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p. 23842407
Received: 1 February 2014; in revised form: 15 April 2014 / Accepted: 17 April 2014 / Published: 25 April 2014
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Abstract: We apply measures of complexity, emergence, and selforganization to an urban traffic model for comparing a traditional trafficlight coordination method with a selforganizing method in two scenarios: cyclic boundaries and nonorientable boundaries. We show that the measures are useful to identify and characterize different dynamical phases. It becomes clear that different operation regimes are required for different traffic demands. Thus, not only is traffic a nonstationary problem, requiring controllers to adapt constantly; controllers must also change drastically the complexity of their behavior depending on the demand. Based on our measures and extending Ashby’s law of requisite variety, we can say that the selforganizing method achieves an adaptability level comparable to that of a living system.
p. 24082432
Received: 9 March 2014; in revised form: 15 April 2014 / Accepted: 24 April 2014 / Published: 29 April 2014
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Abstract: H theorem states that the entropy production is nonnegative and, therefore, the entropy of a closed system should monotonically change in time. In information processing, the entropy production is positive for random transformation of signals (the information processing lemma). Originally, the Htheorem and the information processing lemma were proved for the classical BoltzmannGibbsShannon entropy and for the correspondent divergence (the relative entropy). Many new entropies and divergences have been proposed during last decades and for all of them the Htheorem is needed. This note proposes a simple and general criterion to check whether the Htheorem is valid for a convex divergence H and demonstrates that some of the popular divergences obey no H theorem. We consider systems with n states A_{i} _{} that obey first order kinetics (master equation). A convex function H is a Lyapunov function for all master equations with given equilibrium if and only if its conditional minima properly describe the equilibria of pair transitions A_{i} ⇌ A_{j} _{} . This theorem does not depend on the principle of detailed balance and is valid for general Markov kinetics. Elementary analysis of pair equilibria demonstrate that the popular Bregman divergences like Euclidian distance or ItakuraSaito distance in the space of distribution cannot be the universal Lyapunov functions for the firstorder kinetics and can increase in Markov processes. Therefore, they violate the second law and the information processing lemma. In particular, for these measures of information (divergences) random manipulation with data may add information to data. The main results are extended to nonlinear generalized mass action law kinetic equations.
p. 24332453
Received: 28 February 2014; in revised form: 14 April 2014 / Accepted: 25 April 2014 / Published: 30 April 2014
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Abstract: This work is focused on the thermodynamic optimization of Organic Rankine Cycles (ORCs), coupled with absorption or adsorption cooling units, for combined cooling heating and power (CCHP) generation from biomass combustion. Results were obtained by modelling with the main aim of providing optimization guidelines for the operating conditions of these types of systems, specifically the subcritical or transcritical ORC, when integrated in a CCHP system to supply typical heating and cooling demands in the tertiary sector. The thermodynamic approach was complemented, to avoid its possible limitations, by the technological constraints of the expander, the heat exchangers and the pump of the ORC. The working fluids considered are: n pentane, n heptane, octamethyltrisiloxane, toluene and dodecamethylcyclohexasiloxane. In addition, the energy and environmental performance of the different optimal CCHP plants was investigated. The optimal plant from the energy and environmental point of view is the one integrated by a toluene recuperative ORC, although it is limited to a development with a turbine type expander. Also, the trigeneration plant could be developed in an energy and environmental efficient way with an n pentane recuperative ORC and a volumetric type expander.
p. 24542471
Received: 27 March 2014; in revised form: 25 April 2014 / Accepted: 29 April 2014 / Published: 2 May 2014
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Abstract: A broad view of the nature and potential of computational information geometry in statistics is offered. This new area suitably extends the manifoldbased approach of classical information geometry to a simplicial setting, in order to obtain an operational universal model space. Additional underlying theory and illustrative real examples are presented. In the inﬁnitedimensional case, challenges inherent in this ambitious overall agenda are highlighted and promising new methodologies indicated.
p. 24722487
Received: 13 December 2013; in revised form: 21 April 2014 / Accepted: 25 April 2014 / Published: 6 May 2014
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Abstract: In this paper, we introduce a geometry called F geometry on a statistical manifold S using an embedding F of S into the space R_{X} of random variables. Amari’s α geometry is a special case of F geometry. Then using the embedding F and a positive smooth function G , we introduce (F,G )metric and (F,G )connections that enable one to consider weighted Fisher information metric and weighted connections. The necessary and sufficient condition for two (F,G )connections to be dual with respect to the (F,G )metric is obtained. Then we show that Amari’s 0connection is the only self dual F connection with respect to the Fisher information metric. Invariance properties of the geometric structures are discussed, which proved that Amari’s α connections are the only F connections that are invariant under smooth onetoone transformations of the random variables.
p. 25122529
Received: 27 February 2014; in revised form: 15 April 2014 / Accepted: 7 May 2014 / Published: 9 May 2014
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Abstract: The theoretical development and practical application of three new methods for estimating the entropy parameter M used within the framework of the entropy method proposed by Chiu in the 1980s as a valid alternative to the velocityarea method for measuring the discharge in a river is here illustrated. The first method is based on reproducing the cumulative velocity distribution function associated with a flood event and requires measurements regarding the entire crosssection, whereas, in the second and third method, the estimate of M is based on reproducing the crosssectional mean velocity by following two different procedures. Both of them rely on the entropy parameter M alone and look for that value of M that brings two different estimates of , obtained by using two different M dependentapproaches, as close as possible. From an operational viewpoint, the acquisition of velocity data becomes increasingly simplified going from the first to the third approach, which uses only one surface velocity measurement. The procedures proposed are applied in a case study based on the Ponte Nuovo hydrometric station on the Tiber River in central Italy.
p. 25302548
Received: 6 December 2013; in revised form: 30 April 2014 / Accepted: 5 May 2014 / Published: 9 May 2014
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Abstract: The theory of Shannon entropy was applied to the ChoiWilliams timefrequency 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) and on a dataset of electroencephalographic (EEG) signals recorded in different states (eyesopen, eyesclosed, ictal and nonictal activity). The results have shown that the values of these indexes tend to decrease, with different proportion, when the behavior of the synthetic signals evolved from chaos or randomness to periodicity. Statistical differences (p value < 0.0005) were found between values of these measures comparing eyesopen and eyesclosed states and between ictal and nonictal states in the traditional EEG frequency bands. Finally, this paper has demonstrated that the proposed measures can be useful tools to quantify the different periodic, chaotic and random components in EEG signals.
p. 25492567
Received: 19 November 2013; in revised form: 1 April 2014 / Accepted: 5 May 2014 / Published: 9 May 2014
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Abstract: This study proposes the concept of the local heat loss coefficient and examines the calculation method for the average heat loss coefficient and the average absorber plate temperature. It also presents an exergy analysis model of flat plate collectors, considering nonuniformity in temperature distribution along the absorber plate. The computation results agree well with experimental data. The effects of ambient temperature, solar irradiance, fluid inlet temperature, and fluid mass flow rate on useful heat rate, useful exergy rate, and exergy loss rate are examined. An optimal fluid inlet temperature exists for obtaining the maximum useful exergy rate. The calculated optimal fluid inlet temperature is 69 °C, and the maximum useful exergy rate is 101.6 W. Exergy rate distribution is analyzed when ambient temperature, solar irradiance, fluid mass flow rate, and fluid inlet temperature are set to 20 °C, 800 W/m^{2} , 0.05 kg/s, and 50 °C, respectively. The exergy efficiency is 5.96%, and the largest exergy loss is caused by the temperature difference between the absorber plate surface and the sun, accounting for 72.86% of the total exergy rate.
p. 25682591
Received: 16 January 2014; in revised form: 16 April 2014 / Accepted: 4 May 2014 / Published: 9 May 2014
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Abstract: We introduce a novel Maximum Entropy (MaxEnt) framework that can generate 3D scenes by incorporating objects’ relevancy, hierarchical and contextual constraints in a unified model. This model is formulated by a Gibbs distribution, under the MaxEnt framework, that can be sampled to generate plausible scenes. Unlike existing approaches, which represent a given scene by a single AndOr graph, the relevancy constraint (defined as the frequency with which a given object exists in the training data) require our approach to sample from multiple AndOr graphs, allowing variability in terms of objects’ existence across synthesized scenes. Once an AndOr graph is sampled from the ensemble, the hierarchical constraints are employed to sample the Ornodes (style variations) and the contextual constraints are subsequently used to enforce the corresponding relations that must be satisfied by the Andnodes. To illustrate the proposed methodology, we use desk scenes that are composed of objects whose existence, styles and arrangements (position and orientation) can vary from one scene to the next. The relevancy, hierarchical and contextual constraints are extracted from a set of training scenes and utilized to generate plausible synthetic scenes that in turn satisfy these constraints. After applying the proposed framework, scenes that are plausible representations of the training examples are automatically generated.
p. 25922610
Received: 10 February 2014; in revised form: 17 April 2014 / Accepted: 22 April 2014 / Published: 13 May 2014
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Abstract: Guided selforganization can be regarded as a paradigm proposed to understand how to guide a selforganizing system towards desirable behaviors, while maintaining its nondeterministic dynamics with emergent features. It is, however, not a trivial problem to guide the selforganizing behavior of physically embodied systems like robots, as the behavioral dynamics are results of interactions among their controller, mechanical dynamics of the body, and the environment. This paper presents a guided selforganization approach for dynamic robots based on a coupling between the system mechanical dynamics with an internal control structure known as the attractor selection mechanism. The mechanism enables the robot to gracefully shift between random and deterministic behaviors, represented by a number of attractors, depending on internally generated stochastic perturbation and sensory input. The robot used in this paper is a simulated curved beam hopping robot: a system with a variety of mechanical dynamics which depends on its actuation frequencies. Despite the simplicity of the approach, it will be shown how the approach regulates the probability of the robot to reach a goal through the interplay among the sensory input, the level of inherent stochastic perturbation, i.e., noise, and the mechanical dynamics.
p. 26112628
Received: 6 January 2014; in revised form: 26 April 2014 / Accepted: 28 April 2014 / Published: 13 May 2014
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Abstract: Divergence is a discrepancy measure between two objects, such as functions, vectors, matrices, and so forth. In particular, divergences defined on probability distributions are widely employed in probabilistic forecasting. As the dissimilarity measure, the divergence should satisfy some conditions. In this paper, we consider two conditions: The first one is the scaleinvariance property and the second is that the divergence is approximated by the sample mean of a loss function. The first requirement is an important feature for dissimilarity measures. The divergence will depend on which system of measurements we used to measure the objects. Scaleinvariant divergence is transformed in a consistent way when the system of measurements is changed to the other one. The second requirement is formalized such that the divergence is expressed by using the socalled composite score. We study the relation between composite scores and scaleinvariant divergences, and we propose a new class of divergences called H¨older divergence that satisfies two conditions above. We present some theoretical properties of H¨older divergence. We show that H¨older divergence unifies existing divergences from the viewpoint of scaleinvariance.
p. 26292641
Received: 24 February 2014; in revised form: 6 May 2014 / Accepted: 9 May 2014 / Published: 14 May 2014
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Abstract: Online auctions attract not only legitimate businesses trying to sell their products but also fraudsters wishing to commit fraudulent transactions. Consequently, fraudster detection is crucial to ensure the continued success of online auctions. This paper proposes an approach to detect fraudsters based on the concept of neighbor diversity. The neighbor diversity of an auction account quantifies the diversity of all traders that have transactions with this account. Based on four different features of each trader (i.e. , the number of received ratings, the number of cancelled transactions, k core, and the joined date), four measurements of neighbor diversity are proposed to discern fraudsters from legitimate traders. An experiment is conducted using data gathered from a real world auction website. The results show that, although the use of neighbor diversity on k core or on the joined date shows little or no improvement in detecting fraudsters, both the neighbor diversity on the number of received ratings and the neighbor diversity on the number of cancelled transactions improve classification accuracy, compared to the stateoftheart methods that use k core and center weight.
p. 26422668
Received: 31 March 2014; in revised form: 7 May 2014 / Accepted: 9 May 2014 / Published: 14 May 2014
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Abstract: We study the problem of finding probability densities that match given European call option prices. To allow prior information about such a density to be taken into account, we generalise the algorithm presented in Neri and Schneider (Appl. Math. Finance 2013) to find the maximum entropy density of an asset price to the relative entropy case. This is applied to study the impact of the choice of prior density in two market scenarios. In the first scenario, call option prices are prescribed at only a small number of strikes, and we see that the choice of prior, or indeed its omission, yields notably different densities. The second scenario is given by CBOE option price data for S&P500 index options at a large number of strikes. Prior information is now considered to be given by calibrated Heston, Schöbel–Zhu or Variance Gamma models. We find that the resulting digital option prices are essentially the same as those given by the (nonrelative) Buchen–Kelly density itself. In other words, in a sufficiently liquid market, the influence of the prior density seems to vanish almost completely. Finally, we study variance swaps and derive a simple formula relating the fair variance swap rate to entropy. Then we show, again, that the prior loses its influence on the fair variance swap rate as the number of strikes increases.
p. 26692685
Received: 13 January 2014; in revised form: 17 April 2014 / Accepted: 4 May 2014 / Published: 14 May 2014
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Abstract: The Carnot factor versus enthalpy variation (heat) diagram has been used extensively for the second law analysis of heat transfer processes. With enthalpy variation (heat) as the abscissa and the Carnot factor as the ordinate the area between the curves representing the heat exchanging media on this diagram illustrates the exergy losses due to the transfer. It is also possible to draw the paths of working fluids in steadystate, steadyflow thermodynamic cycles on this diagram using the definition of “the equivalent temperature” as the ratio between the variations of enthalpy and entropy in an analyzed process. Despite the usefulness of this approach two important shortcomings should be emphasized. First, the approach is not applicable for the processes of expansion and compression particularly for the isenthalpic processes taking place in expansion valves. Second, from the point of view of rigorous thermodynamics, the proposed ratio gives the temperature dimension for the isobaric processes only. The present paper proposes to overcome these shortcomings by replacing the actual processes of expansion and compression by combinations of two thermodynamic paths: isentropic and isobaric. As a result the actual (not ideal) refrigeration and power cycles can be presented on equivalent temperature versus enthalpy variation diagrams. All the exergy losses, taking place in different equipments like pumps, turbines, compressors, expansion valves, condensers and evaporators are then clearly visualized. Moreover the exergies consumed and produced in each component of these cycles are also presented. The latter give the opportunity to also analyze the exergy efficiencies of the components. The proposed diagram is finally applied for the second law analysis of an ejector based refrigeration system.
p. 26862698
Received: 1 April 2014; in revised form: 12 May 2014 / Accepted: 13 May 2014 / Published: 14 May 2014
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Abstract: In this note we introduce some divergencebased model selection criteria. These criteria are defined by estimators of the expected overall discrepancy between the true unknown model and the candidate model, using dual representations of divergences and associated minimum divergence estimators. It is shown that the proposed criteria are asymptotically unbiased. The influence functions of these criteria are also derived and some comments on robustness are provided.
p. 26992712
Received: 25 January 2014; Accepted: 12 May 2014 / Published: 14 May 2014
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Abstract: Motivated by the notion of perceptual error, as a core concept of the perceptual control theory, we propose an actionamplitude model for controlled entropic selforganization (CESO). We present several aspects of this development that illustrate its explanatory power: (i) a physical view of partition functions and path integrals, as well as entropy and phase transitions; (ii) a global view of functional compositions and commutative diagrams; (iii) a local geometric view of the Kähler–Ricci flow and timeevolution of entropic action; and (iv) a computational view using various pathintegral approximations.
p. 27132728
Received: 23 February 2014; in revised form: 18 April 2014 / Accepted: 12 May 2014 / Published: 16 May 2014
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Abstract: Powerlaw (PL) formalism is known to provide an appropriate framework for canonical modeling of nonlinear systems. We estimated three stochastically distinct models of constant elasticity of substitution (CES) class functions as nonlinear inverse problem and showed that these PL related functions should have a closed form. The first model is related to an aggregator production function, the second to an aggregator utility function (the Armington) and the third to an aggregator technical transformation function. A qgeneralization of KL information divergence criterion function with a priori consistency constraints is proposed. Related inferential statistical indices are computed. The approach leads to robust estimation and to new findings about the true stochastic nature of this class of nonlinear—up until now—analytically intractable functions. Outputs from traditional econometric techniques (Shannon entropy, NLLS, GMM, ML) are also presented.
p. 27292755
Received: 1 March 2014; in revised form: 28 April 2014 / Accepted: 9 May 2014 / Published: 16 May 2014
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Abstract: A computational procedure is developed to determine initial instabilities within a threedimensional laminar boundary layer and to follow these instabilities in the streamwise direction through to the resulting intermittency exponents within a fully developed turbulent flow. The fluctuating velocity wave vector component equations are arranged into a Lorenztype system of equations. The nonlinear time series solution of these equations at the fifth station downstream of the initial instabilities indicates a sequential outward burst process, while the results for the eleventh station predict a strong sequential inward sweep process. The results for the thirteenth station indicate a return to the original instability autogeneration process. The nonlinear time series solutions indicate regions of order and disorder within the solutions. Empirical entropies are defined from decomposition modes obtained from singular value decomposition techniques applied to the nonlinear time series solutions. Empirical entropic indices are obtained from the empirical entropies for two streamwise stations. The intermittency exponents are then obtained from the entropic indices for these streamwise stations that indicate the burst and autogeneration processes.
p. 27562767
Received: 4 April 2014; in revised form: 28 April 2014 / Accepted: 14 May 2014 / Published: 19 May 2014
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Abstract: The influence of the atomic order on the martensitic transformation entropy change has been studied in a NiMnInCo metamagnetic shape memory alloy through the evolution of the transformation temperatures under hightemperature quenching and postquench annealing thermal treatments. It is confirmed that the entropy change evolves as a consequence of the variations on the degree of L2_{1} atomic order brought by thermal treatments, though, contrary to what occurs in ternary NiMnIn, postquench aging appears to be the most effective way to modify the transformation entropy in NiMnInCo. It is also shown that any entropy change value between around 40 and 5 J/kgK can be achieved in a controllable way for a single alloy under the appropriate aging treatment, thus bringing out the possibility of properly tune the magnetocaloric effect.
p. 27682788
Received: 21 March 2014; in revised form: 8 May 2014 / Accepted: 9 May 2014 / Published: 19 May 2014
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Abstract: In this study, features of the financial returns of the PSI20index, related to market efficiency, are captured using wavelet and entropybased techniques. This characterization includes the following points. First, the detection of long memory, associated with low frequencies, and a global measure of the time series: the Hurst exponent estimated by several methods, including wavelets. Second, the degree of roughness, or regularity variation, associated with the H¨older exponent, fractal dimension and estimation based on the multifractal spectrum. Finally, the degree of the unpredictability of the series, estimated by approximate entropy. These aspects may also be studied through the concepts of nonextensive entropy and distribution using, for instance, the Tsallis qtriplet. They allow one to study the existence of efficiency in the financial market. On the other hand, the study of local roughness is performed by considering wavelet leaderbased entropy. In fact, the wavelet coefficients are computed from a multiresolution analysis, and the wavelet leaders are defined by the local suprema of these coefficients, near the point that we are considering. The resulting entropy is more accurate in that detection than the H¨older exponent. These procedures enhance the capacity to identify the occurrence of financial crashes.
p. 27892819
Received: 28 February 2014; in revised form: 28 April 2014 / Accepted: 4 May 2014 / Published: 21 May 2014
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Abstract: One aspect of intelligence is the ability to restructure your own environment so that the world you live in becomes more beneficial to you. In this paper we investigate how the informationtheoretic measure of agent empowerment can provide a taskindependent, intrinsic motivation to restructure the world. We show how changes in embodiment and in the environment change the resulting behaviour of the agent and the artefacts left in the world. For this purpose, we introduce an approximation of the established empowerment formalism based on sparse sampling, which is simpler and significantly faster to compute for deterministic dynamics. Sparse sampling also introduces a degree of randomness into the decision making process, which turns out to beneficial for some cases. We then utilize the measure to generate agent behaviour for different agent embodiments in a Minecraftinspired three dimensional block world. The paradigmatic results demonstrate that empowerment can be used as a suitable generic intrinsic motivation to not only generate actions in given static environments, as shown in the past, but also to modify existing environmental conditions. In doing so, the emerging strategies to modify an agent’s environment turn out to be meaningful to the specific agent capabilities, i.e. , de facto to its embodiment.
p. 28202838
Received: 30 January 2014; in revised form: 7 May 2014 / Accepted: 13 May 2014 / Published: 21 May 2014
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Abstract: This paper investigates selforganizing binary majority consensus disturbed by faulty nodes with random and persistent failure. We study consensus in ordered and random networks with noise, message loss and delays. Using computer simulations, we show that: (1) explicit randomization by noise, message loss and topology can increase robustness towards faulty nodes; (2) commonlyused faulty nodes with random failure inhibit consensus less than faulty nodes with persistent failure; and (3) in some cases, such randomly failing faulty nodes can even promote agreement.
p. 28392849
Received: 18 February 2014; in revised form: 15 May 2014 / Accepted: 20 May 2014 / Published: 23 May 2014
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Abstract: Using a recently discovered method for producing random symbol sequences with prescribed transition counts, we present an exact null hypothesis significance test (NHST) for mutual information between two random variables, the null hypothesis being that the mutual information is zero (i.e., independence). The exact tests reported in the literature assume that data samples for each variable are sequentially independent and identically distributed (iid). In general, time series data have dependencies (Markov structure) that violate this condition. The algorithm given in this paper is the first exact significance test of mutual information that takes into account the Markov structure. When the Markov order is not known or indefinite, an exact test is used to determine an effective Markov order.
p. 28502868
Received: 27 February 2014; in revised form: 16 May 2014 / Accepted: 20 May 2014 / Published: 23 May 2014
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Abstract: This work presents a formalism to calculate the configurational entropy of mixing based on the identification of noninteracting atomic complexes in the mixture and the calculation of their respective probabilities, instead of computing the number of atomic configurations in a lattice. The methodology is applied in order to develop a general analytical expression for the configurational entropy of mixing of interstitial solutions. The expression is valid for any interstitial concentration, is suitable for the treatment of interstitial shortrange order (SRO) and can be applied to tetrahedral or octahedral interstitial solutions in any crystal lattice. The effect of the SRO of H on the structural properties of the NbH and bcc ZrH solid solutions is studied using an accurate description of the configurational entropy. The methodology can also be applied to systems with no translational symmetry, such as liquids and amorphous materials. An expression for the configurational entropy of a granular system composed by equal sized hard spheres is deduced.
p. 28692889
Received: 22 March 2014; in revised form: 12 May 2014 / Accepted: 21 May 2014 / Published: 23 May 2014
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Abstract: Honeycomb sandwich structures are used in a wide variety of applications. Nevertheless, due to manufacturing defects or impact loads, these structures can be subject to imperfect bonding or debonding between the skin and the honeycomb core. The presence of debonding reduces the bending stiffness of the composite panel, which causes detectable changes in its vibration characteristics. This article presents a new supervised learning algorithm to identify debonded regions in aluminum honeycomb panels. The algorithm uses a linear approximation method handled by a statistical inference model based on the maximumentropy principle. The merits of this new approach are twofold: training is avoided and data is processed in a period of time that is comparable to the one of neural networks. The honeycomb panels are modeled with finite elements using a simplified threelayer shell model. The adhesive layer between the skin and core is modeled using linear springs, the rigidities of which are reduced in debonded sectors. The algorithm is validated using experimental data of an aluminum honeycomb panel under different damage scenarios.
p. 28902903
Received: 26 February 2014; in revised form: 2 April 2014 / Accepted: 20 May 2014 / Published: 23 May 2014
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Abstract: In a recent work, we have reported a study on the figure of merit of a thermoelectric system composed by thermoelectric generators connected electrically and thermally in different configurations. In this work, we are interested in analyzing the output power delivered by a thermoelectric system for different arrays of thermoelectric materials in each configuration. Our study shows the impact of the array of thermoelectric materials in the output power of the composite system. We evaluate numerically the corresponding maximum output power for each configuration and determine the optimum array and configuration for maximum power. We compare our results with other recently reported studies.
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p. 24882511
Received: 17 January 2014; in revised form: 27 February 2014 / Accepted: 22 April 2014 / Published: 8 May 2014
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Abstract: Minimal artificial cells (MACs) are selfassembled chemical systems able to mimic the behavior of living cells at a minimal level, i.e. to exhibit selfmaintenance, selfreproduction and the capability of evolution. The bottomup approach to the construction of MACs is mainly based on the encapsulation of chemical reacting systems inside lipid vesicles, i.e. chemical systems enclosed (compartmentalized) by a doublelayered lipid membrane. Several researchers are currently interested in synthesizing such simple cellular models for biotechnological purposes or for investigating origin of life scenarios. Within this context, the properties of lipid vesicles (e.g., their stability, permeability, growth dynamics, potential to host reactions or undergo division processes…) play a central role, in combination with the dynamics of the encapsulated chemical or biochemical networks. Thus, from a theoretical standpoint, it is very important to develop kinetic equations in order to explore first—and specify later—the conditions that allow the robust implementation of these complex chemically reacting systems, as well as their controlled reproduction. Due to being compartmentalized in small volumes, the population of reacting molecules can be very low in terms of the number of molecules and therefore their behavior becomes highly affected by stochastic effects both in the time course of reactions and in occupancy distribution among the vesicle population. In this short review we report our mathematical approaches to model artificial cell systems in this complex scenario by giving a summary of three recent simulations studies on the topic of primitive cell (protocell) systems.
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