A decision theory for partially consonant belief functions

Partially consonant belief functions (pcb), studied by Walley, are the only class of Dempster-Shafer belief functions that are consistent with the likelihood principle of statistics. Structurally, the set of foci of a pcb is partitioned into non-overlapping groups and within each group, foci are nested. The pcb class includes both probability function and Zadeh’s possibility function as special cases. This paper studies decision making under uncertainty described by pcb. We prove a representation theorem for preference relation over pcb lotteries to satisfy an axiomatic system that is similar in spirit to von Neumann and Morgenstern’s axioms of the linear utility theory. The closed-form expression of utility of a pcb lottery is a combination of linear utility for probabilistic lottery and two-component (binary) utility for possibilistic lottery. In our model, the uncertainty information, risk attitude and ambiguity attitude are separately represented. A tractable technique to extract ambiguity attitude from a decision maker behavior is also discussed.     

On the plausibility transformation method for translating belief function models to probability models

In this paper, we propose the plausibility transformation method for translating Dempster–Shafer (D–S) belief function models to probability models, and describe some of its properties. There are many other transformation methods used in the literature for translating belief function models to probability models. We argue that the plausibility transformation method produces probability models that are consistent with D–S semantics of belief function models, and that, in some examples, the pignistic transformation method produces results that appear to be inconsistent with Dempster’s rule of combination.     

On Jacobi functions and multiplication theorems for integrals of Bessel functions

In this paper, the product of a Jacobi polynomial and function is shown to generate the Jacobi polynomial. This type of expansion was previously known for all the classical orthogonal polynomials except the Jacobi. The result is then used to obtain generalizations to Watson's [10] multiplication theorem involving integrals of Bessel functions. The integrals considered are of the form ∝^∞₀t^1?λJ_v(tx₁) J_μ(tx₂) J_σ(tx₃) J_τ(tx₄) dt.     

On fractional integration formulae for Aleph functions

This paper is devoted to the study of a new special function, which is called, according to the symbol used to represent this function, as an Aleph function. This function is an extension of the I-function, which itself is a generalization of the well-known and familiar G- and H-functions in one variable. In this paper, a notation and complete definition of the Aleph function will be presented. Fractional integration of the Aleph functions, in which the argument of the Aleph function contains a factor t^λ(1 − t)^μ, λ, μ > 0, will be investigated. The results derived are of most general character and include many results given earlier by various authors including Kilbas [10], Kilbas and Saigo [11] and Galué [6] and others. The results obtained form the key formulae for the results on various potentially useful special functions of physical and biological sciences and technology available in the literature.     

On optimal integration algorithms for functions with a known number of extrema

Translated from Programmnoe Obespechenie i Modeli Issledovaniya Operatsii, pp. 177–185, 1986.     

Logics for belief functions on MV-algebras

In this paper we present a generalization of belief functions over fuzzy events. In particular we focus on belief functions defined in the algebraic framework of finite MV-algebras of fuzzy sets. We introduce a fuzzy modal logic to formalize reasoning with belief functions on many-valued events. We prove, among other results, that several different notions of belief functions can be characterized in a quite uniform way, just by slightly modifying the complete axiomatization of one of the modal logics involved in the definition of our formalism.     

A general theory of fuzzy plausibility measures

The purpose of this paper is to present a fuzzification of probability theory, or more precisely to give a fuzzification of plausibility measures first introduced by Shafer in 1976. Although plausibility measures include probability measures as well as possibility measures, it is a typical result of this theory that only a fuzzification of possibility measures is attainable, while a fuzzification of probability measures seems to be impossible. Moreover with regard to fuzzy plausibility measures we specify a concept of mean values and entropies, which can be considered as a direct generalization of the classical notions of mean value and entropy based upon probability measures.     

On lower bounds for integration of multivariate permutation-invariant functions

In this note we study multivariate integration for permutation-invariant functions from a certain Banach space _Ed,α$E_{d,\alpha }$ of Korobov type in the worst case setting. We present a lower error bound which particularly implies that in dimension d$d$ every cubature rule which reduces the initial error necessarily uses at least d+1$d+1$ function values. Since this holds independently of the number of permutation-invariant coordinates, this shows that the integration problem can never be strongly polynomially tractable in this setting. Our assertions generalize results due to Sloan and Wo?niakowski (1997) [3]. Moreover, for large smoothness parameters α$\alpha$ our bound cannot be improved. Finally, we extend our results to the case of permutation-invariant functions from Korobov-type spaces equipped with product weights.     

On the commutant of operators of multiplication by univalent functions

Let be a certain Banach space consisting of continuous functions defined on the open unit disk. Let be a univalent function defined on , and assume that denotes the operator of multiplication by . We characterize the structure of the operator such that . We show that for some function in . We also characterize the commutant of under certain conditions.

     

On the existence of optimal integration formulas for analytic functions

Summary In this paper we prove the existence of quadrature formulas that are optimal with respect to a Hilbert space of analytic functions solving a problem unsuccessfully attacked so far. Although we allow the formulas to be of type (2) the optimal formulas will be of the form (1).Supported in part by N.S.F. Grant G.P.-28111.Supported in part by Sonderforschungsbereich 72 at Institute for Applied Mathematics, University of Bonn and N.S.F. Grant G.P.-18609.     

A process for generating quantitative belief functions

We present a structured methodology for transforming qualitative preference relationships among propositions into appropriate numeric representations. This approach will be useful in the difficult process of knowledge acquisition from experts on the degree of belief in various propositions or the probability of the truthfulness of those propositions. The approach implicitly (through the qualitative assignments) and explicitly (through the vague interval pairwise comparisons) provides for different levels of preference relationships. Among its advantages, it permits the expert to: explore the given problem situation, using linguistic quantifiers; avoid the premature use of numeric measures; and identify input data that are inconsistent with the theory of belief functions.     

Product measures and convergence theorems for product integrals in integration theory for semigroup valued functions

In [11] Sion has investigated measures and integrals having values in uniform semigroups. In this paper a result which gives sufficient conditions for the existence of a unique product of semigroup valued measures is established. Moreover a convergence theorem for product integrals of semigroup valued functions is proved.     

Product measures and convergence theorems for product integrals in integration theory for semigroup valued functions

In [11] Sion has investigated measures and integrals having values in uniform semigroups. In this paper a result which gives sufficient conditions for the existence of a unique product of semigroup valued measures is established. Moreover a convergence theorem for product integrals of semigroup valued functions is proved.     

On Schensted's construction and the multiplication of schur functions

In this paper, certain aspects of Schensted's construction are examined and a generalization is described. From this generalization, the Littlewood-Richardson rule for the multiplication of Schur functions is derived by purely combinatorial methods.     

On the integration of Radon measures and set functions

A theory of integration is described which subsumes as special cases Bourbaki's integration of Radon measures and Caratheodory's integration of set functions.Supported partly by NSF Grant GP-20541     

On a multiplication of semigroup varieties

The role of multiplication of group varieties is well known [2], as well as the role of wreath products in such a construction [1]. In the present paper the idea of exploiting wreath products is used for defining a multiplication for semigroup varieties which differs from the operations given in [3]. In section 5 we also consider some elementary properties of the defined multiplication. Some results of the paper were announced in [9].     

On the fredholm theory of a planar problem with shift for a pair of functions

We establish necessary and sufficient conditions for the Fredholm property of a planar problem with shift and conjugation for a pair of functions and obtain a formula for the determination of its index.