Interpretability assessment of fuzzy knowledge bases: A cointension based approach

Computing with words (CWW) relies on linguistic representation of knowledge that is processed by operating at the semantical level defined through fuzzy sets. Linguistic representation of knowledge is a major issue when fuzzy rule based models are acquired from data by some form of empirical learning. Indeed, these models are often requested to exhibit interpretability, which is normally evaluated in terms of structural features, such as rule complexity, properties on fuzzy sets and partitions. In this paper we propose a different approach for evaluating interpretability that is based on the notion of cointension. The interpretability of a fuzzy rule-based model is measured in terms of cointension degree between the explicit semantics, defined by the formal parameter settings of the model, and the implicit semantics conveyed to the reader by the linguistic representation of knowledge. Implicit semantics calls for a representation of user’s knowledge which is difficult to externalise. Nevertheless, we identify a set of properties - which we call “logical view” - that is expected to hold in the implicit semantics and is used in our approach to evaluate the cointension between explicit and implicit semantics. In practice, a new fuzzy rule base is obtained by minimising the fuzzy rule base through logical properties. Semantic comparison is made by evaluating the performances of the two rule bases, which are supposed to be similar when the two semantics are almost equivalent. If this is the case, we deduce that the logical view is applicable to the model, which can be tagged as interpretable from the cointension viewpoint. These ideas are then used to define a strategy for assessing interpretability of fuzzy rule-based classifiers (FRBCs). The strategy has been evaluated on a set of pre-existent FRBCs, acquired by different learning processes from a well-known benchmark dataset. Our analysis highlighted that some of them are not cointensive with user’s knowledge, hence their linguistic representation is not appropriate, even though they can be tagged as interpretable from a structural point of view.     

MnO x /ZrO2 gel-derived materials for hydrogen peroxide decomposition

Manganese-yttrium-zirconium mixed oxide nanocomposites with three different Mn loadings (5, 15 and 30 wt%) were prepared by sol–gel synthesis. Amorphous xerogels were obtained for each composition. Their structural evolution with the temperature and textural properties were examined by thermogravimetry/differential thermal analysis, X-ray diffraction, diffuse reflectance UV–vis spectroscopy and N₂ adsorption isotherms. Mesoporous materials with high surface area values (70–100 m² g⁻¹) were obtained by annealing in air at 550 °C. They are amorphous or contain nanocrystals of the tetragonal ZrO₂ phase (T-ZrO₂) depending on the Mn amount and exhibit Mn species with oxidation state higher than 2 as confirmed by temperature programmed reduction experiments. T-ZrO₂ is the only crystallizing phase at 700 °C while the monoclinic polymorph and Mn₃O₄ start to appear only after a prolonged annealing at 1,000 °C. The samples annealed at 550 °C were studied as catalysts for H₂O₂ decomposition in liquid phase. Their catalytic activity was higher than that of previously studied Mn/Zr oxide systems prepared by impregnation. Catalytic data were described by a rate equation of Langmuir type. The decrease of catalytic activity with time was related to dissolution of a limited fraction (up to 15%) of Mn into the H₂O₂/H₂O solution.     

Time Decay of Scaling Critical Electromagnetic Schrödinger Flows

We obtain a representation formula for solutions to Schrödinger equations with a class of homogeneous, scaling-critical electromagnetic potentials. As a consequence, we prove the sharp ${L^1 \to L^\infty}$ time decay estimate for the 3D-inverse square and the 2D-Aharonov–Bohm potentials.     

On the lack of dispersion for a class of magnetic Dirac flows

We show that global Strichartz estimates for magnetic Dirac operators generally fail, if the potentials do not decay fast enough at infinity. In order to prove this, we construct some explicit examples of homogeneous magnetic potentials with less than Coulomb decay, that is, with homogeneity-degree more than ?1, such that the magnetic field points to a fixed direction, which does not depend on ${x \in \mathbb{R}^{3}}$ .     

Relativistic Hardy Inequalities in Magnetic Fields

We deal with Dirac operators with external homogeneous magnetic fields. Hardy-type inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality with the same best constant as in the free case. This leaves naturally open an interesting question whether there exist magnetic fields for which a Hardy inequality with a better constant than the usual one, in connection with the well known diamagnetic phenomenon arising in non-relativistic models.     

The Theory of Individual Based Discrete-Time Processes

A general theory is developed to study individual based models which are discrete in time. We begin by constructing a Markov chain model that converges to a one-dimensional map in the infinite population limit. Stochastic fluctuations are hence intrinsic to the system and can induce qualitative changes to the dynamics predicted from the deterministic map. From the Chapman–Kolmogorov equation for the discrete-time Markov process, we derive the analogues of the Fokker–Planck equation and the Langevin equation, which are routinely employed for continuous time processes. In particular, a stochastic difference equation is derived which accurately reproduces the results found from the Markov chain model. Stochastic corrections to the deterministic map can be quantified by linearizing the fluctuations around the attractor of the map. The proposed scheme is tested on stochastic models which have the logistic and Ricker maps as their deterministic limits.     

Structure and crystallization behavior of mixed potassium-lithium niobiosilicate glasses

Potassium-lithium niobiosilicate (KLiNS) glasses with a composition of (27 ? x)K₂O · xLi₂O · 27Nb₂O₅ · 46SiO₂ (x = 0, 3, 12 and 20) have been synthesized by a melt-quenching method. The glass structure and devitrification behavior have been studied by Raman spectroscopy, DTA, and XRD. By increasing the lithium content, less distorted niobium octahedra increase, indicating a niobium clustering. This change strongly affects the crystallization behavior. In the glasses x = 0 and x = 3, just above T_g, only nanocrystals of an unidentified phase are formed, while for x = 12 and x = 20 potassium lithium niobate (KLN) solid solutions with tetragonal tungsten–bronze structure crystallize by bulk nucleation. In these glasses, LiNbO₃ crystallizes at higher temperature by surface nuclei. Ultimately, it is possible to produce nanostructured glasses based on KLN nanocrystals, by partial replacement of K by Li.     

Short-time dynamics in the presence of wave–particles interactions: A perturbative approach

The self-consistent interaction between a beam of charged particles and a wave is considered, within a Vlasov picture. The model is discussed with reference to the case of a Free Electron Laser. Starting with a spatially bunched water-bag distribution, we derive, via perturbative methods, closed analytical expressions for the time evolution of the main macroscopic observables. Predictions of the theory are shown to agree with direct numerical simulations.     

Spectral control for ecological stability

A system made up of N interacting species is considered. Self-reaction terms are assumed of the logistic type. Pairwise interactions take place among species according to different modalities, thus yielding a complex asymmetric disordered graph. A mathematical procedure is introduced and tested to stabilise the ecosystem via an ad hoc rewiring of the underlying couplings. The method implements minimal modifications to the spectrum of the Jacobian matrix which sets the stability of the fixed point and traces these changes back to species–species interactions. Resilience of the equilibrium state appear to be favoured by predator-prey interactions.