scholarly journals Digit-recurrence algorithm for computing Euclidean norm of a 3-D vector

2003 ◽  
Author(s):  
N. Takagi ◽  
S. Kuwahara
Keyword(s):  
Euclidean Norm ◽  
Recurrence Algorithm

Euclidean norm estimates of the inverse of some special block matrices

2016 ◽  
Vol 284 ◽  
pp. 12-23 ◽  
Author(s):  
Ljiljana Cvetković ◽  
Vladimir Kostić ◽  
Ksenija Doroslovački ◽  
Dragana Lj. Cvetković
Keyword(s):  
Euclidean Norm ◽  
Block Matrices ◽  
Norm Estimates

scholarly journals The Pólya-Szegö Principle and the Anisotropic Convex Lorentz-Sobolev Inequality

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Shuai Liu ◽  
Binwu He
Keyword(s):  
Sobolev Inequality ◽  
Euclidean Norm ◽  
Geometric Analogue

An anisotropic convex Lorentz-Sobolev inequality is established, which extends Ludwig, Xiao, and Zhang’s result to any norm from Euclidean norm, and the geometric analogue of this inequality is given. In addition, it implies that the (anisotropic) Pólya-Szegö principle is shown.

scholarly journals Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1162
Author(s):  
Khaled A. AL-Utaibi ◽  
Sadiq H. Abdulhussain ◽  
Basheera M. Mahmmod ◽  
Marwah Abdulrazzaq Naser ◽  
Muntadher Alsabah ◽  
...  
Keyword(s):  
Recurrence Relation ◽  
Coding Theory ◽  
State Of The Art ◽  
Reconstruction Error ◽  
Krawtchouk Polynomials ◽  
Polynomial Size ◽  
Main Challenge ◽  
Recurrence Algorithm ◽  
Symmetry Relations ◽  
Parameter Values

Krawtchouk polynomials (KPs) and their moments are promising techniques for applications of information theory, coding theory, and signal processing. This is due to the special capabilities of KPs in feature extraction and classification processes. The main challenge in existing KPs recurrence algorithms is that of numerical errors, which occur during the computation of the coefficients in large polynomial sizes, particularly when the KP parameter (p) values deviate away from 0.5 to 0 and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefficients of KPs in high orders. In particular, this paper discusses the development of a new algorithm and presents a new mathematical model for computing the initial value of the KP parameter. In addition, a new diagonal recurrence relation is introduced and used in the proposed algorithm. The diagonal recurrence algorithm was derived from the existing n direction and x direction recurrence algorithms. The diagonal and existing recurrence algorithms were subsequently exploited to compute the KP coefficients. First, the KP coefficients were computed for one partition after dividing the KP plane into four. To compute the KP coefficients in the other partitions, the symmetry relations were exploited. The performance evaluation of the proposed recurrence algorithm was determined through different comparisons which were carried out in state-of-the-art works in terms of reconstruction error, polynomial size, and computation cost. The obtained results indicate that the proposed algorithm is reliable and computes lesser coefficients when compared to the existing algorithms across wide ranges of parameter values of p and polynomial sizes N. The results also show that the improvement ratio of the computed coefficients ranges from 18.64% to 81.55% in comparison to the existing algorithms. Besides this, the proposed algorithm can generate polynomials of an order ∼8.5 times larger than those generated using state-of-the-art algorithms.

A Rapid Euclidean Norm Calculation Algorithm that Reduces Overflow and Underflow

2021 ◽  
pp. 95-110
Author(s):  
Takeyuki Harayama ◽  
Shuhei Kudo ◽  
Daichi Mukunoki ◽  
Toshiyuki Imamura ◽  
Daisuke Takahashi
Keyword(s):  
Euclidean Norm ◽  
Calculation Algorithm

scholarly journals The group of isometries on Hardy spaces of the n-ball and the polydisc

1980 ◽  
Vol 21 (2) ◽  
pp. 199-204 ◽  
Author(s):  
Earl Berkson ◽  
Horacio Porta
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Inner Product ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Group Of Isometries

Let C be the complex plane, and U the disc |Z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn;. Bn will be the open unit ball {z ∈ Cn:|z| < 1}, and Un will be the unit polydisc in Cn. For l ≤ p < ∞, p ≠ 2, Gp(Bn) (resp., Gp(Un)) will denote the group of all isometries of Hp(Bn) (resp., Hp(Un)) onto itself, where Hp(Bn) and HP(Un) are the usual Hardy spaces.

Residue Level Inverse Kinematics of Peptide Chains in the Presence of Observation Inaccuracies and Bond Length Changes

2006 ◽  
Vol 129 (3) ◽  
pp. 312-319 ◽  
Author(s):  
Raghavendran Subramanian ◽  
Kazem Kazerounian
Keyword(s):  
Dihedral Angle ◽  
Inverse Kinematics ◽  
Data Bank ◽  
Residue Level ◽  
Peptide Chain ◽  
Euclidean Norm ◽  
Dihedral Angles ◽  
Structural Error ◽  
Length Changes ◽  
Value Decomposition

The process of calculating the dihedral angles of a peptide chain from atom coordinates in the chain is called residue level inverse kinematics. The uncertainties and experimental observation inaccuracies in the atoms’ coordinates handicap this otherwise simple and straightforward process. In this paper, we present and analyze three new efficient methodologies to find all the dihedral angles of a peptide chain for a given conformation. Comparison of these results with the dihedral angle values reported in the protein data bank (PDB) indicates significant improvements. While these improvements benefit most modeling methods in protein analysis, it is in particular, very significant in homology modeling where the dihedral angles are the generalized coordinates (structural variables). The first method presented here fits a best plane through five atoms of each peptide unit. The angle between the successive planes is defined as the dihedral angle. The second method is based on the zero-position analysis method. Successive links in this method rotate by the dihedral angles so as to minimize the structural error between respective atoms in the model conformation with given atoms’ coordinates. Dihedral angle final values correspond to the minimum structural error configuration. In this method, singular value decomposition technique is used to best fit the atoms in the two conformations. The third method is a variant of the second method. In this instead of rotating all the links successively only three links are matched each time to extract the dihedral angle of the middle link. By doing so, the error accumulation on the successive links is reduced. This paper focuses on the Euclidean norm as the measure of merit (structural error) to compare different methods with the PDB. This Euclidean norm is further, minimized by optimizing the geometrical features of the peptide plane.

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