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.

AN EXPLANATION ON FOUR NEW DEFINITIONS OF FRACTIONAL OPERATORS

Fractional calculus has drawn more attentions of mathematicians and engineers in recent years.A lot of new fractional operators were used to handle various practical problems.In this article,we mainly study four new fractional operators,namely the CaputoFabrizio operator,the Atangana-Baleanu operator,the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the k-Prabhakar fractional integral operator.Usually,the theory of the k-Prabhakar fractional integral is regarded as a much broader than classical fractional operator.Here,we firstly give a series expansion of the k-Prabhakar fractional integral by means of the k-Riemann-Liouville integral.Then,a connection between the k-Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown,respectively.In terms of the above analysis,we can obtain this a basic fact that it only needs to consider the k-Prabhakar fractional integral to cover these results from the four new fractional operators.

Blockchain-Assisted Unsupervised Learning Method for Crowdsourcing Reputation Management

Crowdsourcing holds broad applications in information acquisition and dissemination,yet encounters challenges pertaining to data quality assessment and user reputation management.Reputation mechanisms stand as crucial solutions for appraising and updating participant reputation scores,thereby elevating the quality and dependability of crowdsourced data.However,these mechanisms face several challenges in traditional crowdsourcing systems:1)platform security lacks robust guarantees and may be susceptible to attacks;2)there exists a potential for large-scale privacy breaches;and 3)incentive mechanisms relying on reputation scores may encounter issues as reputation updates hinge on task demander evaluations,occasionally lacking a dedicated reputation update module.This paper introduces a reputation update scheme tailored for crowdsourcing,with a focus on proficiently overseeing participant reputations and alleviating the impact of malicious activities on the sensing system.Here,the reputation update scheme is determined by an Empirical Cumulative distribution-based Outlier Detection method(ECOD).Our scheme embraces a blockchain-based crowdsourcing framework utilizing a homomorphic encryption method to ensure data transparency and tamper-resistance.Computation of user reputation scores relies on their behavioral history,actively discouraging undesirable conduct.Additionally,we introduce a dynamic weight incentive mechanism that mirrors alterations in participant reputation,enabling the system to allocate incentives based on user behavior and reputation.Our scheme undergoes evaluation on 11 datasets,revealing substantial enhancements in data credibility for crowdsourcing systems and a reduction in the influence of malicious behavior.This research not only presents a practical solution for crowdsourcing reputation management but also offers valuable insights for future research and applications,holding promise for fostering more reliable and high-quality data collection in crowdsourcing across diverse domains.

On Designs of Decentralized Reputation Management for Permissioned Blockchain Networks

In permissioned blockchain networks,the Proof of Authority(PoA)consensus,which uses the election of authorized nodes to validate transactions and blocks,has beenwidely advocated thanks to its high transaction throughput and fault tolerance.However,PoA suffers from the drawback of centralization dominated by a limited number of authorized nodes and the lack of anonymity due to the round-robin block proposal mechanism.As a result,traditional PoA is vulnerable to a single point of failure that compromises the security of the blockchain network.To address these issues,we propose a novel decentralized reputation management mechanism for permissioned blockchain networks to enhance security,promote liveness,and mitigate centralization while retaining the same throughput as traditional PoA.This paper aims to design an off-chain reputation evaluation and an on-chain reputation-aided consensus.First,we evaluate the nodes’reputation in the context of the blockchain networks and make the reputation globally verifiable through smart contracts.Second,building upon traditional PoA,we propose a reputation-aided PoA(rPoA)consensus to enhance securitywithout sacrificing throughput.In particular,rPoA can incentivize nodes to autonomously form committees based on reputation authority,which prevents block generation from being tracked through the randomness of reputation variation.Moreover,we develop a reputation-aided fork-choice rule for rPoA to promote the network’s liveness.Finally,experimental results show that the proposed rPoA achieves higher security performance while retaining transaction throughput compared to traditional PoA.

Reputation-Based Cooperative Spectrum Sensing Algorithm for Mobile Cognitive Radio Networks

Identifying malicious users accurately in cognitive radio networks(CRNs) is the guarantee for excellent detection performance. However, existing algorithms fail to take the mobility of secondary users into consideration. If applied directly in mobile CRNs, those conventional algorithms would overly punish reliable users at extremely bad or good locations, leading to an obvious decrease in detection performance. To overcome this problem, we divide the whole area of interest into several cells to consider the location diversity of the network. Each user's reputation score is updated after each sensing slot and is used for identifying whether it is malicious or not. If so, it would be removed away. And then our algorithm assigns users in cells with better channel conditions, i.e. larger signal-to-noise ratios(SNRs), with larger weighting coefficients, without requiring the prior information of SNR. Detailed analysis about the validity of our algorithm is presented. The simulation results show that in a CRN with 60 mobile secondary users, among which, 18 are malicious, our solution has an improvement of detection probability by 0.97-d B and 3.57-d B when false alarm probability is 0.1, compared with a conventional trust-value-based algorithm and a trusted collaborative spectrum sensing for mobile CRNs, respectively.

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.

A Reputation-Based AODV Protocol for Blackhole and Malfunction Nodes Detection and Avoidance

Enhancing the security of Wireless Sensor Networks(WSNs)improves the usability of their applications.Therefore,finding solutions to various attacks,such as the blackhole attack,is crucial for the success of WSN applications.This paper proposes an enhanced version of the AODV(Ad Hoc On-Demand Distance Vector)protocol capable of detecting blackholes and malfunctioning benign nodes in WSNs,thereby avoiding them when delivering packets.The proposed version employs a network-based reputation system to select the best and most secure path to a destination.To achieve this goal,the proposed version utilizes the Watchdogs/Pathrater mechanisms in AODV to gather and broadcast reputations to all network nodes to build the network-based reputation system.To minimize the network overhead of the proposed approach,the paper uses reputation aggregator nodes only for forwarding reputation tables.Moreover,to reduce the overhead of updating reputation tables,the paper proposes three mechanisms,which are the prompt broadcast,the regular broadcast,and the light broadcast approaches.The proposed enhanced version has been designed to perform effectively in dynamic environments such as mobile WSNs where nodes,including blackholes,move continuously,which is considered a challenge for other protocols.Using the proposed enhanced protocol,a node evaluates the security of different routes to a destination and can select the most secure routing path.The paper provides an algorithm that explains the proposed protocol in detail and demonstrates a case study that shows the operations of calculating and updating reputation values when nodes move across different zones.Furthermore,the paper discusses the proposed approach’s overhead analysis to prove the proposed enhancement’s correctness and applicability.

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.

Reputation-Based Hierarchically Cooperative Spectrum Sensing Scheme in Cognitive Radio Networks

Cooperative spectrum sensing in cog- nitive radio is investigated to improve the det- ection performance of Primary User （PU）. Meanwhile, cluster-based hierarchical coop- eration is introduced for reducing the overhead as well as maintaining a certain level of sens- ing performance. However, in existing hierar- chically cooperative spectrum sensing algo- rithms, the robustness problem of the system is seldom considered. In this paper, we pro- pose a reputation-based hierarchically coop- erative spectrum sensing scheme in Cognitive Radio Networks （CRNs）. Before spectrum sensing, clusters are grouped based on the location correlation coefficients of Secondary Users （SUs）. In the proposed scheme, there are two levels of cooperation, the first one is performed within a cluster and the second one is carried out among clusters. With the reputa- tion mechanism and modified MAJORITY rule in the second level cooperation, the pro- posed scheme can not only relieve the influ- ence of the shadowing, but also eliminate the impact of the PU emulation attack on a rela- tively large scale. Simulation results show that, in the scenarios with deep-shadowing or mul- tiple attacked SUs, our proposed scheme ach- ieves a better tradeoff between the system robustness and the energy saving compared with those conventionally cooperative sensing schemes.

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.