• Title/Summary/Keyword: square-free discriminant

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EXACT FORMULA FOR JACOBI-EISENSTEIN SERIES OF SQUARE FREE DISCRIMINANT LATTICE INDEX

  • Xiong, Ran
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.481-488
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    • 2020
  • In this paper we give an exact formula for the Fourier coefficients of the Jacobi-Eisenstein series of square free discriminant lattice index. For a special case the discriminant of lattice is prime we show that the Jacobi-Eisenstein series corresponds to a well known Eisenstein series of modular forms.

UNIT GROUPS OF QUOTIENT RINGS OF INTEGERS IN SOME CUBIC FIELDS

  • Harnchoowong, Ajchara;Ponrod, Pitchayatak
    • Communications of the Korean Mathematical Society
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    • v.32 no.4
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    • pp.789-803
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    • 2017
  • Let $K={\mathbb{Q}}({\alpha})$ be a cubic field where ${\alpha}$ is an algebraic integer such that $disc_K({\alpha})$ is square-free. In this paper we will classify the structure of the unit group of the quotient ring ${\mathcal{O}}_K/A$ for each non-zero ideal A of ${\mathcal{O}}_K$.

THE COMPUTATION METHOD OF THE MILNOR NUMBER OF HYPERSURFACE SINGULARITIES DEFINED BY AN IRREDUCIBLE WEIERSTRASS POLYNOMIAL $z^n$+a(x,y)z+b(x,y)=0 in $C^3$ AND ITS APPLICATION

  • Kang, Chung-Hyuk
    • Bulletin of the Korean Mathematical Society
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    • v.26 no.2
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    • pp.169-173
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    • 1989
  • Let V={(x,y,z):f=z$^{n}$ -npz+(n-1)q=0 for n .geq. 3} be a compled analytic subvariety of a polydisc in $C^{3}$ where p=p(x,y) and q=q(x,y) are holomorphic near (x,y)=(0,0) and f is an irreducible Weierstrass polynomial in z of multiplicity n. Suppose that V has an isolated singular point at the origin. Recall that the z-discriminant of f is D(f)=c(p$^{n}$ -q$^{n-1}$) for some number c. Suppose that D(f) is square-free. then we prove that by Theorem 2.1 .mu.(p$^{n}$ -q$^{n-1}$)=.mu.(f)-(n-1)+n(n-2)I(p,q)+1 where .mu.(f), .mu. p$^{n}$ -q$^{n-1}$are the corresponding Milnor numbers of f, p$^{n}$ -q$^{n-1}$, respectively and I(p,q) is the intersection number of p and q at the origin. By one of applications suppose that W$_{t}$ ={(x,y,z):g$_{t}$ =z$^{n}$ -np$_{t}$ $^{n-1}$z+(n-1)q$_{t}$ $^{n-1}$=0} is a smooth family of complex analytic varieties near t=0 each of which has an isolated singularity at the origin, satisfying that the z-discriminant of g$_{t}$ , that is, D(g$_{t}$ ) is square-free. If .mu.(g$_{t}$ ) are constant near t=0, then we prove that the family of plane curves, D(g$_{t}$ ) are equisingular and also D(f$_{t}$ ) are equisingular near t=0 where f$_{t}$ =z$^{n}$ -np$_{t}$ z+(n-1)q$_{t}$ =0.}$ =0.

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RELATIVE CLASS NUMBER ONE PROBLEM OF REAL QUADRATIC FIELDS AND CONTINUED FRACTION OF $\sqrt{m}$ WITH PERIOD 6

  • Lee, Jun Ho
    • 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 Hd(f) of K of discriminant d is the ratio of class numbers 𝒪K and 𝒪f, where 𝒪K is the ring of integers of K and 𝒪f is the order of conductor f given by ℤ + f𝒪K. In 1856, Dirichlet showed that for certain m there exists an infinite number of f such that the relative class number Hd(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.

Utilization of R Program for the Partial Least Square Model: Comparison of SmartPLS and R (부분최소제곱모형을 위한 R 프로그램의 활용: SmartPLS와 R의 비교)

  • Kim, Yong-Tae;Lee, Sang-Jun
    • Journal of Digital Convergence
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    • v.13 no.12
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    • pp.117-124
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    • 2015
  • As the acceptance of statistical analysis has been increased because of Big Data, the needs for an advanced second generation of statistical analysis method like Structural Equation Model are also increasing. This study suggests how R-Program, as open software, can be utilized when Partial Least Square Model, one of the SEMs, is applied to statistical analysis. R is a free software as a part of GNU projects as well as a powerful and useful tool for statistical analysis including Big Data. The study utilized R and SmartPLS, a representative statistical package of PLS-SEM, and analyzed internal consistency reliability, convergent validity, and discriminant validity of the measurement model. The study also analyzed path coefficients and moderator effects of the structural model and compared the results, respectively. The results indicated that R showed the same results with SmartPLS on the measurement model and the structural model. Therefore, the study confirmed that R could be a powerful tool that is alternative to a commercial statistical package in the future.

Nuclear magnetic resonance-based metabolomics analysis and characteristics of beef in different fattening periods

  • Jeong, Jin Young;Baek, Youl-Chang;Ji, Sang Yun;Oh, Young Kyun;Cho, Soohyun;Seo, Hyun-Woo;Kim, Minseok;Lee, Hyun-Jeong
    • Journal of Animal Science and Technology
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    • v.62 no.3
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    • pp.321-333
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    • 2020
  • Beef quality is influenced by the fattening period. Therefore, meat metabolomics profiles from the different fattening periods (e.g., short-term vs. long-term) were analyzed for identify potential indicators using nuclear magnetic resonance. Additionally, blood, free fatty acid, sensory, and mineral compositions in Korean steers were determined. Blood, free fatty acid, and mineral concentrations showed significant differences between short-term and long-term groups that were fed different diets. However, there were no sensory differences in the two fattening groups. Additionally, the metabolic profiles of meats were clearly separated based on multivariate orthogonal partial least square discriminant analysis. Six metabolites of variable importance in the projection plot were identified and showed high sensitivity as candidate markers for meat characteristics. In particular, lactate, carnosine, and creatine could be directly linked to scientific indicators of the fattening stage (31 vs. 28 mo) of meat. Our findings suggest that the metabolomics approach could be a powerful method for the detection of novel signatures underlying the managing period of beef.