BANACH SPACE VALUED QUASI-EVENTUAL MARTINGALES

This article investigates convergence, transforms and q-square summability of banach space valued quasi-eventual martingales. Some basic results of Banach space valued martingales are improved and extended.

BMO BOUNDEDNESS FOR BANACH SPACE VALUED SINGULAR INTEGRALS

In this paper, we consider a class of Banach space valued singular integrals. The Lp boundedness of these operators has already been obtained. We shall discuss their boundedness from BMO to BMO. As applications, we get BMO boundedness for the classic g-function and the Marcinkiewicz integral. Some known results are improved.

Some Mean Convergence Theorems for the Maximum of Normed Double Sums of Banach Space Valued Random Elements

In this correspondence,we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements.Most of the results pertain to random elements which are M-dependent.We expand and improve a number of particular cases in the literature on mean convergence of random elements in Banach spaces.One of the main contributions of the paper is to simplify and improve a recent result of Li,Presnell,and Rosalsky[Journal of Mathematical Inequalities,16,117–126(2022)].A new maximal inequality for double sums of M-dependent random elements is proved which may be of independent interest.The sharpness of the results is illustrated by four examples.

The Hjek-Rnyi Inequlity For Banach Space Valued Random Variable Sequences and Its Application

In this paper we prove the following Hajek Renyi inequality:Let 0<p≤1 ,then for any Banach space B , any L p integrable B valued random variable sequence {D n,n≥1} ,any real number sequence {b n,n≥1} with 0<b n↑∞ ,any integer n≥1 ,there exits a constant C=C p>0 (only depending on p ) such thatP( sup j≥nji=1D ib j≥ε)≤Cε -p (∞j=n+1E‖D j‖ pb p j+nj=1E‖D j‖ pb p n) In the other direction,we prove some strong laws of large numbers and the integrability of the maximal functions for B valued random variable sequences by using this inequality and the Hajeck Renyi inequality we have obtained recently.Some known results are extended and improved.

A. S. Convergence of Two-Parameter Banach Space Valued Martingales and the Radon-Nikodym Property of Banach Spaces

In this paper, we prove that under the F4 condition, any L log+ L bounded two-parameter Banach space valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if the F4 condition is replaced by the weaker local F4 condition.

Endpoints of Multi-Valued Weak Contractions on the Metric Space of Partially Ordered Groups

The present work considers the endpoint in the abstract metric space. It firstly introduces the metric space of partially ordered groups and the metric space of partially ordered modules, respectively;and defines the convergence of sequences and the multi-valued weak contractions, etc., on the introduced space. And then, with the methods of functional analysis and abstract algebra, it successively establishes an endpoint theorem for the metric space of partially ordered groups and an endpoint theorem for the metric space of partially ordered modules. The contributions of this article extend the theory of cone metric space constructed by Huang and Zhang (2007) and some recent results on the fixed point and endpoint theory, such as the endpoint theorem given by Amini-Harandi (2010).

B-Implicit Contractive Conditions and Unique（Common） Fixed Points on Complex Valued Metric Spaces

A class B of complex functions is introduced and several existence theorems of unique(common) fixed points for mappings satisfying a B-implicit contraction are presented.Moreover, the existence results of common fixed points for two mappings on a nonempty set with two complex valued metrics are provided. Our outcomes generalize and improve some known results, especially, for instance, Banach contraction principle, Chatterjea-type fixed point theorem and the corresponding fixed point theorems.

Composition Operators with Linear Fractional Symbols on Vector-Valued Bergman Spaces

Let φ and ψ be linear fractional self\|maps of the unit disk D and X a separable Hilbert space. In this paper we completely characterize the weak compactness of the product operators of a composition operation C φ with another one's adjoint C * ψ on the vector\|valued Bergman space B 1(X) for forms C φC * ψ and C * ψC φ.

Real Interpolation Between B-valued Martingale Hardy Spaces

The interpolation spaces between Banach space valued martingale Hardy spaces, between Hardy and BMO spaces are identified respectively. Some results obtained here are connected closely with the convexity and smoothness of the Banach space which the martingales take values in.

Points of Coincidence and Common Fixed Points for Ⅱ-Expansive Mappings on Complex Valued Metric Spaces

Abstract. We use the two mappings satisfying II-expansive conditions on complex valued metric spaces to construct the convergent sequences and prove that the unique limit of the sequences is the point of coincidence or common fixed point of the two mappings. Also, we discuss the uniqueness of points of coincidence or common fixed points and give the existence theorems of unique fixed points. The obtained results generalize and improve the corresponding conclusions in references.

On the Strong Law of Large Numbers for Non-Independent B-Valued Random Variables

This paper investigates some conditions which imply the strong laws of large numbers for Banach space valued random variable sequences. Some generalizations of the Marcinkiewicz-Zygmund theorem and the Hoffmann-J?rgensen and Pisier theorem are obtained. Key words strong law of large numbers - Banach space valued random variable sequence - p-smoothable Banach space CLC number O 211.4 - O 211.6 Foundation item: Supported by the National Natural Science Foundation of China (10071058)Biography: Gan Shi-xin (1939-), male, Professor, research direction: martingale theory, probability limiting theory and Banach space geometry theory.

COMPLEX INTERPOLATION OF L_(p)(C,H)SPACES WITH RESPECT TO CULLEN-REGULAR

The aim of this paper is to prove a new version of the Riesz-Thorin interpolation theorem on L^(P)(C,H).In the sense of Cullen-regular,we show Hadamard’s three-lines theorem by means of the Maximum modulus principle on a symmetric slice domain.In addition,two applications of the Riesz-Thorin theorem are presented.Finally,we investigate two kinds of Calderón’s complex interpolation methods in LP(C,H).

Representation of measures of noncompactness and its applications related to an initial value problem in Banach spaces

This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).

Tongue diagnosis based on hue-saturation value color space: controlled study of tongue appearance in patients treated with percutaneous coronary intervention for coronary heart disease

Objective To analyze the characteristics of tongue imaging color parameters in patients treated with percutaneous coronary intervention(PCI)and non-PCI for coronary atherosclerotic heart disease(CHD),and to observethe effects of PCI on the tongue images of patients as a basis for the clinical diagnosis and treatment of patientswith CHD.Methods This study used a retrospective cross-sectional survey to analyze tongue photographs and medicalhistory information from 204 patients with CHD between November 2018 and July 2020.Tongue images ofeach subject were obtained using the Z-BOX Series traditional Chinese medicine(TCM)intelligent diagnosisinstruments,the SMX System 2.0 was used to transform the image data into parameters in the HSV color space,and finally the parameters of the tongue image between patients in the PCI-treated and non-PCI-treated groupsfor CHD were analyzed.Results Among the 204 patients,112 were in the non-PCI treatment group(38 men and 74 women;average age of(68.76±9.49)years),92 were in the PCI treatment group(66 men and 26 women;average age of(66.02±10.22)years).In the PCI treatment group,the H values of the middle and tip of the tongue and the overall coating of thetongue were lower(P<0.05),while the V values of the middle,tip,both sides of the tongue,the whole tongueand the overall coating of the tongue were higher(P<0.05).Conclusion The color parameters of the tongue image could reflect the physical state of patients treated withPCI,which may provide a basis for the clinical diagnosis and treatment of patients with CHD.

ON ALMOST SURE CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENT SEQUENCES

We mainly study the almost sure limiting behavior of weighted sums of the form ∑ni=1 aiXi/bn , where {Xn, n ≥ 1} is an arbitrary Banach space valued random element sequence or Banach space valued martingale difference sequence and {an, n ≥ 1} and {bn,n ≥ 1} are two sequences of positive constants. Some new strong laws of large numbers for such weighted sums are proved under mild conditions.

Some Strong Laws of Large Numbers for Blockwise Martingale Difference Sequences in Martingale Type p Banach Spaces

For a blockwise martingale difference sequence of random elements {Vn, n ≥ 1} taking values in a real separable martingale type p （1 ≤ p ≤ 2） Banach space, conditions are provided for strong laws of large numbers of the form limn→∞ Vi/gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p （1〈 p ≤2） Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.

A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such thatan↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.