Статті в журналах: "Fourier transformations"

1

Azooz, A. "Experimentation with Fourier Transformations." International Journal of Electrical Engineering & Education 49, no. 2 (April 2012): 179–96. http://dx.doi.org/10.7227/ijeee.49.2.8.

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2

Treyderowski, Krzysztof, and Christoph Schwarzweller. "Multiplication of Polynomials using Discrete Fourier Transformation." Formalized Mathematics 14, no. 4 (January 1, 2006): 121–28. http://dx.doi.org/10.2478/v10037-006-0015-y.

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Multiplication of Polynomials using Discrete Fourier Transformation In this article we define the Discrete Fourier Transformation for univariate polynomials and show that multiplication of polynomials can be carried out by two Fourier Transformations with a vector multiplication in-between. Our proof follows the standard one found in the literature and uses Vandermonde matrices, see e.g. [27].

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3

Zolesio, Jean-Luc. "Transformations de Fourier discrètes: algorithmes rapides." Annales des Télécommunications 40, no. 9-10 (September 1985): 495–507. http://dx.doi.org/10.1007/bf02998222.

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4

Abe, Sumiyoshi, and John T. Sheridan. "Almost-Fourier and almost-Fresnel transformations." Optics Communications 113, no. 4-6 (January 1995): 385–88. http://dx.doi.org/10.1016/0030-4018(94)00521-u.

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5

Sadek, Bassel A., Elliot W. Martin, and Susan A. Shaheen. "Forecasting Truck Parking Using Fourier Transformations." Journal of Transportation Engineering, Part A: Systems 146, no. 8 (August 2020): 05020006. http://dx.doi.org/10.1061/jtepbs.0000397.

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6

Ginzburg, V. A. "Fourier?langlands transformations on reductive groups." Functional Analysis and Its Applications 22, no. 2 (1988): 143–44. http://dx.doi.org/10.1007/bf01077610.

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7

Ernst, Richard R. "Kernresonanz-Fourier-Transformations-Spektroskopie (Nobel-Vortrag)." Angewandte Chemie 104, no. 7 (July 1992): 817–36. http://dx.doi.org/10.1002/ange.19921040704.

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8

Azhar, Aurizan Himmi, Sugiyanto Sugiyanto, Muhammad Wakhid Musthofa, and Muhamad Zaki Riyanto. "Transformasi Fourier Multiplikatif Dan Aplikasinya Pada Persamaan Diferensial Multiplikatif." Jurnal Derivat: Jurnal Matematika dan Pendidikan Matematika 8, no. 2 (December 20, 2021): 149–60. http://dx.doi.org/10.31316/j.derivat.v8i2.1996.

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This research is a development of multiplicative calculus. This study is about the Fourier multiplicative transformation and its application to the multiplicative differential equation. This study aims to determine the Fourier multiplicative transformation as well as the multiplicative differential equation. This study contains numerical simulations to solve the problem of ordinary multiplicative differential equations of the first order. The methods used in this research are descriptive research methods through the study of literature. The results of this study are the application of multiplicative Fourier transformations to multiplicative differential equations and numerical solutions of ordinary multiplicative differential equations with the Adam Bashforth-Moulton multiplicative method. Keywords: Multiplicative Calculus, Fourier Multiplicative Transformation, Multiplicative Differential Equations, Adams Bashforth Moulton Multiplicative Method

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9

Parfentyev, Nikolay, Schubert Maignan, and Irina Tkachenko. "Delta function of the Fourier transform." E3S Web of Conferences 535 (2024): 01006. http://dx.doi.org/10.1051/e3sconf/202453501006.

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In the analysis of the double window Fourier transform, it is found that both components of the direct transformation contain an equal amount of information sufficient to restore the original time signal. A double conversion diagram has been constructed. For any initial signal, there is a conjugate signal having the same frequency spectrum. Analysis of the harmonic signal transformation showed that at an arbitrary phase of the harmonic, the frequency representation contains two functions: conditionally unipolar with a maximum at the initial frequency and bipolar with a zero value at the specified frequency. For classical Fourier transformations with an infinite limit of integration, these functions turn into two delta functions with the same properties. The practical application of the found patterns is currently limited by the lack of an algorithm that allows the use of data from only one component.

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10

Langenbruch, Michael. "Asymptotic Fourier and Laplace transformations for hyperfunctions." Studia Mathematica 205, no. 1 (2011): 41–69. http://dx.doi.org/10.4064/sm205-1-4.

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11

Blank, R. "Holographic lenses for performing complete Fourier transformations." Optical Engineering 31, no. 3 (1992): 544. http://dx.doi.org/10.1117/12.56093.

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12

Younis, M. S. "Fourier transformations of functions with symmetrical differences." Acta Mathematica Hungarica 51, no. 3-4 (September 1988): 293–99. http://dx.doi.org/10.1007/bf01903336.

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13

Grabow, Jens-Uwe. "Fourier-Transformations-Mikrowellenspektroskopie: Händigkeit durch Rotationskohärenz gefasst." Angewandte Chemie 125, no. 45 (October 2, 2013): 11914–16. http://dx.doi.org/10.1002/ange.201307159.

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14

Fu, Lei. "Calculation of ℓ-adic local Fourier transformations". manuscripta mathematica 133, № 3-4 (1 липня 2010): 409–64. http://dx.doi.org/10.1007/s00229-010-0377-x.

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15

Ciulla, Carlo. "Theory and Applications of Fourier, Laplace, and Z Transformations." Journal of CIEES 2, no. 1 (July 26, 2022): 32–43. http://dx.doi.org/10.48149/jciees.2022.2.1.6.

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This study presents the mathematics for the implementation of direct and inverse Fourier, Laplace, and Z transformations. This research is at the intersection between signal processing, applied mathematics, and software engineering, and it provides a study guide to implementers. Mathematical concepts and details necessary to transform the math into code are provided as theoretical background. Validation is conducted for the cases when the transforms do intersect, when the transforms do not intersect, and when, in Fourier and Z-transformations, the frequency domain encodes a phase shift which is reconstructed as an image space shift. Coherence between the software implementation of the three transformations is confirmed when: 1. The real component of the complex variable s = σ + i ω is σ = 0, which is the case when Fourier and Laplace transforms are the same. 2. When the magnitude of the complex variable z = r e jω is r = 1, which is the case when Fourier and Z transforms are the same. Congruency between software implementation of transformations is confirmed comparing departing image and inverse reconstructed image. The novelty of this research is the presentation style of the theory of direct and inverse Fourier, Laplace, and Z transforms. Details provided in this research make this paper a study guide that is not found elsewhere.

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16

Pavlov V., Andrey. "The Fourier, Laplace Transformations and the Newton Potential." American Journal of Applied Mathematics and Statistics 2, no. 6 (November 5, 2014): 398–401. http://dx.doi.org/10.12691/ajams-2-6-7.

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17

Haider, Q., and L. C. Liu. "Fourier or Bessel transformations of highly oscillatory functions." Journal of Physics A: Mathematical and General 25, no. 24 (December 21, 1992): 6755–60. http://dx.doi.org/10.1088/0305-4470/25/24/026.

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18

Rezaee, S., M. Mohammadpour, and A. R. Soltani. "Stationary Processes via Fourier Transformations: Representation and Estimation." Iranian Journal of Science and Technology, Transactions A: Science 43, no. 3 (June 21, 2018): 937–45. http://dx.doi.org/10.1007/s40995-018-0590-0.

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19

Zotov, A. "Relativistic elliptic matrix tops and finite Fourier transformations." Modern Physics Letters A 32, no. 32 (October 12, 2017): 1750169. http://dx.doi.org/10.1142/s0217732317501693.

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We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the “off-shell” Lax pairs, which do not satisfy the Lax equations in general case but become true Lax pairs under various conditions (reductions). At the level of the off-shell Lax matrix, there is a natural symmetry between the spectral parameter z and relativistic parameter [Formula: see text]. It is generated by the finite Fourier transformation, which we describe in detail. The symmetry allows one to consider z and [Formula: see text] on an equal footing. Depending on the type of integrable reduction, any of the parameters can be chosen to be the spectral one. Then another one is the relativistic deformation parameter. As a by-product, we describe the model of N2 interacting GL(M) matrix tops and/or M2 interacting GL(N) matrix tops depending on a choice of the spectral parameter.

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20

Hudson, R. L. "On Hausdorff-Young inequalities for quantum fourier transformations." Ukrainian Mathematical Journal 49, no. 3 (March 1997): 514–22. http://dx.doi.org/10.1007/bf02487247.

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21

Fan, Hong-Yi, and Jun-Hua Chen. "On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations." Frontiers of Physics 10, no. 1 (February 2015): 1–6. http://dx.doi.org/10.1007/s11467-014-0445-x.

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22

Salamat, Kaushef, and Nousheen Ilyas. "DUALITIES BETWEEN FOURIER SINE AND SOME USEFUL INTEGRAL TRANSFORMATIONS." Journal of Mathematical Sciences & Computational Mathematics 2, no. 4 (July 5, 2021): 542–63. http://dx.doi.org/10.15864/jmscm.2408.

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The most useful technique of the mathematics which are used to finding the solutions of a lot of problems just like bending of beam, electrical network, heat related problems, which occurs in many disciplines of engineering and sciences are the techniques of integral transforms. In our research I discussed the duality between Fourier Sine transforms and some others effective integral transforms (namely Laplace transform, Mahgoub transform, Aboodh transform and Mohand transform). To justify the scope of dualities relation between Fourier Sine transform and other integral transforms (that are mentioned above, I presented the tabular representation of integral transform (namely Laplace transform, Aboodh transform, Mohand transform and Mahgoub transform) of various used functions by using Fourier Sine and other integral transforms dualities relation to signify fruitfulness of such connections. Results showed that these integral transform are strongly related with Fourier Sine transform.

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23

Hu, Liang‐Zie, and George A. McMechan. "Wave‐field transformations of vertical seismic profiles." GEOPHYSICS 52, no. 3 (March 1987): 307–21. http://dx.doi.org/10.1190/1.1442305.

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Vertical seismic profile (VSP) data may be partitioned in a variety of ways by application of wave‐field transformations. These transformations provide insights into the nature of the data and aid in the design of processing operations. Transformations are implemented in a reversible sequence that takes the observed VSP data from the depth‐time (z-t) domain through the slowness‐time intercept (p-τ) domain (by a slant stack), to the slowness‐frequency (p-ω) domain (by a 1-D Fourier transform over τ), to the wavenumber‐frequency (k-ω) domain (by resampling using the Fourier central‐slice theorem), and finally back to the z-t domain (by an inverse 2-D Fourier transform). Multidimensional wave‐field transformations, combined with k-ω, p-ω, and p-τ filtering, can be applied to wave‐field resampling, interpolation, and extrapolation; separation of P-waves and S-waves; separation of upgoing and downgoing waves; and wave‐field decomposition for isolation, identification, and analysis of arrivals.

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24

Sharyn, S. V. "Algebraic and differential properties of polynomial Fourier transformation." Matematychni Studii 53, no. 1 (March 17, 2020): 59–68. http://dx.doi.org/10.30970/ms.53.1.59-68.

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Methods of integral transformations of (generalized) functions are widely used in the solution of initial and boundary value problems for partial differential equations. However, many problems in applied mathematics require a nonlinear generalization of distribution spaces. Besides, an algebraic structure of a space of distributions is desirable, which is needed, for example, in quantum field theory.In the article, we use the adjoint operator method as well as technique of symmetric tensor products to extended the Fourier transformation onto the spaces of so-called polynomial rapidly decreasing test functions and polynomial tempered distributions. In such spaces it is possible to solve some Cauchy problems, for example, infinite dimensional heat equation associated with the Gross Laplacian.Algebraic and differential properties of the polynomial Fourier transformation are investigated. We prove some analogical to classical properties of this map. Unlike to the classic case, the spaces of polynomial test and generalized functions have algebraic structure. We prove that polynomial Fourier transformation acts as homomorphism of appropriate algebras. It is clear that the classical analogue of such property is absent.

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25

AGHILI, S. ALIREZA, DIVYAKANT AGRAWAL, and AMR EL ABBADI. "SEQUENCE SIMILARITY SEARCH USING DISCRETE FOURIER AND WAVELET TRANSFORMATION TECHNIQUES." International Journal on Artificial Intelligence Tools 14, no. 05 (October 2005): 733–54. http://dx.doi.org/10.1142/s0218213005002363.

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In this paper, we study the problem of sequence similarity search. We incorporate vector transformations and apply DFT (Discrete Fourier Transformation) and DWT (Discrete Wavelet Transformation, Haar) dimensionality reduction techniques to reduce the search space/time of sequence similarity range queries. Our empirical results on a number of Prokaryote and Eukaryote DNA contig databases demonstrate up to 50-fold filtration ratio reduction of the search space and up to 13 times faster filtration. The proposed transformation techniques may easily be integrated as a pre-processing phase on top of current similarity search heuristics/techniques such as BLAST, PatternHunter, FastA and QUASAR to efficiently prune non-relevant sequences. We study the precision of applying dimensionality reduction techniques for faster and more efficient range query searches and discuss the imposed trade-offs.

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26

Critchley-Marrows, Joshua, Samanvay Karambhe, Denzil Khan, Elias Vasilikas, and Gareth A. Vio. "Identification of Nonlinearities Using Computational Approaches." Applied Mechanics and Materials 846 (July 2016): 559–64. http://dx.doi.org/10.4028/www.scientific.net/amm.846.559.

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This paper presents an analysis into the computational results for modelling of a two degree-of-freedom nonlinear vibrating structure. Fast Fourier Transformations, Short Time Fourier Transformations, Hilbert Transformations and wavelets are used to model this system. These techniques aim to locate and quantify the nonlinear behaviour of the system. Coulomb friction was detected with a number of these techniques, however other nonlinearities could not be detected. From the analysis conducted, improved computational methods are necessary for the detection of nonlinearities.

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27

Sobchuk, Valentin, and Galina Kharkevych. "DEPENDENCE OF THE QUALITY OF MACHINE TRANSLATION OF THE TEXT ON THE USED FOURIER TRANSFORMATION." Journal of Automation and Information sciences 4 (July 1, 2021): 69–80. http://dx.doi.org/10.34229/1028-0979-2021-4-7.

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Machine translation is widely used in the translation of commercial, technical, scientific information that is connected with the process of globalization and, accordingly, the expansion of the network of business relations. Mathematical methods related to machine translation of the texts have recently received new development due to the intensive development of Fourier transformation theory. Thus, the requirements for filtering accuracy in the processing of contrast signals and images have increased, allowing to create efficient filtering algorithms. Frequency algorithms are the most efficient of all the existing filtering algorithms, i.e., those where the coefficients of decomposition of the noisy signal by Fourier basis are the subject to processing. When using Fourier filtering algorithms, the properties of Fourier transformation play an important role, that depend on belonging to a particular class of differential functions. The necessary condition for the existence of the continuous Fourier transformation is the absolute convergence of some functions by means of which the real studied process is describing. In practice, the so-called “summation functions” are often used as simulated functions, which can be constructed using a linear matrix summation of Fourier series. As for the latter, scientists distinguish between both triangular and rectangular linear matrix methods. This paper is devoted to the study of the convergence conditions of Fourier transformations of both triangular and rectangular linear matrix methods for summing Fourier series. Moreover, this article shows that the rate of convergence of Fourier transformation of the rectangular linear Abel-Poisson method is at times faster than the rate of convergence of the analogous triangular linear Abel-Poisson method. This result can further significantly influence the choice of the more effective Fourier transformation used in the process of machine translation of the text.

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28

Polovynko, Ihor, and Lubomyr Kniazevich. "Improvement of images by using graduate transformations of their Fourier depictions." Technology audit and production reserves 2, no. 2(58) (April 30, 2021): 16–19. http://dx.doi.org/10.15587/2706-5448.2021.230079.

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The object of research is low-quality digital images. The presented work is devoted to the problem of digital processing of low quality images, which is one of the most important tasks of data science in the field of extracting useful information from a large data set. It is proposed to carry out the process of image enhancement by means of tonal processing of their Fourier images. The basis for this approach is the fact that Fourier images are described by brightness values in a wide range of values, which can be significantly reduced by gradation transformations. The work carried out the Fourier transform of the image with the separation of the amplitude and phase. The important role of the phase in the process of forming the image obtained after the implementation of the inverse Fourier transform is shown. Although the information about the signal amplitude is lost during the phase analysis, nevertheless all the main details correspond accurately to the initial image. This suggests that when modifying the Fourier spectra of images, it is necessary to take into account the effect on both the amplitude and the phase of the object under study. The effectiveness of the proposed method is demonstrated by the example of space images of the Earth's surface. It is shown that after the gradation logarithmic Fourier transform of the image and the inverse Fourier transform, an image is obtained that is more contrasting than the original one, will certainly facilitate the work with it in the process of visual analysis. To explain the results obtained, the schedule of the obtained gradation transformation into the Mercator series was carried out. It is shown that the resulting image consists of two parts. The first of them corresponds to the reproduction of the original image obtained by the inverse Fourier transform, and the second performs smoothing of its brightness, similar to the action of the combined method of spatial image enhancement. When using the proposed method, preprocessing is also necessary, which, as a rule, includes operations necessary for centering the Fourier image, as well as converting the original data into floating point format.

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29

Gafiyatov, R. N. "Acoustic waves in two-fraction mixtures of liquid with vapor-gas bubbles." Proceedings of the Mavlyutov Institute of Mechanics 9, no. 1 (2012): 65–68. http://dx.doi.org/10.21662/uim2012.1.011.

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The mathematical model of two-fractional mixture of liquid with vapor-gas bubbles of different gases and sizes with phase transformations is presented. The dispersive equation is received, dispersive curves that determine the propagation of acoustic disturbances was plotted. Calculations on the propagation of impulse pressure perturbations were performed by means of a fast Fourier transformation method.

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30

He, Y., K. Hueske, J. Götze, and E. Coersmeier. "Matrix-Vector Based Fast Fourier Transformations on SDR Architectures." Advances in Radio Science 6 (May 26, 2008): 89–94. http://dx.doi.org/10.5194/ars-6-89-2008.

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Abstract. Today Discrete Fourier Transforms (DFTs) are applied in various radio standards based on OFDM (Orthogonal Frequency Division Multiplex). It is important to gain a fast computational speed for the DFT, which is usually achieved by using specialized Fast Fourier Transform (FFT) engines. However, in face of the Software Defined Radio (SDR) development, more general (parallel) processor architectures are often desirable, which are not tailored to FFT computations. Therefore, alternative approaches are required to reduce the complexity of the DFT. Starting from a matrix-vector based description of the FFT idea, we will present different factorizations of the DFT matrix, which allow a reduction of the complexity that lies between the original DFT and the minimum FFT complexity. The computational complexities of these factorizations and their suitability for implementation on different processor architectures are investigated.

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31

Azana, J., and M. A. Muriel. "Real-time Fourier transformations performed simultaneously over multiwavelength signals." IEEE Photonics Technology Letters 13, no. 1 (January 2001): 55–57. http://dx.doi.org/10.1109/68.903219.

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32

Goginava, Ushangi, Sawsan Omira, Reem Abdel-Latif, and Tasnem AlKrinat. "Matrix transformations of sequences and applications in Fourier analysis." Heliyon 10, no. 8 (April 2024): e29585. http://dx.doi.org/10.1016/j.heliyon.2024.e29585.

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33

Dubrovkin, Joseph. "Linear transformations of multivariate calibration models in near infrared spectroscopy: A comparative study." Journal of Near Infrared Spectroscopy 25, no. 4 (July 30, 2017): 223–30. http://dx.doi.org/10.1177/0967033517723253.

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Анотація:

It was shown that linear transformations are suitable for use in multivariate calibration in near infrared spectroscopy as data compression tools. Partial Least Squares calibration models were built using spectral data transformed by expansion in the series of classical orthogonal polynomials, Fourier and wavelet harmonics. These models allowed effective prediction of the cetane number of diesel fuels, Brix and pol parameters of syrup in sugar production and fat and total protein content in milk. Depending on the compression ratio, prediction errors were no larger than 30% of corresponding errors obtained by the use of the non-transformed models. Although selection of the most suitable transformation depends on the calibration data and on the cross-validation method, in many cases Fourier transform gave satisfactory results.

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34

NILL, FLORIAN. "WEYL ALGEBRAS, FOURIER TRANSFORMATIONS AND INTEGRALS ON FINITE-DIMENSIONAL HOPF ALGEBRAS." Reviews in Mathematical Physics 06, no. 01 (February 1994): 149–66. http://dx.doi.org/10.1142/s0129055x94000092.

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The "Weyl algebras" [Formula: see text], σ, σ′ ∈ {+, –}, over a finite-dimensional Hopf algebra H are defined to be the abstract algebras generated by left or right (σ = ±) multiplication operators Qσ (ψ), ψ ∈ H, and left or right (σ′ = ±) translation operators Pσ′ (a), [Formula: see text] dual of H. It is shown that these algebras are all isomorphic to End (H). Fourier transformations are defined as intertwiners [Formula: see text] implementing a natural isomorphism [Formula: see text]. As a byproduct, this formalism provides a new proof of the invertibility of the antipode and the uniqueness (up to multiplication by a constant) of left and right integrals on finite-dimensional Hopf algebras. If H is a star Hopf algebra, the canonical representations of [Formula: see text] become star representation and Fourier transformation becomes an isometry w.r.t. a natural choice of nondegenerate hermitian forms on H and [Formula: see text]. Some comments on the relation with the C*-approach of Woronowicz and Podleś & Woronowicz are added.

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35

Slobodianuk, Olena. "Abstract law and actual reality: social critique from F. Poullain de la Barre to Charles Fourier." Sententiae 5, no. 1 (April 22, 2002): 130–43. http://dx.doi.org/10.31649/sent05.01.130.

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The article compares the views of François Poullain de la Barre and Charles Fourier on gender equality. The research is based on the background of historical and philosophical transformations associated with the transition from Cartesian rationalism to enlightenment and post-enlightenment thinking. The latter focuses primarily on social criticism and human transformation through the transformation of social relations. Although the author states that some of the provisions of French philosophers have a common origin (the ideological potential of Cartesianism and the modern doctrine of natural law), they interpret the target aspect differently. While Poullain de la Barre seeks to prove the identity of men's and women's abilities, Fourier's task is to find practical grounds that will allow for real equality of the sexes. In addition, the degree of equality achieved by women is derived from the degree of overcoming social (in particular, economic) injustice in general.

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36

Tao, Ngo Quoc. "Extracting invariants based on coordinate transformations." Journal of Computer Science and Cybernetics 9, no. 4 (April 26, 2016): 28–32. http://dx.doi.org/10.15625/1813-9663/9/4/8255.

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This paper deals with some methods extracting invariants for the translation, rotation and scaling and finding invariants which are depended on statistic invariants. It collates them with the Fourier transform. For discrete point set we can use the polar coordinate to construct SCC (star contour code) and give match algorithm for SCC.

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37

Balsa, Jose. "Comparison of Image Compressions: Analog Transformations." Proceedings 54, no. 1 (August 21, 2020): 37. http://dx.doi.org/10.3390/proceedings2020054037.

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A comparison between the four most used transforms, the discrete Fourier transform (DFT), discrete cosine transform (DCT), the Walsh–Hadamard transform (WHT) and the Haar-wavelet transform (DWT), for the transmission of analog images, varying their compression and comparing their quality, is presented. Additionally, performance tests are done for different levels of white Gaussian additive noise.

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38

Zhao, Tieyu, and Yingying Chi. "Quantum Weighted Fractional-Order Transform." Fractal and Fractional 7, no. 3 (March 18, 2023): 269. http://dx.doi.org/10.3390/fractalfract7030269.

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Quantum Fourier transform (QFT) transformation plays a very important role in the design of many quantum algorithms. Fractional Fourier transform (FRFT), as an extension of the Fourier transform, is particularly important due to the design of its quantum algorithm. In this paper, a new reformulation of the weighted fractional Fourier transform (WFRFT) is proposed in order to realize quantum FRFT; however, we found that this reformulation can be applied to other transformations, and therefore, this paper presents the weighted fractional Hartley transform (WFRHT). For the universality of application, we further propose a general weighted fractional-order transform (WFRT). When designing the quantum circuits, we realized the quantum WFRFT via QFT and quantum phase estimation (QPE). Moreover, after extending our design to the WFRHT, we were able to formulate the quantum WFRHT. Finally, in accordance with the research results, we designed the quantum circuit of the general WFRT, and subsequently proposed the quantum WFRT. The research in this paper has great value as a reference for the design and application of quantum algorithms.

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39

Küchenmeister, Jens. "Three-dimensional adaptive coordinate transformations for the Fourier modal method." Optics Express 22, no. 2 (January 14, 2014): 1342. http://dx.doi.org/10.1364/oe.22.001342.

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40

Penmetsa, Ravi C., and Ramana V. Grandhi. "Adaptation of fast Fourier transformations to estimate structural failure probability." Finite Elements in Analysis and Design 39, no. 5-6 (March 2003): 473–85. http://dx.doi.org/10.1016/s0168-874x(02)00104-x.

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41

Pfirsch, Frank, and Michael C. Böhm. "On the accuracy of fourier transformations in crystal orbitat approaches." Chemical Physics 98, no. 1 (August 1985): 89–98. http://dx.doi.org/10.1016/0301-0104(85)80097-5.

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42

Gogolewski, Damian, Paweł Zmarzły, Tomasz Kozior, and Thomas G. Mathia. "Possibilities of a Hybrid Method for a Time-Scale-Frequency Analysis in the Aspect of Identifying Surface Topography Irregularities." Materials 16, no. 3 (January 31, 2023): 1228. http://dx.doi.org/10.3390/ma16031228.

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The article presents research results related to assessing the possibilities of applying modern filtration methods to diagnosing measurement signals. The Fourier transformation does not always provide full information about the signal. It is, therefore, appropriate to complement the methodology with a modern multiscale method: the wavelet transformation. A hybrid combination of two algorithms results in revealing additional signal components, which are invisible in the spectrum in the case of using only the harmonic analysis. The tests performed using both simulated signals and the measured roundness profiles of rollers in rolling bearings proved the advantages of using a complex approach. A combination of the Fourier and wavelet transformations resulted in the possibility to identify the components of the signal, which directly translates into better diagnostics. The tests fill a research gap in terms of complex diagnostics and assessment of profiles, which is very important from the standpoint of the precision industry.

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43

Chakraborty, Avijit, and David Okaya. "Frequency‐time decomposition of seismic data using wavelet‐based methods." GEOPHYSICS 60, no. 6 (November 1995): 1906–16. http://dx.doi.org/10.1190/1.1443922.

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Spectral analysis is an important signal processing tool for seismic data. The transformation of a seismogram into the frequency domain is the basis for a significant number of processing algorithms and interpretive methods. However, for seismograms whose frequency content vary with time, a simple 1-D (Fourier) frequency transformation is not sufficient. Improved spectral decomposition in frequency‐time (FT) space is provided by the sliding window (short time) Fourier transform, although this method suffers from the time‐ frequency resolution limitation. Recently developed transforms based on the new mathematical field of wavelet analysis bypass this resolution limitation and offer superior spectral decomposition. The continuous wavelet transform with its scale‐translation plane is conceptually best understood when contrasted to a short time Fourier transform. The discrete wavelet transform and matching pursuit algorithm are alternative wavelet transforms that map a seismogram into FT space. Decomposition into FT space of synthetic and calibrated explosive‐source seismic data suggest that the matching pursuit algorithm provides excellent spectral localization, and reflections, direct and surface waves, and artifact energy are clearly identifiable. Wavelet‐based transformations offer new opportunities for improved processing algorithms and spectral interpretation methods.

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44

Gitin, Andrey V. "Mathematical Fundamentals of Modern Linear Optics." International Journal of Antennas and Propagation 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/273107.

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All known quantum-mechanical approaches to wave and statistical optics are united into a single theory, using Feynman's path integral as a fundamental principle. In short-wave approximations, this principle, the Fourier transformations, and concepts of the theory reproduce Fermat's principle, the Legendre transformations, and concepts of Hamilton's optics and radiometry in a one-to-one fashion.

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45

Merakos, P. K., K. Masselos, and C. E. Goutis. "Power Efficient Hierarchical Scheduling for DSP Transformations." VLSI Design 14, no. 2 (January 1, 2002): 203–17. http://dx.doi.org/10.1080/10655140290010114.

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In this paper, the problem of scheduling the computation of partial products in transformational Digital Signal Processing (DSP) algorithms, aiming at the minimization of the switching activity in data and address buses, is addressed. The problem is stated as a hierarchical scheduling problem. Two different optimization algorithms, which are based on the Travelling Salesman Problem (TSP), are defined. The proposed optimization algorithms are independent on the target architecture and can be adapted to take into account it. Experimental results obtained from the application of the proposed algorithms in various widely used DSP transformations, like Discrete Cosine Transform (DCT) and Discrete Fourier Transform (DFT), show that significant switching activity savings in data and address buses can be achieved, resulting in corresponding power savings. In addition, the differences between the two proposed methods are underlined, providing envisage for their suitable selection for implementation, in particular transformational algorithms and architectures.

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46

Morimoto, Mitsuo, and Keiko Fujita. "Conical Fourier-Borel transformations for harmonic functionals on the Lie ball." Banach Center Publications 37, no. 1 (1996): 95–113. http://dx.doi.org/10.4064/-37-1-95-113.

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47

Vetterli, Martin, and Henri J. Nussbaumer. "Algorithmes de transformations de Fourier et en cosinus mono et bidimensionnels." Annales des Télécommunications 40, no. 9-10 (September 1985): 466–76. http://dx.doi.org/10.1007/bf02998219.

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48

Wada, Ryoko. "The Fourier-Borel transformations of analytic functionals on the complex sphere." Proceedings of the Japan Academy, Series A, Mathematical Sciences 61, no. 9 (1985): 298–301. http://dx.doi.org/10.3792/pjaa.61.298.

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49

Agafonova, N. Yu. "On Uniform Convergence of Transformations of Fourier Series on Multiplicative Systems." Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 9, no. 1 (2009): 3–8. http://dx.doi.org/10.18500/1816-9791-2009-9-1-3-8.

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50

VanderLugt, A., C. S. Anderson, and P. J. W. Melsa. "Time-delay detection of short pulses by Fresnel and Fourier transformations." Applied Optics 32, no. 20 (July 10, 1993): 3761. http://dx.doi.org/10.1364/ao.32.003761.

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