Among the distance based algorithms in phylogenetic tree reconstruction, the neighbor-joining alg... more Among the distance based algorithms in phylogenetic tree reconstruction, the neighbor-joining algorithm has been a widely used and effective method. We propose a new algorithm which counts the number of consistent quartets for cherry picking with tie breaking. We show that the success rate of the new algorithm is almost equal to that of neighbor-joining. This gives an explanation of the qualitative nature of neighborjoining and that of dissimilarity maps from DNA sequence data. Moreover, the new algorithm always reconstructs correct trees from quartet consistent dissimilarity maps.
We propose a simple and robust numerical algorithm to estimate a time-dependent volatility functi... more We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black–Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect...
This paper proposes a compartment model (SVEIHRM model) based on a system of ordinary differentia... more This paper proposes a compartment model (SVEIHRM model) based on a system of ordinary differential equations to simulate the pandemic of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2).Emergence of mutant viruses gave rise to multiple peaks in the number of confirmed cases. Vaccine developers and WHO suggest individuals to receive multiple vaccinations (the primary and the secondary vaccinations and booster shots) to mitigate transmission of COVID-19. Taking this into account, we include compartments for multiple vaccinations and mutant viruses of COVID-19 in the model. In particular, our model considers breakthrough infection according to the antibody formation rate following multiple vaccinations. We obtain the effective reproduction numbers of the original virus, the Delta, and the Omicron variants by fitting this model to data in Korea. Additionally, we provide various simulations adjusting the daily vaccination rate and the timing of vaccination to investigate the ...
We investigate the structure and properties of an Artinian monomial complete intersection quotien... more We investigate the structure and properties of an Artinian monomial complete intersection quotient A=k[x_1,..., x_n] / (x_1^d, ..., x_n^d) combining the sl_2-representation theory and Schur-Weyl duality. In particular, we construct an explicit basis of A which is compatible with the S_n-module structure when n=3 and d=3, 4, 5.
Let L be a very ample line bundle on a smooth curve C of genus g with 3g+3 < deg L ≤ 2g − 5. T... more Let L be a very ample line bundle on a smooth curve C of genus g with 3g+3 < deg L ≤ 2g − 5. Then L is normally generated if 2 deg L> max{2g+2−4h 1 (C, L),2g − g−1 6 −2h1 (C, L)}. Let C be a triple covering of genus p curve C ′ with C φ → C ′ and D a divisor on C ′ with 4p < deg D < g−1 6 − 2p. Then KC(−φ ∗ D) becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.
Pedagogical Discussion on the concept of Tangent as a Linear Approximation
In the school mathematics the concept of tangent is introduced in several steps in suitable conte... more In the school mathematics the concept of tangent is introduced in several steps in suitable contexts. Students are required to reflect and revise their concepts of tangent in order to apply the improved concept to wider range of contexts. In this paper we consider the tangent as the optimal linear approximation to a curve at a given point and make three discussions on pedagogical aspects of it. First, it provides a method of finding roots of real numbers which can be used as an application of tangent. This may help students improve their affective variables such as interest, attitude, motivation about the learning of tangent. Second, this concept reflects the modern point of view of tangent, the linear approximation of nonlinear problems. Third, it gives precise meaning of two tangent lines appearing two sides of a cusp point of a curve.
Let $C$ be a smooth curve of genus $g\ge 4$ and Clifford index $c$. In this paper, we prove that ... more Let $C$ be a smooth curve of genus $g\ge 4$ and Clifford index $c$. In this paper, we prove that if $C$ is neither hyperelliptic nor bielliptic with $g\ge 2c+5$ and $\mathcal M$ computes the Clifford index of $C$, then either $\deg \mathcal M\le \frac{3c}{2}+3$ or $|\mathcal M|=|g^1_{c+2}+h^1_{c+2}|$ and $g=2c+5$. This strengthens the Coppens and Martens' theorem (\cite{CM}, Corollary 3.2.5). Furthermore, for the latter case (1) $\mathcal M$ is half-canonical unless $C$ is a $\frac{c+2}{2}$-fold covering of an elliptic curve, (2) $\mathcal M(F)$ fails to be normally generated with $\cli(\mathcal M(F))=c$, $h^1(\mathcal M(F))=2$ for $F\in g^1_{c+2}$. Such pairs $(C,\mathcal M)$ can be found on a $K3$-surface whose Picard group is generated by a hyperplane section in $\mathbb P^r$. For such a $(C, \mathcal M)$ on a K3-surface, $\mathcal M$ is normally generated while $\mathcal M(F)$ fails to be normally generated with $\cli(\mathcal M)=\cli(\mathcal M(F))=c$.
Classifications and analysis of articles in Journal series A of Korean Society of Mathematics Education
In this study we classify and analyze 265 papers which had been published in the Journal Series A... more In this study we classify and analyze 265 papers which had been published in the Journal Series A in Korean Society of Mathematics Education from year 2000 to year 2008. We have also studied all the papers in the Journal Series A in Korean Society of Mathematics Education last 46 years based on Professor Lee, Gang-sup's paper 'A Classification and Analysis of the Articles in -From issue 1 to issue 99-'.
Let C be a smooth curve of genus g ≥ 4 and Clifford index c. In this paper, we prove that if C is... more Let C be a smooth curve of genus g ≥ 4 and Clifford index c. In this paper, we prove that if C is neither hyperelliptic nor bielliptic with g ≥ 2c + 5 and M computes the Clifford index of C, then either degM ≤ 3c 2 + 3 or |M| = |g c+2 + h 1 c+2| and g = 2c + 5. This strengthens the Coppens and Martens’ theorem ([CM91], Corollary 3.2.5). Furthermore, for the latter case (1) M is half-canonical unless C is a c+2 2 -fold covering of an elliptic curve, (2) M(F ) fails to be normally generated with Cliff(M(F )) = c, h(M(F )) = 2 for F ∈ g c+2. Such pairs (C,M) can be found on a K3-surface whose Picard group is generated by a hyperplane section in Pr. For such a (C,M) on a K3-surface, M is normally generated while M(F ) fails to be normally generated with Cliff(M) = Cliff(M(F )) = c.
Current trends of education of gifted students and investigation of more efficient management of educational system for gifted students
There is no doubt that the national competitiveness, in 21st century, definitely depends on how e... more There is no doubt that the national competitiveness, in 21st century, definitely depends on how effectively it has been producing high-qualify human resources. It is inevitable that we are required to produce outstanding people who are going to make the use of highly developed scientific technology. Every country has already set to concentrate their all efforts in cultivating competitive human resources, enabling it to strengthen its national competitiveness. We Korea, in order to keep up with it, have arranged legal and systematic basis and are putting spurs to producing competent human resources under the 영재교육진흥법 및 시행령, which took effect from March, 2002. With the lack of experience and short history of Gifted Education, however, it is true that we still have many problems in promoting it in reality, We are asked to improve it by finding out what problems we have in whole area of Gifted Education, such as defining conception, choosing target students, structuring system and managi...
We investigate the structure and properties of an Artinian monomial complete intersection quotien... more We investigate the structure and properties of an Artinian monomial complete intersection quotient $A(n,d)=\mathbf{k} [x_{1}, \ldots, x_{n}] \big / (x_{1}^{d}, \ldots, x_{n}^d)$. We construct explicit homogeneous bases of $A(n,d)$ that are compatible with the $S_{n}$-module structure for $n=3$, all exponents $d \ge 3$ and all homogeneous degrees $j \ge 0$. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible component appearing in the $S_{3}$-module decomposition of homogeneous subspaces. 4, 5$.
First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex a... more First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that phase tropical hypersurfaces with smooth tropicalization, possess naturally a smooth differentiable structure. Moreover, we prove that phase tropical hypersurfaces possess a natural symplectic structure.
We provide a construction of global bases for quantum Borcherds-Bozec algebras and their integrab... more We provide a construction of global bases for quantum Borcherds-Bozec algebras and their integrable highest weight representations.
First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex a... more First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in (C∗)n. Next, we prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin’s decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that a phase tropical hypersurface with smooth tropicalization is naturally a topological manifold. Moreover, we prove that a phase tropical hypersurface is naturally homeomorphic to a symplectic manifold.
Pythagorean theorem is one of mathematical contents which is widely used during human culture hav... more Pythagorean theorem is one of mathematical contents which is widely used during human culture have developed. There are many historial records related to Pythagorean theorem made by Babylonian, Egyptian, and Mesopotamian. The theorem has the important meaning for mathematics education in secondary school education. Along with the importance of the proof itself, diverse proof methods and ideas included in their methods are also important since the methods improve students' ability to think mathematics. Hence, in this paper, we classify and analyze 390 proof methods published in the book "All that Pythagorean theorem" and other materials. Based on the results we derive educational meaning in mathematics with respect to main idea of the proof, the preliminaries of the study, and study skills used for proof.
In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. O... more In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of B(∞) and B(λ) as its application, where B(∞) and B(λ) are the crystals of the negative half part of the quantum Borcherds-Bozec algebra Uq(g) and its irreducible highest weight module V (λ), respectively.
Among the distance based algorithms in phylogenetic tree reconstruction, the neighbor-joining alg... more Among the distance based algorithms in phylogenetic tree reconstruction, the neighbor-joining algorithm has been a widely used and effective method. We propose a new algorithm which counts the number of consistent quartets for cherry picking with tie breaking. We show that the success rate of the new algorithm is almost equal to that of neighbor-joining. This gives an explanation of the qualitative nature of neighborjoining and that of dissimilarity maps from DNA sequence data. Moreover, the new algorithm always reconstructs correct trees from quartet consistent dissimilarity maps.
We propose a simple and robust numerical algorithm to estimate a time-dependent volatility functi... more We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black–Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect...
This paper proposes a compartment model (SVEIHRM model) based on a system of ordinary differentia... more This paper proposes a compartment model (SVEIHRM model) based on a system of ordinary differential equations to simulate the pandemic of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2).Emergence of mutant viruses gave rise to multiple peaks in the number of confirmed cases. Vaccine developers and WHO suggest individuals to receive multiple vaccinations (the primary and the secondary vaccinations and booster shots) to mitigate transmission of COVID-19. Taking this into account, we include compartments for multiple vaccinations and mutant viruses of COVID-19 in the model. In particular, our model considers breakthrough infection according to the antibody formation rate following multiple vaccinations. We obtain the effective reproduction numbers of the original virus, the Delta, and the Omicron variants by fitting this model to data in Korea. Additionally, we provide various simulations adjusting the daily vaccination rate and the timing of vaccination to investigate the ...
We investigate the structure and properties of an Artinian monomial complete intersection quotien... more We investigate the structure and properties of an Artinian monomial complete intersection quotient A=k[x_1,..., x_n] / (x_1^d, ..., x_n^d) combining the sl_2-representation theory and Schur-Weyl duality. In particular, we construct an explicit basis of A which is compatible with the S_n-module structure when n=3 and d=3, 4, 5.
Let L be a very ample line bundle on a smooth curve C of genus g with 3g+3 < deg L ≤ 2g − 5. T... more Let L be a very ample line bundle on a smooth curve C of genus g with 3g+3 < deg L ≤ 2g − 5. Then L is normally generated if 2 deg L> max{2g+2−4h 1 (C, L),2g − g−1 6 −2h1 (C, L)}. Let C be a triple covering of genus p curve C ′ with C φ → C ′ and D a divisor on C ′ with 4p < deg D < g−1 6 − 2p. Then KC(−φ ∗ D) becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.
Pedagogical Discussion on the concept of Tangent as a Linear Approximation
In the school mathematics the concept of tangent is introduced in several steps in suitable conte... more In the school mathematics the concept of tangent is introduced in several steps in suitable contexts. Students are required to reflect and revise their concepts of tangent in order to apply the improved concept to wider range of contexts. In this paper we consider the tangent as the optimal linear approximation to a curve at a given point and make three discussions on pedagogical aspects of it. First, it provides a method of finding roots of real numbers which can be used as an application of tangent. This may help students improve their affective variables such as interest, attitude, motivation about the learning of tangent. Second, this concept reflects the modern point of view of tangent, the linear approximation of nonlinear problems. Third, it gives precise meaning of two tangent lines appearing two sides of a cusp point of a curve.
Let $C$ be a smooth curve of genus $g\ge 4$ and Clifford index $c$. In this paper, we prove that ... more Let $C$ be a smooth curve of genus $g\ge 4$ and Clifford index $c$. In this paper, we prove that if $C$ is neither hyperelliptic nor bielliptic with $g\ge 2c+5$ and $\mathcal M$ computes the Clifford index of $C$, then either $\deg \mathcal M\le \frac{3c}{2}+3$ or $|\mathcal M|=|g^1_{c+2}+h^1_{c+2}|$ and $g=2c+5$. This strengthens the Coppens and Martens' theorem (\cite{CM}, Corollary 3.2.5). Furthermore, for the latter case (1) $\mathcal M$ is half-canonical unless $C$ is a $\frac{c+2}{2}$-fold covering of an elliptic curve, (2) $\mathcal M(F)$ fails to be normally generated with $\cli(\mathcal M(F))=c$, $h^1(\mathcal M(F))=2$ for $F\in g^1_{c+2}$. Such pairs $(C,\mathcal M)$ can be found on a $K3$-surface whose Picard group is generated by a hyperplane section in $\mathbb P^r$. For such a $(C, \mathcal M)$ on a K3-surface, $\mathcal M$ is normally generated while $\mathcal M(F)$ fails to be normally generated with $\cli(\mathcal M)=\cli(\mathcal M(F))=c$.
Classifications and analysis of articles in Journal series A of Korean Society of Mathematics Education
In this study we classify and analyze 265 papers which had been published in the Journal Series A... more In this study we classify and analyze 265 papers which had been published in the Journal Series A in Korean Society of Mathematics Education from year 2000 to year 2008. We have also studied all the papers in the Journal Series A in Korean Society of Mathematics Education last 46 years based on Professor Lee, Gang-sup's paper 'A Classification and Analysis of the Articles in -From issue 1 to issue 99-'.
Let C be a smooth curve of genus g ≥ 4 and Clifford index c. In this paper, we prove that if C is... more Let C be a smooth curve of genus g ≥ 4 and Clifford index c. In this paper, we prove that if C is neither hyperelliptic nor bielliptic with g ≥ 2c + 5 and M computes the Clifford index of C, then either degM ≤ 3c 2 + 3 or |M| = |g c+2 + h 1 c+2| and g = 2c + 5. This strengthens the Coppens and Martens’ theorem ([CM91], Corollary 3.2.5). Furthermore, for the latter case (1) M is half-canonical unless C is a c+2 2 -fold covering of an elliptic curve, (2) M(F ) fails to be normally generated with Cliff(M(F )) = c, h(M(F )) = 2 for F ∈ g c+2. Such pairs (C,M) can be found on a K3-surface whose Picard group is generated by a hyperplane section in Pr. For such a (C,M) on a K3-surface, M is normally generated while M(F ) fails to be normally generated with Cliff(M) = Cliff(M(F )) = c.
Current trends of education of gifted students and investigation of more efficient management of educational system for gifted students
There is no doubt that the national competitiveness, in 21st century, definitely depends on how e... more There is no doubt that the national competitiveness, in 21st century, definitely depends on how effectively it has been producing high-qualify human resources. It is inevitable that we are required to produce outstanding people who are going to make the use of highly developed scientific technology. Every country has already set to concentrate their all efforts in cultivating competitive human resources, enabling it to strengthen its national competitiveness. We Korea, in order to keep up with it, have arranged legal and systematic basis and are putting spurs to producing competent human resources under the 영재교육진흥법 및 시행령, which took effect from March, 2002. With the lack of experience and short history of Gifted Education, however, it is true that we still have many problems in promoting it in reality, We are asked to improve it by finding out what problems we have in whole area of Gifted Education, such as defining conception, choosing target students, structuring system and managi...
We investigate the structure and properties of an Artinian monomial complete intersection quotien... more We investigate the structure and properties of an Artinian monomial complete intersection quotient $A(n,d)=\mathbf{k} [x_{1}, \ldots, x_{n}] \big / (x_{1}^{d}, \ldots, x_{n}^d)$. We construct explicit homogeneous bases of $A(n,d)$ that are compatible with the $S_{n}$-module structure for $n=3$, all exponents $d \ge 3$ and all homogeneous degrees $j \ge 0$. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible component appearing in the $S_{3}$-module decomposition of homogeneous subspaces. 4, 5$.
First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex a... more First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that phase tropical hypersurfaces with smooth tropicalization, possess naturally a smooth differentiable structure. Moreover, we prove that phase tropical hypersurfaces possess a natural symplectic structure.
We provide a construction of global bases for quantum Borcherds-Bozec algebras and their integrab... more We provide a construction of global bases for quantum Borcherds-Bozec algebras and their integrable highest weight representations.
First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex a... more First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in (C∗)n. Next, we prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin’s decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that a phase tropical hypersurface with smooth tropicalization is naturally a topological manifold. Moreover, we prove that a phase tropical hypersurface is naturally homeomorphic to a symplectic manifold.
Pythagorean theorem is one of mathematical contents which is widely used during human culture hav... more Pythagorean theorem is one of mathematical contents which is widely used during human culture have developed. There are many historial records related to Pythagorean theorem made by Babylonian, Egyptian, and Mesopotamian. The theorem has the important meaning for mathematics education in secondary school education. Along with the importance of the proof itself, diverse proof methods and ideas included in their methods are also important since the methods improve students' ability to think mathematics. Hence, in this paper, we classify and analyze 390 proof methods published in the book "All that Pythagorean theorem" and other materials. Based on the results we derive educational meaning in mathematics with respect to main idea of the proof, the preliminaries of the study, and study skills used for proof.
In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. O... more In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of B(∞) and B(λ) as its application, where B(∞) and B(λ) are the crystals of the negative half part of the quantum Borcherds-Bozec algebra Uq(g) and its irreducible highest weight module V (λ), respectively.
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Papers by Young Rock Kim