Scaling Properties of the Period-Adding Sequences in a Multiple Devil’s Staircase

In this letter the scaling properties of the period-adding sequences in a so-called“multiple Devil’s staircase”are reported.It is certified both analytically and numerically that the width of the i-th phase-locked plateau in a sequence scales as In|Δe(i)|∝i,and the position of the plateau scales as In|e_(∞)-e_(i)|∝i.These properties are qualitatively different from those of the period-adding sequences in conventional Devil’s staircases.

GAUSS PERIODS AND CYCLIC CODES FROM CYCLOTOMIC SEQUENCES OF SMALL ORDERS

Let p =ef +1 be an odd prime with positive integers e and f. In this paper, we calculate the values of Gauss periods of order e =3, 4, 6 over a finite field GF(q), where q is a prime with q≠p. As applications, several cyclotomic sequences of order e =3, 4, 6 are employed to construct a number of classes of cyclic codes over GF(q) with prime length. Under certain conditions, the linear complexity and reciprocal minimal polynomials of cyclotomic sequences are calculated, and the lower bounds on the minimum distances of these cyclic codes are obtained.

Interpreting a period-adding bifurcation scenario in neural bursting patterns using border-collision bifurcation in a discontinuous map of a slow control variable

To further identify the dynamics of the period-adding bifurcation scenarios observed in both biological experiment and simulations with differential Chay model, this paper fits a discontinuous map of a slow control variable of Chay model based on simulation results. The procedure of period adding bifurcation scenario from period k to period k ＋ 1 bursting （k = 1, 2, 3, 4） involved in the period-adding cascades and the stochastic effect of noise near each bifurcation point is also reproduced in the discontinuous map. Moreover, dynamics of the border-collision bifurcation is identified in the discontinuous map, which is employed to understand the experimentally observed period increment sequence. The simple discontinuous map is of practical importance in modeling of collective behaviours of neural populations like synchronization in large neural circuits.

Statistical Characteristics of the Complexity for Prime-Periodic Sequences

Using the fact that the factorization of x^N — 1 over GF(2) is especiallyexplicit, we completely establish the distributions and the expected values of the lineal complexityand the k-error linear complexity of the N-periodic sequences respectively,where N is an odd primeand 2 is a primitive root modulo N. The results show that there are a large percentage of sequenceswith both the linear complexity and the k-enor linear complexity not less than N, quite close totheir maximum possible values.

THE DISTRIBUTION FUNCTIONS FOR THE LINEAR COMPLEXITY OF PERIODIC SEQUENCES

Linear complexity is an important standard to scale the randomicity of stream ciphers. The distribution function of a sequence complexity measure gives the function expression for the number of sequences with a given complexity measure value. In this paper, we mainly determine the distribution function of sequences with period over using Discrete Fourier Transform (DFT), where and the characteristics of are odd primes, gcd and is a primitive root modulo The results presented can be used to study the randomness of periodic sequences and the analysis and design of stream cipher.

The k-Error Linear Complexity and the Linear Complexity for pq^n-Periodic Binary Sequences

The k-error linear complexity and the linear complexity of the keystream of a stream cipher are two important standards to scale the randomness of the key stream. For a pq^n-periodic binary sequences where p, q are two odd primes satisfying that 2 is a primitive root module p and q^2 and gcd（p-1, q-1） = 2, we analyze the relationship between the linear complexity and the minimum value k for which the k-error linear complexity is strictly less than the linear complexity.

Existence of periodic orbits and shift-invariant curve sequences near multiple homoclinics

In this paper, the complicated dynamics is studied near a double homoclinic loops with bellows configuration for general systems. For the non-twisted multiple homoclinics, the existence of periodic orbit with the specified route and the existence of shift-invariant curve sequences defined on the cross sections of multiple homoclinics corresponding to any specified one-side infinite sequences are given. In addition, the existence regions are also located.

On the Judgement of Full-Period Sequences and a Novel Congruential Map with Double Modulus on Z(p^n)

This paper studies the judgement problem of full-period maps on Z(p^n) and proposes a novel congruential map with double modulus on Z(p^n). The full-period properties of the sequences generated by the novel map are studied completely. We prove some theorems including full-period judgement theorem on Z(p^n) and validate them by some numerical experiments. In the experiments, full-period sequences are generated by a full-period map on Z(p^n). By the binarization, full-period sequences are transformed into binary sequences. Then, we test the binary sequences with the NIST SP 800-22 software package and make the poker test. The passing rates of the statistical tests are high in NIST test and the sequences pass the poker test. So the randomness and statistic characteristics of the binary sequences are good. The analysis and experiments show that these full-period maps can be applied in the pseudo-random number generation(PRNG), cryptography, spread spectrum communications and so on.

Prediction of the Earlier Strong Aftershocks in the Earthquake Sequences by the Pseudo-Periodicity Method

We propose the pseudo-periodicity method and its quantitative prediction indexes for the occurrence time of earlier strong aftershock. We conducted tests of regressive prediction, and the R-value of the tests is 0.45, indicating that this method is effective for prediction.

THE 2-ERROR LINEAR COMPLEXITY OF 2~n-PERIODIC BINARY SEQUENCES WITH LINEAR COMPLEXITY 2~n-1

Linear complexity and k-error linear complexity of the stream cipher are two important standards to scale the randomicity of keystreams. For the 2n -periodicperiodic binary sequence with linear complexity 2n 1and k = 2,3,the number of sequences with given k-error linear complexity and the expected k-error linear complexity are provided. Moreover,the proportion of the sequences whose k-error linear complexity is bigger than the expected value is analyzed.

ODD-PERIODIC AND PERIODIC COMPLEMENTARY BINARY SEQUENCES

Two or more sequences are called an Odd-Periodic Complementary binary sequences Set (OPCS) if the sum of their respective odd-periodic autocorrelation function is a delta function. In this paper, the definition of OPCS is given and the construction method of OPCS is discussed. The relation of the OPCS with the Periodic Complementary binary sequences Set (PCS) is pointed out, and some new PCSs are obtained based on such relation.

基于Period的三维人体动作识别研究

在剖析现有人体运动建模方法的基础上,提出基于Period的建模方法。首先对关节点的运动规律做出基础假设,即每个关节点的相对运动都被限制在一个半球面上,利用半球模型来表征人体运动数据。然后提出Period的概念,并通过对节点运动路径的近似拟合,以多组Period序列来表示人体运动,使得数据规模大幅减小,最终实现对人体动作的匹配识别。实验表明,该方法能有效提高动作识别的精度和效率。

Sequence stratigraphy, paleogeography, and coal accumulation regularity of major coal-accumulating periods in China

There are 9 major coal-accumulating periods during geological history in China,including the Early Carboniferous,Late Carboniferous-Early Permian,Middle Permian,Late Permian,Late Triassic,Early-Middle Jurassic,Early Cretaceous,Paleogene and Neogene.The coal formed in these periods were developed in different coal-accumulating areas(CAA)including the North China,South China,Northwest China,Northeast China,the Qinghai–Tibet area,and China offshore area.In this paper,we investigated depositional environments,sequence stratigraphy,lithofacies paleogeography and coal accumulation pattern of five major coal-accumulating periods including the Late Carboniferous to Middle Permian of the North China CAA,the Late Permian of the South China CAA,the Late Triassic of the South China CAA,the Early-Middle Jurassic of the North and Northwest China CAA,and the Early Cretaceous in the Northeast China CAA.According to distribution of the coal-bearing strata and the regional tectonic outlines,we have identified distribution range of the coal-forming basins,sedimentary facies types and coal-accumulating models.The sequence stratigraphic frameworks of the major coal-accumulating periods were established based on recognition of a variety of sequence boundaries.The distribution of thick coals and migration patterns of the coal-accumulating centers in the sequence stratigraphic framework were analyzed.The lithofacies paleogeography maps based on third-order sequences were reconstructed and the distribution of coal accumulation centers and coal-rich belts were predicted.

Periodic performance of the chaotic spread spectrum sequence on finite precision

It is well known that the periodic performance of spread spectrum sequence heavily affects the correlative and secure characteristics of communication systems. The chaotic binary sequence is paid more and more attention since it is one kind of applicable spread spectrum sequences. However, there are unavoidable short cyclic problems for chaotic binary sequences in finite precision. The chaotic binary sequence generating methods are studied first. Then the short cyclic behavior of the chaotic sequences is analyzed in detail, which are generated by quantification approaches with finite word-length. At the same time, a chaotic similar function is defined for presenting the cyclic characteristics of the sequences. Based on these efforts, an improved method with scrambling control for generating chaotic binary sequences is proposed. To quantitatively describe the improvement of periodic performance of the sequences, an orthogonal estimator is also defined. Some simulating results are provided. From the theoretical deduction and the experimental results, it is concluded that the proposed method can effectively increase the period and raise the complexity of the chaotic sequences to some extent.

Genetic Diversity of European and Chinese Oilseed Brassica rapa Cultivars from Different Breeding Periods

The Brassica oilseed crops went through two major breeding bottlenecks during the introgression of genes for zero erucic acid and low glucosinolate content, respectively, which may lead to reduced genetic biodiversity of the crop. This study investigates the impact of these bottlenecks on the genetic diversity within and across European and Chinese winter B. rapa cultivars. We compared eight cultivars from Europe and China, representing three different seed qualities from three different breeding periods： （1） high erucic acid, high glucosinolates （＋＋）; （2） zero erucic acid, high glucosinolates（0＋）; （3） zero erucic acid, low glucosonolates （00, canola quality）. Diversity was estimated on 32 plants per cultivar, with 16 simple sequence repeat （SSR） markers covering each of the B. rapa linkage groups. The analysis of molecular variance （AMOVA） showed that genetic variations within cultivars, across cultivars and across regions （Europe and China） were significant, with about 60% of the total variation within cultivars. There was a slight, but non-significant loss in genetic diversity within cultivars when comparing the three breeding periods as indicated by effective number of alleles （2.39, 2.23, and 1.99 for breeding periods 1, 2, and 3, respectively）, Shannon information index （0.93, 0.90, 0.75）, and expected heterozygosity （0.51, 0.49, 0.42）. By cluster analysis （UPGMA dendrogram） and principal coordinate analysis, Chinese and European cultivars were clearly divided into two distinct groups. In conclusion, quality improvement did not significantly reduce the genetic diversity of European and Chinese B. rapa cultivars.

PERIODIC WINDOWS OF NONLINEAR GEAR SYSTEM BASED ON SYMBOLIC DYNAMICS

The periodic window is researched by means of the symbolic dynamics and formal language. Firstly, the proper sampling period is taken and the orbital points of periodic motion are obtained through Poincar6 mapping. Secondly, according to the method of symbolic dynamics of one-dimensional discrete mapping, the symbolic sequence describing the periodic orbit is obtained. Finally, based on the symbolic sequence, the corresponding model of minimal finite automation is constructed and the entropy is obtained by calculating the maximal eigenvalue of Stefan matrix. The results show that the orbits in periodic windows can be strictly marked by using the method of symbolic dynamics, thus a foundation for control of switching between target orbits is provided.

A reduced computational load protein coding predictor using equivalent amino acid sequence of DNA string with period-3 based time and frequency domain analysis

Development of efficient gene prediction algorithms is one of the fundamental efforts in gene prediction study in the area of genomics. In genomic signal processing the basic step of the identification of protein coding regions in DNA sequences is based on the period-3 property exhibited by nucleotides in exons. Several approaches based on signal processing tools and numerical representations have been applied to solve this problem, trying to achieve more accurate predictions. This paper presents a new indicator sequence based on amino acid sequence, called as aminoacid indicator sequence, derived from DNA string that uses the existing signal processing based time-domain and frequency domain methods to predict these regions within the billions long DNA sequence of eukaryotic cells which reduces the computational load by one-third. It is known that each triplet of bases, called as codon, instructs the cell machinery to synthesize an amino acid. The codon sequence therefore uniquely identifies an amino acid sequence which defines a protein. Thus the protein coding region is attributed by the codons in amino acid sequence. This property is used for detection of period-3 regions using amino acid sequence. Physico-chemical properties of amino acids are used for numerical representation. Various accuracy measures such as exonic peaks, discriminating factor, sensitivity, specificity, miss rate, wrong rate and approximate correlation are used to demonstrate the efficacy of the proposed predictor. The proposed method is validated on various organisms using the standard data-set HMR195, Burset and Guigo and KEGG. The simulation result shows that the proposed method is an effective approach for protein coding prediction.

Discrete Pseudo Almost Periodic Solutions for Some Difference Equations

In this work, we study the existence and uniqueness of pseudo almost periodic solutions for some difference equations. Firstly, we investigate the spectrum of the shift operator on the space of pseudo almost periodic sequences to show the main results of this work. For the illustration, some applications are provided for a second order differential equation with piecewise constant arguments.

关于“Structure of the set of periods for the Lorenz map”一文的注记

在“ＳｔｒｕｃｔｕｒｅｏｆｔｈｅｓｅｔｏｆＰｅｏｒｉｄｓｆｏｒｔｈｅＬｏｒｅｎｚｍａｐ”一文中，ＪＬｌｉｂｒｅ得到了，对于Ｌｏｒｅｎｚ映射，著名的Ｓａｒｋｏｖｓｋｉｉ定理仍然成立．即若ｎ∈Ｐ（ｆ）且ｎ＞＞ｋ（ｋ＞１），则ｋ∈Ｐ（ｆ）．然而，ＪＬｌｉｂｒｅ的文中该结论的主要依据有部分错误．本文改正了这些错误，给出了一些引理，并由此证明该结论仍然成立．

PERIODIC COMPLEMENTARY BINARY SEQUENCE PAIRS

A new set of binary sequences-Periodic Complementary Binary Sequence Pair (PCSP) is proposed. A new class of block design-Difference Family Pair (DFP) is also proposed.The relationship between PCSP and DFP, the properties and existing conditions of PCSP and the recursive constructions for PCSP are given.