• Proceedings of the KIEE Conference
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• 2002.11c
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• pp.104-109
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• 2002

A new robust Kalman filter is designed for the linear discrete-time system with norm-bounded parametric uncertainties. Sum quadratic constraint, which describes the uncertainties of the system, is converted into an indefinite quadratic form to be minimized in indefinite inner product space. This minimization problem is solved by the new robust Kalman filter. Since the new filter is obtained by simply modifying the conventional Kalman filter, robust filtering scheme can be more readily designed using the proposed method in comparison with the existing robust Kalman filters. A numerical example demonstrates the robustness and the improvement of the proposed filter compared with the existing filters.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.24 no.4
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• pp.375-385
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• 2020

In this paper, a special indefinite inner product, named hyperbolic scalar product, is used and all acquired results have been raised and proved with the proviso that the space is equipped with this indefinite scalar product. The main objective is to be introduced and applied an indefinite oblique projection method, called Indefinite Lanczos J-biorthogonalizatiom process, which in addition to building a pair of J-biorthogonal bases for two used Krylov subspaces, leads to the introduction of a process for solving large non-J-symmetric linear systems, i.e., Indefinite two-sided Lanczos Algorithm for Linear systems.

• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.99-117
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• 1998

Let $(B, H, p_1)$ be an abstract Wiener space and $F(B)$ the Fresnel class on $(B, H, p_1)$ which consists of functionals F of the form : $$ F(x) = \int_{H} exp{i(h,x)^\sim} df(h), x \in B, $$ where $(\cdot, \cdot)^\sim$ is a stochastic inner product between H and B, and f is in $M(H)$, the space of complex Borel measures on H. We introduce an $L_1$ analytic Fourier-Feynman transforms for functionls in $F(B)$. Furthermore, we introduce a convolution on $F(B)$, and then verify the existence of the $L_1$ analytic Fourier-Feynman transform for the convolution product of two functionals in $F(B)$, and we establish the relationships between the $L_1$ analytic Fourier-Feynman tranform of the convolution product for two functionals in $F(B)$ and the $L_1$ analytic Fourier-Feynman transforms for each functional. Finally, we show that most results in [7] follows from our results in Section 3.

• International Journal of Aeronautical and Space Sciences
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• v.16 no.3
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• pp.437-444
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• 2015

Satellite selection and exclusion techniques have been applied to the global navigation satellite system (GNSS) with the aim of achieving a balance between navigational performance and computational efficiency. Conventional approaches to satellite selection based on the best dilution of precision (DOP) are excessively computational and complicated. This paper proposes a new method that applies a geometric sensitivity index of individual GNSS satellites. The sensitivity index is derived using the inner product of the line of sight (LOS) vector of each satellite. First, the LOS vector is computed, which accounts for the geometry between the satellite and user positions. Second, the inner product of each pair of LOS vectors is calculated, which indicates the proximities of the satellites to one another. The proximity can be determined according to the sensitivity of each satellite. A post-processing test was conducted to verify the reliability of the proposed method. The proposed index and the results of a conventional approach that measures the dilution of precision (DOP) were compared. The test results demonstrate that the proposed index produces results that are within 96% of those of the conventional approach and reduces the computational burden. This index can be utilized to estimate the sensitivity of individual satellites, obtaining a navigation solution. Therefore, the proposed index applies to satellite selection and exclusion as well as to the sensitivity analyses of multiple GNSS applications.

• Journal of applied mathematics & informatics
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• v.5 no.1
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• pp.223-233
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• 1998

This paper is concerned with the problem of existence of solutions to the initial value problem u'(t) = A(t, u(t)), u(a) = z in a probabilistic normed space where $A : [a,b)\;{\times}\;D->E$ is continuous, D is a closed subset of a probabilistic normed space E, and $z\;{\in}\;D$. With a dissipative type condition on A, we estabilish sufficient conditions for this initial value problem to have a solution.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.99-107
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• 1993

A vector space K with scalar product <.,.> is called a Krein space if it can be decomposed as a northogonal sum of a Hilbert space and an anti-space of a Hilbert space. The space K induces a Hilbert space $K_{J}$ in the inner product <.,.> $K_{J}$=<.,.>K, where $J^{2}$=I. the eigenspaces of J are denoted by $K^{+}$$_{J}$, which is a Hilbert space and $K^{-}$$_{J}$, which is an anti-space of a Hilbert space. Then the Krein space K is the orthogonal sum of $K^{+}$$_{J}$ and $K^{-}$$_{J}$. Such a decomposition of K is called a fundamental decomposition. In general, fundamental decompositions are not unique. The norm of the Hilbert space depends on the choice of a fundamental decomposion, but such norms are equivalent. The topology generated by these norms is called the strong or Mackey topology of K. It is used to define all topological notions on the Krein space K with respect to this topology. The Pontryagin index of a Krein space is the dimension of the antispace of a Hilbert space in any such decomposition. the dimension does not depend on the choice of orthogonal decomposition. A Krein space is called a Pontryagin space if it has finite Pontryagin index.dex.yagin index.dex.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.513-525
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• 1996

Let the operator A be self-adjoint with domain, Dom(A), dense in $(H)$ which is a separable Hilbert space with norm $\left\$\mid$ \cdot \right\$\mid$$ and inner product $<\cdot, \cdot>$.

• Honam Mathematical Journal
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• v.45 no.4
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• pp.585-597
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• 2023

We introduce the rotational hypersurface x = x(u, v, s, t) constructed by double rotation in five dimensional Euclidean space 𝔼⁵. We reveal the first and the second fundamental form matrices, Gauss map, shape operator matrix of x. Additionally, defining the i-th curvatures of any hypersurface via Cayley-Hamilton theorem, we compute the curvatures of the rotational hypersurface x. We give some relations of the mean and Gauss-Kronecker curvatures of x. In addition, we reveal Δx=𝓐x, where 𝓐 is the 5 × 5 matrix in 𝔼⁵.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.691-706
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• 2020

Necessary and sufficient conditions are provided under which the weighted Moore-Penrose inverse A^†_MN exists, where A is an adjointable operator between Hilbert C^⁎-modules, and the weights M and N are only self-adjoint and invertible. Relationship between weighted Moore-Penrose inverses A^†_MN is clarified when A is fixed, whereas M and N are variable. Perturbation analysis for the weighted Moore-Penrose inverse is also provided.

• Bulletin of the Korean Mathematical Society
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• v.21 no.1
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• pp.27-30
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• 1984

The concepts of the numerical range of an operator on a Hillbert space and on a Banach space were introduced by Toeplitz in 1918 and Bauer in 1962 respectively. Bauer's paper was concerned only with finite dimensional Banach spaces, but the concept of numerical range that he introduced is available without restriction of the dimension [1, 2]. In this paper, we define a C-algebra spatial numerical range of an operator on C-algebra valued inner product modules introduced by Paschke [4], and give analogous results on these modules as those on Banach spaces.