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p. 40114025
Received: 4 September 2013 / Accepted: 22 September 2013 / Published: 25 September 2013
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Abstract: An entropy for the scalar variable case, parallel to HavrdaCharvat entropy, was introduced by the first author, and the properties and its connection to Tsallis nonextensive statistical mechanics and the Mathai pathway model were examined by the authors in previous papers. In the current paper, we extend the entropy to cover the scalar case, multivariable case, and matrix variate case. Then, this measure is optimized under different types of restrictions, and a number of models in the multivariable case and matrix variable case are obtained. Connections of these models to problems in statistical and physical sciences are pointed out. An application of the simplest case of the pathway model to the interpretation of solar neutrino data by applying standard deviation analysis and diffusion entropy analysis is provided.
p. 40264041
Received: 12 August 2013 / Revised: 17 September 2013 / Accepted: 18 September 2013 / Published: 25 September 2013
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Abstract: We study the phenomenon of periodic pulling which occurs in certain integrated microcircuits of relevant interest in applications, namely the injectionlocked frequency dividers (ILFDs). They are modelled as secondorder driven oscillators working in the subharmonic (secondary) resonance regime, i.e. , when the selfoscillating frequency is close (resonant) to an integer submultiple n of the driving frequency. Under the assumption of weak injection, we find the spectrum of the system’s oscillatory response in the unlocked mode through closedform expressions, showing that such spectrum is doublesided and asymmetric, unlike the singlesided spectrum of systems with primary resonance (n =1). An analytical expression for the amplitude modulation of the oscillatory response is also presented. Numerical results are presented to support theoretical relations derived.
p. 40424065
Received: 15 August 2013 / Revised: 13 September 2013 / Accepted: 16 September 2013 / Published: 25 September 2013
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Abstract: We review a nonparametric version of Amari’s information geometry in which the set of positive probability densities on a given sample space is endowed with an atlas of charts to form a differentiable manifold modeled on Orlicz Banach spaces. This nonparametric setting is used to discuss the setting of typical problems in machine learning and statistical physics, such as blackbox optimization, KullbackLeibler divergence, BoltzmannGibbs entropy and the Boltzmann equation.
p. 40664083
Received: 22 February 2013 / Revised: 10 September 2013 / Accepted: 11 September 2013 / Published: 25 September 2013
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Abstract: Polymers of hydrogen cyanide and their hydrolysis products constitute a plausible, but still poorly understood proposal for early prebiotic chemistry on Earth. HCN polymers are generated by the interplay of more than a dozen distinctive reaction mechanisms and form a highly complex mixture. Here we use a computational model based on graph grammars as a means of exploring the chemical spaces of HCN polymerization and hydrolysis. A fundamental issue is to understand the combinatorial explosion inherent in large, complex chemical systems. We demonstrate that experimental data, here obtained by mass spectrometry, and computationally predicted free energies together can be used to guide the exploration of the chemical space and makes it feasible to investigate likely pathways and chemical motifs even in potentially openended chemical systems.
p. 40844104
Received: 7 June 2013 / Revised: 22 August 2013 / Accepted: 27 August 2013 / Published: 26 September 2013
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Abstract: An informationtheoretical complexity analysis of the S_{N} 2 exchange reaction for CH_{3} Cl + F^{−} is performed in both position and momentum spaces by means of the following composite functionals of the oneparticle density: DL and IJ planes and FisherShannon’s (FS ) and LópezRuizManciniCalbet (LMC ) shape complexities. It was found that all the chemical concepts traditionally assigned to elementary reactions such as the breaking/forming regions (BB/F ), the charge transfer/reorganization and the charge repulsion can be unraveled from the phenomenological analysis performed in this study through aspects of localizability, uniformity and disorder associated with the informationtheoretical functionals. In contrast, no energybased functionals can reveal the above mentioned chemical concepts. In addition, it is found that the TS critical point for this reaction does not show any chemical meaning (other than the barrier height) as compared with the concurrent processes revealed by the informationtheoretical analysis. Instead, it is apparent from this study that a maximum delocalized state could be identified in the transition region which is associated to the charge transfer process as a new concurrent phenomenon associated with the charge transfer region (CT ) for the ioncomplex is identified. Finally it is discussed why most of the chemical features of interest (e.g., CT , BB/F ) are only revealed when some informationtheoretic properties are taken into account, such as localizability, uniformity and disorder.
p. 41054121
Received: 18 August 2013 / Revised: 21 September 2013 / Accepted: 22 September 2013 / Published: 26 September 2013
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Abstract: In a previous paper, we introduced a way to generate a time coordinate system for classical and quantum systems when the potential function has extremal points. In this paper, we deal with the case in which the potential function has no extremal points at all, and we illustrate the method with the harmonic and linear potentials.
p. 41224133
Received: 27 August 2013 / Revised: 22 September 2013 / Accepted: 22 September 2013 / Published: 27 September 2013
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Abstract: The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion (0< r < R) and a matrix (R < r < ∞) being in perfect thermal contact at r = R is considered. The heat conduction in each region is described by the timefractional heat conduction equation with the Caputo derivative of fractional order 0 < a ≤ 2 and 0 < β ≤ 2, respectively. The Laplace transform with respect to time is used. The approximate solution valid for small values of time is obtained in terms of the MittagLeffler, Wright, and Mainardi functions.
p. 41344158
Received: 15 May 2013 / Revised: 14 August 2013 / Accepted: 18 September 2013 / Published: 27 September 2013
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Abstract: A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to a laminar threedimensional boundarylayer flow in a methane fired combustion channel flow environment. The objective of this study is to explore the applicability of a set of low dimensional, coupled, nonlinear differential equations for the prediction of possible deterministic ordered structures within a specific boundarylayer environment. Four increasing channel pressures are considered. The equations are cast into a Lorenztype system of equations, which yields the lowdimensional set of equations. The solutions indicate the presence of several organized flow structures. Singular value decomposition of the nonlinear time series solutions indicate that nearly ninetyeight percent of the fluctuating directed kinetic energy is contained within the first four empirical modes of the decomposition. The empirical entropy computed from these results indicates that these four lowest modes are largely coherent structures with lower entropy rates. Four regions are observed: lowentropy structures over the first four modes; steep increase in entropy over three modes; steady, high entropy over seven modes; and an increase to maximum entropy over the last two modes. A measure, called the empirical exergy, characterizes the extent of directed kinetic energy produced in the nonlinear solution of the deterministic equations used to model the flow environment. The effect of increasing pressure is to produce more distinct ordered structures within the nonlinear time series solutions.
p. 41594187
Received: 26 July 2013 / Revised: 17 September 2013 / Accepted: 22 September 2013 / Published: 27 September 2013
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Abstract: First, this paper recalls a recently introduced method of adaptive monitoring of dynamical systems and presents the most recent extension with a multiscaleenhanced approach. Then, it is shown that this concept of realtime data monitoring establishes a novel nonShannon and nonprobabilistic concept of novelty quantification, i.e., Entropy of Learning, or in short the Learning Entropy. This novel cognitive measure can be used for evaluation of each newly measured sample of data, or even of whole intervals. The Learning Entropy is quantified in respect to the inconsistency of data to the temporary governing law of system behavior that is incrementally learned by adaptive models such as linear or polynomial adaptive filters or neural networks. The paper presents this novel concept on the example of gradient descent learning technique with normalized learning rate.
p. 41884198
Received: 26 August 2013 / Revised: 5 September 2013 / Accepted: 24 September 2013 / Published: 30 September 2013
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Abstract: In this work, we introduce a generalization of the differential polynomial neural network utilizing fractional calculus. Fractional calculus is taken in the sense of the Caputo differential operator. It approximates a multiparametric function with particular polynomials characterizing its functional output as a generalization of input patterns. This method can be employed on data to describe modelling of complex systems. Furthermore, the total information is calculated by using the fractional Poisson process.
p. 41994214
Received: 27 August 2013 / Revised: 24 September 2013 / Accepted: 25 September 2013 / Published: 7 October 2013
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Abstract: As it results from many research works, the majority of real dynamical objects are fractionalorder systems, although in some types of systems the order is very close to integer order. Application of fractionalorder models is more adequate for the description and analysis of real dynamical systems than integerorder models, because their total entropy is greater than in integerorder models with the same number of parameters. A great deal of modern methods for investigation, monitoring and control of the dynamical processes in different areas utilize approaches based upon modeling of these processes using not only mathematical models, but also physical models. This paper is devoted to the design and analogue electronic realization of the fractionalorder model of a fractionalorder system, e.g., of the controlled object and/or controller, whose mathematical model is a fractionalorder differential equation. The electronic realization is based on fractionalorder differentiator and integrator where operational amplifiers are connected with appropriate impedance, with so called Fractional Order Element or Constant Phase Element. Presented network model approximates quite well the properties of the ideal fractionalorder system compared with e.g., domino ladder networks. Along with the mathematical description, circuit diagrams and design procedure, simulation and measured results are also presented.
p. 42154242
Received: 29 August 2013 / Revised: 13 September 2013 / Accepted: 17 September 2013 / Published: 8 October 2013
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Abstract: Diverse online social networks are becoming increasingly interconnected by sharing information. Accordingly, emergent macrolevel phenomena have been observed, such as the synchronous spread of information across different types of social media. Attempting to analyze the emergent global behavior is impossible from the examination of a single social platform, and dynamic influences between different social networks are not negligible. Furthermore, the underlying structural property of networks is important, as it drives the diffusion process in a stochastic way. In this paper, we propose a macrolevel diffusion model with a probabilistic approach by combining both the heterogeneity and structural connectivity of social networks. As realworld phenomena, we explore instances of news diffusion across different social media platforms from a dataset that contains over 386 million web documents covering a onemonth period in early 2011. We find that influence between different media types is varied by the context of information. News media are the most influential in the arts and economy categories, while social networking sites (SNS) and blog media are in the politics and culture categories, respectively. Furthermore, controversial topics, such as political protests and multiculturalism failure, tend to spread concurrently across social media, while entertainment topics, such as film releases and celebrities, are more likely driven by interactions within single social platforms. We expect that the proposed model applies to a wider class of diffusion phenomena in diverse fields and that it provides a way of interpreting the dynamics of diffusion in terms of the strength and directionality of influences among populations.
p. 42434265
Received: 29 July 2013 / Revised: 22 September 2013 / Accepted: 24 September 2013 / Published: 9 October 2013
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Abstract: Maxwell’s Demon conspires to use information about the state of a confined molecule in a Szilard engine (randomly frozen into a state subspace by his own actions) to derive work from a singletemperature heat bath. It is widely accepted that, if the Demon can achieve this at all, he can do so without violating the Second Law only because of a counterbalancing price that must be paid to erase information when the Demon’s memory is reset at the end of his operating cycle. In this paper, Maxwell’s Demon is analyzed within a “referential” approach to physical information that defines and quantifies the Demon’s information via correlations between the joint physical state of the confined molecule and that of the Demon’s memory. On this view, which received early emphasis in Fahn’s 1996 classical analysis of Maxwell’s Demon, information is erased not during the memory reset step of the Demon’s cycle, but rather during the expansion step, when these correlations are destroyed. Dissipation and work extraction are analyzed here for a Demon that operates a generalized quantum mechanical Szilard engine embedded in a globally closed composite, which also includes a work reservoir, a heat bath and the remainder of the Demon’s environment. Memoryengine correlations lost during the expansion step, which enable extraction of work from the Demon via operations conditioned on the memory contents, are shown to be dissipative when this decorrelation is achieved unconditionally so no work can be extracted. Fahn’s essential conclusions are upheld in generalized form, and his quantitative results supported via appropriate specialization to the Demon of his classical analysis, all without external appeal to classical thermodynamics, the Second Law, phase space conservation arguments or Landauer’s Principle.
p. 42664284
Received: 21 June 2013 / Revised: 10 September 2013 / Accepted: 25 September 2013 / Published: 9 October 2013
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Abstract: Ecological multivariate systems offer a suitable data set on which to apply recent advances in information theory and causality detection. These systems are driven by the interplay of various environmental factors: meteorological and hydrological forcing, which are often correlated with each other at different time lags; and biological factors, primary producers and decomposers with both autonomous and coupled dynamics. Here, using conditional spectral Granger causality, we quantify directional causalities in a complex atmosphereplantsoil system involving the carbon cycle. Granger causality is a statistical approach, originating in econometrics, used to identify the presence of linear causal interactions between time series of data, based on prediction theory. We first test to see if there was a significant difference in the causal structure among two treatments where carbon allocation to roots was interrupted by girdling. We then expanded the analysis, introducing radiation and soil moisture. The results showed a complex pattern of multilevel interactions, with some of these interactions depending upon the number of variables in the system. However, no significant differences emerged in the causal structure of above and below ground carbon cycle among the two treatments.
p. 42854299
Received: 29 August 2013 / Revised: 30 September 2013 / Accepted: 30 September 2013 / Published: 10 October 2013
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Abstract: In this paper, we present a methodological framework for conceptual modeling of assembly supply chain (ASC) networks. Models of such ASC networks are divided into classes on the basis of the numbers of initial suppliers. We provide a brief overview of select literature on the topic of structural complexity in assembly systems. Subsequently, the so called Vertex degree index for measuring a structural complexity of ASC networks is applied. This measure, which is based on the Shannon entropy, is well suited for the given purpose. Finally, we outline a generic model of quantitative complexity scale for ASC Networks.
p. 43004309
Received: 26 June 2013 / Accepted: 6 October 2013 / Published: 10 October 2013
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Abstract: Ab initio molecular dynamics simulations were performed to investigate the elasticity of cubic CaSiO_{3} perovskite at high pressure and temperature. All three independent elastic constants for cubic CaSiO_{3} perovskite, C_{11} , C_{12} , and C_{44} , were calculated from the computation of stress generated by small strains. The elastic constants were used to estimate the moduli and seismic wave velocities at the high pressure and high temperature characteristic of the Earth’s interior. The dependence of temperature for sound wave velocities decreased as the pressure increased. There was little difference between the estimated compressional sound wave velocity (V_{P} ) in cubic CaSiO_{3} perovskite and that in the Earth’s mantle, determined by seismological data. By contrast, a significant difference between the estimated shear sound wave velocity (V_{S} ) and that in the Earth’s mantle was confirmed. The elastic properties of cubic CaSiO_{3} perovskite cannot explain the properties of the Earth’s lower mantle, indicating that the cubic CaSiO_{3} perovskite phase is a minor mineral in the Earth’s lower mantle.
p. 43104318
Received: 14 August 2013 / Revised: 25 September 2013 / Accepted: 1 October 2013 / Published: 14 October 2013
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Abstract: Convergence to a period one fixed point is investigated for both logistic and cubic maps. For the logistic map the relaxation to the fixed point is considered near a transcritical bifurcation while for the cubic map it is near a pitchfork bifurcation. We confirmed that the convergence to the fixed point in both logistic and cubic maps for a region close to the fixed point goes exponentially fast to the fixed point and with a relaxation time described by a power law of exponent 1. At the bifurcation point, the exponent is not universal and depends on the type of the bifurcation as well as on the nonlinearity of the map.
p. 43194333
Received: 5 September 2013 / Revised: 25 September 2013 / Accepted: 27 September 2013 / Published: 15 October 2013
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Abstract: In the past, the phasespace elementary cell of a nonquantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the elementarycell volume of a system of noninteracting N particles, changes when an interaction is switched on and the system becomes or evolves to a system of correlated nonBoltzmann particles and derives the appropriate expressions. Even if we assume that nowadays the volume of the elementary cell is equal to the cube of the Planck constant, h3, at least for quantum systems, we show that there is a correspondence between different values of h in the past, with important and, in principle, measurable cosmological and astrophysical consequences, and systems with an effective smaller (or even larger) phasespace volume described by nonextensive generalized statistics.
p. 43344344
Received: 31 August 2013 / Revised: 6 October 2013 / Accepted: 11 October 2013 / Published: 16 October 2013
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Abstract: In this paper, we investigate the finitetime synchronization problem of a novel hyperchaotic complexvariable system which generates 2, 3 and 4scroll attractors. Based on the finitetime stability theory, two control strategies are proposed to realize synchronization of the novel hyperchaotic complexvariable system in finite time. Finally, two numerical examples have been provided to illustrate the effectiveness of the theoretical analysis.
p. 43454375
Received: 5 August 2013 / Revised: 24 September 2013 / Accepted: 9 October 2013 / Published: 16 October 2013
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Abstract: In this study, the balance equation for local entropy density defined on each partition is obtained by the decomposition of the timeevolution operator for local entropy density, on the level of the master equation, by using symmetric and antisymmetric properties for the inversion of partition, density pairs and a given drift velocity. The resultant equation includes the following terms: convection, diffusion, entropy flow due to a thermostat and entropy production. The averaging of the four terms recover the corresponding terms in a balance equation for the macroscopic entropy density of irreversible thermodynamics for a thermostated system. Moreover, an empirical law of order estimation is introduced to explain the limiting behavior of the averaged quantities in the macroscopic limit for the bulk system. The law makes it possible to separate some minor contributions from the major four terms and, for example, to explain the positive entropy production rate in a nonequilibrium state for volumepreserving systems, even if the state is far from steady state. They are numerically confirmed on an invertible, dissipative multibaker chain system, named a circuit model. These properties are independent of partitioning.
p. 43764391
Received: 30 August 2013 / Revised: 23 September 2013 / Accepted: 10 October 2013 / Published: 16 October 2013
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Abstract: Solute transport through homogeneous media has long been assumed to be scaleindependent and can be quantified by the secondorder advectiondispersion equation (ADE). This study, however, observed the opposite in the laboratory, where transport of CuSO4 through relatively homogeneous silicasand columns exhibits subdiffusion growing with the spatial scale. Only at a very small travel distance (approximately 10 cm) and a relatively short temporal scale can the transport be approximated by normal diffusion. This is also the only spatiotemporal scale where the fundamental concept of the “representative element volume” (which defines the scale of homogeneous cells used by the ADEbased hydrologic models) is valid. The failure of the standard ADE motivated us to apply a temperedstable, fractional advectiondispersion equation (TSFADE) to capture the transient anomalous dispersion with exponentially truncated powerlaw latetime tails in CuSO4 breakthrough curves. Results show that the tempering parameter in the TSFADE model generally decreases with an increase of the column length (probably due to the higher probability of long retention processes), while the time index (which is a nonlocal parameter) remains stable for the uniformly packed columns. Transport in sand columns filled with relatively homogeneous silica sand, therefore, is scaledependent, and the resultant transient subdiffusion can be quantified by the TSFADE model.
p. 43924415
Received: 23 August 2013 / Accepted: 9 October 2013 / Published: 16 October 2013
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Abstract: Permutation entropy, introduced by Bandt and Pompe, is a conceptually simple and wellinterpretable measure of time series complexity. In this paper, we propose efficient methods for computing it and related ordinalpatternsbased characteristics. The methods are based on precomputing values of successive ordinal patterns of order d, considering the fact that they are “overlapped” in d points, and on precomputing successive values of the permutation entropy related to “overlapping” successive timewindows. The proposed methods allow for measurement of the complexity of very large datasets in realtime.
p. 44164431
Received: 9 August 2013 / Revised: 16 September 2013 / Accepted: 10 October 2013 / Published: 17 October 2013
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Abstract: Bayesian testing of a point null hypothesis is considered. The null hypothesisis that an observation, x , is distributed according to the normal distribution with a mean ofzero and known variance q2 . The alternative hypothesis is that x is distributed accordingto a normal distribution with an unknown nonzero mean, μ , and variance q^{2} . The testingproblem is formulated as a prediction problem. Bayesian testing based on priors constructedby using conditional mutual information is investigated.
p. 44324483
Received: 3 July 2013 / Revised: 17 September 2013 / Accepted: 22 September 2013 / Published: 17 October 2013
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Abstract: The most successful exorcism of Maxwell’s demon is Smoluchowski’s 1912 observation that thermal fluctuations would likely disrupt the operation of any molecularscale demonic machine. A later tradition sought to exorcise Maxwell’s demon by assessing the entropic cost of the demon’s processing of information. This later tradition fails since these same thermal fluctuations invalidate the molecularscale manipulations upon which the thermodynamics of computation is based. A new argument concerning conservation of phase space volume shows that all Maxwell’s demons must fail.
p. 44844503
Received: 3 June 2013 / Revised: 2 October 2013 / Accepted: 11 October 2013 / Published: 18 October 2013
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Abstract: The information and communication technology (ICT) sector is continuously growing, mainly due to the fast penetration of ICT into many areas of business and society. Growth is particularly high in the area of technologies and applications for communication networks, which can be used, among others, to optimize systems and processes. The ubiquitous application of ICT opens new perspectives and emphasizes the importance of understanding the complex interactions between ICT and other sectors. Complex and interacting heterogeneous systems can only properly be addressed by a holistic framework. Thermodynamic theory, and, in particular, the second law of thermodynamics, is a universally applicable tool to analyze flows of energy. Communication systems and their processes can be seen, similar to many other natural processes and systems, as dissipative transformations that level differences in energy density between participating subsystems and their surroundings. This paper shows how to apply thermodynamics to analyze energy flows through communication networks. Application of the second law of thermodynamics in the context of the Carnot heat engine is emphasized. The use of exergybased lifecycle analysis to assess the sustainability of ICT systems is shown on an example of a radio access network.
p. 45044519
Received: 22 September 2013 / Revised: 11 October 2013 / Accepted: 16 October 2013 / Published: 18 October 2013
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Abstract: There has been considerable technological interest in highentropy alloys (HEAs) since the initial publications on the topic appeared in 2004. However, only several of the alloys investigated are truly singlephase solid solution compositions. These include the FCC alloys CoCrFeNi and CoCrFeMnNi based on 3d transition metals elements and BCC alloys NbMoTaW, NbMoTaVW, and HfNbTaTiZr based on refractory metals. The search for new singlephase HEAs compositions has been hindered by a lack of an effective scientific strategy for alloy design. This report shows that the chemical interactions and atomic diffusivities predicted from ab initio molecular dynamics simulations which are closely related to primary crystallization during solidification can be used to assist in identifying single phase highentropy solid solution compositions. Further, combining these simulations with phase diagram calculations via the CALPHAD method and inspection of existing phase diagrams is an effective strategy to accelerate the discovery of new singlephase HEAs. This methodology was used to predict new singlephase HEA compositions. These are FCC alloys comprised of CoFeMnNi, CuNiPdPt and CuNiPdPtRh, and HCP alloys of CoOsReRu.
p. 45204539
Received: 24 August 2013 / Revised: 27 September 2013 / Accepted: 10 October 2013 / Published: 22 October 2013
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Abstract: Knowing the three dimensional structure of plasma filaments in the uppermost part of the solar atmosphere, known as coronal loops, and especially their length, is an important parameter in the wavebased diagnostics of this part of the Sun. The combination of observations of the Sun from different points of observations in space, thanks to the most recent missions, including the Solar Dynamics Observatory (SDO) and the Solar TErrestrial RElations Observatory (STEREO), allows us to infer information about the geometrical shape of coronal loops in 3D space. Here, we propose a new method to reconstruct the loop shape starting from stereoscopically determined 3D points, which sample the loop length, by principal component analysis. This method is shown to retrieve in an easy way the main parameters that define the loop, e.g., the minor and major axes, the loop plane, the azimuthal and inclination angles, for the special case of a coplanar loop.
p. 45404552
Received: 24 July 2013 / Revised: 14 October 2013 / Accepted: 14 October 2013 / Published: 22 October 2013
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Abstract: Kernel density estimation is a technique for approximating probability distributions. Here, it is applied to the calculation of mutual information on a metric space. This is motivated by the problem in neuroscience of calculating the mutual information between stimuli and spiking responses; the space of these responses is a metric space. It is shown that kernel density estimation on a metric space resembles the knearestneighbor approach. This approach is applied to a toy dataset designed to mimic electrophysiological data.
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p. 39834010
Received: 29 July 2013 / Revised: 17 September 2013 / Accepted: 17 September 2013 / Published: 25 September 2013
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Abstract: We present the main features of the mathematical theory generated by the κ deformed exponential function ${\mathrm{exp}}_{k}\left(x\right)\text{}=\text{}{(\sqrt{1\text{}+\text{}{k}^{2}{x}^{2}}\text{}+\text{}kx)}^{\frac{1}{k}}$ , with 0 ≤ κ < 1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ algebra, we present the associated κ differential and κ integral calculus. Then, we obtain the corresponding κ exponential and κ logarithm functions and give the κ version of the main functions of the ordinary mathematics.
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