• Nonlinear Functional Analysis and Applications
• /
• v.26 no.4
• /
• pp.781-792
• /
• 2021

In this paper, we study well-posed problems and variational inequalities in locally convex Hausdorff topological vector spaces. The necessary and sufficient conditions are obtained for the existence of solutions of variational inequality problems and quasi variational inequalities even when the underlying set K is not convex. In certain cases, solutions obtained are not unique. Moreover, counter examples are also presented for the authenticity of the main results.

• Nonlinear Functional Analysis and Applications
• /
• v.27 no.4
• /
• pp.743-756
• /
• 2022

We obtained results on upper hemi-continuous and pseudo-monotone type two mappings for sets which are not compact. M.S.R. Chowdhury and K.-K. Tan's improved result on Ky Fan's minimax inequality will be used.

• Journal of KIISE:Databases
• /
• v.29 no.5
• /
• pp.358-366
• /
• 2002

Feature-based similarity retrieval become an important research issue in image database systems. The features of image data are useful to discrimination of images. In this paper, we propose the high speed k-Nearest Neighbor search algorithm based on Self-Organizing Maps. Self-Organizing Maps(SOM) provides a mapping from high dimensional feature vectors onto a two-dimensional space and generates a topological feature map. A topological feature map preserves the mutual relations (similarities) in feature spaces of input data, and clusters mutually similar feature vectors in a neighboring nodes. Therefore each node of the topological feature map holds a node vector and similar images that is closest to each node vector. We implemented a k-NN search for similar image classification as to (1) access to topological feature map, and (2) apply to pruning strategy of high speed search. We experiment on the performance of our algorithm using color feature vectors extracted from images. Promising results have been obtained in experiments.

• Proceedings of the KAIS Fall Conference
• /
• 2003.11a
• /
• pp.223-230
• /
• 2003

Feature-based similarity retrieval has become an important research issue in multimedia database systems. The features of multimedia data are useful for discriminating between multimedia objects (e 'g', documents, images, video, music score, etc.). For example, images are represented by their color histograms, texture vectors, and shape descriptors, and are usually high-dimensional data. The performance of conventional multidimensional data structures(e'g', R- Tree family, K-D-B tree, grid file, TV-tree) tends to deteriorate as the number of dimensions of feature vectors increases. The R＊-tree is the most successful variant of the R-tree. In this paper, we propose a SOM-based R＊-tree as a new indexing method for high-dimensional feature vectors.The SOM-based R＊-tree combines SOM and R＊-tree to achieve search performance more scalable to high dimensionalities. Self-Organizing Maps (SOMs) provide mapping from high-dimensional feature vectors onto a two dimensional space. The mapping preserves the topology of the feature vectors. The map is called a topological of the feature map, and preserves the mutual relationship (similarity) in the feature spaces of input data, clustering mutually similar feature vectors in neighboring nodes. Each node of the topological feature map holds a codebook vector. A best-matching-image-list. (BMIL) holds similar images that are closest to each codebook vector. In a topological feature map, there are empty nodes in which no image is classified. When we build an R＊-tree, we use codebook vectors of topological feature map which eliminates the empty nodes that cause unnecessary disk access and degrade retrieval performance. We experimentally compare the retrieval time cost of a SOM-based R＊-tree with that of an SOM and an R＊-tree using color feature vectors extracted from 40, 000 images. The result show that the SOM-based R＊-tree outperforms both the SOM and R＊-tree due to the reduction of the number of nodes required to build R＊-tree and retrieval time cost.

• Journal of the Chungcheong Mathematical Society
• /
• v.25 no.4
• /
• pp.703-709
• /
• 2012

Recently, Kazmi and Khan [7] introduced a kind of equilibrium problem called generalized system (GS) with a single-valued bi-operator F. Next, in [10], the first author considered a generalization of (GS) into a multi-valued circumstance called the multi-valued quasi-generalized system (in short, MQGS). In the current work, we provide an extension of (MQGS) into a system of (MQGS) in general settings. This system is called the generalized multi-valued quasi-generalized system (in short, GMQGS). Using the existence theorem for abstract economy by Kim [8], we prove the existence of solutions for (GMQGS) in the framework of Hausdorff topological vector spaces. As an application, an existence result of a system of generalized vector quasi-variational inequalities is derived.

• Kyungpook Mathematical Journal
• /
• v.13 no.2
• /
• pp.235-240
• /
• 1973

A k-dimensional vector bundle is a bundle ${\xi}=(E,P,B,F^k)$ with fibre $F^k$ satisfying the local triviality, where F is the field of real numbers R or complex numbers C ([1], [2] and [3]). Let $Vect_k(X)$ be the set consisting of all isomorphism classes of k-dimensional vector bundles over the topological space X. Then $Vect_F(X)=\{Vect_k(X)\}_{k=0,1,{\cdots}}$ is a semigroup with Whitney sum (${\S}1$). For a pair (X, A) of topological spaces, a difference isomorphism over (X, A) is a vector bundle morphism ([2], [3]) ${\alpha}:{\xi}_0{\rightarrow}{\xi}_1$ such that the restriction ${\alpha}:{\xi}_0{\mid}A{\longrightarrow}{\xi}_1{\mid}A$ is an isomorphism. Let $S_k(X,A)$ be the set of all difference isomorphism classes over (X, A) of k-dimensional vector bundles over X with fibre $F^k$. Then $S_F(X,A)=\{S_k(X,A)\}_{k=0,1,{\cdots}}$, is a semigroup with Whitney Sum (${\S}2$). In this paper, we shall prove a relation between $Vect_F(X)$ and $S_F(X,A)$ under some conditions (Theorem 2, which is the main theorem of this paper). We shall use the following theorem in the paper. THEOREM 1. Let ${\xi}=(E,P,B)$ be a locally trivial bundle with fibre F, where (B, A) is a relative CW-complex. Then all cross sections S of ${\xi}{\mid}A$ prolong to a cross section $S^*$ of ${\xi}$ under either of the following hypothesis: (H1) The space F is (m-1)-connected for each $m{\leq}dim$ B. (H2) There is a relative CW-complex (Y, X) such that $B=Y{\times}I$ and $A=(X{\times}I)$ ${\cap}(Y{\times}O)$, where I=[0, 1]. (For proof see p.21 [2]).

• Journal of applied mathematics & informatics
• /
• v.7 no.1
• /
• pp.1-28
• /
• 2000

In the present paper, we introduce fundamental results in the KKM theory for G-convex spaces which are equivalent to the Brouwer theorem, the Sperner lemma, and the KKM theorem. Those results are all abstract versions of known corresponding ones for convex subsets of topological vector spaces. Some earlier applications of those results are indicated. Finally, We give a new proof of the Himmelberg fixed point theorem and G-convex space versions of the von Neumann type minimax theorem and the Nash equilibrium theorem as typical examples of applications of our theory.

• The KIPS Transactions:PartB
• /
• v.8B no.5
• /
• pp.515-522
• /
• 2001

Feature-based similarity retrieval become an important research issue in image database systems. The features of image data are useful to discrimination of images. In this paper, we propose the highspeed k-Nearest Neighbor search algorithm based on Self-Organizing Maps. Self-Organizing Map(SOM) provides a mapping from high dimensional feature vectors onto a two-dimensional space. A topological feature map preserves the mutual relations (similarity) in feature spaces of input data, and clusters mutually similar feature vectors in a neighboring nodes. Each node of the topological feature map holds a node vector and similar images that is closest to each node vector. We implemented about k-NN search for similar image classification as to (1) access to topological feature map, and (2) apply to pruning strategy of high speed search. We experiment on the performance of our algorithm using color feature vectors extracted from images. Promising results have been obtained in experiments.

• Journal of Integrative Natural Science
• /
• v.11 no.1
• /
• pp.51-56
• /
• 2018

For topological vector spaces X and Y, let $F_0(X,Y)=\{f{\in}Y^X:f(0)=0\}$. Then it is an extremely large family and the family of linear operators is a very small subfamily of $F_0(X,Y)$. In this paper, we establish the characterizations of $F_0(X,Y)$-matrix families (${l^{\infty}(X)$, ${l^{\infty}(Y)$), ($c_0(X)$, $l^{\infty}(Y)$) and ($c_0(X)$, $l^{\infty}(Y)$).

• Journal of the Korean Institute of Intelligent Systems
• /
• v.11 no.3
• /
• pp.223-230
• /
• 2001

Since self-organizing map (SOM) preserves the topology of ordering in input spaces and trains itself by unsupervised algorithm, it is Llsed in many areas. However, SOM has a shortcoming: structure cannot be easily detcrmined without many trials-and-errors. Structure-adaptive self-orgnizing map (SASOM) which can adapt its structure as well as its weights overcome the shortcoming of self-organizing map: SASOM makes use of structure adaptation capability to place the nodes of prototype vectors into the pattern space accurately so as to make the decision boundmies as close to the class boundaries as possible. In this scheme, the initialization of weights of newly adapted nodes is important. This paper proposes a method which optimizes SASOM with genetic algorithm (GA) to determines the weight vector of newly split node. The leanling algorithm is a hybrid of unsupervised learning method and supervised learning method using LVQ algorithm. This proposed method not only shows higher performance than SASOM in terms of recognition rate and variation, but also preserves the topological order of input patterns well. Experiments with 2D pattern space data and handwritten digit database show that the proposed method is promising.