
Contreras, G. M. (2014). Stochastic volatility models at rho = +/ 1 as second class constrained Hamiltonian systems. Physica A, 405, 289–302.
Abstract: The stochastic volatility models used in the financial world are characterized, in the continuoustime case, by a set of two coupled stochastic differential equations for the underlying asset price S and volatility sigma. In addition, the correlations of the two Brownian movements that drive the stochastic dynamics are measured by the correlation parameter rho (1 <= rho <= 1). This stochastic system is equivalent to the FokkerPlanck equation for the transition probability density of the random variables S and sigma. Solutions for the transition probability density of the Heston stochastic volatility model (Heston, 1993) were explored in Dragulescu and Yakovenko (2002), where the fundamental quantities such as the transition density itself, depend on rho in such a manner that these are divergent for the extreme limit rho = +/ 1. The same divergent behavior appears in Hagan et al. (2002), where the probability density of the SABR model was analyzed. In an option pricing context, the propagator of the bidimensional BlackScholes equation was obtained in Lemmens et al. (2008) in terms of the path integrals, and in this case, the propagator diverges again for the extreme values rho = +/ 1. This paper shows that these similar divergent behaviors are due to a universal property of the stochastic volatility models in the continuum: all of them are second class constrained systems for the most extreme correlated limit rho = +/ 1. In this way, the stochastic dynamics of the rho = +/ 1 cases are different of the rho (1 <= rho <= 1) case, and it cannot be obtained as a continuous limit from the rho not equal +/ 1 regimen. This conclusion is achieved by considering the FokkerPlanck equation or the bidimensional BlackScholes equation as a Euclidean quantum Schrodinger equation. Then, the analysis of the underlying classical mechanics of the quantum model, implies that stochastic volatility models at rho = +/ 1 correspond to a constrained system. To study the dynamics in an appropriate form, Dirac's method for constrained systems (Dirac, 1958, 1967) must be employed, and Dirac's analysis reveals that the constraints are second class. In order to obtain the transition probability density or the option price correctly, one must evaluate the propagator as a constrained Hamiltonian pathintegral (Henneaux and Teitelboim, 1992), in a similar way to the high energy gauge theory models. In fact, for all stochastic volatility models, after integrating over momentum variables, one obtains an effective Euclidean Lagrangian path integral over the volatility alone. The role of the second class constraints is determining the underlying asset price S completely in terms of volatility, so it plays no role in the path integral. In order to examine the effect of the constraints on the dynamics for both extreme limits, the probability density function is evaluated by using semiclassical arguments, in an analogous manner to that developed in Hagan et al. (2002), for the SABR model. (C) 2014 Elsevier B.V. All rights reserved.



Contreras, M., & Hojman, S. A. (2014). Option pricing, stochastic volatility, singular dynamics and constrained path integrals. Physica A, 393, 391–403.
Abstract: Stochastic volatility models have been widely studied and used in the financial world. The Heston model (Heston, 1993) [7] is one of the best known models to deal with this issue. These stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter p which relates the two Brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset. Solutions to the Heston model in the context of option pricing, using a path integral approach, are found in Lemmens et al. (2008) [21] while in Baaquie (2007,1997) [12,13] propagators for different stochastic volatility models are constructed. In all previous cases, the propagator is not defined for extreme cases rho = +/ 1. It is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator. In this paper we study in detail a general class of stochastic volatility models for extreme values rho = +/ 1 and show that in these two cases, the associated classical dynamics corresponds to a system with second class constraints, which must be dealt with using Dirac's method for constrained systems (Dirac, 1958,1967) [22,23] in order to properly obtain the propagator in the form of a Euclidean Hamiltonian path integral (Henneaux and Teitelboim, 1992) [25]. After integrating over momenta, one gets an Euclidean Lagrangian path integral without constraints, which in the case of the Heston model corresponds to a path integral of a repulsive radial harmonic oscillator. In all the cases studied, the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral. (C) 2013 Elsevier B.V. All rights reserved.



Contreras, M., Montalva, R., Pellicer, R., & Villena, M. (2010). Dynamic option pricing with endogenous stochastic arbitrage. Physica A, 389(17), 3552–3564.
Abstract: Only few efforts have been made in order to relax one of the key assumptions of the BlackScholes model: the noarbitrage assumption. This is despite the fact that arbitrage processes usually exist in the real world, even though they tend to be shortlived. The purpose of this paper is to develop an option pricing model with endogenous stochastic arbitrage, capable of modelling in a general fashion any future and underlying asset that deviate itself from its market equilibrium. Thus, this investigation calibrates empirically the arbitrage on the futures on the S&P 500 index using transaction data from September 1997 to June 2009, from here a specific type of arbitrage called “arbitrage bubble”, based on a tstep function, is identified and hence used in our model. The theoretical results obtained for Binary and European call options, for this kind of arbitrage, show that an investment strategy that takes advantage of the identified arbitrage possibility can be defined, whenever it is possible to anticipate in relative terms the amplitude and timespan of the process. Finally, the new trajectory of the stock price is analytically estimated for a specific case of arbitrage and some numerical illustrations are developed. We find that the consequences of a finite and small endogenous arbitrage not only change the trajectory of the asset price during the period when it started, but also after the arbitrage bubble has already gone. In this context, our model will allow us to calibrate the BS model to that new trajectory even when the arbitrage already started. (C) 2010 Elsevier B.V. All rights reserved.



Contreras, M., Pellicer, R., & Villena, M. (2017). Dynamic optimization and its relation to classical and quantum constrained systems. Physica A, 479, 12–25.
Abstract: We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a classical mechanics constrained system, so we must use the Dirac method to analyze it in a correct way. We find that there are two secondclass constraints in the model: one fix the momenta associated with the control variables, and the other is a reminder of the optimal control law. The dynamic evolution of this constrained system is given by the Dirac's bracket of the canonical variables with the Hamiltonian. This dynamic results to be identical to the unconstrained one given by the Pontryagin equations, which are the correct classical equations of motion for our physical optimization problem. In the same Pontryagin scheme, by imposing a closedloop lambdastrategy, the optimality condition for the action gives a consistency relation, which is associated to the HamiltonJacobiBellman equation of the dynamic programming method. A similar result is achieved by quantizing the classical model. By setting the wave function Psi (x, t) = e(is(x,t)) in the quantum Schrodinger equation, a nonlinear partial equation is obtained for the S function. For the righthand side quantization, this is the HamiltonJacobiBellman equation, when S(x, t) is identified with the optimal value function. Thus, the HamiltonJacobiBellman equation in Bellman's maximum principle, can be interpreted as the quantum approach of the optimization problem. (C) 2017 Elsevier B.V. All rights reserved.



Contreras, M., Pellicer, R., Villena, M., & Ruiz, A. (2010). A quantum model of option pricing: When BlackScholes meets Schrodinger and its semiclassical limit. Physica A, 389(23), 5447–5459.
Abstract: The BlackScholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrodinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, shortterm volatility, extreme discontinuities, or serial correlations; the classical nonarbitrage assumption of the BlackScholes model is violated, implying a nonriskfree portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the BlackScholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new BlackScholesSchrodinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrodinger equation in imaginary time for a particle of mass 1/sigma(2) with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the BlackScholes model represent a particular case. Finally, since the Schrodinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the BlackScholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrodinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing. (C) 2010 Elsevier B.V. All rights reserved.



Corral, N., Anrique, N., Fernandes, D., Parrado, C., & Caceres, G. (2012). Power, placement and LEC evaluation to install CSP plants in northern Chile. Renew. Sust. Energ. Rev., 16(9), 6678–6685.
Abstract: Chile is expecting a 5.4% growth in energy consumption per year until 2030, requiring new and better solutions for the upward trend of its electricity demand. This state leads to select and study one of the potential alternatives for electricity generation: concentrated solar power (CSP) plants. Such renewable technology found in Chile a very favorable condition. Recent researches indicate Atacama Desert as one of the best regions for solar energy worldwide, having an average radiation higher than in places where CSP plants are currently implemented, e.g. Spain and USA. The aim of this study is to present an analysis of levelized energy cost (LEC) for different power capacities of CSP plants placed in distinct locations in northern Chile. The results showed that CSP plants can be implemented in Atacama Desert with LECs around 19 (sic)US$/kWh when a gasfired backup and thermal energy storage (TES) systems are fitted. This value increases to approximately 28 (sic)US$/kWh if there is no backup system. (C) 2012 Elsevier Ltd. All rights reserved.



Cortes, C. E., JaraMoroni, P., Moreno, E., & Pineda, C. (2013). Stochastic transit equilibrium. Transp. Res. Pt. BMethodol., 51, 29–44.
Abstract: We present a transit equilibrium model in which boarding decisions are stochastic. The model incorporates congestion, reflected in higher waiting times at bus stops and increasing invehicle travel time. The stochastic behavior of passengers is introduced through a probability for passengers to choose boarding a specific bus of a certain service. The modeling approach generates a stochastic commonlines problem, in which every line has a chance to be chosen by each passenger. The formulation is a generalization of deterministic transit assignment models where passengers are assumed to travel according to shortest hyperpaths. We prove existence of equilibrium in the simplified case of parallel lines (stochastic commonlines problem) and provide a formulation for a more general network problem (stochastic transit equilibrium). The resulting waiting time and network load expressions are validated through simulation. An algorithm to solve the general stochastic transit equilibrium is proposed and applied to a sample network; the algorithm works well and generates consistent results when considering the stochastic nature of the decisions, which motivates the implementation of the methodology on a realsize network case as the next step of this research. (C) 2013 Elsevier Ltd. All rights reserved.



Cortes, M. P., Mendoza, S. N., Travisany, D., Gaete, A., Siegel, A., Cambiazo, V., et al. (2017). Analysis of Piscirickettsia salmonis Metabolism Using GenomeScale Reconstruction, Modeling, and Testing. Front. Microbiol., 8, 15 pp.
Abstract: Piscirickettsia salmonis is an intracellular bacterial fish pathogen that causes piscirickettsiosis, a disease with highly adverse impact in the Chilean salmon farming industry. The development of effective treatment and control methods for piscireckttsiosis is still a challenge. To meet it the number of studies on P. salmonis has grown in the last couple of years but many aspects of the pathogen's biology are still poorly understood. Studies on its metabolism are scarce and only recently a metabolic model for reference strain LF89 was developed. We present a new genomescale model for P. salmonis LF89 with more than twice as many genes as in the previous model and incorporating specific elements of the fish pathogen metabolism. Comparative analysis with models of different bacterial pathogens revealed a lower flexibility in P. salmonis metabolic network. Through constraintbased analysis, we determined essential metabolites required for its growth and showed that it can benefit from different carbon sources tested experimentally in new defined media. We also built an additional model for strain A115972, and together with an analysis of P. salmonis pangenome, we identified metabolic features that differentiate two main species clades. Both models constitute a knowledgebase for P. salmonis metabolism and can be used to guide the efficient culture of the pathogen and the identification of specific drug targets.



Cortez, V., Medina, P., Goles, E., Zarama, R., & Rica, S. (2015). Attractors, statistics and fluctuations of the dynamics of the Schelling's model for social segregation. Eur. Phys. J. B, 88(1), 12 pp.
Abstract: Statistical properties, fluctuations and probabilistic arguments are shown to explain the robust dynamics of the Schelling's social segregation model. With the aid of probability density functions we characterize the attractors for multiple external parameters and conditions. We discuss the role of the initial states and we show that, indeed, the system evolves towards well defined attractors. Finally, we provide probabilistic arguments to explain quantitatively the observed behavior.



Cortez, V., Saravia, G., & Vogel, E. E. (2014). Phase diagram and reentrance for the 3D EdwardsAnderson model using information theory. J. Magn. Magn. Mater., 372, 173–180.
Abstract: Data compressor techniques are used to study the phase diagram of the generalized EdwardsAnderson model in three dimensions covering the full range of mixture between ferromagnetic (concentration 1x) and antiferromagnetic interactions (concentration x). The recently proposed data compressor wlzip is used to recognize criticality by the maximum information content in the files storing the simulation processes. The method allows not only the characterization of the ferromagnetic to paramagnetic (FP) transition (x < 0.22, or x > 0.78) but also it equally well yields the spinglass to paramagnetic (SP) transition (0.22 < x < 0.78). A reentrance of a ferromagnetic phase into the spinglass phase is found in the vicinity of the multicritical point. The differences in the ways to apply the new method to FP and SP transitions are reported. A phase diagram for the entire range of x based entirely on the use of compression techniques is obtained and discussed. The advantages and disadvantages of the method of data compression as compared to other methods to deal with magnetic phase transitions are brought out and explained. (C) 2014 Elsevier B.V. All rights reserved.



Couturier, E., Dumais, J., Cerda, E., & Katifori, E. (2013). Folding of an opened spherical shell. Soft Matter, 9(34), 8359–8367.
Abstract: Thin, doubly curved shells occur commonly in nature and their mechanical properties and modes of deformation are very important for engineering structures of all scales. Although there has been substantial work on the stability and modes of failure of thin shells, relatively little work has been done to understand the conditions that promote continuous large scale deformations. A major impediment to progress in this direction is the inherent difficulty in obtaining analytical expressions for the deformed shapes. In this work we propose a new integrable solution which describes the behavior under load of a thin spherical shell with an opening (aperture) of nfold axial symmetry. We derive a twoparameter family of approximately isometric, constant positive Gaussian curvature shapes that is in excellent agreement with our experimental results of deformed shells (3D scans of compressed pingpong balls) and simulations (tethered membrane simulations minimizing the stretching and bending energy). The integrable solutions that describe those shapes have n symmetrically arranged curvature singularities which correspond to cusps of the folded shape. We examine the properties of the folded shells and observe that in the analytic solutions isometric closure is more easily achieved when the singularities lie away from the center of the aperture. We find that when allowed by the geometry of the aperture and the nature of the load, physical shells expel the curvature singularities into the aperture.



Crutchik, D., Frison, N., Eusebi, A. L., & Fatone, F. (2018). Biorefinery of cellulosic primary sludge towards targeted Short Chain Fatty Acids, phosphorus and methane recovery. Water Res., 136, 112–119.
Abstract: Cellulose from used toilet paper is a major untapped resource embedded in municipal wastewater which recovery and valorization to valuable products can be optimized. Cellulosic primary sludge (CPS) can be separated by upstream dynamic sieving and anaerobically digested to recover methane as much as 4.02 m(3)/capita.year. On the other hand, optimal acidogenic fermenting conditions of CPS allows the production of targeted shortchain fatty acids (SCFAs) as much as 2.92 kg COD/capita . year. Here propionate content can be more than 30% and can optimize the enhanced biological phosphorus removal (EBPR) processes or the higher valuable copolymer of polyhydroxyalkanoates (PHAs). In this work, first a full set of batch assays were used at three different temperatures (37, 55 and 70 degrees C) and three different initial pH (8, 9 and 10) to identify the best conditions for optimizing both the total SCFAs and propionate content from CPS fermentation. Then, the optimal conditions were applied in long term to a Sequencing Batch Fermentation Reactor where the highest propionate production (100120 mg COD/g TVSfed.d) was obtained at 37 degrees C and adjusting the feeding pH at 8. This was attributed to the higher hydrolysis efficiency of the cellulosic materials (up to 44%), which increased the selective growth of Propionibacterium acidopropionici in the fermentation broth up to 34%. At the same time, around 88% of the phosphorus released during the acidogenic fermentation was recovered as much as 0.15 kg of struvite per capita . year. Finally, the potential market value was preliminary estimated for the recovered materials that can triple over the conventional scenario of biogas recovery in existing municipal wastewater treatment plants. (C) 2018 Elsevier Ltd. All rights reserved.



Crutchik, D., Morales, N., VazquezPadin, J. R., & Garrido, J. M. (2017). Enhancement of struvite pellets crystallization in a fullscale plant using an industrial grade magnesium product. Water Sci. Technol., 75(3), 609–618.
Abstract: A fullscale struvite crystallization system was operated for the treatment of the centrate obtained from the sludge anaerobic digester in a municipal wastewater treatment plant. Additionally, the feasibility of an industrial grade Mg(OH) (2) as a cheap magnesium and alkali source was also investigated. The struvite crystallization plant was operated for two different periods: period I, in which an influent with low phosphate concentration (34.0 mg P . L (1)) was fed to the crystallization plant; and period II, in which an influent with higher phosphate concentration (68.0 mg P . L (1)) was used. A high efficiency of phosphorus recovery by struvite crystallization was obtained, even when the effluent treated had a high level of alkalinity. Phosphorus recovery percentage was around 77%, with a phosphate concentration in the effluent between 10.0 and 30.0 mg P .L 1. The experiments gained struvite pellets of 0.5 5.0 mm size. Moreover, the consumption of Mg(OH) (2) was estimated at 1.5 mol Mg added . mol P recovered (1). Thus, industrial grade Mg(OH) (2) can be an economical alternative as magnesium and alkali sources for struvite crystallization at industrial scale.



Crutchik, D., Rodrigues, S., Ruddle, D., & Garrido, J. M. (2018). Evaluation of a lowcost magnesium product for phosphorus recovery by struvite crystallization. J. Chem. Technol. Biotechnol., 93(4), 1012–1021.
Abstract: BACKGROUND: The development of a costeffective process of struvite crystallization requires the selection of appropriate sources of alkali and magnesium. In this study, the effectiveness of two industrial grade products, MgO and Mg(OH)(2), as magnesium and alkali sources to recover phosphorus as struvite were investigated and compared in a first set of experiments. Subsequently, the use of industrial Mg(OH)(2) was compared in two different struvite crystallization systems, an upflow fluidized bed reactor (FBR) and a continuous stirred tank reactor (CSTR) coupled to a settler tank. RESULTS: At the same operational conditions, the consumption of MgO was higher than Mg(OH)(2) consumption. Moreover, industrial Mg(OH)(2) consumption for FBR and the CSTR operation was 1.6 and 1.1 1 mol Mg added mol(1) P precipitated, respectively. This difference was caused by the high mixing intensity and the higher contact time between the Mg(OH)(2) slurry and the influent in the CSTR, favouring the conversion. CONCLUSIONS: Both industrial grade magnesium products are promising options for struvite crystallization. However, Mg(OH)(2) was more effective than the starting material, MgO, to recover phosphorus. Struvite crystallization by adding an industrial grade Mg(OH)(2) could be economically viable with regard to alternative physicochemical P removal processes using metal salts, increasing the attractiveness of this P recovery process. (C) 2017 Society of Chemical Industry.



D'Angelo, G., Di Stefano, G., Navarra, A., Nisse, N., & Suchan, K. (2015). Computing on Rings by Oblivious Robots: A Unified Approach for Different Tasks. Algorithmica, 72(4), 1055–1096.
Abstract: A set of autonomous robots have to collaborate in order to accomplish a common task in a ringtopology where neither nodes nor edges are labeled (that is, the ring is anonymous). We present a unified approach to solve three important problems: the exclusive perpetual exploration, the exclusive perpetual clearing, and the gathering problems. In the first problem, each robot aims at visiting each node infinitely often while avoiding that two robots occupy a same node (exclusivity property); in exclusive perpetual clearing (also known as graph searching), the team of robots aims at clearing the whole ring infinitely often (an edge is cleared if it is traversed by a robot or if both its endpoints are occupied); and in the gathering problem, all robots must eventually occupy the same node. We investigate these tasks in the LookComputeMove model where the robots cannot communicate but can perceive the positions of other robots. Each robot is equipped with visibility sensors and motion actuators, and it operates in asynchronous cycles. In each cycle, a robot takes a snapshot of the current global configuration (Look), then, based on the perceived configuration, takes a decision to stay idle or to move to one of its adjacent nodes (Compute), and in the latter case it eventually moves to this neighbor (Move). Moreover, robots are endowed with very weak capabilities. Namely, they are anonymous, asynchronous, oblivious, uniform (execute the same algorithm) and have no common sense of orientation. In this setting, we devise algorithms that, starting from an exclusive and rigid (i.e. aperiodic and asymmetric) configuration, solve the three above problems in anonymous ringtopologies.



Da Silva, C., Astals, S., Peces, M., Campos, J. L., & Guerrero, L. (2018). Biochemical methane potential (BMP) tests: Reducing test time by early parameter estimation. Waste Manage., 71, 19–24.
Abstract: Biochemical methane potential (BMP) test is a key analytical technique to assess the implementation and optimisation of anaerobic biotechnologies. However, this technique is characterised by long testing times (from 20 to > 100 days), which is not suitable for waste utilities, consulting companies or plants operators whose decisionmaking processes cannot be held for such a long time. This study develops a statistically robust mathematical strategy using sensitivity functions for early prediction of BMP firstorder model parameters, i.e. methane yield (B0) and kinetic constant rate (k). The minimum testing time for early parameter estimation showed a potential correlation with the k value, where (i) slowly biodegradable substrates (k <= 0.1 d(1)) have a minimum testing times of >= 15 days, (ii) moderately biodegradable substrates (0.1 < k < 0.2 d(1)) have a minimum testing times between 8 and 15 days, and (iii) rapidly biodegradable substrates (k > 0.2 d(1)) have testing times lower than 7 days. (C) 2017 Elsevier Ltd. All rights reserved.



Dauelsberg, P., Matus, J. T., Poupin, M. J., LeivaAmpuero, A., Godoy, F., Vega, A., et al. (2011). Effect of pollination and fertilization on the expression of genes related to floral transition, hormone synthesis and berry development in grapevine. J. Plant Physiol., 168(14), 1667–1674.
Abstract: In the present work, the effect of assisted fertilization on anatomical, morphological and gene expression changes occurring in carpels and during early stages of berry development in Vitis vinifera were studied. Inflorescences were emasculated before capfall, immediately manually pollinated (EP) and fruit development was compared to emasculated but nonpollinated (ENP) and selfpollinated inflorescences (NESP). The diameter of berries derived from pollinated flowers (EP and NESP) was significantly higher than from nonpollinated flowers (ENP) at 21 days after emasculation/pollination (DAE), and a rapid increase in the size of the inner mesocarp, together with the presence of an embryolike structure, were observed. The expression of gibberellin oxidases (GA200x and GA2ox), anthranilate synthase (related to auxin synthesis) and cytokinin synthase coding genes was studied to assess the relationship between hormone synthesis and early berry development, while flower patterning genes were analyzed to describe floral transition. Significant expression changes were found for hormonerelated genes, suggesting that their expression at early stages of berry development (13 DAE) is related to cell division and differentiation of mesocarp tissue at a later stage (21 DAE). Expression of hormonerelated genes also correlates with the expression of VvHB13, a gene related to mesocarp expansion, and with an increased repression of floral patterning genes (PISTILLATA and TM6), which may contribute to prevent floral transition inhibiting fruit growth before fertilization takes place. (C) 2011 Elsevier GmbH. All rights reserved.



de Figueiredo, C. M. H., de Mello, C. P., & Ortiz, C. (2000). Edge colouring reduced indifference graphs. Lect. Notes Comput. Sc., 1776, 145–153.
Abstract: The chromatic index problem – finding the minimum number of colours required for colouring the edges of a graph – is still unsolved for indifference graphs, whose vertices can be linearly ordered so that the vertices contained in the same maximal clique are consecutive in this order. Two adjacent vertices are twins if they belong to the same maximal cliques. A graph is reduced if it contains no pair of twin vertices. A graph is overfull if the total number of edges is greater than the product of the maximum degree by [n/2], where n is the number of vertices. We give a structural characterization for neighbourhoodover full indifference graphs proving that a reduced indifference graph cannot be neighbourhoodoverfull. We show that the chromatic index for all reduced indifference graphs is the maximum degree.



de Figueiredo, C. M. H., Meldanis, J., de Mello, C. P., & Ortiz, C. (2003). Decompositions for the edge colouring of reduced indifference graphs. Theor. Comput. Sci., 297(13), 145–155.
Abstract: The chromatic index problemfinding the minimum number of colours required for colouring the edges of a graphis still unsolved for indifference graphs, whose vertices can be linearly ordered so that the vertices contained in the same maximal clique are consecutive in this order. We present new positive evidence for the conjecture: every non neighbourhoodoverfull indifference graph can be edge coloured with maximum degree colours. Two adjacent vertices are twins if they belong to the same maximal cliques. A graph is reduced if it contains no pair of twin vertices. A graph is overfull if the total number of edges is greater than the product of the maximum degree by [n/2], where n is the number of vertices. We give a structural characterization for neighbourhoodoverfull indifference graphs proving that a reduced indifference graph cannot be neighbourhoodoverfull. We show that the chromatic index for all reduced indifference graphs is the maximum degree. We present two decomposition methods for edge colouring reduced indifference graphs with maximum degree colours. (C) 2002 Elsevier Science B.V. All rights reserved.



De la Iglesia, R., ValenzuelaHeredia, D., Andrade, S., Correa, J., & Gonzalez, B. (2012). Composition dynamics of epilithic intertidal bacterial communities exposed to high copper levels. FEMS Microbiol. Ecol., 79(3), 720–727.
Abstract: Copper has a dual role for organisms, both as micronutrient and toxic element. Copper mining activities have an enormous ecological impact because of the extraction process and the consequent release of coppercontaining waste materials to the environment. In northern Chile, mainly in the Chanaral coastal area, this phenomenon is clearly evident. The released waste material has caused a strong modification of the area, and copper enrichment of beaches and rocky shores has provoked a decrease in the richness and diversity of many species of macroorganisms. However, the effects that copper enrichment has on microbial (e.g. bacterial epilithic) communities associated with the rocky shore environment are poorly understood. Using a cultureindependent molecular approach, field sampling and laboratory microcosm experiments, we determined the effects of copper enrichment on bacterial communities inhabiting the rocky shore environment. Field samples showed a strong effect of copper on the structure of the natural bacterial epilithic communities, and microcosm experiments demonstrated rapid changes in bacterial community when copper is added, and reversibility of this effect within 48 h after copper is removed.

