• Kyungpook Mathematical Journal
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• v.62 no.1
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• pp.107-118
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• 2022

In this note we observe that any truncated multi-index sequence has an interpolating measure supported in Euclidean space. It is well known that the consistency of a truncated moment sequence is equivalent to the existence of an interpolating measure for the sequence. When the moment matrix of a moment sequence is nonsingular, the sequence is naturally consistent; a proper perturbation to a given moment matrix enables us to confirm the existence of an interpolating measure for the moment sequence. We also illustrate how to find an explicit form of an interpolating measure for some cases.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.4
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• pp.487-509
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• 2018

The Truncated Moment Problem (TMP) entails finding a positive Borel measure to represent all moments in a finite sequence as an integral; once the sequence admits one or more such measures, it is known that at least one of the measures must be finitely atomic with positive densities (equivalently, a linear combination of Dirac point masses with positive coefficients). On the contrary, there are more general moment problems for which we aim to find a "signed" measure to represent a sequence; that is, the measure may have some negative densities. This type of problem is referred to as the General Truncated Moment Problem (GTMP). The Jordan Decomposition Theorem states that any (signed) measure can be written as a difference of two positive measures, and hence, in the view of this theorem, we are able to apply results for TMP to study GTMP. In this note we observe differences between TMP and GTMP; for example, we cannot have an analogous to the Flat Extension Theorem for GTMP. We then present concrete solutions to lower-degree problems.

• Honam Mathematical Journal
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• v.29 no.4
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• pp.569-576
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• 2007

Let $\cal{H}$ be a separable, infinite dimensional, complex Hilbert space. We call "an operator $\cal{T}$ acting on $\cal{H}$ has a star moment sequence supported on a set K" when there exist nonzero vectors u and v in $\cal{H}$ and a positive Borel measure ${\mu}$ such that <$T^{*j}T^ku$, v> = ${^\int\limits_{K}}\;{{\bar{z}}^j}\;{{\bar{z}}^k}\;d\mu$ for all j, $k\;\geq\;0$. We obtain a characterization to find a representing star moment measure and discuss some related properties.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.241-247
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• 2005

Let ${\gamma}{\equiv}\left{{\gamma}_{ij}\right}(0{\leq}i+j{\leq}2n)$ be a collection of complex numbers with ${\gamma}_{00}>0$ and ${\gamma}_{ji}={\bar{\gamma}}_{ij}$. The truncated complex moment problem for ${\gamma}$ entails finding a positive Borel measure ${\mu}$ supported in the complex plane ${\mathbb{C}}$ such that ${\gamma}_{ij}={\int}{\bar{z}}^{i}z^jd{\mu}(z)(0{\leq}i+j{\leq}2n)$. We solve this truncated moment problem with data in a circle and discuss the behavior of data in an extended moment matrix.

• Journal of Institute of Control, Robotics and Systems
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• v.19 no.4
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• pp.357-363
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• 2013

Most serious patients who have the paralysis of their ankles can't use of their feet freely. But their ankles can be recovered by an ankle bending rehabilitation exercise and a ankle rotating rehabilitation exercise. Recently, the professional rehabilitation therapeutists are much less than stroke patients in number. Therefore, the ankle-rehabilitation robot should be developed. The developed robot can be dangerous because it can't measure the applied bending force and twisting moment of the patients' ankles. In this paper, the six-axis force/moment sensor for the ankle-rehabilitation robot was specially designed the weight of foot and the applied force to foot in rehabilitation exercise. As a test results, the interference error of the six-axis force/moment sensor was less than 2.51%. It is thought that the sensor can be used to measure the bending force and twisting moment of the patients' ankles in rehabilitation exercise.

• East Asian mathematical journal
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• v.40 no.1
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• pp.25-36
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• 2024

Full univariate moment problems have been studied using continued fractions, orthogonal polynomials, spectral measures, and so on. On the other hand, the truncated moment problem has been mainly studied through confirming the existence of the extension of the moment matrix. A few articles on the multivariate moment problem implicitly presented about some results of this note, but we would like to rearrange the important results for the existence of a representing measure of a moment sequence. In addition, new techniques with orthogonal polynomials will be introduced to expand the means of studying truncated moment problems.

• Journal of Institute of Control, Robotics and Systems
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• v.17 no.5
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• pp.484-489
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• 2011

Some patients can't use their hands because of inherent and acquired paralysis of their fingers. Their fingers can recover with rehabilitative training, and the extent of rehabilitation can be judged by grasping a cylindrical-object with their fingers. At present, the cylindrical-object used in hospitals is only a cylinder which cannot measure grasping force of the fingers. Therefore, doctors must judge the extent of rehabilitation by watching patients' fingers as they grasp the cylinder. A cylindrical-type finger force measuring system which can measure the grasping force of patients' fingers should be developed. This paper looks at the development of a cylindrical-type finger force measuring system with two-axis force/moment sensor which can measure grasping force. The two-axis force/moment sensor was designed and fabricated, and the high-speed force measuring device was designed and manufactured by using DSP (digital signal processing). Also, cylindrical-type finger force measuring system was developed using the developed two-axis force/moment sensor and the high-speed force measuring device, and the grasping force tests of men were performed using the developed system. The tests confirm that the average finger forces of right and left hands for men were about 186N and 172N respectively.

• Transactions of the Korean Society of Machine Tool Engineers
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• v.10 no.2
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• pp.1-9
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• 2001

To measure springback ratio of thin sheet or plate, winding bend rig is made. It bends a specimen with keeping its curva-ture constant and measure the bending angles before and after release of bending load. To check the performance of the bend rig, we calculated the bending moment by two ways which are based on simple beam theory. One is that the bending moment is calculated by using the results of bending test, and the other is that the moment is calculated by using the results of tensile tests. The former may entails the effect of the other is that the moment is calculated by using the results of tensile tests. The former may entails the effect of the friction between bending pin of the rig and surface of specimen, but the latter does not contain any effects of the friction since the bending moment is obtained by using tensile tests. Never-theless, the values of the two bending moments shows the same level of bending moment, which implies that the friction does not influence on the presence of friction within the scope of the test performed in this experiment. This phenomenon is explained theoretically by using moment equilibrium.

• Journal of the Korean Mathematical Society
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• v.42 no.4
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• pp.723-747
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• 2005

In this paper, we consider the quartic moment problem suggested by Curto-Fialkow[6]. Given complex numbers $\gamma{\equiv}{\gamma}^{(4)}\;:\;{\gamma00},\;{\gamma01},\;{\gamma10},\;{\gamma01},\;{\gamma11},\;{\gamma20},\;{\gamma03},\;{\gamma12},\;{\gamma21},\;{\gamma30},\;{\gamma04},\;{\gamma13},\;{\gamma22},\;{\gamma31},\;{\gamma40}$, with ${\gamma00},\;>0\;and\;{\gamma}_{ji}={\gamma}_{ij}$ we discuss the conditions for the existence of a positive Borel measure ${\mu}$, supported in the complex plane C such that ${\gamma}_{ij}=\int\;\={z}^i\;z^j\;d{\mu}(0{\leq}i+j{\leq}4)$. We obtain sufficient conditions for flat extension of the quartic moment matrix M(2). Moreover, we examine the existence of flat extensions for nonsingular positive quartic moment matrices M(2).

• Bulletin of the Korean Mathematical Society
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• v.37 no.1
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• pp.91-102
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• 2000

Let ${\gamma}{\equiv}{\gamma}^{(2n)}$ denote a sequence of complex numbers ${\gamma}{00},{\gamma}{01},\cdots,{\gamma}0, 2n,...,{\gamma}{2n},0\;with\; {\gamma}{00}\;>\;0,{\gamma}{ji}={{\overline}{\gamma_{ij}}}$,and let K denote a closed subset of the complex plane C. The truncated K complex moment problem entails finding a positive Borel measure $\mu$ such that ${\gamma}{ij}={\int}{{\overline}{z}}^{i}z^{j}d{\mu}\;(0{\leq}\;i+j\;{\leq}\;2n)$ and supp ${\mu}{\subseteq}\;K$. If n=2, then is called the quartic moment problem. In this paper, we give partial solutions for the singular quartic moment problem with rank M(2)=5 and ${{\overline}{Z}}Z{\in}\;<1,Z,{{\overline}{Z}},Z^{2},{{\overline}{Z}}^2>$.