Abstract:
Two completely new algorithms for generating permutations, shift-cursor algorithm and level algorithm, and their efficient implementations are presented in this paper. One implementation of the shift cursor algorithm gives an optimal solution of the permutation generation problem, and one implementation of the level algorithm can be used to generate random permutations.

Abstract:
This work presents a new, simple O(log^2|G|) algorithm, the Fibonacci cube algorithm, for producing random group elements in black box groups. After the initial O(log^2|G|) group operations, epsilon-uniform random elements are produced using O((log 1/epsilon)log|G|) operations each. This is the first major advance over the ten year old result of Babai [Babai91], which had required O(log^5|G|) group operations. Preliminary experimental results show the Fibonacci cube algorithm to be competitive with the product replacement algorithm. The new result leads to an amusing reversal of the state of affairs for permutation group algorithms. In the past, the fastest random generation for permutation groups was achieved as an application of permutation group membership algorithms and used deep knowledge about permutation representations. The new black box random generation algorithm is also valid for permutation groups, while using no knowledge that is specific to permutation representations. As an application, we demonstrate a new algorithm for permutation group membership that is asymptotically faster than all previously known algorithms.

Abstract:
It is proved in [21] that there is a constant $c$ such that each transitive permutation group of degree $d\ge 2$ can be generated by $\lfloor cd/\sqrt{\log{d}}\rfloor$ elements. In this paper, we explicitly estimate $c$.

Abstract:
An algorithm written in SAS/IML is presented that can perform an exact permutation test for a two-sample comparison. All possible permutations are considered. The Baumgartner-Wei -Schindler statistic is exemplarily used as the test statistic for the permutation test.

Abstract:
In this paper, r-permutation factor circulant matrix is defined based on the permutation factor circulant matrix , and a fast algorithm for conditions of solution and solution of r-permutation factor circulant matrix linear equations AX=b are presented. When r-permutation factor circulant matrix are nonsingular, this algorithm computes the single solution of r-permutation factor circulant matrix linear equations , that is , there exists a unique r-permutation factor circulant matrix*, which the only solution of AX=b is the first column of C; When r-permutation factor circulant matrix are singular, it computes the special solution and general of r-permutation factor circulant matrix linear equations, which there is a unique r-permutation factor circulant matrix *and * makes the first column X1 of C is a special solution of AX=b, but also the * is the general solution of AX=b, here Z is an n arbitrary-dimensional column vectors.(* Indicates a formula, please see the full text)

Abstract:
Randomness number generation plays a key role in network, information security and IT applications. In this paper, a permutation and complementary algorithm is proposed to use vector complementary and permuta-tion operations to extend n-variable Logic function space from 2^{2n} functions to 2^{2n} * 2^{n}! configurations for variant logic framework. Each configuration contains ^{2n} functions can be shown in a 2^{2n-1}*2^{2n-1}
matrix. A set of visual results can be represented by their symmetric properties in W, F and C codes respec-tively to provide the essential support on the variant logic framework.

Abstract:
We propose a new discrete symmetry in the generation space of the fundamental fermions, consistent with the observed fermion mass spectrum. In the case of the quarks, the symmetry leads to the unique prediction of a flat CKM matrix at high energy. We explore the possibility that evolution due to quantum corrections leads to the observed hierarchical form of the CKM matrix at low energies.

Abstract:
A permutation-based algorithm is introduced for the representation of closed RNA secondary structures. It is an efficient ‘loopless’ algorithm, which generates the permutations on base-pairs of ‘k-noncrossing’ setting partitions. The proposed algorithm reduces the computational complexity of known similar techniques in O(n), using minimal change ordering and transposing of not adjacent elements.

Abstract:
Flowshop Scheduling is used to determine the optimal sequence of n jobs to be processed on m machines in the same order.The permutation flowshop represents a particular case of the flowshop scheduling problem having as goal the deployment of an optimal schedule for N jobs on M machines. Solving the flowshop problem consists in scheduling n jobs (i= 1…..n) on m machines (j=1….m). A job consists in m operations and the jth operation of each job must be processed on machine j. So, one job can start on machine j if it is completed on machine j-1 and if machine j is free. Each operation has a known processing time pij. For the permutation flowshop the operating sequences of the jobs are the same on every machine. If one job is at the ith position on machine 1, then this job will be at the ith position on all the machines. Such problems are NP-Complete and hence optimal solutions are not guaranteed but heuristics have been shown to produce good working solutions.NEH (Nawaz, Enscore, Ham) Algorithm is an efficient algorithm that works by minimizing the makespan for Permutation flowshop Scheduling Problems PFSP. The proposed algorithm is obtained by modifying the NEH algorithm and produces improved quality solutions (i.e. makespan) with algorithmic complexity same as the original algorithm.

Abstract:
It proves the symmetrical permutation's circle structure and it's count.It puts forward and designs the certain symmetrical permutation that made block cipher algorithm's hardware space decrease and the speed of encryption/decryption promote.This paper shows that Hi-times substitute makes permutation structure more complicity,so symmetrical permutation can be chosen as the diffusion module in block cipher algorithm design.