Parameterized Littlewood-Paley Operators and Their Commutators on Lebesgue Spaces with Variable Exponent

In this paper, by applying the technique of the sharp maximal function and the equivalent representation of the norm in the Lebesgue spaces with variable exponent, the boundedness of the parameterized Litflewood-Paley operators, including the parameterized Lusin area integrals and the parameterized Littlewood-Paley gλ^＊- functions, is established on the Lebesgue spaces with variable exponent. Furthermore, the boundedness of their commutators generated respectively by BMO functions and Lipschitz functions are also obtained.

Parameterized Littlewood-Paley Operators on Weighted Herz Spaces

The strong type and weak type estimates of parameterized Littlewood-Paley operators on the weighted Herz spaces Kq α,p（ω1,ω2） are considered. The boundednessof the commutators generated by BMO functions and parameterized Littlewood-Paley operators are also obtained.

Parameterized Littlewood-Paley Operators and Area Integrals on Weak Hardy Spaces

In this paper, we establish the boundedness of parameterized Littlewood-Paley operator μλ^＊,p and parameterized area integralμΩ^σSp with kernel satisfying the logarithmic type Lipschitz condition on the weak Hardy space.

Universal Index Formula of Parameterized Combination Operator for Eco-environmental Quality Evaluation

[Objective] The aim was to evaluate the regional eco-environmental quality by using the universal index formula of parameterization combination operator based normalized index values.[Method] Through setting reference values and normalized transformation formulae for typical ecological environmental indexes appropriately,the difference among the standard normalized values would become very small after normalized transformation,and the ecological environmental indices expressed by normalized values can be equivalent to normalized indices.Under certain optimization conditions,shuffled frog leaping based on immune evolutionary particle swarm optimization algorithm was applied to optimize the parameters in parameterization combination operator formula,and the universal index formula suited to eco-environmental quality assessment was established finally.[Result] The universal index formula of parameterization combination operator,appropriate for any m(1≤m≤23) ecological environmental indices,was used to assess the eco-environmental quality of towns surrounding Headland Reservoir,and the results were in full accordance with those of unascertained measure method,that is,the eco-environmental quality of five towns around Headland Reservoir was the fourth grade.[Conclusion]The universal index formula of parameterization combination operator,suited to eco-environmental quality evaluation,is simple and intuitive in form,easy in computation and universal in application.

THE CENTRAL BMO SPACES AND LITTLEWOOD-PALEY OPERATORS

The authors give a characterization of central bounded mean oscillation space CBMO2(Rγ) in terms of the central Carleson measure. Using this character, the authors establish the CBMO2(Rγ)-boundedness for several classes of general Littlewood-Paley operators.

The Boundedness of Littlewood-Paley Operators with Rough Kernels on Weighted (L^q , L^p )~α (R^n) Spaces

In this paper, we shall deal with the boundedness of the Littlewood-Paley operators with rough kernel. We prove the boundedness of the Lusin-area integral μΩs and Littlewood-Paley functions μΩ and μλ^＊ on the weighted amalgam spaces （Lω^q,L^p）^α（R^n）as 1〈q≤α〈p≤∞.

Optimizing near-carbon-free nuclear energy systems:advances in reactor operation digital twin through hybrid machine learning algorithms for parameter identification and state estimation

Accurate and efficient online parameter identification and state estimation are crucial for leveraging digital twin simulations to optimize the operation of near-carbon-free nuclear energy systems.In previous studies,we developed a reactor operation digital twin(RODT).However,non-differentiabilities and discontinuities arise when employing machine learning-based surrogate forward models,challenging traditional gradient-based inverse methods and their variants.This study investigated deterministic and metaheuristic algorithms and developed hybrid algorithms to address these issues.An efficient modular RODT software framework that incorporates these methods into its post-evaluation module is presented for comprehensive comparison.The methods were rigorously assessed based on convergence profiles,stability with respect to noise,and computational performance.The numerical results show that the hybrid KNNLHS algorithm excels in real-time online applications,balancing accuracy and efficiency with a prediction error rate of only 1%and processing times of less than 0.1 s.Contrastingly,algorithms such as FSA,DE,and ADE,although slightly slower(approximately 1 s),demonstrated higher accuracy with a 0.3%relative L_2 error,which advances RODT methodologies to harness machine learning and system modeling for improved reactor monitoring,systematic diagnosis of off-normal events,and lifetime management strategies.The developed modular software and novel optimization methods presented offer pathways to realize the full potential of RODT for transforming energy engineering practices.

LITTLEWOOD-PALEY THEOREM FOR SCHR DINGER OPERATORS

Let H be a Schroedinger operator on R^n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.

LITTLEWOOD-PALEY OPERATORS AND MARCINKIEWICZ INTEGRAL ON GENERALIZED CAMPANATO SPACES

Littlewood-Paley operators and Marcinkiewicz integral on generalized Campanula spaces ερ,φ are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in ερ,φ.

Boundedness of the Littlewood-Paley Operator,the Function g(f),on BMO(T)

Littlewood-Paley operator,the function g(f), is considered as an operator on BMO(T). It is proved that if f E BMO (T), then g(f)∈BMO (T) and there is a constant C that is independent of f such that ‖g(f)‖_* ≤C‖f‖_*. Moreover,we have got the further results.

Integral Operator Solving Process of the Boundary Value Problem of Abstract Kinetic Equation with the First Kind of Critical Parameter and Generalized Periodic Boundary Conditions

In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.

ADC-GERT network parameter estimation model for mission effectiveness of joint operation system

Effectiveness evaluation of the joint operation system is an important basis for the demonstration and development of weapon equipment.With the consideration that existing models of system effectiveness evaluation seldom describe the structural relationship among equipment clearly as well as reflect the dynamic,the analog-to-digital converter-graphical evaluation and review technique(ADC-GERT)network parameter estimation model is proposed based on the ADC model and the joint operation system structure.Firstly,analysis of the joint operation system structure and operation process is conducted to build the GERT network,where equipment subsystems are nodes and activities are directed arches.Then the mission effectiveness of equipment subsystems is calculated by the ADC model.The probability transfer parameters are modified by the mission effectiveness of equipment subsystems based on the Bayesian theorem,with the ADC-GERT network parameter estimation model constructed.Finally,a case study is used to validate the efficiency and dynamic of the ADC-GERT network parameter estimation model.

Unifying the theory of integration within normal-,Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators

By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators （which considers normally ordered, antinormally ordered and Weyl ordered product of operators as its special cases）. The s-ordered operator expansion （denoted by s…s ） formula of density operators is derived, which isρ=2/1-s∫d^2β/π〈-β｜ρ｜β〉sexp{2/s-1（s｜β｜^2-β＊α＋βa-αα）}s The s-parameterized quantization scheme is thus completely established.

Characterizations of product Hardy space associated to Schrodinger operators

Let L1 and L2 be the Schrodinger operators on Rn and Rm, respectively. By using different maximal functions and Littlewood-Paley g function on distinct variables, in this paper,some characterizations for functions in the product Hardy space HL1,L21(R^n×R^m) associated to operators L1 and L2 are obtained.

Multilinear Commutators of Sublinear Operators on Triebel-Lizorkin Spaces

In this sublinear operators paper, the boundedness of multilinear commutators related to with Lipschitz function on Triebel-Lizorkin spaces is given. As an application, we prove that the multilinear commutators of Littlewood-Paley operator and Bochner-Riesz operator are bounded on Triebel-Lizorkin spaces.

Optimization of Control Loops and Operating Parameters for Three-Phase Separators Used in Oilfield Central Processing Facilities

In this study,the Stokes formula is used to analyze the separation effect of three-phase separators used in a Oilfield Central Processing Facility.The considered main influencing factors include(but are not limited to)the typical size of oil and water droplets,the residence time and temperature of fluid and the dosage of demulsifier.Using the“Specification for Oil and Gas Separators”as a basis,the control loops and operating parameters of each separator are optimized Considering the Halfaya Oilfield as a testbed,it is shown that the proposed approach can lead to good results in the production stage.

Recast combination functions of coordinate and momentum operators into their ordered product forms

By using the parameter differential method of operators,we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings,which is more ecumenical,simpler,and neater than the existing ways.These products are very useful in obtaining some new differential relations and useful mathematical integral formulas.Further,we derive the normally ordered form of the operator(fQ+gP)^-n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate-momentum representation.In addition,general mutual transformation rules of the normal and anti-normal orderings,which have good universality,are derived and hence the anti-normal ordering of(fQ+gP)^-n is also obtained.Finally,the application of some new identities is given.

Estimation of a Type of Form-Invariant Combined Signals under Autoregressive Operators

We focus on a type of combined signals whose forms remain invariant under the autoregressive operators. To extract the true signal from the autoregressive noise, we develop a strategy to separate parameters and use a two-step least squares approach to estimate the autoregressive parameters directly and then further give the estimate of the signal parameters. This method overcomes the difficulty that the autoregressive noise remains unknown in other methods. It can effectively separate the noise and extract the true signal. The algorithm is linear. The solution of the problem is computationally cheap and practical with high accuracy.

The s-parameterized Weyl-Wigner correspondence in the entangled form and its applications

Based on the theory of integration within s-ordering of operators and the bipartite entangled state representation we introduce s-parameterized Weyl-Wigner correspondence in the entangled form. Some of its applications in quantum optics theory are presented as well.

Exponential stabilization of distributed parameter switched systems under dwell time constraints

The exponential stabilization problem for finite dimensional switched systems is extended to the infinite dimensional distributed parameter systems in the Hilbert space. Based on the semigroup theory, by applying the multiple Lyapunov function method, the exponential stabilization conditions are derived. These conditions are given in the form of linear operator inequalities where the decision variables are operators in the Hilbert space; while the stabilization properties depend on the switching rule. Being applied to the two-dimensional heat switched propagation equations with the Dirichlet boundary conditions, these linear operator inequalities are transformed into standard linear matrix inequalities. Finally, two examples are given to illustrate the effectiveness of the proposed results.