Approximation Theoretic Aspects of Probabilistic Representations for Bi-continuous C-semigroups

By means of Riemann-Stieltjes stochastic process, moment-generating functions and operator-valued mathematical expectation, the problem of probabilistic approximation for bi-continuous C-semigroups is studied and the general probabilistic approximation of exponential formulas and the generation theorems are given.

EXPONENTIALLY BOUNDED C-SEMIGROUP AND INTEGRATED SEMIGROUP WITH NONDENSELY DEFINED GENERATORS I: APPROXIMATION:

In this paper, by Laplace transform version of the Trotter-Kato approximation theorem and the integrated C-semigroup introduced by Myadera, the authors obtained some Trotter-Kato approximation theorems on exponentially bounded C-semigroups, where the range of C (and so the domain of the generator) may not be dense. The authors deduced the corresponding results on exponentially bounded integrated semigroups with nondensely generators. The results of this paper extended and perfected the results given by Lizama, Park and Zheng.

ADDITIVE PERTURBATIONS OF LOCAL C-SEMIGROUPS

In this paper, we apply the contraction mapping theorem to establish some bounded and unbounded additive perturbation theorems concerning local C-semigroups. Some growth conditions of perturbations of local C-semigroups axe also established.

MULTIPLICATIVE PERTURBATIONS OF LOCAL α-TIMES INTEGRATED C-SEMIGROUPS

We establish some left and right multiplicative perturbations of a local α-timesintegrated C-semigroup S（·） on a complex Banach space X with non-densely defined genera- tor, which can be applied to obtain some additive perturbation results concerning S（·）. Some growth conditions of the perturbations of S（·） are also established.

LOCAL C-SEMIGROUPS AND LOCAL C-COSINE FUNCTIONS

This paper is concerned with the local C-semigroups and local C-cosine functions without the assumption that the image of C is dense in a Banach space X, First, the sufficient and necessary conditions for a local C-semigroup and a C-cosine function to be the restriction of a global C-semigroup and a global C-cosine function to an interval are given, respectively, Secondly, it is characterized for a closed operator to be the generator of a local C-semigroup and a local C-cosine function, respectively.

SUBGENERATORS OF C-SEMIGROUPS ON l^2

This paper gives some results about the subgenerators of special C-semigroups on l^2, partly answers the problem mentioned in [1].

THE EXPONENTIAL-TYPE REPRESENTATION FORMULAS FOR C-SEMIGROUPS

Some exponential type representation formulas for C-semigroups are given in Banach space. Moreover, we obtain a corresponding Voronovskaja - type asymptotic formula.

Exponentially Bounded C-semigroup and the Cauchy Initial Value Problems in Complete Random Normed Modules

In this paper,the notion of C-semigroup of continuous module homomorphisms on a complete random normal(RN)module is introduced and investigated.The existence and uniqueness of solution to the Cauchy problem with respect to exponentially bounded C-semigroups of continuous module homomorphisms in a complete RN module are established.

HOLOMORPHIC INTERGATED MILD C-EXISTENCE FAMILIES

By defined holomorphic n-times integrated mild C-existence families and Csemigroups, their relationship with holomorphic mild C1-exustence families and holomorphic C1 -semigroups is discussed, respectively. For exponentially bounded cases, this papergives several Hille-Yosida type conditions for an operator to have (or generate) one of thesefamilies of operators and generalize the corresponding results in [1], [2] and [3]. These families by the holomorphic solvability of the abstract Cauchy problem is also characterized.

Stability of a Class of Linear Systems on Hilbert Space

Consider the linear control systems x′(t)=Ax(t)+Bu(t)(t>0), x(0)=x_0 , where A is the generator of an exponentially stable C-semigroup on a Hilbert space X, B is a bounded operator from the Hilbert space Y to X. In the condition that the resolvent set A is not empty and the range of C is dense in X, we obtain that the extended controllability map is the unique self-adjoint solution to the Lyapunov equation. Moreover, sufficient conditions for asymptotically stability of C-semigroup are given.

Structure of the weakly left C-semigroups

Left (right) C-semigroups are the regular semigroups satisfying the following condition:eSSe (SeeS) for every e∈E (the set of idempotents of S). Refs. [1, 2] give the left(right) semi-spined product structure and the left (right) △-product structure of a left(right) C-semigroup, respectively. It is easy to see that the above condition defining a reg-ular semigroup S to be a left (right) C-semigroup may be replaced by the following condi-tion:

A Perturbation Theorem for Exponentially Bounded C-semigroups

Let (X;‖ ‖) be a Banach space. By B(X) we denote the set of all bounded linear operators in X. Let C∈B(X) be injective.A strongly continuous family of bounded operators {S(t); t≥0} is called an exponentially bounded C-semigroup (hereinafter abbrevi-

The generators of exponentially bounded C-semigroups and theHille-Yosida spaces

Let（X,||·||）be a Banach space and let C be an injective bounded linear operator inX.A strongly continuous family of bounded linear operators{S（t）;t≥0}is called anexponentially bounded C-semigroup（hereinafter abbreviated to C-semigroup）on X,if S（0）=C,S（t）S（s）=S（t+s）C,t,s≥0,and ||S（t）||≤Meat,t≥0.

Bright-white hydrogels for on-demand passive cooling

Passive cooling permits thermal management of near-zero energy consumption and low CO_(2)emissions.Despite significant progress of passive radiative coolers,comfortable and steady temperatures can hardly be achieved due to their inadequate daytime cooling power(below 0.2 k W m^(-2))yet over-cooling at night.Here,we provide a bright-white hydrogel that enables ondemand passive cooling by virtue of adaptive water evaporation and high solar reflectance up to 86.1%.Notably,theoretical cooling power determined by the evaporating rate can reach 1.25 k W m^(-2)in daytime but decreases dramatically at night.Hence sub-ambient temperature reduction of 11-13℃at noon yet nearly none at night are realized,with the diurnal temperature difference narrowed significantly.Moreover,effective cooling using colored hydrogels,and transition from evaporative cooling to solar heating have been demonstrated.This novel evaporative cooling approach will pave the way for smart passive coolers of high efficiency,colorful appearance,and low cost.

Polymerization-induced microphase separation of polymer-polyoxometalate nanocomposites for anhydrous solid state electrolytes

Solid-state electrolytes(SSEs)with high ionic conductivity,mechanical stability,and high thermal stability,as well as the stringent requirement of application in high-temperature fuel cells and lithium-ion batteries is receiving increasing attention.Polymer nanocomposites(PNCs),combining the advantages of inorganic materials with those of polymeric materials,offer numerous opportunities for SSEs design.In this work,we report a facile and general one-pot approach based on polymerization-induced microphase separation(PIMS)to generate PNCs with bi-continuous microphases.This synthetic strategy transforms a homogeneous liquid precursor consisting of polyoxometalates(POMs,H_(3)PW_(12)O_(40),Li_(7)[V_(15)O_(36)(CO_(3))]),poly(ethylene glycol)(PEG)macro-chain-transfer agent,styrene and divinylbenzene monomers,into a robust and transparent monolith.The resulting POMs are uniformly dispersed in the PEG block(PEG/POM)to form a conducting pathway that successfully realizes the effective transfer of protons and lithium ions,while the highly cross-linked polystyrene domains(P(S-co-DVB))as mechanical support provide outstanding mechanical properties and thermal stability.As the POM loading ratio up to 35 wt%,the proton conductivity of nanocomposite reaches as high as 5.99×10^(-4) S/cm at 100℃ in anhydrous environment,which effectively promotes proton transfer under extreme environments.This study broadens the application of fuel cells and lithium-ion batteries in extreme environments.

Effect of Microdomain Structure on the Mechanical Behavior of Binary Blends

The relationship between microphase structure and mechanical response of the binary blends consisting of polystyrene-block-polyisoprene-block-polystyrene copolymer and low molecular weight polystyrene has been investigated. Low molecular weight polystyrene was chosen to obtain uniformly solubilized nano-blends without macrophase separation. The specimens were solution-cast by adding different amounts of homo-polystyrene to acquire different microphase structures. Small angle X-ray scattering（SAXS）, transmission electron microscopy（TEM） and atom force microscopy（AFM） have been used to study the microdomain and grain structure. It is observed that the structural changes in d-spacing and grain size on account of different amounts of polystyrene alter the mechanical behavior in both monotonic tensile and cyclic tests. The elastic and the Mullins effects are strongly sensitive to the changes in d-spacing and grain sizes. Moreover, the sample with bi-continuous structure shows the largest tensile strength and Mullins effect. In addition, the Mooneye-Rivlin phenomenological model was used to evaluate and explore the relationship between the polymer topological networks and the rubber elasticity of these styrenic nano-blends.

Entire vectors for unbounded normal operators and abstract Cauchy problems

LET A be a closed operator on a Banach space X, and C~∞ (A)=∩_n~∞=1,D(A^n). The C~∞ vec-tor x is an entire vector for A if sum from n=0 to ∞ (t^n)/(n!)||A^n x||【∞ (1)for all t】0. We will write ε(A) for the set of all entire vectors for A. It is well known that the set of entire vectors for a self-adjoint operator is dense. Inthis note, we generalize this result to the situation of (unbounded) normal operators. We

C-Wellposedness of the Complete Second Order Abstract Cauchy Problem and Applications

In this paper,we first give a sufficient and necessary condition for M= to generate an exponentially bounded =-semigroup and discuss its relations to the C-wellposedness of the complete second order abstract Cauchy problem （（ACP2） for short） in some sense.Then we use these results and those in [1] to discuss the C-（exponential） wellposedness of a kind of （ACP2） with application backgrounds,and develop the results in [2].

THE ABSTRACT CAUCHY PROBLEM AND A GENERALIZATION OF THE LUMER-PHILLIPS THEOREM

For injective, bounded operator C on a Banach space X , the author defines the C -dissipative operator, and then gives Lumer-Phillips characterizations of the generators of quasi-contractive C -semigroups, where a C -semigroup T(·) is quasi-contractive if ‖T(t)x‖‖Cx‖ for all t0 and x∈X . This kind of generators guarantee that the associate abstract Cauchy problem u′(t,x)=Au(t,x) has a unique nonincreasing solution when the initial data is in C(D(A)) (here D(A) is the domain of A ). Also, the generators of quasi isometric C -semigroups are characterized.