Generalized descriptive set theory and classification theory

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.

Segimon Comas : importància i contextualització històrica d'un universitari i acadèmic del set-cents

Contextualització històrica dels inicis de l'Acadèmia de Bones Lletres al segle XVIII, i un dels seus artífexs, Segimon Comas.

Fuzzy set Theory and Quantum Chemistry

Es discuteixen breument algunes consideracions sobre l'aplicació de la Teoria delsConjunts 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 espotencien tractant-les amb les eines associades als esmentats Conjunts difusos. La funciódensitat es pren com a exemple de l'ús de distribucions de possibilitat al mateix temps queles distribucions de probabilitat quàntiques

When a data set can be considered compositional?

Traditionally, compositional data has been identified with closed data, and the simplex has been considered as the natural sample space of this kind of data. In our opinion, the emphasis on the constrained nature ofcompositional data has contributed to mask its real nature. More crucial than the constraining property of compositional data is the scale-invariant property of this kind of data. Indeed, when we are considering only few parts of a full composition we are not working with constrained data but our data are still compositional. We believe that it is necessary to give a more precisedefinition of composition. This is the aim of this oral contribution

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

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

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

Effect of basis set superposition error on the electron density of molecular complexes

The effect of basis set superposition error (BSSE) on molecular complexes is analyzed. The BSSE causes artificial delocalizations which modify the first order electron density. The mechanism of this effect is assessed for the hydrogen fluoride dimer with several basis sets. The BSSE-corrected first-order electron density is obtained using the chemical Hamiltonian approach versions of the Roothaan and Kohn-Sham equations. The corrected densities are compared to uncorrected densities based on the charge density critical points. Contour difference maps between BSSE-corrected and uncorrected densities on the molecular plane are also plotted to gain insight into the effects of BSSE correction on the electron density

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

C-H···O H-bonded complexes: how does basis set superposition error change their potential-energy surfaces?

Geometries, vibrational frequencies, and interaction energies of the CNH⋯O3 and HCCH⋯O3 complexes are calculated in a counterpoise-corrected (CP-corrected) potential-energy surface (PES) that corrects for the basis set superposition error (BSSE). Ab initio calculations are performed at the Hartree-Fock (HF) and second-order Møller-Plesset (MP2) levels, using the 6-31G(d,p) and D95++(d,p) basis sets. Interaction energies are presented including corrections for zero-point vibrational energy (ZPVE) and thermal correction to enthalpy at 298 K. The CP-corrected and conventional PES are compared; the unconnected PES obtained using the larger basis set including diffuse functions exhibits a double well shape, whereas use of the 6-31G(d,p) basis set leads to a flat single-well profile. The CP-corrected PES has always a multiple-well shape. In particular, it is shown that the CP-corrected PES using the smaller basis set is qualitatively analogous to that obtained with the larger basis sets, so the CP method becomes useful to correctly describe large systems, where the use of small basis sets may be necessary

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

How does basis set superposition error change the potential surfaces for hydrogen-bonded dimers?

We describe a simple method to automate the geometric optimization of molecular orbital calculations of supermolecules on potential surfaces that are corrected for basis set superposition error using the counterpoise (CP) method. This method is applied to the H-bonding complexes HF/HCN, HF/H2O, and HCCH/H2O using the 6-31G(d,p) and D95 + + (d,p) basis sets at both the Hartree-Fock and second-order Møller-Plesset levels. We report the interaction energies, geometries, and vibrational frequencies of these complexes on the CP-optimized surfaces; and compare them with similar values calculated using traditional methods, including the (more traditional) single point CP correction. Upon optimization on the CP-corrected surface, the interaction energies become more negative (before vibrational corrections) and the H-bonding stretching vibrations decrease in all cases. The extent of the effects vary from extremely small to quite large depending on the complex and the calculational method. The relative magnitudes of the vibrational corrections cannot be predicted from the H-bond stretching frequencies alone