• Title, Summary, Keyword: Quadrature Formulas

### QUADRATURE FORMULAS FOR WAVELET COEFFICIENTS

• Kwon, Soon-Geol
• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.911-925
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• 1997
• We derive quadrature formulas for approximating wavelet coefficients for smooth functions from equally spaced point values with arbitrarily high degree of accuracy. Wa also estimate the error of quadrature formulas.

### Ostrowski's Inequality for Monotonous Mappings and Applications

• Dragomir, Sever Silvestru
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.1
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• pp.127-135
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• 1999
• An inequality of Ostrowski's type for monotonous nondecreasing mappings is given. Applications for quadrature formulas are pointed out.

### VARIANTS OF NEWTON'S METHOD USING FIFTH-ORDER QUADRATURE FORMULAS: REVISITED

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1195-1209
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• 2009
• In this paper, we point out some errors in a recent paper by Cordero and Torregrosa [7]. We prove the convergence of the variants of Newton's method for solving the system of nonlinear equations using two different approaches. Several examples are given, which illustrate the cubic convergence of these methods and verify the theoretical results.

### A New Unified Scheme Computing the Quadrature Weights, Integration and Differentiation Matrix for the Spectral Method

• Kim, Chang-Joo;Park, Joon-Goo;Sung, Sangkyung
• Journal of Electrical Engineering and Technology
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• v.10 no.3
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• pp.1188-1200
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• 2015
• A unified numerical method for computing the quadrature weights, integration matrix, and differentiation matrix is newly developed in this study. For this purpose, a spline-like interpolation using piecewise continuous polynomials is converted into a global spline interpolation formula, with which the quadrature formulas can be derived from integration and differentiation of the transformed function in an exact manner. To prove the usefulness of the suggested approach, both the Lagrange and tension spline interpolations are represented in exactly the same form as global spline interpolation. The applicability of the proposed method on arbitrary nodes is illustrated using two different sets of nodes. A series of validations using three test functions is conducted to show the flexibility in selecting computational nodes with the present method.

### Vibration of mitred and smooth pipe bends and their components

• Redekop, D.;Chang, D.
• Structural Engineering and Mechanics
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• v.33 no.6
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• pp.747-763
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• 2009
• In this work, the linear vibration characteristics of $90^{\circ}$ pipe bends and their cylindrical and toroidal shell components are studied. The finite element method, based on shear-deformation shell elements, is used to carry out a vibration analysis of metallic multiple $90^{\circ}$ mitred pipe bends. Single, double, and triple mitred bends are considered, as well as a smooth bend. Sample natural frequencies and mode shapes are given. To validate the procedure, comparison of the natural frequencies is made with existing results for cylindrical and toroidal shells. The influence of the multiplicity of the bend, the boundary conditions, and the various geometric parameters on the natural frequency is described. The differential quadrature method, based on classical shell theory, is used to study the vibration of components of these bends. Regression formulas are derived for cylindrical shells (straight pipes) with one or two oblique edges, and for sectorial toroidal shells (curved pipes, pipe elbows). Two types of support are considered for each case. The results given provide information about the vibration characteristics of pipe bends over a wide range of the geometric parameters.

### A new Tone's method in APOLLO3® and its application to fast and thermal reactor calculations

• Mao, Li;Zmijarevic, Igor
• Nuclear Engineering and Technology
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• v.49 no.6
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• pp.1269-1286
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• 2017
• This paper presents a newly developed resonance self-shielding method based on Tone's method in $APOLLO3^{(R)}$ for fast and thermal reactor calculations. The new method is based on simplified models, the narrow resonance approximation for the slowing down source and Tone's approximation for group collision probability matrix. It utilizes mathematical probability tables as quadrature formulas in calculating effective cross-sections. Numerical results for the ZPPR drawer calculations in 1,968 groups show that, in the case of the double-column fuel drawer, Tone's method gives equivalent precision to the subgroup method while markedly reducing the total number of collision probability matrix calculations and hence the central processing unit time. In the case of a single-column fuel drawer with the presence of a uranium metal material, Tone's method obtains less precise results than those of the subgroup method due to less precise heterogeneous-homogeneous equivalence. The same options are also applied to PWR UOX, MOX, and Gd cells using the SHEM 361-group library, with the objective of analyzing whether this energy mesh might be suitable for the application of this methodology to thermal systems. The numerical results show that comparable precision is reached with both Tone's and the subgroup methods, with the satisfactory representation of intrapellet spatial effects.

### Design of Closed-Form QMF Filters with Maximally Flat and Half-Band Characteristics in the Frequency Domain (주파수 영역에서 최대평탄과 하프대역 특성을 갖는 폐쇄형 QMF 필터들의 설계)

• Jeon, Joon-Hyeon
• Journal of the Institute of Electronics Engineers of Korea SP
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• v.44 no.4
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• pp.70-77
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• 2007
• Two kinds of QMF(Quadrature Mirror Filter) pairs are used in JPEG2000 standard, which don't have QMF distortions. However, the QMF pairs have the main disadvantages such that there are gentle roll-off rate, ripples in the passband and unequal band decomposition. In this paper, Maxflat(maximally flat) QMF pairs with a half-band gain are proposed for overcoming these problems. Maxflat QMF pairs are realized due to generalized closed-form formulas, and the filters have maximally flat response in the passband/stopband as well as sharp roll-off rate in the transition band. Comparing proposed filters and JPEG2000's filters in frequency domain, it is found that proposed filters have better performance JPEG2000's filters. Moreover, Maxflat QMF pairs show stopband-attenuation exceeding 200 dB almost everywhere.

### Domain decomposition technique to simulate crack in nonlinear analysis of initially imperfect laminates

• Ghannadpour, S. Amir M.;Karimi, Mona
• Structural Engineering and Mechanics
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• v.68 no.5
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• pp.603-619
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• 2018
• In this research, an effective computational technique is carried out for nonlinear and post-buckling analyses of cracked imperfect composite plates. The laminated plates are assumed to be moderately thick so that the analysis can be carried out based on the first-order shear deformation theory. Geometric non-linearity is introduced in the way of von-Karman assumptions for the strain-displacement equations. The Ritz technique is applied using Legendre polynomials for the primary variable approximations. The crack is modeled by partitioning the entire domain of the plates into several sub-plates and therefore the plate decomposition technique is implemented in this research. The penalty technique is used for imposing the interface continuity between the sub-plates. Different out-of-plane essential boundary conditions such as clamp, simply support or free conditions will be assumed in this research by defining the relevant displacement functions. For in-plane boundary conditions, lateral expansions of the unloaded edges are completely free while the loaded edges are assumed to move straight but restricted to move laterally. With the formulation presented here, the plates can be subjected to biaxial compressive loads, therefore a sensitivity analysis is performed with respect to the applied load direction, along the parallel or perpendicular to the crack axis. The integrals of potential energy are numerically computed using Gauss-Lobatto quadrature formulas to get adequate accuracy. Then, the obtained non-linear system of equations is solved by the Newton-Raphson method. Finally, the results are presented to show the influence of crack length, various locations of crack, load direction, boundary conditions and different values of initial imperfection on nonlinear and post-buckling behavior of laminates.