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This paper presents 6-point subdivision schemes with cubic precision. We first derive a relation between the 4-point interpolatory subdivision and the quintic B-spline refinement. By using the relation, we further propose the counterparts of cubic and quintic B-spline refinements based on 6-point interpolatory subdivision schemes. It is proved that the new family of 6-point combined subdivision schemes has higher smoothness and better polynomial reproduction property than the B-spline counterparts. It is also showed that, both having cubic precision, the well-known Hormann-Sabin's family increase the degree of polynomial generation and smoothness in exchange of the increase of the support width, while the new family can keep the support width unchanged and maintain higher degree of polynomial generation and smoothness.

This work is a continuation of the earlier article [1]. We establish new numerical methods for solving systems of Volterra integral equations with cardinal splines. The unknown functions are expressed as a linear combination of horizontal translations of certain cardinal spline functions with small compact supports. Then a simple system of equations on the coefficients is acquired for the system of integral equations. It is relatively straight forward to solve the system of unknowns and an approximation of the original solution with high accuracy is achieved. Several cardinal splines are applied in the paper to enhance the accuracy. The sufficient condition for the existence of the inverse matrix is examined and the convergence rate is investigated. We demonstrated the value of the methods using several examples.

Quantum Cloud Computing is the technology which has the capability to shape the future of computing. In “Platform as a Service (PaaS)” type of cloud computing, the development environment is delivered as a service. In this paper, a multi-user broadcast protocol in network is developed with the mode of one master and N slaves together with a sequence of single photons. It can be applied to a multi-node network, in which a single photon sequence can be sent to all the slave nodes simultaneously. In broadcast communication networks, these single photons encode classical information directly through noisy quantum communication channels. The results show that this protocol can realize the secret key generation and sharing of multiple nodes. The protocol we propose is also proved to be unconditionally secure in theory, which indicates its feasibility in theoretical application.

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