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"Permutations"
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Prime suspects : the anatomy of integers and permutations
Integers and permutations--two of the most basic mathematical objects--are born of different fields and analyzed with separate techniques. Yet when the Mathematical Sciences Investigation team of crack forensic mathematicians, led by Professor Gauss, begins its autopsies of the victims of two seemingly unrelated homicides, Arnie Integer and Daisy Permutation, they discover the most extraordinary similarities between the structures of each body.
Book
Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations
Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.
Journal Article
The permutation polynomial with the formula b(xq + ax + δ)i + c(xq + ax + δ)j + uxq + vx and their compositional inverses over q2
The field containing exactly q elements is referred to as q, where q is a power of a prime. The class of permutations polynomials (PP) with the formula f(x) = b(xq+ax+δ)i+c(xq+ax+δ)j+uxq+vx and its compositional inverse over Fq is examined in this work, where b, c ∈ q, a, δ, u and v ∈ q2 with a1+q = 1, (avq + u)(au − v) ≠ 0 and uq + v = a(u + avq).
Journal Article
Random Permutation Set
For exploring the meaning of the power set in evidence theory, a possible explanation of power set is proposed from the view of Pascal’s triangle and combinatorial number. Here comes the question: what would happen if the combinatorial number is replaced by permutation number? To address this issue, a new kind of set, named as random permutation set (RPS), is proposed in this paper, which consists of permutation event space (PES) and permutation mass function (PMF). The PES of a certain set considers all the permutation of that set. The elements of PES are called the permutation events. PMF describes the chance of a certain permutation event that would happen. Based on PES and PMF, RPS can be viewed as a permutation-based generalization of random finite set. Besides, the right intersection (RI) and left intersection (LI) of permutation events are presented. Based on RI and LI, the right orthogonal sum (ROS) and left orthogonal sum (LOS) of PMFs are proposed. In addition, numerical examples are shown to illustrate the proposed conceptions. The comparisons of probability theory, evidence theory, and RPS are discussed and summarized. Moreover, an RPS-based data fusion algorithm is proposed and applied in threat assessment. The experimental results show that the proposed RPS-based algorithm can reasonably and efficiently deal with uncertainty in threat assessment with respect to threat ranking and reliability ranking.
Journal Article
2×22×2 monotone grid classes are finitely based
In this note, we prove that all 2×22×2 monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain 2×22×2 (generalized) grid classes having two monotone cells in the same row.
Journal Article
A Bijection on Classes Enumerated by the Schr\\\oder Numbers
We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine are known to be enumerated by the Schr\"oder numbers. In this paper, we give a bijection between these sortable permutations of length nn and Schr\"oder paths -- the lattice paths from (0,0)(0,0) to (n−1,n−1)(n−1,n−1) composed of East steps (1,0)(1,0), North steps (0,1)(0,1), and Diagonal steps (1,1)(1,1) that travel weakly below the line y=xy=x.
Journal Article
A NEW CHARACTERISATION FOR QUARTIC RESIDUACITY OF $\\mathbf {2}
Let p be a prime with
$p\\equiv 1\\pmod {4}$
. Gauss first proved that
$2$
is a quartic residue modulo p if and only if
$p=x^2+64y^2$
for some
$x,y\\in \\Bbb Z$
and various expressions for the quartic residue symbol
$(\\frac {2}{p})_4$
are known. We give a new characterisation via a permutation, the sign of which is determined by
$(\\frac {2}{p})_4$
. The permutation is induced by the rule
$x \\mapsto y-x$
on the
$(p-1)/4$
solutions
$(x,y)$
to
$x^2+y^2\\equiv 0 \\pmod {p}$
satisfying
$1\\leq x < y \\leq (p-1)/2$
.
Journal Article
Composite multi-scale phase reverse permutation entropy and its application to fault diagnosis of rolling bearing
Permutation entropy has been used as a powerful nonlinear dynamic tool for randomness measurement of time series and has been used in the area of condition monitoring and early failure fault detection of rolling bearing. However, the detail size relationship between adjacent amplitudes of signal is ignored in the calculation process of the original permutation entropy algorithms. The reverse permutation entropy was developed as a new nonlinear dynamic parameter through introducing distance information to time series with different lengths to improve the performance and stability of permutation entropy. Since the single-scale permutation entropy or reverse permutation entropy cannot completely reflect the complexity features of time series, in this paper, the phase reverse permutation entropy is proposed by introducing phase information into reverse permutation entropy to improve the detection ability of signal dynamic changes as much as possible. Based on phase reverse permutation entropy, the composite multi-scale phase reverse permutation entropy is proposed to extract the complexity information hidden in different time scales and overcome the defects of traditional coarse-grained multi-scale. Also, phase reverse permutation entropy is compared with reverse permutation entropy through simulation data and the result shows that the introduced phase information can increase the sensitivity of phase reverse permutation entropy in mutation characteristics detection of signal. After that, a new fault diagnosis method of rolling bearing was proposed based on composite multi-scale phase reverse permutation entropy for fault feature extraction and the whale optimization algorithm support vector machine for failure mode identification. Finally, the proposed fault diagnosis method was applied to the experimental data analysis of rolling bearing by comparing it with the composite multi-scale permutation entropy, the multi-scale permutation entropy, as well as multi-scale phase reverse permutation entropy based fault diagnosis approaches. The comparison results shows that the proposed method can effectively the fault location and severity of rolling bearings and reaches the highest fault recognition rate among the mentioned methods above.
Journal Article