BOUNDEDNESS OF FRACTIONAL INTEGRALS IN HARDY SPACES WITH NON-DOUBLING MEASURE—In Memory of Professor Sun Yongsheng

Let μ be a Borel measure on R^d which may be non doubling. The only condition that μ must satisfy is μ（Q） ≤ col（Q）^n for any cube Q belong to R^d with sides parallel to the coordinate axes and for some fixed n with 0 〈 n ≤ d. The purpose of this paper is to obtain a boundedness property of fractional integrals in Hardy spaces H^1（μ）.

Measurement of integral diffraction coefficients of crystals on beamline 4B7 of Beijing Synchrotron Radiation Facility

Integral diffraction coefficients of the crystal are the essential data of a crystal spectrometer which is extensively used to measure quantitative x-ray spectra of high temperature plasmas in kilo-electron-volt region. An experimental method has been developed to measure the integral diffraction coefficients of crystals on beamline 4B7 of Beijing Synchrotron Radiation Facility. The integral diffraction coefficients of several crystals including polyethylene terephthalate （PET）, thallium acid phthalate （T1AP） and rubidium acid phthalate （RAP） crystals have been measured in the x-ray energy range 2100-5600 eV and compared with the calculations of the ＇Darwin Prins＇ and the ＇Mosaic＇ models. It is shown that the integral diffraction coefficients of these crystals are between the calculations of the ＇Darwin Prins＇ and the ＇Mosaic＇ models, but more close to the ＇Darwin Prins＇ model calculations.

Emotional inference by means of Choquet integral and λ-fuzzy measurement in consideration of ambiguity of human mentality

Research on human emotions has started to address psychological aspects of human nature and has advanced to the point of designing various models that represent them quantitatively and systematically. Based on the findings, a method is suggested for emotional space formation and emotional inference that enhance the quality and maximize the reality of emotion-based personalized services. In consideration of the subjective tendencies of individuals, AHP was adopted for the quantitative evaluation of human emotions, based on which an emotional space remodeling method is suggested in reference to the emotional model of Thayer and Plutchik, which takes into account personal emotions. In addition, Sugeno fuzzy inference, fuzzy measures, and Choquet integral were adopted for emotional inference in the remodeled personalized emotional space model. Its performance was evaluated through an experiment. Fourteen cases were analyzed with 4.0 and higher evaluation value of emotions inferred, for the evaluation of emotional similarity, through the case studies of 17 kinds of emotional inference methods. Matching results per inference method in ten cases accounting for 71% are confirmed. It is also found that the remaining two cases are inferred as adjoining emotion in the same section. In this manner, the similarity of inference results is verified.

AN ENDPOINT ESTIMATE FOR THE COMMUTATORS OF SINGULAR INTEGRALS WITH NON DOUBLING MEASURES

A weak type endpoint estimate is established for commutators of the Calderon-Zygmund singular integrals on Rd with non doubling measures.

Fuzzy integral based measurement and assessment of medium and small enterprises'product innovation

A fuzzy integral based way of measurement assessment has been established by using linguistic varia-bles and combining fuzzy integral with hierarchy analysis for measurement of medium and small enterprises'product innovation. The conclusions drawn from analyses made with 20 medium and small enterprises providebases for governments to formulate applicable policies and for medium and small enterprises to enhance theirproduct innovation.

Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New Probability Measures Generalizing the Gamma-Type and Gaussian-Type Distributions

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, and where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1F1 and the hypergeometric function 1F2. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1F1 and 1F2. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (x2) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.

Monotone Set-Valued Function Defined by Set-Valued Choquet Integral

Structural characteristics and absolute continuities of monotone set-valued function defined by set- valued Choquet integral are discussed. Similar to the single-valued monotone set function, several important structural characteristics of set-valued function are defined and have been proven the same as those in the original set functions, such as null-additivity, weakly null-additivity, order continuity, strong order continuity and property（S）. A counterexample shows that order continuity and strong order continuity of the original set functions are no longer kept in a monotone set-valued function when Choquet integrably bounded assumption is abandoned. Four kinds of absolute continuities are defined for set-valued function, and all been proven valid with respect to the original set functions.

SOLVABILITY OF AN IMPLICIT FRACTIONAL INTEGRAL EQUATION VIA A MEASURE OF NONCOMPACTNESS ARGUMENT

In this paper,we study the existence of solutions to an implicit functional equation involving a fractional integral with respect to a certain function,which generalizes the Riemann-Liouville fractional integral and the Hadaniard fractional integral.We establish an existence result to such kind of equations using a generalized version of Darbo＇s theorem associated to a certain measure of nonconipactness.Some examples are presented.

Boundedness for Multilinear Commutators of Marcinkiewicz Integral on Morrey-Herz Spaces with Non Doubling Measures

In this paper, the authors establish the boundedness of multilinear commutators generated by a Marcinkiewicz integral operator and a RBMO（μ） function on homogeneous Morrey-Herz spaces with non doubling measures.

Supplier Evaluation Model Based on Fuzzy Measures and Choquet Integral

To identify interactions among evaluation criteria and describe their importance,a new identification method making use of a fuzzy measure is presented.The relative weight and interaction degree of every evaluation criteria pair are obtained by using the diamond pairwise comparison method.Based on comparison results,the maximum eigenvector method of analytic hierarchy process （AHP）,the hierarchical clustering method,and the phi（s） transformation are utilized to generate values of the fuzzy measure for each subset of the evaluation criterion set.Overall evaluation on each supplier is aggregated by Choquet integral with respect to the fuzzy measure.Finally,an illustrative example demonstrates the practical feasibility and validity of the proposed method.

Boundedness of Multilinear Singular Integrals and Their Commutators on Morrey-Herz Spaces with Non-doubling Measures

Let μ be a non-negative Radon measure on R^d which satisfies some growth conditions. The boundedness of multilinear Calderon-Zygmund singular integral operator T and its commutators with RBMO functions on Morrey-Herz spaces are obtained if T is bounded from L^1（μ）χ…χ L^1（μ） to L1/m,∞（μ）.

Monotone Measures Defined by Pan-Integral

Given a pan-space ?and a nonnegative measurable function f on measurable space (X, A), the pan-integral of f with respect to monotone measure μ and pan-operation ?determines a new monotone measure ?on?(X, A). Such the new monotone measure ?is absolutely continuous with respect to the monotone measure μ. We show that the new monotone measure preserves some important structural characteristics of the original monotone measures, such as continuity from below, subadditivity, null-additivity, weak null-additivity and (S) property. Since the pan-integral based on a pair of pan-operations ?covers the Sugeno integral (based on ) and the Shilkret integral (based on ), therefore, the previous related results for the Sugeno integral are covered by the results presented here, in the meantime, some special results related the Shilkret integral are also obtained.

Iterated Commutators for Multilinear Singular Integral Operators on Morrey Space with Non-Doubling Measures

Let μ be a non-negative Radon measure on Rd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r)) ≤ Crn?for all x∈ Rd, r > 0 and some fixed n ∈ (0,d]. This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces?.Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and 1 qj ≤ pj ∞ with 1/p = 1/p1 + ... + 1/pm and 1/q = 1/q1+ ... + 1/qm.

F-Rough Integrals and Its Measurements

F-rough integrals is defined on the basis of the dual of function one direction S-rough sets,which has dynamic characteristics.Using F-rough integrals,the concepts of expansion measurement-expansion degree and expansion ratio are given.By expansion degree and expansion ratio the changing extent can be expressed with numbers,and the recognition principle of attribute effect on function equivalences is got.

Set-Valued Stochastic Integrals with Respect to Finite Variation Processes

In a Euclidean space Rd, the Lebesgue-Stieltjes integral of set-valued stochastic processes with respect to real valued finite variation process is defined directly by employing all integrably bounded selections instead of taking the decomposable closure appearing in some existed references. We shall show that this kind of integral is measurable, continuous in t under the Hausdorff metric and L2-bounded.

ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES

In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

FOURIER TRANSFORMATION AND SINGULAR INTEGRALS ON SELF-SIMILAR MEASURE

This paper serves two purposes. One is to modify Strichartz's results with respect to the asymptotic averages of the Fourier transform of μ on , self-similar measure defined by Hutchinson. Another purpose is to consider a singular integral operator on μ and show that this op- erator is of type (p,p)(1<p<∞).

APPROXIMATION OF SPHERICAL MEANS OF MULTIPLE FOURIER INTEGRALS AND FOURIER SERIES ON SET OF FULL MEASURE

In this paper we consider the approximation for functions in some subspaces of L^2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.

THE INTEGRAL FORMULA FOR CALCULATINGTHE HAUSDORFF MEASURE OF SOME FRACTAL SETS

It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.

A NEW INTEGRATION WITH RESPECT TO A VECTOR MEASURE AND ITS PROPERTIES

We introduce a new integration with respect to a vector measure, which may be considered the generalization of a line integration with respect to coordinates in classical calculus, we disscuss its properties and investigate the relationship between an integration with respect to a vector measure and one with respect to the variation of the vector measure, which is similar to the relationship between a line integration.with respect to coordinates and one with respect to arc length.