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.

A novel GM(1,N) model based on interval gray number and its application to research on smog pollution

Kybernetes ◽  
2019 ◽  
Vol 49 (3) ◽  
pp. 753-778
Author(s):  
Pingping Xiong ◽  
Zhiqing He ◽  
Shiting Chen ◽  
Mao Peng
Keyword(s):  
Prediction Model ◽  
Practical Significance ◽  
Upper And Lower Bounds ◽  
Gray Model ◽  
Number Sequence ◽  
Mathematical Methods ◽  
Public Attention ◽  
Content Type ◽  
Model Based ◽  
Definition Of

Purpose In recent years, domestic smog has become increasingly frequent and the adverse effects of smog have increasingly become the focus of public attention. It is a way to analyze such problems and provide solutions by mathematical methods. Design/methodology/approach This paper establishes a new gray model (GM) (1,N) prediction model based on the new kernel and degree of grayness sequences under the case that the interval gray number distribution information is known. First, the new kernel and degree of grayness sequences of the interval gray number sequence are calculated using the reconstruction definition of the kernel and degree of grayness. Then, the GM(1,N) model is formed based on the above new sequences to simulate and predict the kernel and degree of the grayness of the interval gray number sequence. Finally, the upper and lower bounds of the interval gray number are deduced based on the calculation formulas of the kernel and degree of grayness. Findings To verify further the practical significance of the model proposed in this paper, the authors apply the model to the simulation and prediction of smog. Compared with the traditional GM(1,N) model, the new GM(1,N) prediction model established in this paper has better prediction effect and accuracy. Originality/value This paper improves the traditional GM(1,N) prediction model and establishes a new GM(1,N) prediction model in the case of the known distribution information of the interval gray number of the smog pollutants concentrations data.

Periodic Timetabling with Integrated Routing: Toward Applicable Approaches

2020 ◽  
Vol 54 (6) ◽  
pp. 1714-1731
Author(s):  
Philine Schiewe ◽  
Anita Schöbel
Keyword(s):  
Real World ◽  
Public Transportation ◽  
Computation Time ◽  
Upper And Lower Bounds ◽  
Problem Size ◽  
Classic Problem ◽  
Passenger Travel ◽  
Periodic Timetabling ◽  
Public Transportation Planning ◽  
Passenger Routing

Periodic timetabling is an important, yet computationally challenging, problem in public transportation planning. The usual objective when designing a timetable is to minimize passenger travel time. However, in most approaches, it is ignored that the routes of the passengers depend on the timetable, so handling their routing separately leads to timetables that are suboptimal for the passengers. This has recently been recognized, but integrating the passenger routing in the optimization is computationally even harder than solving the classic periodic timetabling problem. In our paper, we develop an exact preprocessing method for reducing the problem size and a heuristic reduction approach in which only a subset of the passengers is considered. It provides upper and lower bounds on the objective value, such that it can be adjusted with respect to quality and computation time. Together, we receive an approach that is applicable for real-world problems. We experimentally evaluate the performance of the approach on a benchmark example and on three close-to-real-world instances. Furthermore, we prove that the ratio between the classic problem without routing and the problem with integrated routing is bounded under weak and realistic assumptions.

scholarly journals Construction of generalized rotations and quasi-orthogonal matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 107-113
Author(s):  
Luis Verde-Star
Keyword(s):  
Image Processing ◽  
Hadamard Matrices ◽  
Complex Numbers ◽  
Data Communications ◽  
Matrix Representations ◽  
Orthogonal Matrices ◽  
Orthogonal Designs ◽  
The Matrix ◽  
The One

Abstract We propose some methods for the construction of large quasi-orthogonal matrices and generalized rotations that may be used in applications in data communications and image processing. We use certain combinations of constructions by blocks similar to the one used by Sylvester to build Hadamard matrices. The orthogonal designs related with the matrix representations of the complex numbers, the quaternions, and the octonions are used in our construction procedures.

scholarly journals Complex Hadamard Matrices from Sylvester Inverse Orthogonal Matrices

2009 ◽  
Vol 16 (04) ◽  
pp. 387-405 ◽  
Author(s):  
Petre Diţă
Keyword(s):  
Unit Circle ◽  
Hadamard Matrices ◽  
Orthogonal Matrices ◽  
Novel Method ◽  
Free Parameters

A novel method to obtain parametrizations of complex inverse orthogonal matrices is provided. These matrices are natural generalizations of complex Hadamard matrices which depend on complex parameters. The method we use is via doubling the size of inverse complex conference matrices. When the free parameters take values on the unit circle the inverse orthogonal matrices transform into complex Hadamard matrices, and in this way we find new parametrizations of Hadamard matrices for dimensions n = 8, 10, and 12.

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

Orthogonal Matrices with Zero Diagonal. II

1971 ◽  
Vol 23 (5) ◽  
pp. 816-832 ◽  
Author(s):  
P. Delsarte ◽  
J. M. Goethals ◽  
J. J. Seidel
Keyword(s):  
General Class ◽  
Special Class ◽  
Hadamard Matrices ◽  
Orthogonal Matrices ◽  
Adjacency Matrices ◽  
Class Of Matrices

C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v ≡ 2 (mod 4), and the skew C-matrices, of order v ≡ 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, λ)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tournaments introduced by Szekeres [24] and by Reid and Brown [21].

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. 

Helping Hadamard conjecture to become a theorem. Part 1

2018 ◽  
pp. 2-13 ◽  
Author(s):  
N. A. Balonin ◽  
M. B. Sergeev
Keyword(s):  
Practical Importance ◽  
Hadamard Matrices ◽  
Difficult Problem ◽  
Discrete Mathematics ◽  
Symmetric Matrices ◽  
Discrete Component ◽  
Video Information ◽  
Procrustes Algorithm ◽  
Conditional Extremes ◽  
A Chain

Introduction:Hadamard conjecture about the existence of specific square matrices was formulated not by Hadamard but by other mathematicians in the early 20th century. Later, this problem was revised by Ryser together with Bruck and Chowla, and also by Hall, one of the founders of discrete mathematics. This is a problem of the boundary mixed type, as it includes both the continuous and discrete components. The combinatorial approach used in the framework of the discrete component has run its course by the end of the century. The article discusses an alternative based on both concepts.Purpose:To analyze the reasons why the conjecture about the existence of Hadamard matrices of all orders n = 4t is considered unproven, and to propose possible ways to prove it.Methods:Transition, by lowering the order n = 4t– 2, to two-level quasiorthogonal matrices with elements 1 and –b whose existence on all specified orders is not a difficult problem due to the possible irrationality of their entries. Subsequent construction of a chain of transformations to matrix orders n = 4t – 1, n = 4t, n = 4t + 1.Results:It is proved that Gauss points on an x2+ 2y2+ z2= n spheroid are in one-to-one correspondence with symmetric Hadamard matrices (constructed on the basis of the Balonin — Seberry arrays), covering up the gaps on the unsolvable orders 140, 112, etc. known in Williamson’s array theory. Solution tables are found and systematized, which include so-called «best» three-block matrices L (p, q), where p ≥ q is the number of non-conjugated symmetric matrices of the order in question, and q is the number of block-symmetric matrices which coincide with Williamson’s solutions. The iterative Procrustes algorithm which reduces the norm of the maximum entry in a matrix is proposed for obtaining Hadamard matrices by searching for local and global conditional extremes of the determinant.Practical relevance:The obtained Hadamard matrices and quasi-orthogonal matrices of orders n = 4t – 2, n = 4t – 1, n = 4t + 1 are of immediate practical importance for the problems of noise-resistant coding, compression and masking of video information.

Helping Hadamard conjecture to become a theorem. Part 2

2019 ◽  
pp. 2-10 ◽  
Author(s):  
N. A. Balonina ◽  
M. B. Sergeeva
Keyword(s):  
Practical Importance ◽  
Hadamard Matrices ◽  
Difficult Problem ◽  
Discrete Mathematics ◽  
Symmetric Matrices ◽  
Discrete Component ◽  
Video Information ◽  
Procrustes Algorithm ◽  
Conditional Extremes ◽  
A Chain

Introduction:Hadamard conjecture about the existence of specific square matrices was formulated not by Hadamard but by other mathematicians in the early 20th century. Later, this problem was revised by Ryser together with Bruck and Chowla, and also by Hall, one of the founders of discrete mathematics. This is a problem of the boundary mixed type, as it includes both the continuous and discrete components. The combinatorial approach used in the framework of the discrete component has run its course by the end of the century. The article discusses an alternative based on both concepts.Purpose:To analyze the reasons why the conjecture about the existence of Hadamard matrices of all ordersn =4tis considered unproven, and to propose possible ways to prove it.Methods:Transition, by lowering the ordern =4t— 2, to two-level quasiorthogonal matrices with elements 1 and –bwhose existence on all specified orders is not a difficult problem due to the possible irrationality of their entries. Subsequent construction of a chain of transformations to matrix ordersn= 4t–1,n= 4t,n= 4t+ 1.Results:It is proved that Gauss points on anx2 + 2y2 +z2 =nspheroid are in one-to-one correspondence with symmetric Hadamard matrices (constructed on the basis of the Balonin — Seberry arrays), covering up the gaps on the unsolvable orders 140, 112, etc. known in Williamson’s array theory. Solution tables are found and systematized, which include so-called «best» three-block matricesL(p,q), wherep³qis the number of non-conjugated symmetric matrices of the order in question, andqis the number of block-symmetric matrices which coincide with Williamson’s solutions. The iterative Procrustes algorithm which reduces the norm of the maximum entry in a matrix is proposed for obtaining Hadamard matrices by searching for local and global conditional extremes of the determinant.Practical relevance:The obtained Hadamard matrices and quasi-orthogonal matrices of ordersn =4t– 2,n =4t– 1,n =4t +1 are of immediate practical importance for the problems of noise-resistant coding, compression and masking of video information.

scholarly journals Eigenvalue Optimisation on Flat Tori and Lattice Points in Anisotropically Expanding Domains

2019 ◽  
Vol 72 (4) ◽  
pp. 967-987
Author(s):  
Jean Lagacé
Keyword(s):  
Boundary Conditions ◽  
Lower Bounds ◽  
Unit Volume ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Lattice Points ◽  
Injectivity Radius ◽  
Upper And Lower Bounds ◽  
Weyl Asymptotics ◽  
Flat Tori

AbstractThis paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

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