Fuzzy set Theory and Quantum Chemistry

Es discuteixen breument algunes consideracions sobre l'aplicació de la Teoria dels Conjunts difusos a la Química quàntica. Es demostra aqui que molts conceptes químics associats a la teoria són adequats per ésser connectats amb l'estructura dels Conjunts difusos. També s'explica com algunes descripcions teoriques dels observables quàntics es potencien tractant-les amb les eines associades als esmentats Conjunts difusos. La funció densitat es pren com a exemple de l'ús de distribucions de possibilitat al mateix temps que les distribucions de probabilitat quàntiques

A Homogeneous Set-Theoretical Frame for Clustering Fuzzy Relational Data

Our purpose is to provide a set-theoretical frame to clustering fuzzy relational data basically based on cardinality of the fuzzy subsets that represent objects and their complementaries, without applying any crisp property. From this perspective we define a family of fuzzy similarity indexes which includes a set of fuzzy indexes introduced by Tolias et al, and we analyze under which conditions it is defined a fuzzy proximity relation. Following an original idea due to S. Miyamoto we evaluate the similarity between objects and features by means the same mathematical procedure. Joining these concepts and methods we establish an algorithm to clustering fuzzy relational data. Finally, we present an example to make clear all the process

Quantified set inversion with applications to control

This paper describes a new reliable method, based on modal interval analysis (MIA) and set inversion (SI) techniques, for the characterization of solution sets defined by quantified constraints satisfaction problems (QCSP) over continuous domains. The presented methodology, called quantified set inversion (QSI), can be used over a wide range of engineering problems involving uncertain nonlinear models. Finally, an application on parameter identification is presented

Prediction of the bulking phenomenon in wastewater treatment plants

The control and prediction of wastewater treatment plants poses an important goal: to avoid breaking the environmental balance by always keeping the system in stable operating conditions. It is known that qualitative information — coming from microscopic examinations and subjective remarks — has a deep influence on the activated sludge process. In particular, on the total amount of effluent suspended solids, one of the measures of overall plant performance. The search for an input–output model of this variable and the prediction of sudden increases (bulking episodes) is thus a central concern to ensure the fulfillment of current discharge limitations. Unfortunately, the strong interrelation between variables, their heterogeneity and the very high amount of missing information makes the use of traditional techniques difficult, or even impossible. Through the combined use of several methods — rough set theory and artificial neural networks, mainly — reasonable prediction models are found, which also serve to show the different importance of variables and provide insight into the process dynamics

Second-order Møller-Plesset perturbation theory without basis set superposition error : II : open-shell systems

The basis set superposition error-free second-order MØller-Plesset perturbation theory of intermolecular interactions was studied. The difficulties of the counterpoise (CP) correction in open-shell systems were also discussed. The calculations were performed by a program which was used for testing the new variants of the theory. It was shown that the CP correction for the diabatic surfaces should be preferred to the adiabatic ones

Basis set superposition error-counterpoise corrected potential energy surfaces : application to hydrogen peroxide ... X (X=F-, Cl-, Br-, Li+, Na+) complexes

Møller-Plesset (MP2) and Becke-3-Lee-Yang-Parr (B3LYP) calculations have been used to compare the geometrical parameters, hydrogen-bonding properties, vibrational frequencies and relative energies for several X- and X+ hydrogen peroxide complexes. The geometries and interaction energies were corrected for the basis set superposition error (BSSE) in all the complexes (1-5), using the full counterpoise method, yielding small BSSE values for the 6-311 + G(3df,2p) basis set used. The interaction energies calculated ranged from medium to strong hydrogen-bonding systems (1-3) and strong electrostatic interactions (4 and 5). The molecular interactions have been characterized using the atoms in molecules theory (AIM), and by the analysis of the vibrational frequencies. The minima on the BSSE-counterpoise corrected potential-energy surface (PES) have been determined as described by S. Simón, M. Duran, and J. J. Dannenberg, and the results were compared with the uncorrected PES

A chemical Hamiltonian approach study of the basis set superposition error changes on electron densities and one- and two-center energy components

A comparision of the local effects of the basis set superposition error (BSSE) on the electron densities and energy components of three representative H-bonded complexes was carried out. The electron densities were obtained with Hartee-Fock and density functional theory versions of the chemical Hamiltonian approach (CHA) methodology. It was shown that the effects of the BSSE were common for all complexes studied. The electron density difference maps and the chemical energy component analysis (CECA) analysis confirmed that the local effects of the BSSE were different when diffuse functions were present in the calculations

A quantum molecular similarity analysis of changes in molecular electron density caused by basis set flotation and electric field application

Quantum molecular similarity (QMS) techniques are used to assess the response of the electron density of various small molecules to application of a static, uniform electric field. Likewise, QMS is used to analyze the changes in electron density generated by the process of floating a basis set. The results obtained show an interrelation between the floating process, the optimum geometry, and the presence of an external field. Cases involving the Le Chatelier principle are discussed, and an insight on the changes of bond critical point properties, self-similarity values and density differences is performed

Ab initio benchmark study for the oxidative addition of CH4 to Pd: importance of basis-set flexibility and polarization

To obtain a state-of-the-art benchmark potential energy surface (PES) for the archetypal oxidative addition of the methane C-H bond to the palladium atom, we have explored this PES using a hierarchical series of ab initio methods (Hartree-Fock, second-order Møller-Plesset perturbation theory, fourth-order Møller-Plesset perturbation theory with single, double and quadruple excitations, coupled cluster theory with single and double excitations (CCSD), and with triple excitations treated perturbatively [CCSD(T)]) and hybrid density functional theory using the B3LYP functional, in combination with a hierarchical series of ten Gaussian-type basis sets, up to g polarization. Relativistic effects are taken into account either through a relativistic effective core potential for palladium or through a full four-component all-electron approach. Counterpoise corrected relative energies of stationary points are converged to within 0.1-0.2 kcal/mol as a function of the basis-set size. Our best estimate of kinetic and thermodynamic parameters is -8.1 (-8.3) kcal/mol for the formation of the reactant complex, 5.8 (3.1) kcal/mol for the activation energy relative to the separate reactants, and 0.8 (-1.2) kcal/mol for the reaction energy (zero-point vibrational energy-corrected values in parentheses). This agrees well with available experimental data. Our work highlights the importance of sufficient higher angular momentum polarization functions, f and g, for correctly describing metal-d-electron correlation and, thus, for obtaining reliable relative energies. We show that standard basis sets, such as LANL2DZ+ 1f for palladium, are not sufficiently polarized for this purpose and lead to erroneous CCSD(T) results. B3LYP is associated with smaller basis set superposition errors and shows faster convergence with basis-set size but yields relative energies (in particular, a reaction barrier) that are ca. 3.5 kcal/mol higher than the corresponding CCSD(T) values

Basis set and electron correlation effects on initial convergence for vibrational nonlinear optical properties of conjugated organic molecules

The level of ab initio theory which is necessary to compute reliable values for the static and dynamic (hyper)polarizabilities of three medium size π-conjugated organic nonlinear optical (NLO) molecules is investigated. With the employment of field-induced coordinates in combination with a finite field procedure, the calculations were made possible. It is stated that to obtain reasonable values for the various individual contributions to the (hyper)polarizability, it is necessary to include electron correlation. Based on the results, the convergence of the usual perturbation treatment for vibrational anharmonicity was examined

One- and two-center physical space partitioning of the energy in the density functional theory

A conceptually new approach is introduced for the decomposition of the molecular energy calculated at the density functional theory level of theory into sum of one- and two-atomic energy components, and is realized in the "fuzzy atoms" framework. (Fuzzy atoms mean that the three-dimensional physical space is divided into atomic regions having no sharp boundaries but exhibiting a continuous transition from one to another.) The new scheme uses the new concept of "bond order density" to calculate the diatomic exchange energy components and gives them unexpectedly close to the values calculated by the exact (Hartree-Fock) exchange for the same Kohn-Sham orbitals

Estudi de mètodes de classificació borrosa i la seva aplicació a l'agrupació de zones geogràfiques en base a diverses característiques incertes

Aquesta memòria està estructurada en sis capítols amb l'objectiu final de fonamentar i desenvolupar les eines matemàtiques necessàries per a la classificació de conjunts de subconjunts borrosos. El nucli teòric del treball el formen els capítols 3, 4 i 5; els dos primers són dos capítols de caire més general, i l'últim és una aplicació dels anteriors a la classificació dels països de la Unió Europea en funció de determinades característiques borroses. En el capítol 1 s'analitzen les diferents connectives borroses posant una especial atenció en aquells aspectes que en altres capítols tindran una aplicació específica. És per aquest motiu que s'estudien les ordenacions de famílies de t-normes, donada la seva importància en la transitivitat de les relacions borroses. La verificació del principi del terç exclòs és necessària per assegurar que un conjunt significatiu de mesures borroses generalitzades, introduïdes en el capítol 3, siguin reflexives. Estudiem per a quines t-normes es verifica aquesta propietat i introduïm un nou conjunt de t-normes que verifiquen aquest principi. En el capítol 2 es fa un recorregut general per les relacions borroses centrant-nos en l'estudi de la clausura transitiva per a qualsevol t-norma, el càlcul de la qual és en molts casos fonamental per portar a terme el procés de classificació. Al final del capítol s'exposa un procediment pràctic per al càlcul d'una relació borrosa amb l'ajuda d'experts i de sèries estadístiques. El capítol 3 és un monogràfic sobre mesures borroses. El primer objectiu és relacionar les mesures (o distàncies) usualment utilitzades en les aplicacions borroses amb les mesures conjuntistes crisp. Es tracta d'un enfocament diferent del tradicional enfocament geomètric. El principal resultat és la introducció d'una família parametritzada de mesures que verifiquen unes propietats de caràcter conjuntista prou satisfactòries. L'estudi de la verificació del principi del terç exclòs té aquí la seva aplicació sobre la reflexivitat d'aquestes mesures, que són estudiades amb una certa profunditat en alguns casos particulars. El capítol 4 és, d'entrada, un repàs dels principals resultats i mètodes borrosos per a la classificació dels elements d'un mateix conjunt de subconjunts borrosos. És aquí on s'apliquen els resultats sobre les ordenacions de les famílies de t-normes i t-conormes estudiades en el capítol 1. S'introdueix un nou mètode de clusterització, canviant la matriu de la relació borrosa cada vegada que s'obté un nou clúster. Aquest mètode permet homogeneïtzar la metodologia del càlcul de la relació borrosa amb el mètode de clusterització. El capítol 5 tracta sobre l'agrupació d'objectes de diferent naturalesa; és a dir, subconjunts borrosos que pertanyen a diferents conjunts. Aquesta teoria ja ha estat desenvolupada en el cas binari; aquí, el que es presenta és la seva generalització al cas n-ari. Més endavant s'estudien certs aspectes de les projeccions de la relació sobre un cert espai i el recíproc, l'estudi de cilindres de relacions predeterminades. Una aplicació sobre l'agrupació de les comarques gironines en funció de certes variables borroses es presenta al final del capítol. L'últim capítol és eminentment pràctic, ja que s'aplica allò estudiat principalment en els capítols 3 i 4 a la classificació dels països de la Unió Europea en funció de determinades característiques borroses. Per tal de fer previsions per a anys venidors s'han utilitzat sèries temporals i xarxes neuronals. S'han emprat diverses mesures i mètodes de clusterització per tal de poder comparar els diversos dendogrames que resulten del procés de clusterització. Finalment, als annexos es poden consultar les sèries estadístiques utilitzades, la seva extrapolació, els càlculs per a la construcció de les matrius de les relacions borroses, les matrius de mesura i les seves clausures.

Implementation and application of basis set superposition error-correction schemes to the theoretical modeling of weak intermolecular interactions

This thesis deals with the so-called Basis Set Superposition Error (BSSE) from both a methodological and a practical point of view. The purpose of the present thesis is twofold: (a) to contribute step ahead in the correct characterization of weakly bound complexes and, (b) to shed light the understanding of the actual implications of the basis set extension effects in the ab intio calculations and contribute to the BSSE debate. The existing BSSE-correction procedures are deeply analyzed, compared, validated and, if necessary, improved. A new interpretation of the counterpoise (CP) method is used in order to define counterpoise-corrected descriptions of the molecular complexes. This novel point of view allows for a study of the BSSE-effects not only in the interaction energy but also on the potential energy surface and, in general, in any property derived from the molecular energy and its derivatives A program has been developed for the calculation of CP-corrected geometry optimizations and vibrational frequencies, also using several counterpoise schemes for the case of molecular clusters. The method has also been implemented in Gaussian98 revA10 package. The Chemical Hamiltonian Approach (CHA) methodology has been also implemented at the RHF and UHF levels of theory for an arbitrary number interacting systems using an algorithm based on block-diagonal matrices. Along with the methodological development, the effects of the BSSE on the properties of molecular complexes have been discussed in detail. The CP and CHA methodologies are used for the determination of BSSE-corrected molecular complexes properties related to the Potential Energy Surfaces and molecular wavefunction, respectively. First, the behaviour of both BSSE-correction schemes are systematically compared at different levels of theory and basis sets for a number of hydrogen-bonded complexes. The Complete Basis Set (CBS) limit of both uncorrected and CP-corrected molecular properties like stabilization energies and intermolecular distances has also been determined, showing the capital importance of the BSSE correction. Several controversial topics of the BSSE correction are addressed as well. The application of the counterpoise method is applied to internal rotational barriers. The importance of the nuclear relaxation term is also pointed out. The viability of the CP method for dealing with charged complexes and the BSSE effects on the double-well PES blue-shifted hydrogen bonds is also studied in detail. In the case of the molecular clusters the effect of high-order BSSE effects introduced with the hierarchical counterpoise scheme is also determined. The effect of the BSSE on the electron density-related properties is also addressed. The first-order electron density obtained with the CHA/F and CHA/DFT methodologies was used to assess, both graphically and numerically, the redistribution of the charge density upon BSSE-correction. Several tools like the Atoms in Molecules topologycal analysis, density difference maps, Quantum Molecular Similarity, and Chemical Energy Component Analysis were used to deeply analyze, for the first time, the BSSE effects on the electron density of several hydrogen bonded complexes of increasing size. The indirect effect of the BSSE on intermolecular perturbation theory results is also pointed out It is shown that for a BSSE-free SAPT study of hydrogen fluoride clusters, the use of a counterpoise-corrected PES is essential in order to determine the proper molecular geometry to perform the SAPT analysis.

Modelling, control and supervision for a class of hybrid systems

The aim of this thesis is to narrow the gap between two different control techniques: the continuous control and the discrete event control techniques DES. This gap can be reduced by the study of Hybrid systems, and by interpreting as Hybrid systems the majority of large-scale systems. In particular, when looking deeply into a process, it is often possible to identify interaction between discrete and continuous signals. Hybrid systems are systems that have both continuous, and discrete signals. Continuous signals are generally supposed continuous and differentiable in time, since discrete signals are neither continuous nor differentiable in time due to their abrupt changes in time. Continuous signals often represent the measure of natural physical magnitudes such as temperature, pressure etc. The discrete signals are normally artificial signals, operated by human artefacts as current, voltage, light etc. Typical processes modelled as Hybrid systems are production systems, chemical process, or continuos production when time and continuous measures interacts with the transport, and stock inventory system. Complex systems as manufacturing lines are hybrid in a global sense. They can be decomposed into several subsystems, and their links. Another motivation for the study of Hybrid systems is the tools developed by other research domains. These tools benefit from the use of temporal logic for the analysis of several properties of Hybrid systems model, and use it to design systems and controllers, which satisfies physical or imposed restrictions. This thesis is focused in particular types of systems with discrete and continuous signals in interaction. That can be modelled hard non-linealities, such as hysteresis, jumps in the state, limit cycles, etc. and their possible non-deterministic future behaviour expressed by an interpretable model description. The Hybrid systems treated in this work are systems with several discrete states, always less than thirty states (it can arrive to NP hard problem), and continuous dynamics evolving with expression: with Ki ¡ Rn constant vectors or matrices for X components vector. In several states the continuous evolution can be several of them Ki = 0. In this formulation, the mathematics can express Time invariant linear system. By the use of this expression for a local part, the combination of several local linear models is possible to represent non-linear systems. And with the interaction with discrete events of the system the model can compose non-linear Hybrid systems. Especially multistage processes with high continuous dynamics are well represented by the proposed methodology. Sate vectors with more than two components, as third order models or higher is well approximated by the proposed approximation. Flexible belt transmission, chemical reactions with initial start-up and mobile robots with important friction are several physical systems, which profits from the benefits of proposed methodology (accuracy). The motivation of this thesis is to obtain a solution that can control and drive the Hybrid systems from the origin or starting point to the goal. How to obtain this solution, and which is the best solution in terms of one cost function subject to the physical restrictions and control actions is analysed. Hybrid systems that have several possible states, different ways to drive the system to the goal and different continuous control signals are problems that motivate this research. The requirements of the system on which we work is: a model that can represent the behaviour of the non-linear systems, and that possibilities the prediction of possible future behaviour for the model, in order to apply an supervisor which decides the optimal and secure action to drive the system toward the goal. Specific problems can be determined by the use of this kind of hybrid models are: - The unity of order. - Control the system along a reachable path. - Control the system in a safe path. - Optimise the cost function. - Modularity of control The proposed model solves the specified problems in the switching models problem, the initial condition calculus and the unity of the order models. Continuous and discrete phenomena are represented in Linear hybrid models, defined with defined eighth-tuple parameters to model different types of hybrid phenomena. Applying a transformation over the state vector : for LTI system we obtain from a two-dimensional SS a single parameter, alpha, which still maintains the dynamical information. Combining this parameter with the system output, a complete description of the system is obtained in a form of a graph in polar representation. Using Tagaki-Sugeno type III is a fuzzy model which include linear time invariant LTI models for each local model, the fuzzyfication of different LTI local model gives as a result a non-linear time invariant model. In our case the output and the alpha measure govern the membership function. Hybrid systems control is a huge task, the processes need to be guided from the Starting point to the desired End point, passing a through of different specific states and points in the trajectory. The system can be structured in different levels of abstraction and the control in three layers for the Hybrid systems from planning the process to produce the actions, these are the planning, the process and control layer. In this case the algorithms will be applied to robotics ¡V a domain where improvements are well accepted ¡V it is expected to find a simple repetitive processes for which the extra effort in complexity can be compensated by some cost reductions. It may be also interesting to implement some control optimisation to processes such as fuel injection, DC-DC converters etc. In order to apply the RW theory of discrete event systems on a Hybrid system, we must abstract the continuous signals and to project the events generated for these signals, to obtain new sets of observable and controllable events. Ramadge & Wonham¡¦s theory along with the TCT software give a Controllable Sublanguage of the legal language generated for a Discrete Event System (DES). Continuous abstraction transforms predicates over continuous variables into controllable or uncontrollable events, and modifies the set of uncontrollable, controllable observable and unobservable events. Continuous signals produce into the system virtual events, when this crosses the bound limits. If this event is deterministic, they can be projected. It is necessary to determine the controllability of this event, in order to assign this to the corresponding set, , controllable, uncontrollable, observable and unobservable set of events. Find optimal trajectories in order to minimise some cost function is the goal of the modelling procedure. Mathematical model for the system allows the user to apply mathematical techniques over this expression. These possibilities are, to minimise a specific cost function, to obtain optimal controllers and to approximate a specific trajectory. The combination of the Dynamic Programming with Bellman Principle of optimality, give us the procedure to solve the minimum time trajectory for Hybrid systems. The problem is greater when there exists interaction between adjacent states. In Hybrid systems the problem is to determine the partial set points to be applied at the local models. Optimal controller can be implemented in each local model in order to assure the minimisation of the local costs. The solution of this problem needs to give us the trajectory to follow the system. Trajectory marked by a set of set points to force the system to passing over them. Several ways are possible to drive the system from the Starting point Xi to the End point Xf. Different ways are interesting in: dynamic sense, minimum states, approximation at set points, etc. These ways need to be safe and viable and RchW. And only one of them must to be applied, normally the best, which minimises the proposed cost function. A Reachable Way, this means the controllable way and safe, will be evaluated in order to obtain which one minimises the cost function. Contribution of this work is a complete framework to work with the majority Hybrid systems, the procedures to model, control and supervise are defined and explained and its use is demonstrated. Also explained is the procedure to model the systems to be analysed for automatic verification. Great improvements were obtained by using this methodology in comparison to using other piecewise linear approximations. It is demonstrated in particular cases this methodology can provide best approximation. The most important contribution of this work, is the Alpha approximation for non-linear systems with high dynamics While this kind of process is not typical, but in this case the Alpha approximation is the best linear approximation to use, and give a compact representation.