Quotient space model based on algebraic structure

In the quotient space theory of granular computing,the universe structure is assumed to be a topology,therefore,its application is still limited.In this study,based on the quotient space model,the universe structure is assumed as an algebra instead of a topology.As to obtain the algebraic quotient operator,the granulation must be uniquely determined by a congruence relation,and all the congruence relations form a complete semi-order lattice,which is the theoretical basis of granularities ' completeness.When the given equivalence relation is not a congruence relation,it defines the concepts of upper quotient and lower quotient,and discusses some of their properties which demonstrate that falsity preserving principle and truth preserving principle are still valid.Finally,it presents the algorithms and example of upper quotient and lower quotient.The work extends the quotient space theory from structure,and provides theoretical basis for the combination of the quotient space theory and the algebra theory.

The rough representation and measurement of quotient structure in algebraic quotient space model

Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.

A Resistant Quantum Key Exchange Protocol and Its Corresponding Encryption Scheme

The emergence of quantum computer will threaten the security of existing public-key cryptosystems, including the Diffie Hellman key exchange protocol, encryption scheme and etc, and it makes the study of resistant quantum cryptography very urgent. This motivate us to design a new key exchange protocol and eneryption scheme in this paper. Firstly, some acknowledged mathematical problems was introduced, such as ergodic matrix problem and tensor decomposition problem, the two problems have been proved to NPC hard. From the computational complexity prospective, NPC problems have been considered that there is no polynomial-time quantum algorithm to solve them. From the algebraic structures prospective, non-commutative cryptography has been considered to resist quantum. The matrix and tensor operator we adopted also satisfied with this non-commutative algebraic structures, so they can be used as candidate problems for resisting quantum from perspective of computational complexity theory and algebraic structures. Secondly, a new problem was constructed based on the introduced problems in this paper, then a key exchange protocol and a public key encryption scheme were proposed based on it. Finally the security analysis, efficiency, recommended parameters, performance evaluation and etc. were also been given. The two schemes has the following characteristics, provable security,security bits can be scalable, to achieve high efficiency, quantum resistance, and etc.

Generalized Dirac Equation with Vector Structural Coefficient and Its Generalized Solutions in Biquaternions Algebra

On the base of differential biquatemions algebra and theories of generalized functions the biquaternionic wave equation of general type is considered under vector representation of its structural coefficient. Its generalized decisions in the space of tempered generalized functions are constructed. The elementary twistors and twistor fields are built and their properties are investigated. Introduction. The proposed by V.P. Hamilton quatemions algebra [1] and its complex extension - biquaternions algebra are very convenient mathematical tool for the description of many physical processes. At presence these algebras have been actively used in in the work of various authors to solve a number of problems in electrodynamics, quantum mechanics, solid mechanics and field theory. The properties of these algebras are actively studied in the framework of the theory of Clifford algebras. In the papers [2, 3] the differential algebra of biquatemions has been elaborated for construction of generalized solutions of the biquaternionic wave （biwave） equations. The particular types of biwave equations were considered, which are equivalent to the systems of Maxwell and Dirac equations and their generalizations, their biquaternionic decisions also were constructed. Here the biwave equation is considered with vector structural coefficient which is biquaternionic generalization of Dirac equations. Their generalized solutions in the space of tempered distributions are defined and their properties are researched.

Bisimulation Lattice of Asymmetric Chi Calculus with Mismatch

This paper carries out a systematic investigation into the bisimulation lattice of asymmetric chi calculus with a mismatch combinator. It is shown that all the sixty three L bisimilarities collapse to twelve distinct relations and they form a bisimulation lattice with respect to set inclusion. The top of the lattice coincides with the barbed bisimilarity.

On some mathematical aspects of the Heisenberg relation

The Heisenberg commutation relation, QP P Q = ihI, is the most fundamental relation of quantum mechanics. Heisenberg's encoding of the ad-hoc quantum rules in this simple relation embodies the character-istic indeterminacy and uncertainty of quantum theory. Representations of the Heisenberg relation in various mathematical structures are discussed. In particular, after a discussion of unbounded operators affiliated with finite von Neumann algebras, especially, factors of Type Ⅱ1 , we answer the question of whether or not the Heisenberg relation can be realized with unbounded self-adjoint operators in the algebra of operators affiliated with a factor of type Ⅱ1 .