes comprise the noble gases, mercury vapour, inorganic gases and hydrocarbons up to propane. The involved solvents comprise alkanes, cycloalkanes, nitromethane, polythene, ... read more aromatics, dimethylsulfoxide and perfluoromethyl-cyclohexane. It appears, that this theory describes the solubilities satisfactory. The description of the entropies of solution is less good. The enhanced solubilities of BF3, CO2, Cl2 and C2H2 in some solvents are ascribed to electron donor-acceptor interaction. The association constants for donor-acceptor complex formation are tabulated. show less ...