scholarly journals INTERPOLATING WAVELETS ON THE SPHERE

2019 ◽  
Vol 5 (2) ◽  
pp. 3
Author(s):  
Nikolai I. Chernykh
Keyword(s):  
Continuous Function ◽  
Tensor Product ◽  
Periodic Function ◽  
Unit Sphere ◽  
Main Idea ◽  
Lifting Scheme ◽  
Spherical Coordinates ◽  
One Dimensional ◽  
Several Variables ◽  
Interpolating Wavelets

There are several works where bases of wavelets on the sphere (mainly orthogonal and wavelet-like bases) were constructed. In all such constructions, the authors seek to preserve the most important properties of classical wavelets including constructions on the basis of the lifting-scheme. In the present paper, we propose one more construction of wavelets on the sphere. Although two of three systems of wavelets constructed in this paper are orthogonal, we are more interested in their interpolation properties. Our main idea consists in a special double expansion of the unit sphere in \(\mathbb{R}^3\) such that any continuous function on this sphere defined in spherical coordinates is easily mapped into a \(2\pi\)-periodic function on the plane. After that everything becomes simple, since the classical scheme of the tensor product of one-dimensional bases of functional spaces works to construct bases of spaces of functions of several variables.

scholarly journals Special Finite Elements with Adaptive Strain Field on the Example of One-Dimensional Elements

2021 ◽  
Vol 11 (2) ◽  
pp. 609
Author(s):  
Tadeusz Chyży ◽  
Monika Mackiewicz
Keyword(s):  
Finite Elements ◽  
Strain Field ◽  
Main Idea ◽  
Deformation Field ◽  
Geometrical Parameters ◽  
Single Element ◽  
One Dimensional ◽  
New Type ◽  
Self Adaptation ◽  
Adaptation Method

The conception of special finite elements called multi-area elements for the analysis of structures with different stiffness areas has been presented in the paper. A new type of finite element has been determined in order to perform analyses and calculations of heterogeneous, multi-coherent, and layered structures using fewer finite elements and it provides proper accuracy of the results. The main advantage of the presented special multi-area elements is the possibility that areas of the structure with different stiffness and geometrical parameters can be described by single element integrated in subdivisions (sub-areas). The formulation of such elements has been presented with the example of one-dimensional elements. The main idea of developed elements is the assumption that the deformation field inside the element is dependent on its geometry and stiffness distribution. The deformation field can be changed and adjusted during the calculation process that is why such elements can be treated as self-adaptive. The application of the self-adaptation method on strain field should simplify the analysis of complex non-linear problems and increase their accuracy. In order to confirm the correctness of the established assumptions, comparative analyses have been carried out and potential areas of application have been indicated.

Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet

2016 ◽  
Vol 14 (01) ◽  
pp. 1650004 ◽  
Author(s):  
M. Tahami ◽  
A. Askari Hemmat ◽  
S. A. Yousefi
Keyword(s):  
Integral Equation ◽  
Tensor Product ◽  
Numerical Solution ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Wavelet Basis ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
One Dimensional

In one-dimensional problems, the Legendre wavelets are good candidates for approximation. In this paper, we present a numerical method for solving two-dimensional first kind Fredholm integral equation. The method is based upon two-dimensional linear Legendre wavelet basis approximation. By applying tensor product of one-dimensional linear Legendre wavelet we construct a two-dimensional wavelet. Finally, we give some numerical examples.

Semi-supervised learning using autodidactic interpolation on sparse representation-based multiple one-dimensional embedding

2019 ◽  
Vol 17 (03) ◽  
pp. 1950013 ◽  
Author(s):  
Hao Deng ◽  
Chao Ma ◽  
Lijun Shen ◽  
Chuanwu Yang
Keyword(s):  
Sparse Representation ◽  
Euclidean Distance ◽  
Main Idea ◽  
Classification Problem ◽  
Classification Performance ◽  
One Dimensional ◽  
Adaptive Interpolation ◽  
Shortest Distance ◽  
The Common ◽  
Sample Set

In this paper, we present a novel semi-supervised classification method based on sparse representation (SR) and multiple one-dimensional embedding-based adaptive interpolation (M1DEI). The main idea of M1DEI is to embed the data into multiple one-dimensional (1D) manifolds satisfying that the connected samples have shortest distance. In this way, the problem of high-dimensional data classification is transformed into a 1D classification problem. By alternating interpolation and averaging on the multiple 1D manifolds, the labeled sample set of the data can enlarge gradually. Obviously, proper metric facilitates more accurate embedding and further helps improve the classification performance. We develop a SR-based metric, which measures the affinity between samples more accurately than the common Euclidean distance. The experimental results on several databases show the effectiveness of the improvement.

SOME REMARKS ON FRACTIONAL INTEGRAL OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050005
Author(s):  
JIA YAO ◽  
YING CHEN ◽  
JUNQIAO LI ◽  
BIN WANG
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Positive Order ◽  
One Dimensional ◽  
Katugampola Fractional Integral ◽  
Hadamard Fractional Integral

In this paper, we make research on Katugampola and Hadamard fractional integral of one-dimensional continuous functions on [Formula: see text]. We proved that Katugampola fractional integral of bounded and continuous function still is bounded and continuous. Box dimension of any positive order Hadamard fractional integral of one-dimensional continuous functions is one.

Critical behavior of percolation and Markov fields on branching planes

1993 ◽  
Vol 30 (3) ◽  
pp. 538-547 ◽  
Author(s):  
C. Chris Wu
Keyword(s):  
Continuous Function ◽  
Critical Behavior ◽  
Mean Field ◽  
Percolation Model ◽  
Homogeneous Tree ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Percolation Probability ◽  
Triangle Condition ◽  
Markov Fields

For an independent percolation model on, whereis a homogeneous tree andis a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probabilityθ(p) is a continuous function ofpat the critical pointpc, and the critical exponents,γ,δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields onare also obtained.

scholarly journals Dunkl harmonic analysis and fundamental sets of continuous functions on the unit sphere

2015 ◽  
Vol 6 (3) ◽  
Author(s):  
Roman A. Veprintsev
Keyword(s):  
Continuous Function ◽  
Harmonic Analysis ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient

AbstractWe establish a necessary and sufficient condition on a continuous function on [-1,1] under which the family of functions on the unit sphere 𝕊

Function Extrapolation at One Inaccessible Point Using Converging Lines

2015 ◽  
Author(s):  
Yiming Zhang ◽  
Nam Ho Kim ◽  
Chanyoung Park ◽  
Raphael T. Haftka
Keyword(s):  
Drag Coefficient ◽  
Prediction Accuracy ◽  
Main Idea ◽  
Bayesian Theory ◽  
High Dimensional ◽  
Coefficient Function ◽  
Two Dimensional ◽  
Post Processing ◽  
One Dimensional ◽  
Selection Of

Focus of this paper is on the prediction accuracy of multidimensional functions at an inaccessible point. The paper explores the possibility of extrapolating a high-dimensional function using multiple one-dimensional converging lines. The main idea is to select samples along lines towards the inaccessible point. Multi-dimensional extrapolation is thus transformed into a series of one-dimensional extrapolations that provide multiple estimates at the inaccessible point. We demonstrate the performance of converging lines using Kriging to extrapolate a two-dimensional drag coefficient function. Post-processing of extrapolation results from different lines based on Bayesian theory is proposed to combine the multiple predictions. Selection of lines is also discussed. The method of converging lines proves to be more robust and reliable than two-dimensional Kriging surrogate for the example.

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