On simple bases of k-valued logic

The paper derives an asymptotic iterated (two-fold) logarithm of the number of simple bases in P_k, as well as the maximal order of the functions prime for each precomplete class in P₃.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 471–482, April, 1969.The author wishes to take this opportunity to express his gratitude to S. V. Yablonskii under whose direction this paper was written.     

On the convergence of some possibilistic clustering algorithms

In this paper, an analysis of the convergence performance is conducted for a class of possibilistic clustering algorithms (PCAs) utilizing the Zangwill convergence theorem. It is shown that under certain conditions the iterative sequence generated by a PCA converges, at least along a subsequence, to either a local minimizer or a saddle point of the objective function of the algorithm. The convergence performance of more general PCAs is also discussed.     

On the equivalence between logic programming semantics and argumentation semantics

In the current paper, we re-examine the connection between formal argumentation and logic programming from the perspective of semantics. We observe that one particular translation from logic programs to instantiated argumentation (the one described by Wu, Caminada and Gabbay) is able to serve as a basis for describing various equivalences between logic programming semantics and argumentation semantics. In particular, we are able to show equivalence between regular semantics for logic programming and preferred semantics for formal argumentation. We also show that there exist logic programming semantics (L-stable semantics) that cannot be captured by any abstract argumentation semantics.     

On the uniqueness of possibilistic measure of uncertainty and information

It is demonstrated, through a series of theorems, that the U-uncertainty (introduced by Higashi and Klir in 1982) is the only possibilistic measure of uncertainty and information that satisfies possibilistic counterparts of axioms of the well established Shannon and hartley measures of uncertainty and information. Two complementary forms of the possibilistic counterparts of the probabilistic branching (or grouping) axiom, which is usually used in proofs of the uniqueness of the Shannon measure, are introduced in this paper for the first time. A one-to-one correspondence between possibility distributions and basic probabilistic assignments (introduced by Shafer in his mathematical theory of evidence) is instrumental in most proofs in this paper. The uniqueness proof is based on possibilistic formulations of axioms of symmetry, expansibility, additivity, branching, monotonicity, and normalization.     

Properties of transformation matrices between the normal bases in orders of cubic field

In this paper, the properties of transformation matrices between the normal bases in orders of algebraic number fields of degree 3 are shown.     

Approximation by radial bases and neural networks

In this paper, we study approximation by radial basis functions including Gaussian, multiquadric, and thin plate spline functions, and derive order of approximation under certain conditions. Moreover, neural networks are also constructed by wavelet recovery formula and wavelet frames.     

On the possibilistic mean value and variance of multiplication of fuzzy numbers

In this paper, we introduce the definitions of the possibilistic mean, variance and covariance of multiplication of fuzzy numbers, and show some properties of these definitions. Then, we apply these definitions to build the possibilistic models of portfolio selection under the situations involving uncertainty over the time horizon, by considering the portfolio selection problem from the point of view of possibilistic analysis. Moreover, numerical experiments with real market data indicate that our approach results in better portfolio performance.     

On the generalized Ball bases

The Ball basis was introduced for cubic polynomials by Ball, and two different generalizations for higher degree m polynomials have been called the Said–Ball and the Wang–Ball basis, respectively. In this paper, we analyze some shape preserving and stability properties of these bases. We prove that the Wang–Ball basis is strictly monotonicity preserving for all m. However, it is not geometrically convexity preserving and is not totally positive for m>3, in contrast with the Said–Ball basis. We prove that the Said–Ball basis is better conditioned than the Wang–Ball basis and we include a stable conversion between both generalized Ball bases. The Wang–Ball basis has an evaluation algorithm with linear complexity. We perform an error analysis of the evaluation algorithms of both bases and compare them with other algorithms for polynomial evaluation.     

On the equivalence of brushlet and wavelet bases

We prove that the Meyer wavelet basis and a class of brushlet systems associated with exponential type partitions of the frequency axis form a family of equivalent (unconditional) bases for the Besov and Triebel-Lizorkin function spaces. This equivalence is then used to obtain new results on nonlinear approximation with brushlets in Triebel-Lizorkin spaces.     

On k-dimensional graphs and their bases

Periodica Mathematica Hungarica -     

On additive involutions and Hamel bases

We provide an example of a discontinuous involutory additive function ${a: \mathbb{R}\to \mathbb{R}}$ such that ${a(H) \setminus H \ne \emptyset}$ for every Hamel basis ${H \subset \mathbb{R}}$ and show that, in fact, the set of all such functions is dense in the topological vector space of all additive functions from ${\mathbb{R}}$ to ${\mathbb{R}}$ with the Tychonoff topology induced by ${\mathbb{R}^{\mathbb{R}}}$ .     

On the distribution of Lachlan nonsplitting bases

 We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base (LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees w,v ≤ a, if a ≤ w or; v or; b then either a ≤ w or; b or a ≤ v or; b. In this paper we investigate the relationship between bounding and nonbounding of Lachlan nonsplitting bases and the high /low hierarchy. We prove that there is a non-Low₂ c.e. degree which bounds no Lachlan nonsplitting base. Received: 11 October 1999 / Published online: 7 May 2002