Some Finite Two-Dimensional Contractions

1968 ◽  
Vol 19 (1) ◽  
pp. 91-104
Author(s):  
R. D. Mills
Keyword(s):  
Velocity Potential ◽  
Axisymmetric Flow ◽  
Elliptic Functions ◽  
Two Dimensional ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Form Representation ◽  
Axial Velocity Distribution ◽  
Similar Series

SummaryGeneral solutions for two-dimensional incompressible potential flow occurring between two equipotential planes perpendicular to the x-axis are given. The first form is the two-dimensional analogue of Thwaites’s solution for axisymmetric flow and allows the calculation of the flow when the axial velocity distribution is specified as a Fourier cosine series in x. The second form of solution, obtained by “inverting” the first form, allows the calculation of the flow when the shape of the “boundary streamline” is specified by a similar series in the velocity potential ϕ.It is shown how the second form of solution may be utilised to design contracting channels between equipotential planes. The computation of the contraction shapes and velocities is straightforward. In particular, contractions are derived from smoothing conditions similar to those used by Thwaites, and from a flow having a single (ϕ, y) step-discontinuity. It is shown in the Appendix that the latter flow possesses a closed form representation in terms of elliptic functions.

APPROXIMATING THE DENSITY OF THE TIME TO RUIN VIA FOURIER-COSINE SERIES EXPANSION

2016 ◽  
Vol 47 (1) ◽  
pp. 169-198 ◽  
Author(s):  
Zhimin Zhang
Keyword(s):  
Risk Model ◽  
Series Expansions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
One Dimensional ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Compound Poisson Risk Model ◽  
Fourier Cosine Series ◽  
Approximation Errors

AbstractIn this paper, the density of the time to ruin is studied in the context of the classical compound Poisson risk model. Both one-dimensional and two-dimensional Fourier-cosine series expansions are used to approximate the density of the time to ruin, and the approximation errors are also obtained. Some numerical examples are also presented to show that the proposed method is very efficient.

scholarly journals A Mode-Matching Solution for TE-Backscattering from an Arbitrary 2D Rectangular Groove in a PEC

2020 ◽  
Vol 20 (3) ◽  
pp. 159-163 ◽  
Author(s):  
Mehdi Bozorgi
Keyword(s):  
Singular Integrals ◽  
Logarithmic Singularity ◽  
Linear Equations ◽  
Mode Matching ◽  
Electric Conductor ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Highly Oscillatory Integrals ◽  
Rectangular Groove

In this paper, the simple yet effective mode-matching technique is utilized to compute TE-backscattering from a 2D filled rectangular groove in an infinite perfect electric conductor (PEC). The tangential magnetic fields inside and outside of the groove are represented as the sums of infinite series of cosine harmonics (half-range Fourier cosine series). By applying the continuity of the tangential magnetic field, these modes are matched on the groove to obtain the series coefficients by solving a system of linear equations. For this purpose, some oscillatory logarithmic singular integrals involving Hankel and trigonometric functions are solved numerically, starting by removing the logarithmic singularity via integration by parts. In the following, the new well-behaved highly oscillatory integrals are computed using efficient methods, and several comparisons are made to demonstrate the validity and ability of the presented procedure.

P2438Deep learning model for atrial fibrillation detection based on single beat morphology

2019 ◽  
Vol 40 (Supplement_1) ◽  
Author(s):  
S W E Baalman ◽  
F E Schroevers ◽  
A Oakley ◽  
L A Ramos ◽  
R R Lopes ◽  
...  
Keyword(s):  
Atrial Fibrillation ◽  
Deep Learning ◽  
Sinus Rhythm ◽  
Learning Model ◽  
Heart Beat ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Specificity And Sensitivity ◽  
Fourier Cosine Series ◽  
Data Points

Abstract Background The electrocardiogram (ECG) is commonly used, but most recent rhythm discrimination algorithms still lack both specificity and sensitivity. Deep learning techniques have shown promising results in the classification of physiological signals like ECGs. Purpose To develop and test a deep learning (DL) model to discriminate between atrial fibrillation (AF) and sinus rhythm (SR). Methods For the development of the DL model we used 1499 ECGs sampled at 500 Hz of patients diagnosed with AF. All ECGs were labeled by two experienced investigators. Only ECGs labeled as SR or AF were included in the dataset. To simplify the learning process, solely the first ECG channel was used. The ECG waveforms were preprocessed using the Fourier cosine series to correct for baseline wander. Input data was generated by normalizing and scaling all different heartbeats by centralizing the R peak, leading to 15744 single heart beat samples of 80 data points (figure A). Multiple feedforward architectures were tested with different numbers of layers, filters and activation functions. The models were trained by equally splitting the data (50%SR, 50%AF) in a training (65%), validation (25%) and test set (15%). The best performing model was chosen based on the accuracy. Results A total of 1469 ECGs (1061 (72%)SR, 408 (28%)AF) were included. The model with the best performance was a feedforward model consisting three dense layers with ReLU activation and four dense layers with Linear activation. Training of the model was performed in 32 epochs. Validation of the model resulted in an accuracy of 96% (figure B), precision of 95% and recall of 96%. Conclusions The morphology based deep learning model developed in this study was able to discriminate atrial fibrillation from sinus rhythm with a fairly high accuracy using a limited size dataset and only one lead.

An Exact Series Solution for the Vibration of Mindlin Rectangular Plates with Elastically Restrained Edges

2013 ◽  
Vol 572 ◽  
pp. 489-493 ◽  
Author(s):  
Kai Xue ◽  
Jiu Fa Wang ◽  
Qiu Hong Li ◽  
Wei Yuan Wang ◽  
Ping Wang
Keyword(s):  
Series Solution ◽  
Rectangular Plates ◽  
Analysis Method ◽  
Original Function ◽  
Cross Sectional ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

An analysis method has been proposed for the vibration analysis of the Mindlin rectangular plates with general elastically boundary supports, in which the vibration displacements and the cross-sectional rotations of the mid-plane are sought as the linear combination of a double Fourier cosine series and auxiliary series functions. The use of these supplementary functions is to solve the potential discontinuity associated with the x-derivative and y-derivative of the original function along the four edges, so this method can be applied to get the exact solution. Finally the numerical results are presented to validate the correct of the method.

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