2,597 research outputs found
The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications
We completely describe the functional graph associated to iterations of
Chebyshev polynomials over finite fields. Then, we use our structural results
to obtain estimates for the average rho length, average number of connected
components and the expected value for the period and preperiod of iterating
Chebyshev polynomials
Chebyshev Action on Finite Fields
Given a polynomial f and a finite field F one can construct a directed graph
where the vertices are the values in the finite field, and emanating from each
vertex is an edge joining the vertex to its image under f. When f is a
Chebyshev polynomial of prime degree, the graphs display an unusual degree of
symmetry. In this paper we provide a complete description of these graphs, and
also provide some examples of how these graphs can be used to determine the
decomposition of primes in certain field extensions
CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters
The rise of graph-structured data such as social networks, regulatory
networks, citation graphs, and functional brain networks, in combination with
resounding success of deep learning in various applications, has brought the
interest in generalizing deep learning models to non-Euclidean domains. In this
paper, we introduce a new spectral domain convolutional architecture for deep
learning on graphs. The core ingredient of our model is a new class of
parametric rational complex functions (Cayley polynomials) allowing to
efficiently compute spectral filters on graphs that specialize on frequency
bands of interest. Our model generates rich spectral filters that are localized
in space, scales linearly with the size of the input data for
sparsely-connected graphs, and can handle different constructions of Laplacian
operators. Extensive experimental results show the superior performance of our
approach, in comparison to other spectral domain convolutional architectures,
on spectral image classification, community detection, vertex classification
and matrix completion tasks
A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions
In this paper, the fractional order of rational Bessel functions collocation
method (FRBC) to solve Thomas-Fermi equation which is defined in the
semi-infinite domain and has singularity at and its boundary condition
occurs at infinity, have been introduced. We solve the problem on semi-infinite
domain without any domain truncation or transformation of the domain of the
problem to a finite domain. This approach at first, obtains a sequence of
linear differential equations by using the quasilinearization method (QLM),
then at each iteration solves it by FRBC method. To illustrate the reliability
of this work, we compare the numerical results of the present method with some
well-known results in other to show that the new method is accurate, efficient
and applicable
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