• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.61-74
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• 2014

This paper describes an iterative method for orthogonal projections $AA^+$ and $A^+A$ of an arbitrary matrix A, where $A^+$ represents the Moore-Penrose inverse. Convergence analysis along with the first and second order error estimates of the method are investigated. Three numerical examples are worked out to show the efficacy of our work. The first example is on a full rank matrix, whereas the other two are on full rank and rank deficient randomly generated matrices. The results obtained by the method are compared with those obtained by another iterative method. The performance measures in terms of mean CPU time (MCT) and the error bounds for computing orthogonal projections are listed in tables. If $Z_k$, k = 0,1,2,... represents the k-th iterate obtained by our method then the sequence of the traces {trace($Z_k$)} is a monotonically increasing sequence converging to the rank of (A). Also, the sequence of traces {trace($I-Z_k$)} is a monotonically decreasing sequence converging to the nullity of $A^*$.

• Communications for Statistical Applications and Methods
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• v.27 no.5
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• pp.523-533
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• 2020

This paper suggests three available methods for finding nonnegative estimates of variance components of the random effects in mixed models. The three proposed methods based on the concepts of projections are called projection method I, II, and III. Each method derives sums of squares uniquely based on its own method of projections. All the sums of squares in quadratic forms are calculated as the squared lengths of projections of an observation vector; therefore, there is discussion on the decomposition of the observation vector into the sum of orthogonal projections for establishing a projection model. The projection model in matrix form is constructed by ascertaining the orthogonal projections defined on vector subspaces. Nonnegative estimates are then obtained by the projection model where all the coefficient matrices of the effects in the model are orthogonal to each other. Each method provides its own system of linear equations in a different way for the estimation of variance components; however, the estimates are given as the same regardless of the methods, whichever is used. Hartley's synthesis is used as a method for finding the coefficients of variance components.

• International Journal of CAD/CAM
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• v.6 no.1
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• pp.117-123
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• 2006

Recently with the development of 3D modeling and digitizing tools, more and more models have been created, which leads to the necessity of the technique of 3D mode retrieval system. In this paper we investigate a new method for 3D model retrieval based on orthogonal projections. We assume that 3D models are composed of trigonal meshes. Algorithms process first by a normalization step in which the 3D models are transformed into the canonical coordinates. Then each model is orthogonally projected onto six surfaces of the projected cube which contains it. A following step is feature extraction of the projected images which is done by Moment Invariants and Polar Radius Fourier Transform. The feature vector of each 3D model is composed of the features extracted from projected images with different weights. Our System validates that this means can distinguish 3D models effectively. Experiments show that our method performs quit well.

• Bulletin of the Korean Mathematical Society
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• v.22 no.2
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• pp.128-128
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• 1985

• Communications for Statistical Applications and Methods
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• v.26 no.4
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• pp.337-346
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• 2019

This paper discusses a method for obtaining nonnegative estimates for variance components in a random effects model. A variance component should be positive by definition. Nevertheless, estimates of variance components are sometimes given as negative values, which is not desirable. The proposed method is based on two basic ideas. One is the identification of the orthogonal vector subspaces according to factors and the other is to ascertain the projection in each orthogonal vector subspace. Hence, an observation vector can be denoted by the sum of projections. The method suggested here always produces nonnegative estimates using projections. Hartley's synthesis is used for the calculation of expected values of quadratic forms. It also discusses how to set up a residual model for each projection.

• Communications of the Korean Mathematical Society
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• v.36 no.2
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• pp.327-342
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• 2021

We consider a generalization of the L^q-spectrum with respect to two Borel probability measures on ℝⁿ having the same compact support, and also study their behavior under orthogonal projections of measures onto an m-dimensional subspace. In particular, we try to improve the main result of Bahroun and Bhouri [4]. In addition, we are interested in studying the behavior of the generalized lower and upper L^q-spectrum with respect to two measures on "sliced" measures in an (n - m)-dimensional linear subspace. The results in this article establish relations with the L^q-spectrum with respect to two Borel probability measures and its projections and generalize some well-known results.

• Honam Mathematical Journal
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• v.6 no.1
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• pp.59-83
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• 1984

• Korean Management Science Review
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• v.21 no.2
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• pp.43-60
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• 2004

Every iteration of interior-point methods of large scale optimization requires computing at least one orthogonal projection. In the practice, symmetric variants of the Gaussian elimination such as Cholesky factorization are accepted as the most efficient and sufficiently stable method. In this paper several specific implementation issues of the symmetric factorization that can be applied for solving such equations are discussed. The code called McSML being the result of this work is shown to produce comparably sparse factors as another implementations in the $MATLAB^{***}$ environment. It has been used for computing projections in an efficient implementation of self-regular based interior-point methods, McIPM. Although primary aim of developing McSML was to embed it into an interior-point methods optimizer, the code may equally well be used to solve general large sparse systems arising in different applications.

• Journal of Ginseng Research
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• v.37 no.3
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• pp.341-348
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• 2013

Identification of the origins of Panax ginseng has been issued in Korea scientifically and economically. We describe a metabolomics approach used for discrimination and prediction of ginseng roots from different origins in Korea. The fresh ginseng roots from six ginseng cooperative associations (Gangwon, Gaeseong, Punggi, Chungbuk, Jeonbuk, and Anseong) were analyzed by UPLC-MS-based approach combined with orthogonal projections to latent structure-discriminant analysis multivariate analysis. The ginsengs from Gangwon and Gaeseong were easily differentiated. We further analyzed the metabolomics results in subgroups. Punggi, Chungbuk, Jeonbuk, and Anseong ginseng could be easily differentiated by the first two orthogonal components. As a validation of the discrimination model, we performed blind prediction tests of sample origins using an external test set. Our model predicted their geographical origins as 99.7% probability. The robust discriminatory power and statistical validity of our method suggest its general applicability for determining the origins of P. ginseng samples.

• Journal of Communications and Networks
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• v.5 no.4
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• pp.328-343
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• 2003

Orthogonal pulse-shape modulation using Hermite pulses for ultra-wideband communications is reviewed. Closedform expressions of cross-correlations among Hermite pulses and their corresponding transmit and receive waveforms are provided. These show that the pulses lose orthogonality at the receiver in the presence of differentiating antennas. Using these expressions, an algebraic model is established based on the projections of distorted receive waveforms onto the orthonormal basis given by the set of normalized orthogonal Hermite pulses. Using this new matrix model, a number of pulse-shape modulation schemes are analyzed and a novel orthogonal design is proposed. In the proposed orthogonal design, transmit waveforms are constructed as combinations of elementary Hermites with weighting coefficients derived by employing the Gram-Schmidt (QR) factorization of the differentiating distortion model’s matrix. The design ensures orthogonality of the vectors at the output of the receiver bank of correlators, without requiring compensation for the distortion introduced by the antennas. In addition, a new set of elementary Hermite Pulses is proposed which further enhances the performance of the new design while enabling a simplified hardware implementation.