Parallel Implementation of Successive Convex Relaxation Methods for Quadratic Optimization Problems

As computing resources continue to improve, global solutions for larger size quadrically constrained optimization problems become more achievable. In this paper, we focus on larger size problems and get accurate bounds for optimal values of such problems with the successive use of SDP relaxations on a parallel computing system called Ninf (Network-based Information Library for high performance computing).     

Distributing the Encryption and Decryption of a Block Cipher

In threshold cryptography, the goal is to distribute the computation of basic cryptographic primitives across a number of nodes in order to relax trust assumptions on individual nodes, as well as to introduce a level of fault-tolerance against node compromise. Most threshold cryptography has previously looked at the distribution of public key primitives, particularly threshold signatures and threshold decryption mechanisms. In this paper, we look at the application of threshold cryptography to symmetric primitives, and in particular the encryption or decryption of a symmetric key block cipher. We comment on some previous work in this area and then propose a model for shared encryption / decryption of a block cipher. We will present several approaches to enable such systems and will compare them.AMS classification: 94A60, 94A62, 68P25     

Computing eigenvectors of block tridiagonal matrices based on twisted block factorizations

New methods for computing eigenvectors of symmetric block tridiagonal matrices based on twisted block factorizations are explored. The relation of the block where two twisted factorizations meet to an eigenvector of the block tridiagonal matrix is reviewed. Based on this, several new algorithmic strategies for computing the eigenvector efficiently are motivated and designed. The underlying idea is to determine a good starting vector for an inverse iteration process from the twisted block factorizations such that a good eigenvector approximation can be computed with a single step of inverse iteration.     

Bäcklund transformation and multi-soliton solutions for a (2 + 1)-dimensional Korteweg-de Vries system via symbolic computation

In this paper, the (2 + 1)-dimensional Korteweg-de Vries system is symbolically investigated. By the bilinear method, the N-soliton solution is presented. Then, based on the Bäcklund transformation in bilinear form, a new Bäcklund transformation is obtained and new representation of the N-soliton solution is derived. A class of novel multi-soliton solutions are obtained by the new Bäcklund transformation and the availability of symbolic computation is demonstrated.     

Dark solitonic excitations and collisions from a fourth-order dispersive nonlinear Schrödinger model for the alpha helical protein

Recent protein observations motivate the dark-soliton study to explain the energy transfer in the proteins. In this paper we will investigate a fourth-order dispersive nonlinear Schrödinger equation, which governs the Davydov solitons in the alpha helical protein with higher-order effects. Painlevé analysis is performed to prove the equation is integrable. Through the introduction of an auxiliary function, bilinear forms and dark N-soliton solutions are constructed with the Hirota method and symbolic computation. Asymptotic analysis on the two-soliton solutions indicates that the soliton collisions are elastic. Decrease of the coefficient of higher-order effects can increase the soliton velocities. Graphical analysis on the two-soliton solutions indicates that the head-on collision between the two solitons, overtaking collision between the two solitons and collision between a moving soliton and a stationary one are all elastic. Collisions among the three solitons are all pairwise elastic.     

Verified convex hull and distance computation for octree-encoded objects

This paper discusses algorithms for computing verified convex hull and distance enclosure for objects represented by axis-aligned or unaligned octrees. To find a convex enclosure of an octree, the concept of extreme vertices of boxes on its boundary has been used. The convex hull of all extreme vertices yields an enclosure of the object. Thus, distance algorithms for convex polyhedra to obtain lower bounds for the distance between two octrees can be applied. Since using convex hulls makes it possible to avoid the unwanted wrapping effect that results from repeated decompositions, it also opens a way to dynamic distance algorithms for moving objects.     

Factors affecting the process of proficient mental addition and subtraction: case studies of flexible and inflexible computers

The relationship between mental computation and number sense is complex: mental computation can facilitate number sense when students are encouraged to be flexible, but flexibility and number sense is neither sufficient nor necessary for accuracy in mental computation. It is possible for familiarity with a strategy to compensate for a lack of number sense and inefficient processes. This study reports on six case studies exploring Year 3 students’ procedures for and understanding of mental addition and subtraction, and understanding of number sense and other cognitive, metacognitive, and affective factors associated with mental computation. The case studies indicate that the mental computation process is composed of four stages in which cognitive, metacognitive and affective factors operate differently for flexible and inflexible computers. The authors propose a model in which the differences between computer types are seen in terms of the application of different knowledges in number facts, numeration, effect of operation on number, and beliefs and metacognition on strategy choice and strategy implementation.     

Darboux transformation and explicit solutions for the integrable sixth-order KdV equation for nonlinear waves

Korteweg-de Vries (KdV)-type equations can describe some physical phenomena in fluids, nonlinear optics, quantum mechanics, plasmas, etc. In this paper, with the aid of symbolic computation, the integrable sixth-order KdV equation is investigated. Darboux transformation (DT) with an arbitrary parameter is presented. Explicit solutions are derived with the DT. Relevant properties are graphically illustrated, which might be helpful to understand some physical processes in fluids, plasmas, optics and quantum mechanics.