Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

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Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation System

Volume 2E: Turbomachinery ◽

10.1115/gt2020-14754 ◽

2020 ◽

Author(s):

Sen Zhang ◽

Dingxi Wang ◽

Yi Li ◽

Hangkong Wu ◽

Xiuquan Huang

Keyword(s):

Fourier Transform ◽

Stability Analysis ◽

Condition Number ◽

Inverse Fourier Transform ◽

Real Life ◽

Equation System ◽

Time Instant ◽

Transform Matrix ◽

Spectral Equation ◽

The Impact

Abstract The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

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The Bernstein Projector Determined by a Weak Associate Class of Good Cosets

International Mathematics Research Notices ◽

10.1093/imrn/rnab161 ◽

2021 ◽

Author(s):

Yeansu Kim ◽

Loren Spice ◽

Sandeep Varma

Keyword(s):

Fourier Transform ◽

Characteristic Function ◽

Lie Algebra ◽

Equivalence Relation ◽

Characteristic Zero ◽

Closed Subset ◽

Inverse Fourier Transform ◽

Reductive Group ◽

Associate Class ◽

Weak Associate

Abstract Let ${\text G}$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$, and let $G={\text G}(F)$. In [ 15], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for ${\text G}$ in the sense of A. Moy and G. Prasad. Following [ 15], we attach to the set $\overline{\mathfrak{s}}$ of good $K$-types in a weak associate class of positive-depth unrefined minimal $K$-types a ${G}$-invariant open and closed subset $\mathfrak{g}_{\overline{\mathfrak{s}}}$ of the Lie algebra $\mathfrak{g} = {\operatorname{Lie}}({\text G})(F)$, and a subset $\tilde{{G}}_{\overline{\mathfrak{s}}}$ of the admissible dual $\tilde{{G}}$ of ${G}$ consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak{s}}$. Then $\tilde{{G}}_{\overline{\mathfrak{s}}}$ is the union of finitely many Bernstein components of ${G}$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak{s}}}$ that it determines. We show that $E_{\overline{\mathfrak{s}}}$ vanishes outside the Moy–Prasad ${G}$-domain ${G}_r \subset{G}$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak{s}}}$ to ${G}_r\,$, pushed forward via the logarithm to the Moy–Prasad ${G}$-domain $\mathfrak{g}_r \subset \mathfrak{g}$, agrees on $\mathfrak{g}_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak{g}_{\overline{\mathfrak{s}}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan, and Y. Varshavsky in [8] for the depth-$r$ Bernstein projector.

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On the Inverse Fourier Transform of the Planck-Einstein law

10.20944/preprints202109.0369.v1 ◽

2021 ◽

Author(s):

Alireza Jamali

Keyword(s):

Fourier Transform ◽

Inverse Fourier Transform ◽

Maximum Frequency ◽

Principle Of Maximum

After proposing a natural metric for the space in which particles spin which implements the principle of maximum frequency, E=hf is generalised and its inverse Fourier transform is calculated.

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BASES FOR Sk(Γ1(4)) AND FORMULAS FOR EVEN POWERS OF THE JACOBI THETA FUNCTION

International Journal of Number Theory ◽

10.1142/s1793042113500693 ◽

2013 ◽

Vol 09 (08) ◽

pp. 1973-1993 ◽

Cited By ~ 1

Author(s):

SHINJI FUKUHARA ◽

YIFAN YANG

Keyword(s):

Eisenstein Series ◽

Theta Function ◽

Congruence Subgroup ◽

Negative Integer ◽

Cusp Forms ◽

Jacobi Theta Function

We find a basis for the space Sk(Γ1(4)) of cusp forms of weight k for the congruence subgroup Γ1(4) in terms of Eisenstein series. As an application, we obtain formulas for r2k(n), the number of ways to represent a non-negative integer n as sums of 2k integer squares.

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LEGENDRE POLYNOMIALS AS THE NORMALIZATION OF PHOTON-SUBTRACTED SQUEEZED STATES

Modern Physics Letters A ◽

10.1142/s021773230902996x ◽

2009 ◽

Vol 24 (20) ◽

pp. 1597-1603 ◽

Cited By ~ 10

Author(s):

HONG-YI FAN ◽

LI-YUN HU ◽

XUE-XIANG XU

Keyword(s):

Legendre Polynomial ◽

Legendre Polynomials ◽

Hermite Polynomial ◽

Vacuum State ◽

Squeezed States ◽

Squeezed State ◽

Normalization Factor ◽

Squeezed Vacuum State ◽

Squeezed Vacuum ◽

Excitation State

By converting the photon-subtracted squeezed state (PSSS) to a squeezed Hermite-polynomial excitation state we find that the normalization factor of PSSS is an m-order Legendre polynomial of the squeezing parameter, where m is the number of subtracted photons. Some new relations about the Legendre polynomials are obtained by this analysis. We also show that the PSSS can also be treated as a Hermite-polynomial excitation on squeezed vacuum state.

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Quantitative characterization, classification and reconstruction of oocyst shapes of Eimeria species from cattle

Parasitology ◽

10.1017/s003118209700187x ◽

1998 ◽

Vol 116 (1) ◽

pp. 21-28 ◽

Cited By ~ 17

Author(s):

C. SOMMER

Keyword(s):

Fourier Transform ◽

Quantitative Data ◽

Inverse Fourier Transform ◽

Morphological Variability ◽

Eimeria Species ◽

Equal Weight ◽

Agglomerative Clustering ◽

Multivariate Statistical ◽

Mean Values ◽

Size And Shape

This study reports on morphological variability of Eimeria species, which may be given either by drawings or as quantitative data. The drawings may be used to facilitate identification by eye of ‘unknown’ Eimeria specimens, whereas quantitative data may serve as a reference set for identification by multivariate statistical techniques. The morphology of 810 Eimeria specimens was defined in binary (b/w) digital images by pixels of their oocyst outline. A Fourier transform of pixel positions yielded size and shape features. To classify coccidia, the quantitative data were employed in an agglomerative clustering by average linkage algorithm with equal weight assigned to size and shape. An inverse Fourier transform served to reconstruct oocyst outlines, i.e. outlines of average shape and size, from mean values of features in resulting clusters. Clusters were subsequently identified based on their average morphology by comparison with drawings of species in an earlier taxonomical work. Five hundred oocyst outlines were simulated for each cluster representing a species, and shape/size variability was presented in contour diagrams. Differences in species shapes, and correspondence in length and width, were seen after reconstruction by inverse Fourier transform and comparison with earlier studies.

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Theta function identities from optical neural network transformations

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171293001000 ◽

1993 ◽

Vol 16 (4) ◽

pp. 805-810

Author(s):

E. Elizalde ◽

A. Romeo

Keyword(s):

Neural Network ◽

Theta Function ◽

Heisenberg Model ◽

Symmetric Functions ◽

New Approach ◽

Jacobi Theta Function ◽

Cylindrically Symmetric ◽

Optical Neural Network ◽

Theta Function Identities

We take a new approach to the generation of Jacobi theta function identities. It is complementary to the procedure which makes use of the evaluation of Parseval-like identities for elementary cylindrically-symmetric functions on computer holograms. Our method is more simple and explicit than this one, which was an outcome of the construction of neurocomputer architectures through the Heisenberg model.

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Random regression models to estimate genetic parameters for milk production of Guzerat cows using orthogonal Legendre polynomials

Pesquisa Agropecuária Brasileira ◽

10.1590/s0100-204x2014000500007 ◽

2014 ◽

Vol 49 (5) ◽

pp. 372-383 ◽

Cited By ~ 4

Author(s):

Maria Gabriela Campolina Diniz Peixoto ◽

Daniel Jordan de Abreu Santos ◽

Rusbel Raul Aspilcueta Borquis ◽

Frank Ângelo Tomita Bruneli ◽

João Cláudio do Carmo Panetto ◽

...

Keyword(s):

Milk Production ◽

Legendre Polynomial ◽

Legendre Polynomials ◽

Regression Models ◽

Environmental Effect ◽

Genetic Parameters ◽

Genetic Effect ◽

Random Regression ◽

Additive Genetic Effect ◽

Fifth Order

The objective of this work was to compare random regression models for the estimation of genetic parameters for Guzerat milk production, using orthogonal Legendre polynomials. Records (20,524) of test-day milk yield (TDMY) from 2,816 first-lactation Guzerat cows were used. TDMY grouped into 10-monthly classes were analyzed for additive genetic effect and for environmental and residual permanent effects (random effects), whereas the contemporary group, calving age (linear and quadratic effects) and mean lactation curve were analized as fixed effects. Trajectories for the additive genetic and permanent environmental effects were modeled by means of a covariance function employing orthogonal Legendre polynomials ranging from the second to the fifth order. Residual variances were considered in one, four, six, or ten variance classes. The best model had six residual variance classes. The heritability estimates for the TDMY records varied from 0.19 to 0.32. The random regression model that used a second-order Legendre polynomial for the additive genetic effect, and a fifth-order polynomial for the permanent environmental effect is adequate for comparison by the main employed criteria. The model with a second-order Legendre polynomial for the additive genetic effect, and that with a fourth-order for the permanent environmental effect could also be employed in these analyses.

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