Quantification of shape by use of Fourier analysis: the Mississippian blastoid genus Pentremites

Paleobiology ◽  
1977 ◽  
Vol 3 (3) ◽  
pp. 288-299 ◽  
Author(s):  
Johnny Arlton Waters
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
External Morphology ◽  
Taxonomic Character ◽  
Fossil Species ◽  
Growth Series ◽  
Wide Range ◽  
Lawrence County ◽  
Psychological Evidence ◽  
Important Character

Psychological evidence suggests that the visual outline of an object is the most important character for discriminating differences in external morphology. External morphology is an important taxonomic character in describing living and fossil species, for example, the blastoid Pentremites. Comparison of different views of the same specimen utilizing Fourier series suggests that the skeleton of Pentremites commonly is not rotationally symmetrical and that the asymmetry is not associated with any specific ray. Analysis of a growth series indicates that the amplitudes of the second and third harmonics are significantly correlated with growth, which is demonstrated to be anisometric. Definable changes in the lateral outline can be attributed to changes in harmonic amplitudes and a wide range of morphological forms comparable to known taxa can be generated by systematically varying the amplitudes of the first four harmonics. The population used in this study probably represents Pentremites robustus Lyon and is from Bangor Limestone in the abandoned Moulton Quarry, Lawrence County, Alabama.

Theory of Nonlinear Fourier Analysis: The Construction of Quasiperiodic Fourier Series for Nonlinear Wave Motion

2020 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Kdv Equation ◽  
Wave Turbulence ◽  
Two Dimensional ◽  
Kp Equation ◽  
Wave Dynamics ◽  
Wide Range ◽  
Stokes Waves ◽  
Parallel Dynamics

Abstract I give a description of nonlinear water wave dynamics using a recently discovered tool of mathematical physics I call nonlinear Fourier analysis (NLFA). This method is based upon and is an application of a theorem due to Baker [1897, 1907] and Mumford [1984] in the field of algebraic geometry and from additional sources by the author [Osborne, 2010, 2018, 2019]. The theory begins with the Kadomtsev-Petviashvili (KP) equation, a two dimensional generalization of the Korteweg-deVries (KdV) equation: Here the NLFA method is derived from the complete integrability of the equation by finite gap theory or the inverse scattering transform for periodic/quasiperiodic boundary conditions. I first show, for a one-dimensional, plane wave solution, that the KP equation can be rotated to a solution of the KdV equation, where the coefficients of KdV are now functions of the rotation angle. I then show how the rotated KdV equation can be used to compute the spectral solutions of the KP equation itself. Finally, I write the spectral solutions of the KP equation as a finite gap solution in terms of Riemann theta functions. By virtue of the fact that I am able to write a theta function formulation of the KP equation, it is clear that the wave dynamics lie on tori and constitute parallel dynamics on the tori in the integrable cases and non-parallel dynamics on the tori for certain perturbed quasi-integrable cases. Therefore, we are dealing with a Kolmogorov-Arnold-Moser KAM theory for nonlinear partial differential wave equations. The nonlinear Fourier series have particular nonlinear Fourier modes, including: sine waves, Stokes waves and solitons. Indeed the theoretical formulation I have developed is a kind of exact two-dimensional “coherent wave turbulence” or “integrable wave turbulence” for the KP equation, for which the Stokes waves and solitons are the coherent structures. I discuss how NLFA provides a number of new tools that apply to a wide range of problems in offshore engineering and coastal dynamics: This includes nonlinear Fourier space and time series analysis, nonlinear Fourier wave field analysis, a nonlinear random phase approximation, the study of nonlinear coherent functions and nonlinear bi and tri spectral analysis.

Nonlinear Fourier Analysis with Sine Wave, Stokes Wave and Rogue Wave Basis Functions: A Paradigm Change in the Understanding of Nonlinear Waves

2020 ◽  
Author(s):  
Alfred Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Rogue Wave ◽  
Random Phase ◽  
Stokes Wave ◽  
Random Wave ◽  
Linear Dispersion ◽  
Wide Range ◽  
Stokes Waves ◽  
Wave Trains

<p>I give a new perspective for the description of nonlinear water wave trains using mathematical methods I refer to as nonlinear Fourier analysis (NLFA). I discuss how this approach holds for one-space and one time dimensions (1+1) and for two-space and one time dimensions (2+1) to all orders of approximation. I begin with the nonlinear Schroedinger (NLS) equation in 1+1 dimensions: Here the NLFA method is derived from the complete integrability of the equation by the periodic inverse scattering transform. I show how to compute the nonlinear Fourier series that exactly solve 1+1 NLS. I then show how to extend the order of 1+1 NLS to the Dysthe and the extended Dysthe equations. I also show how to include directional spreading in the formulation so that I can address the 2+1 NLS, the 2+1 Dysthe and the 2+1 Trulsen-Dysthe equations. This hierarchy of equations extends formally all the way to the Zakharov equations in the infinite order limit. Each order and extension from 1+1 to 2+1 dimensions is characterized by its own modulational dispersion relation that is required at each order of the NLFA formalism. NLFA is characterized by its own fundamental nonlinear Fourier series, which has particular nonlinear Fourier modes: sine waves, Stokes waves and breather trains. We are all familiar with sine waves (known for centuries) and Stokes waves (known since the Stokes paper in 1847). Breather trains have become known over the past three decades as a major source of rogue or freak waves in the ocean: Breather packets are known to pulse up and down during their evolution. At the moment of the maximum amplitude the largest wave in a breather packet is often referred to as a “rogue” or “freak” wave. Such extreme packets are known to be “coherent structures" so that pure linear dispersion does not occur as in a linear packet. Instead the breather packets have components that are phase locked with each other and hence remain coherent and are “long lived” just as vortices do in classical turbulence. Because the breathers live for a long time, the notion of risk based upon linear dispersion, as used in the oil and shipping industries, must be revised upwards. I discuss how to apply NLFA to (1) nonlinearly Fourier analyze time series, (2) to analyze wave fields from radar, lidar and synthetic aperture radar measurements, (3) how to treat NLFA to describe nonlinear, random wave trains using a kind of nonlinear random phase approximation and (4) how to compute the nonlinear power spectrum in terms of the parameters used to describe the rogue wave Fourier modes in a random wave train. Thus the emphasis here is to discuss a number of new tools for nonlinear Fourier analysis in a wide range of problems in the field of ocean surface waves.</p>

Stability-Optimized Clearance Configuration of Fluid-Film Bearings

2006 ◽  
Vol 129 (1) ◽  
pp. 106-111 ◽  
Author(s):  
Koichi Matsuda ◽  
Shinya Kijimoto ◽  
Yoichi Kanemitsu
Keyword(s):  
Fourier Series ◽  
Load Capacity ◽  
Fluid Film ◽  
Eccentricity Ratio ◽  
Rotating Speed ◽  
Fluid Film Bearings ◽  
Wide Range ◽  
Jeffcott Rotor ◽  
The Stability ◽  
Frequency Ratios

The whirl instability occurs at higher rotating speeds for a full circular fluid-film journal bearing, and many types of clearance configuration have been proposed to solve this instability problem. A clearance configuration of fluid-film journal bearings is optimized in a sense of enhancing the stability of the full circular bearing at high rotational speeds. A performance index is chosen as the sum of the squared whirl-frequency ratios over a wide range of eccentricity ratios, and a Fourier series is used to represent an arbitrary clearance configuration of fluid-film bearings. An optimization problem is then formulated to find the Fourier coefficients to minimize the index. The designed bearing has a clearance configuration similar to that of an offset two-lobe bearing for smaller length-to-diameter ratios. It is shown that the designed bearing cannot destabilize the Jeffcott rotor at any high rotating speed for a wide range of eccentricity ratio. The load capacity of the designed bearings is nearly in the same magnitude as that of the full circular bearing for smaller length-to-diameter ratios. The whirl-frequency ratios of the designed bearing are very sensitive to truncating higher terms of the Fourier series for some eccentricity ratio. The designed bearings successfully enhance the stability of a full circular bearing and are free from the whirl instability.

NUMERICAL ANALYSES OF LEAF AND FRUIT EXTERNAL MORPHOLOGY IN Moringa oleifera LAM

2015 ◽  
Vol 77 (13) ◽  
Author(s):  
Daniel Andrawus Zhigila ◽  
Sulaiman Mohammed ◽  
Felix Ayodele Oladele ◽  
Fatima B. J. Sawa
Keyword(s):  
Moringa Oleifera ◽  
Crop Improvement ◽  
Morphological Diversity ◽  
External Morphology ◽  
Numerical Analyses ◽  
Qualitative And Quantitative ◽  
Quantitative Characters ◽  
Diversity Assessment ◽  
Wide Range ◽  
Improvement Programme

The wide range of uses of Moringa oleifera in recent time has witnessed increasing demand of its foliar and seed products in nutritional, medical and ecological applications. The upsurge of demand for these products needs to be balanced with new varieties of improved performance to meet the supply chain. To achieve this, morphological diversity assessment is prerequisite for future crop improvement programme. Therefore, numerical analyses of the external morphology of leaf and fruit of thirty accessions of Moringa oleifera were assessed. The study was carried out on both qualitative and quantitative characters to assess the diversity at morphological level to establish the phenetic relationships and the delimitation of accessions. Relationship studies showed considerable correlation between the leaf and fruits characters that produced clear and reproducible threats and were selected for diversity study. Numerical analysis of the qualitative and quantitative characters clustered the accessions into five groups – operational taxonomic units (OTUs) 1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30 were clustered in group one; OTUs 6 and 8 were clustered in group two and three respectively; OTUs 15 and 16 in group four and OUT 23 in group five cluster membership. Principal Component Analysis was carried out to augment the Cluster Analysis which showed large morphological diversity existing in accessions of Moringa oleifera hence, infraspecific classification is hereby proposed.  These analysis particularly traits related to leaf and fruits yield can also be utilised for crop improvement programme.

Nonlinear Fourier Analysis for Shallow Water Waves

2021 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Shallow Water ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Wave Equations ◽  
Superposition Principle ◽  
Broad Band ◽  
Kp Equation ◽  
Linear Superposition

Abstract I consider nonlinear wave motion in shallow water as governed by the KP equation plus perturbations. I have previously shown that broad band, multiply periodic solutions of the KP equation are governed by quasiperiodic Fourier series [Osborne, OMAE 2020]. In the present paper I give a new procedure for extending this analysis to the KP equation plus shallow water Hamiltonian perturbations. We therefore have the remarkable result that a complex class of nonlinear shallow water wave equations has solutions governed by quasiperiodic Fourier series that are a linear superposition of sine waves. Such a formulation is important because it was previously thought that solving nonlinear wave equations by a linear superposition principle was impossible. The construction of these linear superpositions in shallow water in an engineering context is the goal of this paper. Furthermore, I address the nonlinear Fourier analysis of experimental data described by shallow water physics. The wave fields dealt with here are fully two-dimensional and essentially consist of the linear superposition of generalized cnoidal waves, which nonlinearly interact with one another. This includes the class of soliton solutions and their associated Mach stems, both of which are important for engineering applications. The newly discovered phenomenon of “fossil breathers” is also characterized in the formulation. I also discuss the exact construction of Morison equation forces on cylindrical piles in terms of quasiperiodic Fourier series.

A Bandpass Active Filter for Fourier Analysis Laboratory

1995 ◽  
Vol 32 (4) ◽  
pp. 350-353
Author(s):  
Marcelo Basilio Joaquim
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Signal Analysis ◽  
Active Filter ◽  
Experimental Demonstration ◽  
High Q ◽  
State Variable ◽  
Analysis Laboratory ◽  
Tuning Frequency

A bandpass active filter for Fourier analysis laboratory In this note is presented a modified bandpass state-variable active filter which exhibits constant gain, high-Q, where its tuning frequency can easily be varied. This circuit proves to be useful for experimental demonstration of the Fourier series in communications or signal analysis laboratories.

scholarly journals On Fourier Series of Fuzzy-Valued Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Uğur Kadak ◽  
Feyzi Başar
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Uniform Convergence ◽  
Level Sets ◽  
Closed Interval ◽  
Complex Form ◽  
Dirichlet Kernel ◽  
Extension Principle ◽  
Science And Engineering ◽  
Interest In Science

Fourier analysis is a powerful tool for many problems, and especially for solving various differential equations of interest in science and engineering. In the present paper since the utilization of Zadeh’s Extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct a fuzzy-valued function on a closed interval via related membership function. We derive uniform convergence of a fuzzy-valued function sequences and series with level sets. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, Fourier series of periodic fuzzy-valued functions is defined and its complex form is given via sine and cosine fuzzy coefficients with an illustrative example. Finally, by using the Dirichlet kernel and its properties, we especially examine the convergence of Fourier series of fuzzy-valued functions at each point of discontinuity, where one-sided limits exist.

Paleohaemoproteus burmacis gen. n., sp. n. (Haemospororida: Plasmodiidae) from an Early Cretaceous biting midge (Diptera: Ceratopogonidae)

Parasitology ◽  
2005 ◽  
Vol 131 (1) ◽  
pp. 79-84 ◽  
Author(s):  
G. POINAR ◽  
S. R. TELFORD
Keyword(s):  
Early Cretaceous ◽  
Malaria Parasite ◽  
Developmental Stages ◽  
Abdominal Cavity ◽  
Malarial Parasite ◽  
Fossil Species ◽  
Extinct Species ◽  
Biting Midge ◽  
Extant Species ◽  
Wide Range

Paleohaemoproteus burmacis gen. n., sp. n. (Haemospororida: Plasmodiidae) is described from the abdominal cavity of a female biting midge (Diptera: Ceratopogonidae) preserved in 100 million year old amber from Myanmar (Burma). The description is based on the developmental stages of oocysts and sporozoites. The fossil species differs from extant species of Haemoproteus by its wide range of oocyst sizes, small sporozoites and occurrence in an extinct species of biting midge. Numerous sporozoites in the abdominal cavity suggest that the biting midge was an effective vector of this malarial parasite. Characters of the biting midge suggest that the host was a large, cold-blooded vertebrate. This is the earliest record of a malaria parasite and first indication that Early Cretaceous reptiles were infected with haemosporidial parasites.

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