• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.911-926
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• 2022

Let u be a function on a locally finite graph G = (V, E) and Ω be a bounded subset of V. Let 𝜀 > 0, p > 2 and 0 ≤ λ < λ₁(Ω) be constants, where λ₁(Ω) is the first eigenvalue of the discrete Laplacian, and h : V → ℝ be a function satisfying h ≥ 0 and $h{\not\equiv}0$. We consider a perturbed Yamabe equation, say $$\{\begin{array}{lll}-{\Delta}u-{\lambda}u={\mid}u{\mid}^{p-2}u+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω and ∂Ω denote the interior and the boundary of Ω, respectively. Using variational methods, we prove that there exists some positive constant 𝜀₀ > 0 such that for all 𝜀 ∈ (0, 𝜀₀), the above equation has two distinct solutions. Moreover, we consider a more general nonlinear equation $$\{\begin{array}{lll}-{\Delta}u=f(u)+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ and prove similar result for certain nonlinear term f(u).

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.863-872
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• 2023

Let n ⩾ 2 be an integer, we denote the smallest integer b such that gcd {(ⁿ_k) : b < k < n - b} > 1 as b(n). For any prime p, we denote the highest exponent α such that p^α | n as v_p(n). In this paper, we partially answer a question asked by Hong in 2016. For a composite number n and a prime number p with p | n, let n = a_mp^m + r, 0 ⩽ r < p^m, 0 < a_m < p. Then we have $$v_p\(\text{gcd}\{\(n\\k\)\;:\;b(n)1\}\)=\{\array{1,&&a_m=1\text{ and }r=b(n),\\0,&&\text{otherwise.}}$$

• Journal of English Language & Literature
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• v.57 no.5
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• pp.871-887
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• 2011

In Lacanian terms, the real, which is a non-representative Ding an sich, is indirectly approachable only in and through language. This 'speaking of the real' is made possible through a restoration of the missing link between one signifier, S1 and another signifier, S₂, as is manifested in the Lacanian formula of metaphor. In Freudian terms of textual metaphor, the missing link is restored by substituting a new edition for an old edition of one's historical text of life. This is what this essay means by the metaphorical/dualistic structure of the analytic/literary text. And this is a way of talking about an intertextuality between literature and psychoanalysis in the sense of the 'text as psyche' and the 'psyche as text.' Applying the 'signifying substitution' to the Oedipus complex, the Oedipal child can find a meaning(s), "my erotic indulgement with my Mom is wrong" by metaphorically substituting S₂: the Name of the Father for S1: the Desire of the Mother. This meaning leads to the constitution of the human subject and the formation of the incest taboo, one of the most significant distinctive features of the human being as distinguished from the animals. We can see a similar metaphorical structure of S1-S2 taking place in the literary texts such as Macbeth and "Dover Beach": in the course of the stage of life being substituted for the primal scene in the former, and the plain of Tucydides for a bed scene in the latter, respectively.

• Journal of information and communication convergence engineering
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• v.20 no.4
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• pp.309-316
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• 2022

This investigation is aimed at automatic text summarization on large-scale Vietnamese datasets. Vietnamese articles were collected from newspaper websites and plain text was extracted to build the dataset, that included 1,101,101 documents. Next, a new single-document extractive text summarization model was proposed to evaluate this dataset. In this summary model, the k-means algorithm is used to cluster the sentences of the input document using different text representations, such as BoW (bag-of-words), TF-IDF (term frequency - inverse document frequency), Word2Vec (Word-to-vector), Glove, and FastText. The summary algorithm then uses the trained k-means model to rank the candidate sentences and create a summary with the highest-ranked sentences. The empirical results of the F1-score achieved 51.91% ROUGE-1, 18.77% ROUGE-2 and 29.72% ROUGE-L, compared to 52.33% ROUGE-1, 16.17% ROUGE-2, and 33.09% ROUGE-L performed using a competitive abstractive model. The advantage of the proposed model is that it can perform well with O(n,k,p) = O(n^(k+2/p)) + O(nlog₂n) + O(np) + O(nk²) + O(k) time complexity.

• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.689-702
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• 2019

Let 𝓐 and 𝓑 be real (or complex) algebras. We investigate the stability of a sequence F = {f₀, f₁, ⋯, f_n, ⋯ } of mappings from 𝓐 into 𝓑 satisfying the higher functional equation: $$f_n(x+y+zw)=f_n(x)+f_n(y)+\;{\normalsize\sum\limits_{\tiny{i+j=n \atop i{\leq}j}}}\;[f_i(z)f_j(w)+c_{ij}f_i(w)f_j(z)]$$ for each n = 0, 1, ⋯ and all x, y, z, w ∈ 𝓐, where $$c_{ij}=\left\{1{\text{ if }}i{\neq}j,\\0{\text{ if }}i=j.\right.$$.

• Journal of the Korean Mathematical Society
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• v.38 no.3
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• pp.595-611
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• 2001

Let Q(n,1) be the set of even unimodular positive definite integral quadratic forms in n-variables. Then n is divisible by 8. For A[X] in Q(n,1), the theta series $\theta$(sub)A(z) = ∑(sub)X∈Z(sup)n e(sup)$\pi$izA[X] (Z∈h (※Equations, See Full-text) the complex upper half plane) is a modular form of weight n/2 for the congruence group Γ$_1$(8) = {$\delta$∈SL$_2$(Z)│$\delta$≡()mod 8} (※Equation, See Full-text). If n$\geq$24 and A[X], B{X} are tow quadratic forms in Q(n,1), the quotient $\theta$(sub)A(z)/$\theta$(sub)B(z) is a modular function for Γ$_1$(8). Since we identify the field of modular functions for Γ$_1$(8) with the function field K(X$_1$(8)) of the modular curve X$_1$(8) = Γ$_1$(8)＼h(sup)* (h(sup)* the extended plane of h) with genus 0, we can express it as a rational function of j(sub) 1,8 over C which is a field generator of K(X$_1$(8)) and defined by j(sub)1,8(z) = $\theta$$_3$(2z)/$\theta$$_3$(4z). Here, $\theta$$_3$ is the classical Jacobi theta series.

• The KIPS Transactions:PartB
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• v.12B no.6 s.102
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• pp.703-714
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• 2005

Researches in text categorization have been confined to whole-document-level classification, probably due to lacks of full-text test collections. However, full-length documents availably today in large quantities pose renewed interests in text classification. A document is usually written in an organized structure to present its main topic(s). This structure can be expressed as a sequence of sub-topic text blocks, or passages. In order to reflect the sub-topic structure of a document, we propose a new passage-level or passage-based text categorization model, which segments a test document into several Passages, assigns categories to each passage, and merges passage categories to document categories. Compared with traditional document-level categorization, two additional steps, passage splitting and category merging, are required in this model. By using four subsets of Routers text categorization test collection and a full-text test collection of which documents are varying from tens of kilobytes to hundreds, we evaluated the proposed model, especially the effectiveness of various passage types and the importance of passage location in category merging. Our results show simple windows are best for all test collections tested in these experiments. We also found that passages have different degrees of contribution to main topic(s), depending on their location in the test document.

• Journal of KIISE:Computing Practices and Letters
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• v.16 no.8
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• pp.873-877
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• 2010

CAPTCHA is a security tool that prevents the automatic sign-up by a spam or a robot. This CAPTCHA usually depends on the smart readability of humans. However, the common and plain CAPTCHA with text-based system is not difficult to be solved by intelligent web-bot and machine learning tools. In this paper, we propose a new sub-image based CAPTCHA system totally different from the text based system. Our system offers a set of cropped sub-image from a whole digital picture and asks user to identify the correct orientation. Though there are some nice machine learning tools for this job, but they are useless for a cropped sub-images, which was clearly revealed by our experiment. Experiment showed that our sub-image based CAPTCHA is easy to human solver, but very hard to all kinds of machine learning or AI tools. Also our CAPTCHA is easy to be generated automatical without any human intervention.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.741-754
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• 2023

In this paper, we study the existence and nonexistence of solutions for a class of Hamiltonian strongly degenerate elliptic system with subcritical growth $$\left{\array{-{\Delta}_{\lambda}u-{\mu}v={\mid}v{\mid}^{p-1}v&&\text{in }{\Omega},\\-{\Delta}_{\lambda}v-{\mu}u={\mid}u{\mid}^{q-1}u&&\text{in }{\Omega},\\u=v=0&&\text{ on }{\partial}{\Omega},}$$ where p, q > 1 and Ω is a smooth bounded domain in ℝ^N, N ≥ 3. Here Δ_λ is the strongly degenerate elliptic operator. The existence of at least a nontrivial solution is obtained by variational methods while the nonexistence of positive solutions are proven by a contradiction argument.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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• pp.1181-1209
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• 2021

In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following Schrödinger-Poisson-Kirchhoff system $$\;\{\begin{array}{lll}-(1+b{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{{\mathbb{R}}^3}}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+k(x){\phi}u={\lambda}f(x)u+{\mid}u{\mid}^4u,&&\text{in }{\mathbb{R}}^3,\\-{\Delta}{\phi}=k(x)u^2,&&\text{in }{\mathbb{R}}^3,\end{array}$$ where b > 0, V (x), k(x) and f(x) are positive continuous smooth functions; 0 < λ < λ₁ and λ₁ is the first eigenvalue of the problem -∆u + V(x)u = λf(x)u in H. With the help of the constraint variational method, we obtain that the Schrödinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ₁. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.