The Loeb Space of Denumerable Infinite Dimensional Probability Product Measure Spaces

Let {(Xi, Si, μi) : i ℃ N} be a sequence of probability measure spaces and (*Xi, L(*Si), L(*μi)) be the Loeb measure space with respect to (Xi, Si, μi) for i ℃ N. Let X =× Xi, S = ×Si,μ = ×μi. We prove that × L(*Si) CL(*S) and in embedding meaning.

OBSTACLE PROBLEMS ON RCD(K,N)METRIC MEASURE SPACES

In this paper,we solve the obstacle problems on metric measure spaces with generalized Ricci lower bounds.We show the existence and Lipschitz continuity of the solutions,and then we establish some regularities of the free boundaries.

Multilinear Commutators of θ-Type Calderón-Zygmund Operators on Non-Homogeneous Metric Measure Spaces

In this paper, the boundedness in Lebesgue spaces of commutators and multilinear commutators generated by θ-type Calderon-Zygmund operators with RBMO（μ） functions on non-homogeneous metric measure spaces is obtained.

Hardy spaces H^p over non-homogeneous metric measure spaces and their applications

Let(X,d,)be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions.Let ρ∈(1,∞),0<p≤1≤q≤∞,p≠q,γ∈[1,∞)and ∈∈(0,∞).In this paper,the authors introduce the atomic Hardy space Hp,q,γ atb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ∈(μ)via the discrete coefficient K(ρ),p B,S,and prove that the Calder′on-Zygmund operator is bounded from Hp,q,γ,δmb,ρ(μ)(or Hp,q,γatb,ρ(μ))into Lp(μ),and from Hp,q,γ+1atb,ρ(ρ+1)(μ)into H p,q,γ,12(δ-νp+ν)mb,ρ(μ).The boundedness of the generalized fractional integral Tβ(β∈(0,1))from Hp1,q,γ,θmb,ρ(μ)(or Hp1,q,γatb,ρ(μ))into Lp2(μ)with 1/p2=1/p1-β is also established.The authors also introduce theρ-weakly doubling condition,withρ∈(1,∞),of the measure and construct a non-doubling measure satisfying this condition.If isρ-weakly doubling,the authors further introduce the Campanato space Eα,qρ,η,γ(μ)and show that Eα,qρ,η,γ(μ)is independent of the choices ofρ,η,γand q;the authors then introduce the atomic Hardy space Hp,q,γatb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ(μ),which coincide with each other;the authors finally prove that Hp,q,γatb,ρ(μ)is the predual of E1/p-1,1ρ,ρ,1(μ).Moreover,if is doubling,the authors show that Eα,qρ,η,γ(μ)and the Lipschitz space Lipα,q(μ)(q∈[1,∞)),or Hp,q,γatb,ρ(μ)and the atomic Hardy space Hp,q at(μ)(q∈(1,∞])of Coifman and Weiss coincide.Finally,if(X,d,)is an RD-space(reverse doubling space)with(X)=∞,the authors prove that Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ)and Hp,q at(μ)coincide for any q∈(1,2].In particular,when(X,d,):=(RD,||,dx)with dx being the D-dimensional Lebesgue measure,the authors show that spaces Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ),Hp,q,γatb,ρ(μ)and Hp,q,γ,mb,ρ(μ)all coincide with Hp(RD)for any q∈(1,∞).

Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces

Let(X, d, μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderón-Zygmund operator and b:=(b1,..., bm) be a finite family of RBMO(μ) functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator T∏bgenerated by T and b are obtained.

Inequalities for Eigenvalues of a System of Equations of Elliptic Operator in Weighted Divergence Form on Metric Measure Space

Let A be a symmetric and positive definite(1,1)tensor on a bounded domain Ω in an ndimensional metric measure space(R^n,<,>,e^-φdv).In this paper,we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form{LA,φu+α▽(divu)-▽φdivu]=-su,inΩ,u∣aΩ=0,where LA,φ=div(A▽(·))-(A▽φ,▽(·)),α is a nonnegative constant and u is a vector-valued function.Some universal inequalities for eigenvalues of this problem are established.Moreover,as applications of these results,we give some estimates for the upper bound of sk+1 and the gap of sk+1-sk in terms of the first k eigenvalues.Our results contain some results for the Lam′e system and a system of equations of the drifting Laplacian.

Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature

Let(X,d,μ)be a metric measure space with non-negative Ricci curvature.This paper is concerned with the boundary behavior of harmonic function on the(open)upper half-space X×R_(+).We derive that a function f of bounded mean oscillation(BMO)is the trace of harmonic function u(x,t)on X×R_(+),u(x,0)=f(x),whenever u satisfies the following Carleson measure condition supx_(B),r_(B) ∫_(0)^(r_(B) ) f_(B(x_(B),r_(B))) |t■u(x,t)|^(2)dμ(x)dt/t≤C<∞,where ■=(■_(x),■_(t))denotes the total gradient and B(x_(B),r_(B)) denotes the(open)ball centered at x_(B) with radius r_(B).Conversely,the above condition characterizes all the harmonic functions whose traces are in BMO space.

Endpoint Estimates of Generalized Homogeneous Littlewood–Paley g-functions over Non-Homogeneous Metric Measure Spaces

Let （x, d,u） be a metric measure the upper doubling conditions. Under the weak space satisfying both the geometrically doubling and reverse doubling condition, the authors prove that the generalized homogeneous Littlewood-Paley g-function gr （r ∈ [2, ∞）） is bounded from Hardy space H1（u） into L1（u）. Moreover, the authors show that, if f ∈ RBMO（u）, then [gr（f）]r is either infinite everywhere or finite almost everywhere, and in the latter case, [gr（f）]r belongs to RBLO（u） with the norm no more than ｜｜f｜｜ RBMO（u） multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of gr from RBMO（u） into RBLO（u）. The vector valued Calderon-Zygmund theory over （X, d, u） is also established with details in this paper.

Two-weight Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces

In this paper,the authors establish the two-weight boundedness of the local fractional maximal operators and local fractional integrals on Gaussian measure spaces associated with the local weights.More precisely,the authors first obtain the two-weight weak-type estimate for the locala fractional maximal operators of orderαfrom L^(p)(v)to L^(q,∞)(u)with 1≤p≤q<∞under a condition of(u,v)∈∪b>a A_(p,q,a)^(b') ,and then obtain the two-weight weak-type estimate for the local fractional integrals.In addition,the authors obtain the two-weight strong-type boundedness of the local fractional maximal operators under a condition of(u,v)∈M_(p,q,a)^(6a+9√da^2) and the two-weight strong-type boundedness of the local fractional integrals.These estimates are established by the radialization method and dyadic approach.

A Singular Moser-Trudinger Inequality on Metric Measure Space

Let(X,d,μ)be a metric space with a Borel-measureμ,supposeμsatisfies the Ahlfors-regular condition,i.e.birs≤μu(Br(x))≤b2rs,VBr(x)CX,r>0,where bi,b2 are two positive constants and s is the volume growth exponent.In this paper,we mainly study two things,one is to consider the best constant of the Moser-Trudinger inequality on such metric space under the condition that s is not less than 2.The other is to study the generalized Moser-Trudinger inequality with a singular Weight.

De Lellis-Topping type inequalities on smooth metric measure spaces

We obtain some De Lellis-Topping type inequalities on the smootla metric measure spaces, some of them are as generalization of De Lellis-Topping type inequality that was proved by X. Cheng [Ann. Global Anal. Geom., 2013, 43： 153-160].

Fast Measurement of Dielectric Conductivity for Space Application by Surface Potential Decay Method

Surface potential decay of polymers for electrical insulation can help to determine the dark conductivity for spacecraft charging analysis. Due to the existence of radiation-induced conductivity, it decays fast in the first few hours after irradiation and exponentially slowly for the remaining time. The measurement of dark conductivity with this method usually takes the slow part and needs a couple of days. Integrating the Fowler formula into the deep dielectric charging equations, we obtain a new expression for the fast decay part. The experimental data of different materials, dose rates and temperatures are fitted by the new expression. Both the dark conductivity and the radiation-induced conductivity are derived and compared with other methods. The result shows a good estimation of dark conductivity and radiation-induced conductivity in high-resistivity polymers, which enables a fast measurement of dielectric conductivity within about 600 rain after irradiation.

ON GRAPH MEASURES IN BANACH SPACES AND DESCRIPTION OF ESSENTIAL SPECTRA OF MULTIDIMENSIONAL TRANSPORT EQUATION

In this paper,we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness.Moreover,we apply the obtained results to discuss the incidence of some perturbation results realized in [2] on the behavior of essential spectra of such closed densely defined linear operators on Banach spaces.These results are exploited to investigate the essential spectra of a multidimensional neutron transport operator on L1 spaces.

The Loeb Algebras of Absolutely Continuous Measure

It is proved that if μ and v be finite measures on a measurable space (X, S) and v is absolutely continuous with respect to v, then it is holds that L(* S, * μ) L(* S, *v), while L(*S, *μ) and L(*S, V) are the Loeb algebras with respect to measure spaces (X, S,μ) and (X,S,v).

Light pen whole space coordinate measuring system based on a tracking turntable

Light pen coordinate measuring system(LPCMS)is a kind of portable coordinate measuring technique based on vision metrology.In classical LPCMS,the measuring range is limited to the camera’s field of view.To overcome this defect,a new LPCMS is designed in this paper to fulfil whole space coordinate measurement.The camera is installed on a turntable instead of a tripod,so that the camera can rotate to track the movement of the light pen.The new system can be applied to large scale onsite measurement,and therefore it notably extends the application of LPCMS.To guarantee the accuracy of the new system,a method to calibrate the parameters of the tracking turntable is also proposed.Fixing the light pen at a stationary position,and changing the azimuth angles of the turntable’s two shafts,so that the camera can capture the images of the light pen from different view angles.According to the invariant spatial relationship between the camera and the pedestal of the tracking turntable,a system of nonlinear equations can be established to solve the parameters of the turntable.Experimental results show that the whole space coordinate measuring accuracy of the new system can reach 0.25 mm within 10 m.It can be concluded that the newly designed system can significantly expand the measuring range of LPCMS without losing too much accuracy.

Integrated framework for space-and energy-efficient retrofitting in multifunctional buildings:A synergy of agent-based modeling and performance-based modeling

This research investigates retrofitting strategies for multifunctional spaces within educational buildings,employing agent-based and performance-based modeling to support decision-making.An experimental matrix was developed,reflecting three usage scenarios(reading,exhibition,lecture)across four retrofitting schemes.An agent-based model was developed to delineate intricate human behaviors in space and examined the self-organizing behaviors of 30 agents for each scheme in every scenario,evaluating six metrics on spatial efficiency and visual experience.Calibrated models,derived from real data and processed through DesignBuilder software,evaluated three metrics:energy use,thermal comfort,and visual comfort.The research then incorporated metrics from the agent-based model and performance simulation to develop a method for discussing the decision-making process in retrofit strategies.The findings indicate that the optimal retrofitting solution for multifunctional spaces is heavily influenced by the distribution of usage scenarios.Given the substantial influence of space metrics on selecting the optimal retrofit scheme,the proposed framework effectively facilitates decision-making for building retrofits by providing a holistic evaluation of both spatial and energy criteria.

Direct measurement of the linear energy transfer of ions in silicon for space application

The single event effect(SEE) is an important consideration in electronic devices used in space environments because it can lead to spacecraft anomalies and failures. The linear energy transfer(LET) of ions is commonly investigated in studies of SEE. The use of a thin detector is an economical way of directly measuring the LET in space. An LET telescope consists of a thin detector as the front detector(D1), along with a back detector that indicates whether D1 was penetrated. The particle radiation effect monitor(PREM) introduced in this paper is designed to categorize the LET into four bins of 0.2–0.4, 0.4–1.0, 1.0–2.0 and 2.0–20 Me V·cm^2/mg, and one integral bin of LET>20 Me V·cm^2/mg. After calibration with heavy ions and Geant4 analysis, the LET boundaries of the first four bins are determined to be 0.236, 0.479, 1.196, 2.254, and 17.551 Me V·cm^2/mg, whereas that of the integral bin is determined to be LET>14.790 Me V·cm^2/mg. The acceptances are calculated by Geant4 analysis as 0.452, 0.451, 0.476, 0.446, and 1.334, respectively. The LET accuracy is shown to depend on the thickness of D1; as D1 is made thinner, the accuracy of the measured values increases.

Coordinated target localization base on pseudo measurement for clustered space robot

This paper presents a coordinated target localization method for clustered space robot.According to the different measuring capabilities of cluster members,the master-slave coordinated relative navigation strategy for target localization with respect to slavery space robots is proposed;then the basic mathematical models,including coordinated relative measurement model and cluster centralized dynamics,are established respectively.By employing the linear Kalman flter theorem,the centralized estimator based on truth measurements is developed and analyzed frstly,and with an intention to inhabit the initial uncertainties related to target localization,the globally stabilized estimator is designed through introduction of pseudo measurements.Furthermore,the observability and controllability of stochastic system are also analyzed to qualitatively evaluate the convergence performance of pseudo measurement estimator.Finally,on-orbit target approaching scenario is simulated by using semi-physical simulation system,which is used to verify the convergence performance of proposed estimator.During the simulation,both the known and unknown maneuvering acceleration cases are considered to demonstrate the robustness of coordinated localization strategy.

A novel high precision Doppler frequency estimation method based on the third-order phase-locked loop

In deep space exploration,many engineering and scientific requirements require the accuracy of the measured Doppler frequency to be as high as possible.In our paper,we analyze the possible frequency measurement points of the third-order phase-locked loop(PLL)and find a new Doppler measurement strategy.Based on this finding,a Doppler frequency measurement algorithm with significantly higher measurement accuracy is obtained.In the actual data processing,compared with the existing engineering software,the accuracy of frequency of 1 second integration is about 5.5 times higher when using the new algorithm.The improved algorithm is simple and easy to implement.This improvement can be easily combined with other improvement methods of PLL,so that the performance of PLL can be further improved.

The Relations between Discrete Frames and Continuous Frames with Respect to （N, μ）

The discrete frame and the continuous frame in a Hilbert space are discussed. By the tool excess of a sequence, a sufficient and necessary condition is presented under which a discrete frame is equivalent to a continuous frame with respect to the whole natural numbers with a positive Borel measure. And some examples are given to illuminate the condition.