• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.123-135
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• 2023

The goal of this article is to introduce the concept of pseudo-weighted Browder spectrum when the underlying Hilbert space is not necessarily separable. To attain this goal, the notion of α-pseudo-Browder operator has been introduced. The properties and the relation of the weighted spectrum, pseudo-weighted spectrum, weighted Browder spectrum, and pseudo-weighted Browder spectrum have been investigated by extending analogous properties of their corresponding essential pseudo-spectrum and essential pseudo-weighted spectrum. The weighted spectrum, pseudo-weighted spectrum, weighted Browder, and pseudo-weighted Browder spectrum of the sum of two bounded linear operators have been characterized in the case when the Hilbert space (not necessarily separable) is a direct sum of its closed invariant subspaces. This exploration ends with a characterization of the pseudo-weighted Browder spectrum of the sum of two bounded linear operators defined over the arbitrary Hilbert spaces under certain conditions.

• The Korean Journal of Applied Statistics
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• v.24 no.6
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• pp.995-1006
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• 2011

A time series can be decomposed into simple components with a multiscale method. Empirical mode decomposition(EMD) is a recently invented multiscale method in Huang et al. (1998). It is natural to apply a classical prediction method such a vector autoregressive(AR) model to the obtained simple components instead of the original time series; in addition, a prediction procedure combining a classical prediction model to EMD and Hilbert spectrum is proposed in Kim et al. (2008). In this paper, we suggest to adopt principal component analysis(PCA) to the prediction procedure that enables the efficient selection of input variables among obtained components by EMD. We discuss the utility of adopting PCA in the prediction procedure based on EMD and Hilbert spectrum and analyze the daily worm account data by the proposed PCA adopted prediction method.

• Journal of Mechanical Science and Technology
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• v.20 no.11
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• pp.1781-1789
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• 2006

Fourier methods are not generally an appropriate approach in the investigation of faults signals with transient components. This work presents the application of a new signal processing technique, the Hilbert-Huang transform and its marginal spectrum, in analysis of vibration signals and faults diagnosis of gear. The Empirical mode decomposition (EMD), Hilbert-Huang transform (HHT) and marginal spectrum are introduced. Firstly, the vibration signals are separated into several intrinsic mode functions (IMFs) using EMD. Then the marginal spectrum of each IMF can be obtained. According to the marginal spectrum, the wear fault of the gear can be detected and faults patterns can be identified. The results show that the proposed method may provide not only an increase in the spectral resolution but also reliability for the faults diagnosis of the gear.

• Earthquakes and Structures
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• v.16 no.2
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• pp.197-208
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• 2019

The objective of this study is to propose new seismic intensity parameters based on the Hilbert spectrum and to associate them with the seismic damage potential. In recent years the assessment of even more seismic features derived from the seismic acceleration time-histories was associated with the structural damage. For a better insight into the complex seismic acceleration time-history, Hilbert-Huang Transform (HHT) analysis is utilized for its processing, and the Hilbert spectrum is obtained. New proposed seismic intensity parameters based on the Hilbert spectrum are derived. The aim is to achieve a significant estimation of the seismic damage potential on structures from the proposed new intensity parameters confirmed by statistical methods. Park-Ang overall structural damage index is used to describe the postseismic damage status of structures. Thus, a set of recorded seismic accelerograms from all over the word is applied on a reinforced concrete frame structure, and the Park-Ang indices through nonlinear dynamic analysis are provided and considered subsequently as reference numerical values. Conventional seismic parameters, with well-known seismic structural damage interrelation, are evaluated for the same set of excitations. Statistical procedures, namely correlation study and multilinear regression analysis, are applied on the set of the conventional parameters and the set of proposed new parameters separately, to confirm their interrelation with the seismic structural damage. The regression models are used for the evaluation of the structural damage indices for every set of parameters, respectively. The predicted numerical values of the structural damage indices evaluated from the two sets of seismic intensity parameters are inter-compared with the reference values. The numerical results confirm the ability of the proposed Hilbert spectrum based new seismic intensity parameters to approximate the postseismic structural damage with a smaller Standard Error of Estimation than this accomplished of the conventional ones.

• Korean Journal of Mathematics
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• v.13 no.1
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• pp.43-50
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• 2005

We introduce the Joint spectrum on the complex Banach space and on the complex Hilbert space and the tensor product spectrums on the tensor product spaces. And we will show ${\sigma}[P(T_1,T_2,{\ldots},T_n)]={\sigma}(T_1{\otimes}T_2{\otimes}{\cdots}{\otimes}T_n)$ on $X_1{\overline{\otimes}}X_2{\overline{\otimes}}{\cdots}{\overline{\otimes}}X_n$ for a polynomial P.

• Journal of applied mathematics & informatics
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• v.38 no.3_4
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• pp.271-276
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• 2020

Let R be a commutative ring with identity, R[X] the polynomial ring over R, ∗ a radical operation on R and ⋆ a radical operation of finite character on R[X]. In this paper, we give Hilbert basis theorem for rings with ∗-Noetherian spectrum. More precisely, we show that if (I^∗R[X])^⋆ = (IR[X])^⋆ and (I^∗R[X])^⋆ ∩ R = I^∗ for all ideals I of R, then R has ∗-Noetherian spectrum if and only if R[X] has ⋆-Noetherian spectrum. This is a generalization of a well-known fact that R has Noetherian spectrum if and only if R[X] has Noetherian spectrum.

• Earthquakes and Structures
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• v.18 no.4
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• pp.507-517
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• 2020

This study aims to present new frequency-related seismic intensity parameters (SIPs) based on the Hilbert-Huang Transform (HHT) analysis. The proposed procedure is utilized for the processing of several seismic accelerograms. Thus, the entire evaluated Hilbert Spectrum (HS) of each considered seismic velocity time-history is investigated first, and then, a delimited area of the same HS around a specific frequency is explored, for the proposition of new SIPs. A first application of the suggested new parameters is to reveal the interrelation between them and the structural damage of a reinforced concrete frame structure. The index of Park and Ang describes the structural damage. The fundamental frequency of the structure is considered as the mentioned specific frequency. Two statistical methods, namely correlation analysis and multiple linear regression analysis, are used to identify the relationship between the considered SIPs and the corresponding structural damage. The results confirm that the new proposed HHT-based parameters are effective descriptors of the seismic damage potential and helpful tools for forecasting the seismic damages on buildings.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.1-13
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• 2022

The main aim of the article is to introduce new generalizations of Fredholm and Browder classes of spectra when the underlying Hilbert space is not necessarily separable and study their properties. To achieve the goal the notions of 𝛼-Browder operators, 𝛼-B-Fredholm operators, 𝛼-B-Browder operators and 𝛼-Drazin invertibility have been introduced. The relation of these classes of operators with their corresponding weighted spectra has been investigated. An equivalence of 𝛼-Drazin invertible operators with 𝛼-Browder operators and 𝛼-B-Browder operators has also been established. The weighted Browder spectrum of the sum of two bounded linear operators has been characterised in the case when the Hilbert space (not necessarily separable) is a direct sum of its closed invariant subspaces.

• Structural Monitoring and Maintenance
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• v.7 no.1
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• pp.13-25
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• 2020

One of the most important aspects in structural health monitoring is the detection of fatigue damage. Structural components such as heavy-duty bolts work under high dynamic loads, and thus are prone to accumulate fatigue damage and cracks may originate. Those heavy-duty bolts are used, for example, in wind power generation and mining equipment. Therefore, the investigation of new and more effective monitoring technologies attracts a great interest. In this study the acoustic emission (AE) technology was employed to detect incipient damage during fatigue testing of a M36 bolt. Initial results showed that the AE signals have a high level of background noise due to how the load is applied by the fatigue testing machine. Thus, an advanced signal processing method in the time-frequency domain, the Hilbert-Huang Spectrum (HHS), was applied to reveal AE components buried in background noise in form of high-frequency peaks that can be associated with damage progression. Accordingly, the main contribution of the present study is providing insights regarding the detection of incipient damage during fatigue testing using AE signals and providing recommendations for further research.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.597-602
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• 1997

Let H be a separable complex Hilbert space and L(H) be the *-algebra of all bounded linear operators on H. For $T \in L(H)$, we construct a pair of semi-positive definite operators $$ $\mid$T$\mid$_r = (T^*T)^{\frac{1}{2}} and $\mid$T$\mid$_l = (TT^*)^{\frac{1}{2}}. $$ An operator T is called a semi-hyponormal operator if $$ Q_T = $\mid$T$\mid$_r - $\mid$T$\mid$_l \geq 0. $$ In this paper, by using a technique introduced by Berberian [1], we show that the approximate point spectrum $\sigma_{ap}(T)$ of a semi-hyponomal operator T is empty.