• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1559-1568
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• 2015

The relative class number $H_d(f)$ of a real quadratic field $K=\mathbb{Q}(\sqrt{m})$ of discriminant d is the ratio of class numbers of $O_f$ and $O_K$, where $O_K$ denotes the ring of integers of K and $O_f$ is the order of conductor f given by $\mathbb{Z}+fO_K$. In a recent paper of A. Furness and E. A. Parker the relative class number of $\mathbb{Q}(\sqrt{m})$ has been investigated using continued fraction in the special case when $(\sqrt{m})$ has a diagonal form. Here, we extend their result and show that there exists a conductor f of relative class number 1 when the continued fraction of $(\sqrt{m})$ is non-diagonal of period 4 or 5. We also show that there exist infinitely many real quadratic fields with any power of 2 as relative class number if there are infinitely many Mersenne primes.

• East Asian mathematical journal
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• v.37 no.5
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• pp.613-617
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• 2021

Abstract. For a positive square-free integer m, let K = ℚ($\sqrt{m}$) be a real quadratic field. The relative class number H_d(f) of K of discriminant d is the ratio of class numbers 𝒪_K and 𝒪_f, where 𝒪_K is the ring of integers of K and 𝒪_f is the order of conductor f given by ℤ + f𝒪_K. In 1856, Dirichlet showed that for certain m there exists an infinite number of f such that the relative class number H_d(f) is one. But it remained open as to whether there exists such an f for each m. In this paper, we give a result for existence of real quadratic field ℚ($\sqrt{m}$) with relative class number one where the period of continued fraction expansion of $\sqrt{m}$ is 6.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.709-716
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• 1996

Nielsen coincidence theory is concerned with the determinatin of a lower bound of the minimal number MC[f,g] of coincidence points for all maps in the homotopy class of a given map (f,g) : X $\to$ Y. The Nielsen Nielsen number $N_R(f,g)$ (similar to [9]) is introduced in [3], which is a lower bound for the number of coincidence points in the relative homotopy class of (f,g) and $N_R(f,g) \geq N(f,g)$.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.559-578
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• 2019

We list all subfields of cyclotomic function fields over rational function fields with class number three. We also determine all the imaginary abelian extensions with relative class number three, explicitly.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.171-181
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• 1995

Nielsen coincidence theory is concerned with the estimation of a lower bound for the number of coincidences of two maps $f,g: X \longrightarrow Y$. For this purpose the so-called Nielsen number N(f,g) is introduced, which is a lower bound for the number of coincidences ([1]). The relative Nielsen number N(f : X,A) in the fixed point theory is introduced in [3], which is a lower bound for the number of fixed points for all maps in the relative homotopy class of f:(X,A) $\longrightarrow$ (X,A), and its estimation is given in [5].

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.245-252
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• 1996

The relative Nielsen number N(f;X,A) was introduced in 1986. It gives us a better, and ideally sharp, lower bound for the minimum number MF[f;X,A] of fixed points in the homotopy class of the map $f;(X,A) \to (X,A)$. Similarly, we also can think about the Nielsen map $f:(X,A) \to (X,A)$. Similarly, we also can be think about the Nielsen root theory. In this paper, we introduce a relative root Nielsen number N(f;X,A,c) of $f:(X,A) \to (Y,B)$ and show some basic properties.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.77-80
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• 2006

Let K be a number field, $K_n$ its ray class field with conductor n and L a Galois extension of K containing $K_n$. We prove that $L/K_n$ has a relative integral basis (RIB) under certain simple condition. Also we reduce the problem of the existence of a RIB to a quadratic extension of $K_n$ under some condition.

• Journal of Forest and Environmental Science
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• v.33 no.2
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• pp.97-104
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• 2017

Investigation was carried out on response of Tetrapleura tetraptera (Schum. and Thonn.) to soil, water and light with the view of its domestication and introduction to different ecological regions. The experiment was arranged in a factorial experiment of $3{\times}3{\times}3$ in a completely randomized design (CRD) with three replicates. The factors were: soil textural class (Loamy sand, Sand and Sandy clay loam), watering regime (daily, twice a week and once a week) and light intensity (100%, 75% and 50%). Soil textural classes had significant influence on collar diameter, stem height, number of leaflets, root/shoot ratio and relative growth rate of Tetrapleura seedlings. Seedlings grown on loamy sand recorded the highest mean value- 2.28 mm for collar diameter, stem height- 12.9 cm, number of leaflets- 19.9, chlorophyll b- $0.34mg\;mL^{-1}$, leaf relative water content- 27.4% and relative growth rate- $0.037mg\;g^{-1}\;day^{-1}$. Watering regime had significant influence on the collar diameter of Tetrapleura. Seedlings watered daily recorded the highest mean value- 2.25 mm for collar diameter. Light intensity significantly influenced collar diameter and root/shoot ratio. Seedlings exposed to 100% light intensity recorded higher mean value for collar diameter- 2.28 mm and root/shoot ratio- 1.481 cm. The interaction between soil textural class and light intensity significantly affected collar diameter, stem height and number of leaflets. Higher mean value for collar diameter (2.47 mm) stem height (13.25 cm) and number of leaflets (21.16) were recorded while the interaction between soil textural class, light intensity and watering regime was significant for only number of leaflets. Tetrapleura exhibited some level of tolerance to different soil texture, drought and light intensity. Therefore, Tetrapleura has the potentials to be raised in different ecological zones characterized by difference in soil, rainfall and amount of sunshine.

• Korean Journal of Veterinary Research
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• v.50 no.3
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• pp.179-185
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• 2010

The present study was undertaken to establish reference values for the composition blood lymphocyte populations and compare forty three Hanwoo neonatal calves (KC) with twenty one Holstein calves (HC) by blood cell count and immunophynotying. The percentages of CD2+, CD4+, CD8+, CD26+, ACT2+, MHC class, MHC class II and WC1+ T cells, B cells were determined by flow cytometry. The number of lymphocyte and monocyte in HC were higher than those of KC. However, the number of neutrophils was higher in HC than KC. The proportions of CD2+, CD4+, CD8+, MHC class, and WC1+ lymphocytes remained relatively stable during the study period, while there was a moderate increase in the relative percentage of CD26+, ACT2+, MHC class II and B cell from birth to approximately 3 weeks of age. Marked differences in the relative proportions of the lymphocyte subpopulations were noted between the individual calves. The present study shows that the T-cell subpopulations are present in peripheral blood of KC at levels comparable with HC, while the MHC class II and B cell population of KC increases significantly with age. The absolute number of WBC in KC was due to the decrease of absolute number of neutrophil rather than the increase of lymphocyte. The results indicated that KC have significantly higher number of neutrophils, and proportion of MHC class II and B cell than HC.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.815-827
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• 2019

It is well known that the denominator of the Dedekind sum s(c, d) divides 2 gcd(d, 3)d and that no smaller denominator independent of c can be expected. In contrast, here we prove that we usually get a smaller denominator in S(H, d), the sum of the s(c, d)'s over all the c's in a subgroup H of order n > 1 in the multiplicative group $(\mathbb{Z}/d\mathbb{Z})^*$. First, we prove that for p > 3 a prime, the sum 2S(H, p) is a rational integer of the same parity as (p-1)/2. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of S(H, d) for non necessarily prime d's. We show that its denominator is a divisor of some explicit divisor of 2d gcd(d, 3).