Quantum Computation in a Ising Spin Chain Taking into Account Second Neighbor Couplings

We consider the realization of a quantum computer in a chain of nuclear spins coupled by an Ising interaction. Quantum algorithms can be performed with the help of appropriate radio-frequency pulses. In addition to the standard nearest-neighbor Ising coupling, we also allow for a second neighbor coupling. It is shown, how to apply the 2π k method in this more general setting, where the additional coupling eventually allows to save a few pulses. We illustrate our results with two numerical simulations: the Shor prime factorization of the number 4 and the teleportation of a qubit along a chain of 3 qubits. In both cases, the optimal Rabi frequency (to suppress non-resonant effects) depends primarily on the strength of the second neighbor interaction.     

n次单位根在代数问题中的应用

主要论述n次单位根在一些组合数求和、多项式的整除、因式分解中的应用,说明了n次单位根法可拓宽解题思路,是一种方便、快捷的数学解题方法.     

Reconstruction of the shape of object with near field measurements in a half-plane

We consider a mathematical problem modelling some characteristics of near field optical microscope.We take a monofrequency line source to illuminate a sample with constant index of refraction and use the scattered field data measured near the sample to reconstruct the shape of it. Mixed reciprocity relation and factorization method are applied to solve our problem.Some numerical examples to show the feasibility of the method are presented.     

ON GROWTH FACTORS OF THE LU FACTORIZATION

In this article, we derive upper bounds of different growth factors for the LU factorization, which are dominated by A11(k)-1A12(k),A21(k)A11(k)-1, where A11(k), A12(k), A21(k), A22(k) are sub-matrices of A. We also derive upper bounds of growth factors for the Cholesky factorization. Numerical examples are presented to verify our findings.     

Approximate colored range and point enclosure queries

In this paper, we formulate two classes of problems, the colored range query problems and the colored point enclosure query problems to model multi-dimensional range and point enclosure queries in the presence of categorical information. Many of these problems are difficult to solve using traditional data structural techniques. Based on a new framework of combining sketching techniques and traditional data structures, we obtain two sets of results in solving the problems approximately and efficiently. In addition, the framework can be employed to attack other related problems by finding the appropriate summary structures.     

A unification of some matrix factorization results

A unification of some factorization results regarding products of positive-definite matrices, commutators, and products of involutions is presented. This topic was first presented by Sourour for fields with, in the latter two classes, sufficiently many elements in terms of the order of the matrix being factored. The current presentation is valid for matrices over any field with at least four elements, and is independent of the order of the matrix being factored.     

A modification of a solution concept of the linear programming problem with interval coefficients in the constraints

We propose a modification of a concept of solving a linear programming problem with interval coefficients in the constraints. The original concept imposes the decision maker a way comparing intervals, our modification—which is an interactive approach, comprising the original one as its special case—gives him more freedom in the choice of the preference relation. Thanks to this flexibility he may be able to find a solution which better suits his needs and attitude towards risk. What is more, our modification corrects a serious drawback of the original method.     

Majorization, range inclusion, and factorization for bounded linear operators

In this paper, relationships among the concepts, majorization, range inclusion, and factorization, are studied in a general setting for bounded linear operators. Some applications of these concepts are given.

     

Rank reduction, factorization and conjugation

We investigate the rank reduction procedure, related factorizations and conjugation algorithms. An exact characterization of the full rank factorization produced by the rank reduction algorithm is given. This result is then used to derive matrix decompositions and conjugation procedures.