Kraus operator solutions to a fermionic master equation describing a thermal bath and their matrix representation

We solve the fermionic master equation for a thermal bath to obtain its explicit Kraus operator solutions via the fermionic state approach. The normalization condition of the Kraus operators is proved. The matrix representation for these solutions is obtained, which is incongruous with the result in the book completed by Nielsen and Chuang [Quan- tum Computation and Quantum Information, Cambridge University Press, 2000]. As especial cases, we also present the Kraus operator solutions to master equations for describing the amplitude-decay model and the diffusion process at finite temperature.

Triangular Matrix Representations of Rings of Generalized Power Series

Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].

Multi-degree reduction of NURBS curves based on their explicit matrix representation and polynomial approximation theory

NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form. And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves, which can be widely used in computer graphics and industrial design.

On Fine Matrix Representations of Nilpotent Operators

In this paper we introduce the concepts of "O-state" and "I--state" for nilpotent operators and give a fine matrix representation for each nilpotent operator according to its state.Let X’, Y be infinite dimensional Banach spaces over complex field A. Denote by Y} the set of all bounded linear operators from X to Y. If X=Y, we write instead of B(X, F). For T^B(X"), let -D(T), ker T, R(T} denote the domain, kerner and the range of T respectively. For a subset MdX, M denotes the closure of M. Let H be a Hilbert space and MdH, then ML denotes the annihilator of If.

OPERATOR AND MATRIX REPRESENTATION FOR THE GENERALIZED INVERSE A_(T,S)^(2)

This paper presents the matrix representation for extension of inverse of restriction of a linear operator to a subspace, on the basis of which we establish useful representations in operator and matrix form for the generalized inverse A(T,S)^(2) and give some of their applications.

On the First-degree Algebraic Equation of the Generalized Quaternion

In this paper, by using the matrix representation of the generalized quaternion algebra, we discussed solution problem for two classes of the first_degree algebraic equation of the generalized quaternion and obtained critical conditions on existence of a unique solution, infinitely many solutions or nonexistence any solution for the two classes algebraic equation.

α-Automated Reasoning Method Based on Lattice-Valued Propositional Logic LP(X)

This paper is focused on automated reasoning based on classical propositional logic and lattice valued propositional logic LP(X) . A new method of automated reasoning is given, and the soundness and completeness theorems of this method are proved.

3D Object Recognition Based on Linear Lie Algebra Model

A surface model called the fibre bundle model and a 3D object model based on linear Lie algebra model are proposed. Then an algorithm of 3D object recognition using the linear Lie algebra models is presented. It is a convenient recognition method for the objects which are symmetric about some axis. By using the presented algorithm, the representation matrices of the fibre or the base curve from only finite points of the linear Lie algebra model can be obtained. At last some recognition results of practicalities are given.

The Quaternion Matrix Equation ∑A^iXB_i=E

Let H_F be the generalized quaternion division algebra over a field F with charF≠2. In this paper, the ad joint matrix of any n×n matrix over H_F[λ] is defined and its properties is discussed. By using the adjoint matrix and the method of representation matrix, this paper obtains several necessary and sufficieut conditions for the existence of a solution or a unique solution to the matrix equation sum from n=i to k A^iXB_i=E over H_F, and gives some explicit formulas of solutions.

On the Properties of Cheng Projection

This paper mainly investigates some properties of Cheng projection,which was proposed recently by Prof.Cheng to reduce the dimension of vector.As a linear transformation from the original vector space to the target vector space,the matrix representation of Cheng projection is given.Then,the structure matrix of Cheng projection,called the Cheng projection matrix,is obtained.Algebraic properties of Cheng projection are derived via its projection matrix.Furthermore,the product and norm of Cheng projection matrices are concerned.

Universal Similarity Factorization Equalities over Generalized Clifford Algebras

For any element a in a generalized 2^n-dimensional Clifford algebra Lln （F） over an arbitrary field F of characteristic not equal to two, it is shown that there exits a universal invertible matrix Pn over Lln（F） such that Pn^-1DnPn= φ（α）∈F^2n×2n, where φ（a） is a matrix representation of α over and Dα is a diagonal matrix consisting of a or its conjugate.

Hyponormal Toeplitz Operators on the Polydisk

By bounded vector-valued functions and block matrix representations of Hankel operators, we completely characterize the hyponormality of Toeplitz operators on the Hardy space of the polydisk.

Knowledge Graph Embedding for Hyper-Relational Data

Knowledge graph representation has been a long standing goal of artificial intelligence. In this paper,we consider a method for knowledge graph embedding of hyper-relational data, which are commonly found in knowledge graphs. Previous models such as Trans（E, H, R） and CTrans R are either insufficient for embedding hyper-relational data or focus on projecting an entity into multiple embeddings, which might not be effective for generalization nor accurately reflect real knowledge. To overcome these issues, we propose the novel model Trans HR, which transforms the hyper-relations in a pair of entities into an individual vector, serving as a translation between them. We experimentally evaluate our model on two typical tasks—link prediction and triple classification.The results demonstrate that Trans HR significantly outperforms Trans（E, H, R） and CTrans R, especially for hyperrelational data.