MAXIMAL FUNCTION CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BOTH NON-NEGATIVE SELF-ADJOINT OPERATORS SATISFYING GAUSSIAN ESTIMATES AND BALL QUASI-BANACH FUNCTION SPACES

Assume that L is a non-negative self-adjoint operator on L^(2)(ℝ^(n))with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space onℝ^(n) satisfying some mild assumptions.Let HX,L(ℝ^(n))be the Hardy space associated with both X and L,which is defined by the Lusin area function related to the semigroup generated by L.In this article,the authors establish various maximal function characterizations of the Hardy space HX,L(ℝ^(n))and then apply these characterizations to obtain the solvability of the related Cauchy problem.These results have a wide range of generality and,in particular,the specific spaces X to which these results can be applied include the weighted space,the variable space,the mixed-norm space,the Orlicz space,the Orlicz-slice space,and the Morrey space.Moreover,the obtained maximal function characterizations of the mixed-norm Hardy space,the Orlicz-slice Hardy space,and the Morrey-Hardy space associated with L are completely new.

THE ABSENCE OF SINGULAR CONTINUOUS SPECTRUM FOR PERTURBED JACOBI OPERATORS

This paper is mainly about the spectral properties of a class of Jacobi operators(H_(c,b)u)(n)=c_(n)u(n+1)+c_(n-1)u(n-1)+b_(n)u(n),.where∣c_(n)−1∣=O(n^(−α))and b_(n)=O(n^(−1)).We will show that,forα≥1,the singular continuous spectrum of the operator is empty.

Effect of continuous nursing on treatment compliance and side effect management of coronary heart disease

BACKGROUND There are relatively few studies on continuing care of coronary heart disease(CHD),and its research value needs to be further clarified.AIM To investigate the effect of continuous nursing on treatment compliance and side effect management in patients with CHD.METHODS This is a retrospective study with patients from January 2021 to 2023.The study was divided into two groups with 30 participants in each group.Self-rating anxiety scale(SAS)and Self-rating depression scale(SDS)were used to assess patients'anxiety and depression,and medical coping questionnaire was used to assess patients'coping styles.The pelvic floor dysfunction questionnaire(PFDI-20)was used to assess the status of pelvic floor function,including bladder symptoms,intestinal symptoms,and pelvic symptoms.RESULTS SAS score decreased from 57.33±3.01before treatment to 41.33±3.42 after treatment,SDS score decreased from 50.40±1.45 to 39.47±1.57.The decrease of these two indexes was statistically significant(P<0.05).PFDI-20 scores decreased from the mean 16.83±1.72 before treatment to 10.47±1.3the mean after treatment,which was statistically significant(P<0.05).CONCLUSION The results of this study indicate that pioneering research in continuous care of CHD has a positive impact on improving patients'treatment compliance,reducing anxiety and depression levels,and improving coping styles and pelvic floor functional status.

Effects of Continuous Precision Nursing Model on Knowledge, Attitudes, and Practices (KAP) Behavior and Cardiac Function in Patients after Percutaneous Coronary Angiography and Stent Implantation

Objective:To explore the impact of a continuous precision nursing model on patients’Knowledge,Attitudes,and Practices(KAP)and cardiac function during the nursing process of patients undergoing percutaneous coronary angiography and stent implantation.Methods:Ninety patients who underwent percutaneous coronary angiography and stent implantation in our hospital from April 2022 to April 2023 were selected and randomly divided into the control group(45 cases),in which routine nursing support was carried out during the treatment process,and the observation group(45 cases),in which continuous precision nursing model was carried out during the treatment process.Comparisons were made between the two groups of patients on their KAP,cardiac function,and quality of life during recovery.Results:There was no difference in the left ventricular ejection fraction(LVEF),cardiac output(CO),and cardiac index(CI)levels before intervention.After the intervention,the levels of cardiac function in the observation group were higher than those of the control group(P<0.05).There was no difference in the Exercise of Self-Care Agency(ESCA)self-care ability scale scores before the intervention.After the intervention,the observation group had higher ESCA scores than the control group(P<0.05).Conclusion:Implementation of a continuous precision nursing model in the care of patients undergoing percutaneous coronary angiography and stent implantation improved the patient’s cardiac function,and KAP,and promoted recovery.

Inverse Differential Operators in Time and Space

When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to accomplish this, but variations on this theme exist, and we examine a few such variations. The mathematical analysis of is sought in the form if such an inverse operator exists, but physics is defined by both mathematical formula and ontological formalism, as I show for an example based on the Dirac equation. Finally, I contrast these “standard” approaches with a novel exact inverse operator for field equations.

Semigroup of Weakly Continuous Operators Associated to a Generalized Schrödinger Equation

In this work, we prove the existence and uniqueness of the solution of the generalized Schrödinger equation in the periodic distributional space P’. Furthermore, we prove that the solution depends continuously respect to the initial data in P’. Introducing a family of weakly continuous operators, we prove that this family is a semigroup of operators in P’. Then, with this family of operators, we get a fine version of the existence and dependency continuous theorem obtained. Finally, we provide some consequences of this study.

Group of Weakly Continuous Operators Associated to a Generalized Schrödinger Type Homogeneous Model

In this work, we prove the existence and uniqueness of the solution of the generalized Schrödinger type homogeneous model in the periodic distributional space P’. Furthermore, we prove that the solution depends continuously respect to the initial data in P’. Introducing a family of weakly continuous operators, we prove that this family is a group of operators in P’. Then, with this family of operators, we get a fine version of the existence and dependency continuous theorem obtained. Finally, we give some remarks derived from this study.

Wigner function of optical cumulant operator and its dissipation in thermo-entangled state representation

To conveniently calculate the Wigner function of the optical cumulant operator and its dissipation evolution in a thermal environment, in this paper, the thermo-entangled state representation is introduced to derive the general evolution formula of the Wigner function, and its relation to Weyl correspondence is also discussed. The method of integration within the ordered product of operators is essential to our discussion.

The Construction and Analysis of Linear Ring Spaces

This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi-linearity is discussed, in addition to the examination of properties and classifications of linear ring spaces. Particularly, the ring of holomorphic functions on a region of the complex plane is examined, and the manner in which it generates an iterated linear ring space under the complex derivative operator. This notion is then generalized to all rings with nth order linear and surjective operators. Basic operator theory regarding the classifications of linear ring maps is also covered.

From Hölder Continuous Solutions of 3D Incompressible Navier-Stokes Equations to No-Finite Time Blowup on [ 0,∞ ]

This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.

Inverse Spectral Problem for Sturm-Liouville Operator with Boundary and Jump Conditions Dependent on the Spectral Parameter

In this paper, the inverse spectral problem of Sturm-Liouville operator with boundary conditions and jump conditions dependent on the spectral parameter is investigated. Firstly, the self-adjointness of the problem and the eigenvalue properties are given, then the asymptotic formulas of eigenvalues and eigenfunctions are presented. Finally, the uniqueness theorems of the corresponding inverse problems are given by Weyl function theory and inverse spectral data approach.

EXACT CONTROLLABILITY AND CONTINUOUS DEPENDENCE OF FRACTIONAL NEUTRAL INTEGRO-DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

In the present paper,with the help of the resolvent operator and some analytic methods,the exact controllability and continuous dependence are investigated for a fractional neutral integro-differential equations with state-dependent delay.As an application,we also give one example to demonstrate our results.

ESSENTIAL NORMS OF PRODUCTS OF WEIGHTED COMPOSITION OPERATORS AND DIFFERENTIATION OPERATORS BETWEEN BANACH SPACES OF ANALYTIC FUNCTIONS

We obtain several estimates of the essential norms of the products of differen- tiation operators and weighted composition operators between weighted Banach spaces of analytic functions with general weights. As applications, we also give estimates of the es- sential norms of weighted composition operators between weighted Banach space of analytic functions and Bloch-type spaces.

ENTIRE FUNCTIONS SHARING ONE SMALL FUNCTION CM WITH THEIR SHIFTS AND DIFFERENCE OPERATORS

In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf（z） be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f（z）, f（z＋c）, and △n cf（z） share 0 CM, then f（z＋c）≡Af（z）, where A（≠0） is a complex constant. Moreover, let a（z）, b（z）（ O） ∈ S（f） be periodic entire functions with period c and if f（z） - a（z）, f（z ＋ c） - a（z）, △cn f（z） - b（z） share 0 CM, then f（z ＋ c） ≡ f（z）.

ON THE BOUNDEDNESS AND THE NORM OF A CLASS OF INTEGRAL OPERATORS

The boundedness and the norm of a class of integral operators Ta,b,c on Lλ^P spaces are studied in this paper. The author not only gives the sufficient and necessary condition for the boundedness of Ta,b,c on Lλ^P,but also obtains its accurate norm on Lλ^P for some range under the condition of c = n ＋ a ＋ b.

PRODUCTS OF WEIGHTED COMPOSITION AND DIFFERENTIATION OPERATORS INTO WEIGHTED ZYGMUND AND BLOCH SPACES

We characterize boundedness and compactness of products of differentiation op- erators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.

The construction and approximation of some neural networks operators

In this paper, the technique of approximate partition of unity is used to construct a class of neural networks operators with sigmoidal functions. Using the modulus of continuity of function as a metric, the errors of the operators approximating continuous functions defined on a compact interval are estimated. Furthmore, Bochner-Riesz means operators of double Fourier series are used to construct networks operators for approximating bivariate functions, and the errors of approximation by the operators are estimated.

Elements and Experiments of Shenzhen Continuous Operating Reference Stations (SZCORS) System

Real-time kinematic GPS precise positioning has been playing an increasing role in both surveying and navigation. Based on the city’s fibre LAN network, Shenzhen Continuous Operating Reference Stations (SZCORS) system has been established and consists of GPS reference stations, system control center, user’s data center, and real-time datacom network. The SZCORS system provides users the real-time centimeter positioning or post-processing millimeter positioning. This paper makes discussion on the structure of SZCORS system. Some experiments have been made to test the usablity, and then the data has been analyzed.

SANDWICH-TYPE RESULTS FOR A CLASS OF CONVEX INTEGRAL OPERATORS

Let H（U） be the space of analytic functions in the unit disk U. For the integral operator

Composition Operators From B0 to QK and QK,0 Spaces

When φ is an analytic map of the unit disk D into itself,and X is a Banach space of analytic functions on D,define the composition operator Cφ by Cφ(f) = f οφ,for f ∈ X.In this paper,we study the boundedness and compactness of composition operators from the space B0 to QK and QK,0.