• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.41-63
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• 2024

In this paper, we study complete Riemannian immersions into a semi-Riemannian warped product obeying suitable curvature constraints. Under appropriate differential inequalities involving higher order mean curvatures, we establish rigidity and nonexistence results concerning these immersions. Applications to the cases that the ambient space is either an Einstein manifold, a steady state type spacetime or a pseudo-hyperbolic space are given, and a particular investigation of entire graphs constructed over the fiber of the ambient space is also made. Our approach is based on a parabolicity criterion related to a linearized differential operator which is a divergence-type operator and can be regarded as a natural extension of the standard Laplacian.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1323-1336
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• 2015

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}$(S), and the other definition yields an undirected graph ${\overline{\Gamma}}$(S). It is shown that ${\Gamma}$(S) is not necessarily connected, but ${\overline{\Gamma}}$(S) is always connected and diam$({\overline{\Gamma}}(S)){\leq}3$. For a ring R define a directed graph ${\mathbb{APOG}}(R)$ to be equal to ${\Gamma}({\mathbb{IPO}}(R))$, where ${\mathbb{IPO}}(R)$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph ${\overline{\mathbb{APOG}}}(R)$ to be equal to ${\overline{\Gamma}}({\mathbb{IPO}}(R))$. We show that R is an Artinian (resp., Noetherian) ring if and only if ${\mathbb{APOG}}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that ${\overline{\mathbb{APOG}}}(R)$ is a complete graph if and only if either $(D(R))^2=0,R$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that ${\mathbb{IPO}}(R)=\{0,m,m^2,R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n{\times}n}(R)$ where $n{\geq} 2$.

• Asia pacific journal of information systems
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• v.29 no.4
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• pp.838-855
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• 2019

Online product reviews are a vital source for companies in that they contain consumers' opinions of products. The earlier methods of opinion mining, which involve drawing semantic information from text, have been mostly applied in one dimension. This is not sufficient in itself to elicit reviewers' comprehensive views on products. In this paper, we propose a novel approach in opinion mining by projecting online consumers' reviews in a multidimensional framework to improve review interpretation of products. First of all, we set up a new framework consisting of six dimensions based on a marketing management theory. To calculate the distances of review sentences and each dimension, we embed words in reviews utilizing Google's pre-trained word2vector model. We classified each sentence of the reviews into the respective dimensions of our new framework. After the classification, we measured the sentiment degrees for each sentence. The results were plotted using a radar graph in which the axes are the dimensions of the framework. We tested the strategy on Amazon product reviews of the iPhone and Galaxy smartphone series with a total of around 21,000 sentences. The results showed that the radar graphs visually reflected several issues associated with the products. The proposed method is not for specific product categories. It can be generally applied for opinion mining on reviews of any product category.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.1
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• pp.7-14
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• 2001

Let G be a connected graph. A pebbling move on a graph G is taking two pebbles off one vertex and placing one of them on an adjacent vertex. The pebbling number of a connected graph G, f(G), is the least n such that any distribution of n pebbles on the vertices of G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbling moves. In this paper, the pebbling numbers of the lexicographic products of some graphs are computed.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1409-1418
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• 2021

In this paper, we study domination in the zero-divisor graph of a *-ring. We first determine the domination number, the total domination number, and the connected domination number for the zero-divisor graph of the product of two *-rings with componentwise involution. Then, we study domination in the zero-divisor graph of a Rickart *-ring and relate it with the clique of the zero-divisor graph of a Rickart *-ring.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.247-259
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• 2018

Factorable surfaces, i.e. graphs associated with the product of two functions of one variable, constitute a wide class of surfaces in differential geometry. Such surfaces in the pseudo-Galilean space with zero Gaussian and mean curvature were obtained in [2]. In this study, we provide new results relating to the factorable surfaces with non-zero constant Gaussian and mean curvature.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1155-1162
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• 2010

In this paper, we find the maximum exponent of D ${\times}$ E, the cartesian product of two digraphs D and E on n, n + 2 vertices, respectively for an even integer $n\geq4$. We also characterize the extrema cases.

• Journal of the Korean Mathematical Society
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• v.52 no.3
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• pp.523-535
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• 2015

We study surfaces in Euclidean space which are obtained as the sum of two curves or that are graphs of the product of two functions. We consider the problem of finding all these surfaces with constant Gauss curvature. We extend the results to non-degenerate surfaces in Lorentz-Minkowski space.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.663-680
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• 2024

The zero divisor graph is the most basic way of representing an algebraic structure as a graph. For any commutative ring R, each element is a vertex on the zero divisor graph and two vertices are defined as adjacent if and only if the product of those vertices equals zero. In this research, we determine some topological indices such as the Wiener index, the edge-Wiener index, the hyper-Wiener index, the Harary index, the first Zagreb index, the second Zagreb index, and the Gutman index of zero divisor graph of integers modulo prime power and its direct product.

• Kyungpook Mathematical Journal
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• v.45 no.1
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• pp.45-53
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• 2005

A basis of the cycle space C (G) is d-fold if each edge occurs in at most d cycles of C(G). The basis number, b(G), of a graph G is defined to be the least integer d such that G has a d-fold basis for its cycle space. MacLane proved that a graph G is planar if and only if $b(G)\;{\leq}\;2$. Schmeichel showed that for $n\;{\geq}\;5,\;b(K_{n}\;{\bullet}\;P_{2})\;{\leq}\;1\;+\;b(K_n)$. Ali proved that for n, $m\;{\geq}\;5,\;b(K_n\;{\bullet}\;K_m)\;{\leq}\;3\;+\;b(K_n)\;+\;b(K_m)$. In this paper, we give an upper bound for the basis number of the semi-strong product of a bipartite graph with a cycle.