Contribution to application of energy measure of fuzzy sets

The works of De Luca & Termini continued by, for example, Knopfmacher, Loo and Gottwald, are the most important on the topic of determination of measures of fuzzy sets. The matter is to evaluate how fuzzy a fuzzy set is. There are two general concepts of measures of fuzzy set, i.e. entropy and energy measures.We show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations.Applications of the use of energy measure discussed in detail include decision making, fuzzy process control and prediction in fuzzy systems.     

Fuzzy points and local properties of fuzzy topological spaces

We prove that three theorems of C.K. Wong on local properties of fuzzy topology are wrong, thereby we discuss his notions of fuzzy point, C₁, separability, and local compactness.     

Fast and Parallel Interval Arithmetic

Infimum-supremum interval arithmetic is widely used because of ease of implementation and narrow results. In this note we show that the overestimation of midpoint-radius interval arithmetic compared to power set operations is uniformly bounded by a factor 1.5 in radius. This is true for the four basic operations as well as for vector and matrix operations, over real and over complex numbers. Moreover, we describe an implementation of midpoint-radius interval arithmetic entirely using BLAS. Therefore, in particular, matrix operations are very fast on almost any computer, with minimal effort for the implementation. Especially, with the new definition it is seemingly the first time that full advantage can be taken of the speed of vector and parallel architectures. The algorithms have been implemented in the Matlab interval toolbox INTLAB.     

Ill-Conditionedness Need not be Componentwise Near to Ill-Posedness for Least Squares Problems

The condition number of a problem measures the sensitivity of the answer to small changes in the input, where small refers to some distance measure. A problem is called ill-conditioned if the condition number is large, and it is called ill-posed if the condition number is infinity. It is known that for many problems the (normwise) distance to the nearest ill-posed problem is proportional to the reciprocal of the condition number. Recently it has been shown that for linear systems and matrix inversion this is also true for componentwise distances. In this note we show that this is no longer true for least squares problems and other problems involving rectangular matrices. Problems are identified which are arbitrarily ill-conditioned (in a componentwise sense) whereas any componentwise relative perturbation less than 1 cannot produce an ill-posed problem. Bounds are given using additional information on the matrix.     

Scaling of the Irreducible SO(3)-Invariants of Velocity Correlations in Turbulence

The scaling behavior of the SO(3) irreducible amplitudes $d_n^1 (r)$ of velocity structure tensors is numerically examined for Navier–Stokes turbulence. Here, l characterizes the irreducible representation by the index of the corresponding Legendre polynomial, and n denotes the tensorial rank, i.e., the order of the moment. For moments of different order n but with the same representation index l extended self-similarity (ESS) towards large scales is found. Intermittency seems to increase with l. We estimate that a crossover behavior between different inertial subrange scaling regimes in the longitudinal and transversal structure functions will hardly be detectable for achievable Reynolds numbers.     

Cuspidality and the growth of Fourier coefficients: small weights

We characterize Siegel cusp forms in the space of Siegel modular forms of small weight \(k \ge n+4\) on the congruence subgroups \(\Gamma ^n_0(N)\) of any degree n and any level N, by a suitable growth of their Fourier coefficients (e.g., by the well known Hecke bound) at any one of the cusps. For this, we use the formalism of Jacobi forms and the ‘Witt-operator’ on modular forms.