Finite Fourier Analysis Research Articles

Arithmetic progressions in certain subsets of finite fields

In this note, we focus on how many arithmetic progressions we have in certain subsets of finite fields. For this purpose, we consider the sets Sp={t2:t∈Fp} and Cp={t3:t∈Fp}, and we use the results on Gauss and Kummer sums. We prove that for any integer k≥3 and for an odd prime number p, the number of k-term arithmetic progressions in Sp is given byp22k+R, where|R|≤(k−24−k−22k−1)⋅p32+ck⋅p and ck is a computable constant depending only on k. The proof also uses finite Fourier analysis and certain types of Weil estimates. Also, we obtain some formulas that give the exact number of arithmetic progressions of length ℓ in the set Sp when ℓ∈{3,4,5} and p is an odd prime number. For ℓ=4,5, our formulas are based on the number of points on certain elliptic curves, and the error term is best possible due to the Sato-Tate conjecture.

A family of geometric operators on triangles with two complex variables

Given a plane triangle $\Delta$, one can construct a new triangle $\Delta'$ whose vertices are intersections of two cevian triples of $\Delta$. We extend the family of operators $\Delta\mapsto\Delta'$ by complexifying the defining two cevian parameters and study its rich structure from arithmetic-geometric viewpoints. We also find another useful parametrization of the operator family via finite Fourier analysis and apply it to investigate area-preserving operators on triangles.

Cyclotomic polynomials at roots of unity

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $\Phi_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for $\Phi_n(x)$ at the other roots of unity. This allows one to explicitly evaluate $\Phi_n(e^{2\pi i/m})$ with $m\in \{3,4,5,6,8,10,12\}$. We use this evaluation with $m=5$ to give a simple reproof of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $\Phi_n(x)$. We also obtain a formula for $\Phi_n'(e^{2\pi i/m}) / \Phi_n(e^{2\pi i/m})$ with $n \ne m$, which is effectively applied to $m \in \{3,4,6\}$. Furthermore, we compute the resultant of two cyclotomic polynomials in a novel very short way.

Contractors for flows

We answer a question raised by Lovász and B. Szegedy [Contractors and connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a contractor for the graph parameter counting the number of B B -flows of a graph, where B B is a subset of a finite Abelian group closed under inverses. We prove our main result using the duality between flows and tensions and finite Fourier analysis.

Contractors for flows

Abstract We answer a question raised by Lovasz and B. Szegedy [Contractors and connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a contractor for the graph parameter counting the number of B -flows of a graph, where B is a subset of a finite Abelian group closed under inverses. We prove our main result using the duality between flows and tensions in the context of finite Fourier analysis.

Parity, eulerian subgraphs and the Tutte polynomial

Identities obtained by elementary finite Fourier analysis are used to derive a variety of evaluations of the Tutte polynomial of a graph G at certain points ( a , b ) where ( a − 1 ) ( b − 1 ) ∈ { 2 , 4 } . These evaluations are expressed in terms of eulerian subgraphs of G and the size of subgraphs modulo 2 , 3 , 4 or 6. In particular, a graph is found to have a nowhere-zero 4-flow if and only if there is a correlation between the event that three subgraphs A , B , C chosen uniformly at random have pairwise eulerian symmetric differences and the event that ⌊ | A | + | B | + | C | 3 ⌋ is even. Some further evaluations of the Tutte polynomial at points ( a , b ) where ( a − 1 ) ( b − 1 ) = 3 are also given that illustrate the unifying power of the methods used. The connection between results of Matiyasevich, Alon and Tarsi and Onn is highlighted by indicating how they may all be derived by the techniques adopted in this paper.

Calculation of spheroidal wave functions

This paper discusses the calculation of prolate spheroidal wave functions for integral mode numbers and real spheroidal parameter. The functions have applications in electromagnetism, acoustics, quantum mechanics and finite Fourier analysis. The objective is the use of large values of mode number and spheroidal parameter. This capability allows the solution of the Helmholtz equation when the spheroid is large compared with the wave-length. The paper develops the theory, describes demonstration software and gives several numerical examples. Multiple-precision calculations are also explored. For many purposes the calculation of the eigenvalues and the regular angular function can be regarded as completely solved. Accurate results may be obtained for the radial functions, but the scope is limited by numerical cancellation and slow convergence.

Underlying Hydrophobic Sequence Periodicity of Protein Tertiary Structure

Hydropathy plots or window averages over local stretches of the sequence of residue hydrophobicity have revealed patterns related to various protein tertiary structural features. This has enabled identification of regions of the sequence that are at the surface or within the interior of globular soluble proteins, regions located within the lipid bilayer of transmembrane proteins, portions of the sequence that characterize repeating motifs, as well as motifs that usefully characterize different protein structural families. This, therefore, provides one example of the generally expressed maxim that “sequence determines structure”. On the other hand, a number of previous investigations have shown the rapidly varying values of residue hydrophobicity along the sequence to be distributed approximately randomly. So one might question just how much of the sequence actually determines structure. It is, therefore, of interest to extract that part of this rapidly varying distribution of residue hydrophobicity that is responsible for the longer wavelength variations that correlate with protein tertiary structural features and to determine their prevalence within the entire distribution. This is accomplished by a finite Fourier analysis of the sequence of residue hydrophobicity and of a new measure of residue distance from the protein interior. Calculations are performed on a number of globins, immunoglobulins, cuprodoxins, and papain-like structures. The spectral power of the Fourier amplitudes of the frequencies extracted, whose inverse transforms underlie the windowed values of residue hydrophobicity is shown to be a small fraction of the total power of the hydrophobicity distribution and thereby consistent with a distribution that might appear to be predominantly random. The wide range of sequence identity between proteins having the same fold, all exhibiting similar small fractions of power amplitude that correlate with the longer wavelength inside-to- outside excursions of the amino acid residues, supports the general contention that close sequence identity is an expression of a close evolutionary relationship rather than an expression of structural similarity. Practical implications of the present analysis for protein structure prediction and engineering are also described.

Teaching time-series analysis. II. Wave height and water surface elevation probability distributions

This paper describes the second of a two-part series of pedagogical exercises to introduce students to methods of time-series analysis. While these exercises are focused on the analysis of wind generated surface gravity waves, they are cross-disciplinary in nature and can be applied to other fields dealing with random signal analysis. Two computer laboratory exercises are presented which enable students to understand many of the facets of random signal analysis with less difficulty and more understanding than standard classroom instruction alone. The first pedagogical exercise, described in the previous article, uses mathematical software on which the students execute the manual arithmetic operations of a finite Fourier analysis on a complex wave record. The results are then compared to those obtained by a fast Fourier transform. This article, the second of this two-part pedagogical series, addresses analysis of a complex sea using observed and theoretical wave height and water surface elevation probability distributions and wave spectra. These results are compared to a fast Fourier transform analysis, thus providing a link back to the first exercise.

Teaching time-series analysis. I. Finite Fourier analysis of ocean waves

The introduction of students to methods of time-series analysis is a pedagogical challenge, since the availability of easily manipulated computer software presents an attractive alternative to an understanding of the computations, as well as their assumptions and limitations. A two-part pedagogical tutorial exercise is offered as a hands-on laboratory to complement classroom discussions or as a reference for students involved in independent research projects. The exercises are focused on the analysis of ocean waves, specifically wind-generated surface gravity waves. The exercises are cross-disciplinary in nature and can be extended to any other field dealing with random signal analysis. The first exercise introduces the manual arithmetic steps of a finite Fourier analysis of a wave record, develops a spectrum, and compares these results to the results obtained using a fast Fourier transform (FFT). The second part of the exercise, described in the subsequent article, takes a longer wave record and addresses the theoretical and observed wave probability distributions of wave heights and sea surface elevations. These results are then compared to a FFT, thus linking the two pedagogical laboratory exercise parts for a more complete understanding of both exercises.

The finite Fourier analysis and B-splines

The finite Fourier analysis and B-splines

Binary sequences. II. Homogeneity and symmetry

Finite binary sequences are analysed according to their content of homogeneity and symmetry. For this purpose two series of measures are defined, using cyclic transformations and their inversions. The homogeneity and symmetry measures were found to be equal for special cases. The invariance properties of the different measures were analysed and related to the theory of symmetry groups of dieders. Further, the relationships of the measures to the theory of finite Fourier analysis, correlation theory, and the entropy concept of information theory are outlined. For the concatenation of two sequences X and Y, “superadditivity”, i.e. H( XY) ⩾ H( X) + H( Y), holds for homogeneity and symmetry in most cases, and for a modified definition of homogeneity in every case—just complementary to the subadditivity of various complexity measures known from the literature.

63—SOME ASPECTS OF WORSTED-YARN STRUCTURE. PART IV: THE APPLICATION OF FOURIER ANALYSIS TO THE STUDY OF SINGLE-FIBRE CONFIGURATIONS IN A SERIES OF WORSTED YARNS

A study is reported of some aspects of the configuration of single wool fibres in some worsted yarns. By applying the technique of finite Fourier analysis to traces of fibres obtained by utilizing the dyed-tracer-fibre technique, it is demonstrated that the paths of single fibres in worsted yarns can be broken down into a wide range of periodicities, some of which are related to spinning parameters and others to fibre characteristics.