• East Asian mathematical journal
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• v.22 no.1
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• pp.17-35
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• 2006

Some additive reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in Banach spaces are given. Applications for complex-valued functions are considered as well.

• East Asian mathematical journal
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• v.23 no.1
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• pp.59-73
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• 2007

A new reverse of the generalised triangle inequality that complements the classical results of Diaz and Metcalf is obtained. Applications for inner product spaces and for complex numbers are provided.

• East Asian mathematical journal
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• v.22 no.1
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• pp.1-16
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• 2006

Some new reverses of the Schwarz inequality in inner product spaces and applications for vector-valued sequnces and integrals are given.

• Journal of Digital Contents Society
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• v.13 no.3
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• pp.309-315
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• 2012

Active data are the input data which are changed its membership as Vector Quantization codebook generation algorithm is processed. In the process of VQ codebook generation algorithm performed, the actual active data out of the entire input data will be less presented as the process is performed. Therefore, if we can accurately find the active data and only if we are going to do VQ codebook generation on the active data, then we can significantly reduce the overall generation time. In this paper, we presented the triangle inequality based algorithm to select the active data. Experimental results show that our algorithm is superior to other methods in terms of the VQ codebook generation time.

• Kyungpook Mathematical Journal
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• v.15 no.1
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• pp.9-11
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• 1975

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1587-1598
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• 2013

In this paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation $$\frac{{\partial}u}{{\partial}t}={\triangle}u-b(x,t)u^{\sigma}$$ under general geometric flow on complete noncompact manifolds, where 0 < ${\sigma}$ < 1 is a real constant and $b(x,t)$ is a function which is $C^2$ in the $x$-variable and $C^1$ in the$t$-variable. As an application, we get an interesting Harnack inequality.

• Journal for History of Mathematics
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• v.20 no.4
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• pp.105-122
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• 2007

In this paper we analyze various studies on $Erd\ddot{o}s-Mordell$ inequality(from a point O inside a given triangle ABC the perpendiculars OP, OQ, OR are drawn to its sides, then OA+OB+OC${\ge}$2(OP+OQ+OR)). We find out three trends of studies on $Erd\ddot{o}s-Mordell$ inequality: studies on various proof methods, studies on related means, studies on $Erd\ddot{o}s-Mordell$ type inequalities.

• Journal of Digital Contents Society
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• v.16 no.3
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• pp.373-379
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• 2015

In this paper, we propose an online VQ Codebook generation method for updating an existing VQ Codebook in real-time and adding to an existing cluster with newly created text data which are news paper, web pages, blogs, tweets and IoT data like sensor, machine. Without degrading the performance of the batch VQ Codebook to the existing data, it was able to take advantage of the newly added data by using a triangle inequality which modifying the VQ Codebook progressively show a high degree of accuracy and speed. The result of applying to test data showed that the performance is similar to the batch method.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2000.11a
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• pp.467-470
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• 2000

Vector quantization (VQ) is widely used in many high-quality and high-rate data compression applications such as speech coding, audio coding, image coding and video coding. When the size of a VQ codebook is large, the computational complexity for the full codeword search method is a significant problem for many applications. A number of complexity reduction algorithms have been proposed and investigated using such properties of the codebook as the triangle inequality. This paper proposes a new structured VQ search algorithm that is based on a multi-stage structure for searching for the best codeword. Even using only two stages, a significant complexity reduction can be obtained without any loss of quality.

• Management Science and Financial Engineering
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• v.20 no.1
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• pp.17-19
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• 2014

In this paper, we consider an interesting variant of the inverse minimum traveling salesman problem. Given an instance (G, w) of the minimum traveling salesman problem defined on a metric space, we fix a specified Hamiltonian cycle $HC_0$. The task is then to adjust the edge cost vector w to w' so that the new cost vector w' satisfies the triangle inequality condition and $HC_0$ can be returned by the minimum spanning tree algorithm in the TSP-instance defined with w'. The objective is to minimize the total deviation between the original and the new cost vectors with respect to the $L_1$-norm. We call this problem the inverse metric traveling salesman problem against the minimum spanning tree algorithm and show that it is closely related to the inverse metric spanning tree problem.