scholarly journals Three new lengths for cyclic Legendre pairs

2021 ◽  
pp. 2-7
Author(s):  
Nikolay Balonin ◽  
Dragomir Dokovic
Keyword(s):  
Autocorrelation Function ◽  
Cyclic Group ◽  
Hadamard Matrices ◽  
Related Structure ◽  
Practical Relevance ◽  
Ring Of Integers ◽  
Video Information ◽  
On Line ◽  
Periodic Autocorrelation ◽  
Periodic Autocorrelation Function

Introduction: It is conjectured that the cyclic Legendre pairs of odd lengths >1 always exist. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1, and whose periodic autocorrelation function adds up to the constant value −2 (except at the origin). Here G is a finite cyclic group and Z is the ring of integers. These conditions are fundamental and the closely related structure of Hadamard matrices having a two circulant core and double border is incompletely described in literature, which makes its study especially relevant. Purpose: To describe the two-border two-circulant-core construction for Legendre pairs having three new lengths. Results: To construct new Legendre pairs we use the subsets X={x∈G: a(x)=–1} and Y={x∈G: b(x)=–1} of G. There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v=77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing thereby the number of undecided cases to 17. In the last section of the paper we list some new examples of cyclic Legendre pairs for lengths v≤123. Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, and compression and masking of video information. Programs for search of Hadamard matrices and a library of constructed matrices are used in the mathematical network “mathscinet.ru” together with executable on-line algorithms

scholarly journals Symmetric Hadamard Matrices of Orders 268, 412, 436 and 604

2018 ◽  
pp. 2-8 ◽  
Author(s):  
N. A. Balonin ◽  
D. Z. Ðokovic'
Keyword(s):  
Hash Function ◽  
Hadamard Matrices ◽  
Orbit Method ◽  
Practical Relevance ◽  
Video Information ◽  
The Past ◽  
Difference Families ◽  
Feasible Sets ◽  
On Line ◽  
First Time

Purpose.To investigate more fully, than what was done in the past, certain families of symmetric Hadamard matrices of small orders by using the so called propus construction.Methods.Orbit method for the search of three cyclic blocks to construct Hadamard matrices of propus type. This method speeds up the classical search of required sequences by distributing them into different bins using a hash-function.Results. Our main result is that we have constructed, for the first time, symmetric Hadamard matrices of order 268, 412, 436 and 604. The necessary difference families are constructed by restricting the search to the families which admit a nontrivial multiplier. A wide collection of new symmetric Hadamard matrices was obtained and tabulated, according to the feasible sets of parameters.Practical relevance.Hadamard matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in the mathematical network “Internet” together with executable on-line algorithms. 

Conference matrices from Legendre C-pairs

2020 ◽  
pp. 2-10
Author(s):  
Nikolay Balonin ◽  
Dragomir Ðoković
Keyword(s):  
Prime Power ◽  
Identity Element ◽  
Finite Abelian Group ◽  
Difference Sets ◽  
Practical Relevance ◽  
Difference Family ◽  
Ring Of Integers ◽  
Video Information ◽  
On Line ◽  
Universal Structure

Introduction: There are just a few known methods for the construction of symmetric C-matrices, due to the lack of a universal structure for them. This obstruction is fundamental, in addition, the structure of C-matrices with a double border is incompletely described in literature, which makes its study especially relevant. The purpose: To describe the two-border two-circulant construction in detail with the proposal of the concept of C-pairs Legendre. Results: The paper deals with C-matrices of order n=2v+2 with two borders and extends the so called generalized Legendre pairs, v odd, to a wider class of Legendre C-pairs with even and odd v, defined on a finite abelian group G of order v. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1 except that a(e)=0, where e is the identity element of G and Z is  the ring of integers. To characterize the Legendre C-pairs we use the subsets X={xÎG: a(x)=–1} and Y={xÎG: b(x)=–1} of G. We show that a(x−1)=(−1)v a(x) for all x. For odd v we show that X and Y form a difference family, which is not true for even v. These difference families are precisely the so called Szekeres difference sets, used originally for the construction of skew-Hadamard matrices. We introduce the subclass of the special Legendre C-pairs and prove that they exist whenever 2v+1 is a prime power. In the last two sections of the paper we list examples of special cyclic Legendre C-pairs for lengths v<70. Practical relevance: C-matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of conference matrices and a library of constructed matrices are used in the mathematical network “mathscinet.ru” together with executable on-line algorithms.

16-QAM Sequences with Good Periodic Autocorrelation Function

2019 ◽  
Vol E102.A (12) ◽  
pp. 1697-1700
Author(s):  
Fanxin ZENG ◽  
Yue ZENG ◽  
Lisheng ZHANG ◽  
Xiping HE ◽  
Guixin XUAN ◽  
...  
Keyword(s):  
Autocorrelation Function ◽  
Periodic Autocorrelation ◽  
Periodic Autocorrelation Function

Circles on lattices and Hadamard matrices

2019 ◽  
pp. 2-9 ◽  
Author(s):  
N. A. Balonin ◽  
M. B. Sergeev ◽  
J. Seberry ◽  
O. I. Sinitsyna
Keyword(s):  
Hadamard Matrices ◽  
Lattice Points ◽  
Practical Significance ◽  
Upper And Lower Bounds ◽  
Problem Size ◽  
Orthogonal Matrices ◽  
Video Information ◽  
Scientific Schools ◽  
Gauss Theorem ◽  
Triangular Numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

scholarly journals On the isomorphism class of the ring of all integers of a cyclic wildly ramified extension of degree p II

1988 ◽  
Vol 111 ◽  
pp. 165-171 ◽  
Author(s):  
Yoshimasa Miyata
Keyword(s):  
Number Field ◽  
Cyclic Group ◽  
Algebraic Number ◽  
Isomorphism Class ◽  
Algebraic Number Field ◽  
Ring Of Integers

Let k be an algebraic number field with the ring of integers ok = o and let G be a cyclic group of order p, an odd prime.

scholarly journals Modeling and Forecasting Periodic Time Series data with Fourier Autoregressive Model

2019 ◽  
pp. 1367-1373
Author(s):  
Abass I. Taiwo ◽  
Timothy Olabisi Olatayo ◽  
Adedayo Funmi Adedotun ◽  
Kazeem kehinde Adesanya
Keyword(s):  
Time Series ◽  
Autocorrelation Function ◽  
Autoregressive Model ◽  
Estimation Method ◽  
Information Criteria ◽  
Series Data ◽  
Percentage Error ◽  
Model Forecast ◽  
Modeling And Forecasting ◽  
Periodic Autocorrelation Function

Most frequently used models for modeling and forecasting periodic climatic time series do not have the capability of handling periodic variability that characterizes it. In this paper, the Fourier Autoregressive model with abilities to analyze periodic variability is implemented. From the results, FAR(1), FAR(2) and FAR(2) models were chosen based on Periodic Autocorrelation function (PeACF) and Periodic Partial Autocorrelation function (PePACF). The coefficients of the tentative model were estimated using a Discrete Fourier transform estimation method. FAR(1) models were chosen as the optimal model based on the smallest values of Periodic Akaike (PAIC) and Bayesian Information criteria (PBIC). The residual of the fitted models was diagnosed to be white noise. The in-sample forecast showed a close reflection of the original rainfall series while the out-sample forecast exhibited a continuous periodic forecast from January 2019 to December 2020 with relatively small values of Periodic Root Mean Square Error (PRMSE), Periodic Mean Absolute Error (PMAE) and Periodic Mean Absolute Percentage Error (PMAPE). The comparison of FAR(1) model forecast with AR(3), ARMA(2,1), ARIMA(2,1,1) and SARIMA( 1,1,1)(1,1,1)12 model forecast indicated that FAR(1) outperformed the other models as it exhibited a continuous periodic forecast. The continuous monthly periodic rainfall forecast indicated that there will be rapid climate change in Nigeria in the coming yearly and Nigerian Government needs to put in place plans to curtail its effects.

On-Line Grinding Chatter Detection Based on Minimum Entropy Deconvolution and Autocorrelation Function

2021 ◽  
Author(s):  
Dan He ◽  
Zexing Ni ◽  
Xiufeng Wang
Keyword(s):  
Autocorrelation Function ◽  
Line Detection ◽  
Detection Accuracy ◽  
Minimum Entropy ◽  
Chatter Detection ◽  
Vibration Signals ◽  
Grinding Chatter ◽  
Grinding Conditions ◽  
On Line ◽  
Periodic Impulse

Abstract On-line detection of chatter is one of the key techniques to avoid the harmful effects caused by chatter in grinding process. The key to chatter detection is to capture reliable chatter features and thresholds. To achieve this, it is important to make clear and extract the essential characteristics of the grinding chatter signal, which has not yet been well studied. In this paper, we are going to investigate the essential characteristics of the grinding chatter signal and propose a new approach for on-line detection of grinding chatter. The proposed approach for on-line detection of grinding chatter is based on minimum entropy deconvolution and autocorrelation function, in which the minimum entropy deconvolution is employed to deconvolve the effect of transmission path, and further to restore the essential characteristics of the chatter signals. To eliminate the interference of the non-periodic impulse signals in the measured vibration signals, an autocorrelation function is introduced. Kurtosis is employed to indicate chatter according to the changes of the processed signal. The validity of the proposed method is demonstrated through the measured vibration signals obtained from grinding processes and the presented chatter detection index is independent from the grinding conditions with excellent detection accuracy and permissible computational efficiency. This demonstrates the effectiveness of proposed method in on-line implementation.

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