Relatively Compact Sets on Abstract Wiener Space

In this note, we obtain a sufficient and necessary condition for a set in an abstract Winner space （X, H, μ） to be relatively compact in L^2（X, μ）. Meanwhile, we give a sufficient condition for relative compactness in L^P（X, μ） for p〉1. We also provide an example of Da Prato-Malliavin Nualart to show the result.

Influence of relative compaction and degree of saturation on the deformation characteristics of bentonite under freeze-thaw cycles

Bentonite,consisting of clay minerals of the montmorillonite group,has been widely used as an adsorbent and backfill material in nuclear waste disposal and groundwater remediation.It is challenging to use bentonite as a filling material in cold regions since bentonite is highly sensitive to thermal environmental changes,during which its bulk volume and microstructure change significantly.In this study,a series of one-dimensional and three-dimensional freeze-thaw tests were carried out within a closed system to investigate the influencing factors of the deformation of bentonite under freeze-thaw cycles.Results show that the initial soil water content greatly impacts bentonite's deformation during freeze-thaw cycles.For an initial higher degree of saturation(Sr),the expansion caused by the formation of ice lenses has a greater impact than the shrinkage induced by dehydration,ice-cementation,and so on.Conversely,bentonite tends to shrink at a lower degree of saturation during freezing.And the critical degree of saturation that determines bentonite's behavior of frost heave or frost shrinkage seems to be roughly 0.8.As the number of freeze-thaw cycles rises,initially uncompacted bentonite clay becomes more compacted,and initially compacted bentonite clay remains unchanged.

Fixed Point Theorems for Increasing Operators of None－continuity and None－compactness Conditions

In this paper, the fixed point theorems of composite set-valued increasing operators are given. As a corollary, the fixed point theorem for increasing operator of none-continuity and nonecompactness conditions is also given. Some relevant results are improved and generalized.

Relative Strong F Compactness and Relative Ultra-F Compactness in L-topological Spaces

In this paper, two concepts of relative compactness-the relative strong fuzzy compactness and the relative ultra-fuzzy compactness are defined in L-topological spaces for an arbitrary L-set. Properties of relative strong fuzzy sets and relative ultra-fuzzy compact sets are studied in detail and some characteristic theorems are given. Some examples are illustrated.

THE GENERALIZED EINSTEIN RELATION ON A COMPACT ORIENTABLE RIMANNIAN MANIFOLD

A generalized Einstein relation is established for a drifted Brownian motion on a compact orientable Remannian manifole. The original form of such relation comes from an explicit formula of diffusion coeffcient by using the autocorrelation function of the velocity of the Einstein fluctuation theory of the diffusion process

An Operator Arzelà-Ascoli Theorem

The author generalizes the Arzela-Ascoli theorem to the setting of matrix order unit spaces, extending the work of Antonescu-Christensen on unital C^＊-algebras. This gives an affirmative answer to a question of Antonescu and Christensen.

On Weak Well-posedness of the Nearest Point and Mutually Nearest Point Problems in Banach Spaces

Let G be a nonempty closed subset of a Banach space X.Let B(X)be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and B_(G)(X)={A∈B(X):A∩G=φ},where the closure is taken in the metric space(B(X),H).For x∈X and F∈B_(G)(X),we denote the nearest point problem inf{||x-g||:g∈G}by min(x,G)and the mutually nearest point problem inf{||f-g||:f∈ F,g∈G}by min(F,G).In this paper,parallel to well-posedness of the problems min(a:,G)and mm(F,G)which are defined by De Blasi et al.,we further introduce the weak well-posedness of the problems min(x,G)and min(F,G).Under the assumption that the Banach space X has some geometric properties,we prove a series of results on weak well-posedness of min(x,G)and min(F,G).We also give two sufficient conditions such that two classes of subsets of X are almost Chebyshev sets.