Fractal Fractional Order Operators in Computational Techniques for Mathematical Models in Epidemiology

New fractional operators, the COVID-19 model has been studied in this paper. By using different numericaltechniques and the time fractional parameters, the mechanical characteristics of the fractional order model areidentified. The uniqueness and existence have been established. Themodel’sUlam-Hyers stability analysis has beenfound. In order to justify the theoretical results, numerical simulations are carried out for the presented methodin the range of fractional order to show the implications of fractional and fractal orders.We applied very effectivenumerical techniques to obtain the solutions of themodel and simulations. Also, we present conditions of existencefor a solution to the proposed epidemicmodel and to calculate the reproduction number in certain state conditionsof the analyzed dynamic system. COVID-19 fractional order model for the case of Wuhan, China, is offered foranalysis with simulations in order to determine the possible efficacy of Coronavirus disease transmission in theCommunity. For this reason, we employed the COVID-19 fractal fractional derivative model in the example ofWuhan, China, with the given beginning conditions. In conclusion, again the mathematical models with fractionaloperators can facilitate the improvement of decision-making for measures to be taken in the management of anepidemic situation.

抗纤维化与促“H”型血管核壳结构生物支架修复临界骨缺损

背景:在骨组织愈合过程中,促进新生骨的血管化是加速骨组织修复的常用策略,然而骨缺损修复过程中过快的纤维化进程常常被忽略。目的:设计并制备一种具有调节新生骨痂内纤维化及血管化进程的核壳结构仿生支架,表征其物理学特征,验证其体内外抗纤维化与促成骨性能。方法:制备具有调节新生骨痂内纤维化及血管化进程的核壳结构仿生支架,外层壳结构由聚己内酯静电纺丝纳米纤维负载成纤维细胞活化蛋白抑制剂组成,内层核结构由甲基丙烯酸酐明胶水凝胶负载去铁胺组成,表征静电纺丝和水凝胶的物理学特征,利用活死染色与CCK-8实验验证材料的生物相容性。通过成纤维细胞体外实验分析支架壳结构的抗纤维化作用,通过MC3T3-E1细胞体外实验分析支架壳结构中成纤维细胞活化蛋白抑制剂的促成骨作用,通过人脐静脉内皮细胞分析支架核结构中去铁胺的促血管形成作用。通过大鼠临界骨缺损实验评估核壳结构仿生支架的骨修复作用。结果与结论:①核壳结构仿生支架具有良好的生物相容性,静电纺丝技术制备的聚己内酯电纺纤维具有疏水性,在物理层面上有效阻隔外源性纤维组织长入,电纺纤维膜在2周内可有效释放抗纤维化药物成纤维细胞活化蛋白抑制剂,释放的抗纤维化药物可以抑制骨缺损周围成纤维细胞的生长、黏附,有效降低成纤维相关蛋白的表达,同时可促进MC3T3-E1细胞成骨蛋白的表达,加快其矿化速度;支架核结构中的去铁胺可促进人脐静脉内皮细胞的迁移、成管能力,并促进其强烈表达“H”血管特征性蛋白。②在SD大鼠的股骨制造临界骨缺损模型中,相较于聚己内酯膜,核壳结构仿生支架实现了骨缺损的有效修复。③结果表明,核壳结构仿生支架在预防骨痂过度纤维化的同时促进新生骨痂内“H”血管形成,能有效避免骨不连的发生,加速临界骨缺损修复进程。

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

温肾通络止痛方对小鼠H型骨微血管内皮细胞/骨髓间充质干细胞共培养体系的影响

背景:骨骼依赖血管与骨细胞之间的密切连接来维持完整性,骨骼处在生理低氧环境中,因此在低氧环境下研究血管生成与骨生成具有重要意义。目的:探究在低氧环境下温肾通络止痛方对H型骨微血管内皮细胞/骨髓间充质干细胞共培养体系的影响及相关机制。方法:运用酶消化法及流式分选技术,分选小鼠H型血管特异型骨微血管内皮细胞;应用骨髓贴壁法分离获取小鼠骨髓间充质干细胞;采用Transwell小室以及缺氧培养工作站建立两种细胞低氧共培养体系,使用50,100μg/m L质量浓度温肾通络止痛方冻干粉进行干预;通过划痕迁移实验与管形成实验评估共培养体系内H型骨微血管内皮细胞的成血管功能;通过碱性磷酸酶染色与茜素红染色评估共培养体系内骨髓间充质干细胞的成骨分化能力;使用Western Blotting与q-PCR检测共培养体系内H型骨微血管内皮细胞中PDGF/PI3K/AKT信号轴相关分子的蛋白表达及mRNA表达。结果与结论:(1)与常氧组相比,低氧组H型骨微血管内皮细胞划痕迁移率及新生血管长度均显著升高(P<0.05),骨髓间充质干细胞成骨分化能力增强(P<0.05);PDGF/PI3K/AKT信号轴相关分子的蛋白、mRNA表达升高(P<0.05);(2)与低氧组相比,经不同质量浓度温肾通络止痛方干预后H型骨微血管内皮细胞划痕迁移率及新生血管长度进一步提高(P<0.05),骨髓间充质干细胞成骨分化能力更强(P<0.05),PDGF/PI3K/AKT信号轴相关分子的蛋白、mRNA表达也进一步升高(P<0.05)。结果表明:低氧条件下H型血管生成及骨生成可能与PDGF/PI3K/AKT信号轴有关,而温肾通络止痛方可能通过激活PDGF/PI3K/AKT信号轴促进“H型血管生成-骨生成”作用从而防治骨质疏松症。

GENERALIZED FORELLI-RUDIN TYPE OPERATORS BETWEEN SEVERAL FUNCTION SPACES ON THE UNIT BALL OF C^(N)

In this paper,we investigate sufficient and necessary conditions such that generalized Forelli-Rudin type operators T_(λ,τ,k),S_(λ,τ,k),Q_(λ,τ,k)and R_(λ,τ,k)are bounded between Lebesgue type spaces.In order to prove the main results,we first give some bidirectional estimates for several typical integrals.

An Intelligent MCGDM Model in Green Suppliers Selection Using Interactional Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Sets

Green supplier selection is an important debate in green supply chain management(GSCM),attracting global attention from scholars,especially companies and policymakers.Companies frequently search for new ideas and strategies to assist them in realizing sustainable development.Because of the speculative character of human opinions,supplier selection frequently includes unreliable data,and the interval-valued Pythagorean fuzzy soft set(IVPFSS)provides an exceptional capacity to cope with excessive fuzziness,inconsistency,and inexactness through the decision-making procedure.The main goal of this study is to come up with new operational laws for interval-valued Pythagorean fuzzy soft numbers(IVPFSNs)and create two interaction operators-the intervalvalued Pythagorean fuzzy soft interaction weighted average(IVPFSIWA)and the interval-valued Pythagorean fuzzy soft interaction weighted geometric(IVPFSIWG)operators,and analyze their properties.These operators are highly advantageous in addressing uncertain problems by considering membership and non-membership values within intervals,providing a superior solution to other methods.Moreover,specialist judgments were calculated by the MCGDM technique,supporting the use of interaction AOs to regulate the interdependence and fundamental partiality of green supplier assessment aspects.Lastly,a statistical clarification of the planned method for green supplier selection is presented.

Weighted multilinear p-adic Hardy operators and commutators on p-adic Herz-type spaces

In this paper,we introduce the weighted multilinear p-adic Hardy operator and weighted multilinear p-adic Ces`aro operator,we also obtain the boundedness of these two operators on the product of p-adic Herz spaces and p-adic Morrey-Herz spaces,the corresponding operator norms are also established in each case.Moreover,the boundedness of commutators of these two operators with symbols in central bounded mean oscillation spaces and Lipschitz spaces on p-adic Morrey-Herz spaces are also given.

Maximal operators of pseudo-differential operators with rough symbols

Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.

THE SCHUR TEST OF COMPACT OPERATORS

Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space,but not every infinite matrix corresponds to an operator.The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators,which is considered a respectable mathematical accomplishment.In this paper,we prove the compact version of the Schur test.Moreover,we provide the Schur test for the Schatten class S_(2).It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers.We finally provide the Schur test for compact operators from l_(p) into l_(q).

BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS

In this article,we investigate the(big) Hankel operator H_(f) on the Hardy spaces of bounded strongly pseudoconvex domains Ω in C^(n).We observe that H_(f ) is bounded on H~p(Ω)(1 <p <∞) if f belongs to BMO and we obtain some characterizations for Hf on H^(2)(Ω) of other pseudoconvex domains.In these arguments,Amar's L^(p)-estimations and Berndtsson's L^(2)-estimations for solutions of the ■_(b)-equation play a crucial role.In addition,we solve Gleason's problem for Hardy spaces H^(p)(Ω)(1 ≤p≤∞) of bounded strongly pseudoconvex domains.

SOME PROPERTIES OF THE INTEGRATION OPERATORS ON THE SPACES F(p,q,s)

We study the closed range property and the strict singularity of integration operators acting on the spaces F(p,pα-2,s).We completely characterize the closed range property of the Volterra companion operator I_(g)on F(p,pα-2,s),which generalizes the existing results and answers a question raised in[A.Anderson,Integral Equations Operator Theory,69(2011),no.1,87-99].For the Volterra operator J_(g),we show that,for 0<α≤1,J_(g)never has a closed range on F(p,pα-2,s).We then prove that the notions of compactness,weak compactness and strict singularity coincide in the case of J_(g)acting on F(p,p-2,s).

Porous-DeepONet:Learning the Solution Operators of Parametric Reactive Transport Equations in Porous Media

Reactive transport equations in porous media are critical in various scientific and engineering disciplines,but solving these equations can be computationally expensive when exploring different scenarios,such as varying porous structures and initial or boundary conditions.The deep operator network(DeepONet)has emerged as a popular deep learning framework for solving parametric partial differential equations.However,applying the DeepONet to porous media presents significant challenges due to its limited capability to extract representative features from intricate structures.To address this issue,we propose the Porous-DeepONet,a simple yet highly effective extension of the DeepONet framework that leverages convolutional neural networks(CNNs)to learn the solution operators of parametric reactive transport equations in porous media.By incorporating CNNs,we can effectively capture the intricate features of porous media,enabling accurate and efficient learning of the solution operators.We demonstrate the effectiveness of the Porous-DeepONet in accurately and rapidly learning the solution operators of parametric reactive transport equations with various boundary conditions,multiple phases,and multiphysical fields through five examples.This approach offers significant computational savings,potentially reducing the computation time by 50–1000 times compared with the finite-element method.Our work may provide a robust alternative for solving parametric reactive transport equations in porous media,paving the way for exploring complex phenomena in porous media.

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.

PRODUCT TYPE OPERATORS ACTING BETWEEN WEIGHTED BERGMAN SPACES AND BLOCH TYPE SPACES

For analytic functions u,ψin the unit disk D in the complex plane and an analytic self-mapφof D,we describe in this paper the boundedness and compactness of product type operators T_(u,ψ,φ)f(z)=u(z)f(φ(z))+ψ(z)f'(φ(z)),z∈D,acting between weighted Bergman spaces induced by a doubling weight and a Bloch type space with a radial weight.

On the global L^(1)-boundedness of Fourier integral operators with rough amplitude and phase functions

Let T_(ϕ,a)be a Fourier integral operator with amplitude a and phase functions ϕ.In this paper,we study the boundedness of Fourier integral operator of rough amplitude a∈L^(∞)S_(ρ)^(m)and rough phase functionsϕ∈L^(m)ϕ^(2)with some measure condition.We prove the global L^(1)boundedness for T_(ϕ,a),when 1/<ρ≤1 and m<ρ-n+1/2.Our theorem improves some known results.

TOEPLITZ OPERATORS BETWEEN WEIGHTED BERGMAN SPACES OVER THE HALF-PLANE

In this paper,by characterizing Carleson measures,we investigate a class of bounded Toeplitz operator between weighted Bergman spaces with Békolléweights over the half-plane for all index choices.

PERIODIC SYSTEMS WITH TIME DEPENDENT MAXIMAL MONOTONE OPERATORS

We consider a first order periodic system in R^(N),involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation.We prove the existence theorems for both the convex and nonconvex problems.We also show the existence of extremal periodic solutions and provide a strong relaxation theorem.Finally,we provide an application to nonlinear periodic control systems.

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.

Distributional Chaoticity of the Minimal Subshift of Shift Operators

This paper focus on the chaotic properties of minimal subshift of shift operators. It is proved that the minimal subshift of shift operators is uniformly distributional chaotic, distributional chaotic in a sequence, distributional chaotic of type k ( k∈{ 1,2,2 1 2 ,3 } ), and ( 0,1 ) -distribution.

Study of Axisymmetric Infinite Guide Lined with Locally Reacting Material without Flow Using DtN Operators

The present work proposed a new method for the modeling by the finite element method of the acoustic propagation problems in infinite axisymmetric cylindrical guides lined with locally reacting absorbent materials without flow. The method deals with the development of an efficient transparent boundary condition based on DtN operators. The method developed in this study is successfully applied to a straight axisymmetric lined guide by imposing a mode on one of the artificial boundaries of the truncated guide. The results are in good agreement with analytical solutions. Applying the method for a non-uniform axisymmetric lined guide which is a complex case, proved its effectiveness and the results compared to those of PML layers are in very good agreement.