• The Pure and Applied Mathematics
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• 제2권1호
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• pp.17-24
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• 1995

We begin with a brief survey of some of the known results dealing with Bloch constants. Bloch's theorem asserts that there is a constant B$\_$1.C/(1, 0) such that if f is holomorphic in the open unit disk D and normalized by │f'(0)│$\geq$1, then the Riemann surface of f contains an unramified disk of radius at least B$\_$1.C/(1, 0) (see[7,p.14]).(omitted)

• The Pure and Applied Mathematics
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• 제18권4호
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• pp.345-351
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• 2011

Taylor's formula is a powerful tool in analysis. In this study, we assume that an operator is m-times Fr$\acute{e}$chet-differentiable and satisfies a Lipschitz condition. We then obtain some Taylor formulas using only the Lipschitz constants. Applications are also provided.

• Bulletin of the Korean Mathematical Society
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• 제40권4호
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• pp.583-591
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• 2003

For a weighted composition operator $uC_{\varphi}$ on C(X), we determine its essential norm and the constant for its Hyers-Ulam stability, in terms of the set $\varphi(\{x\;\in\;X\;:\;$\mid$u(x)$\mid$\;\geq\;r\})$ (r > 0).

• Communications of the Korean Mathematical Society
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• 제13권2호
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• pp.435-443
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• 1998

Some mathematical constants have been used in evaluating series involving the Zeta function, the origin of which can be traced back to an over two centries old theorem of Christian Goldbach. We show some of the series involving the Zeta function can be evaluated in terms of the Catalan's constant by using the theory of the double Gamma function.

• East Asian mathematical journal
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• 제18권1호
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• pp.75-84
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• 2002

The third important Euler-Mascheroni constant $\gamma$, like $\pi$ and e, is involved in representations, evaluations, and purely relationships among other mathematical constants and functions, in various ways. The main object of this note is to summarize some known series representaions for $\gamma$ with comments for their proofs, and to point out that one of those series representaions for $\gamma$ seems to be incorrectly recorded. A brief historical comment for $\gamma$ is also provided.

• Bulletin of the Korean Mathematical Society
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• 제40권2호
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• pp.341-349
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• 2003

Let {$x_{nk}\;$\mid$1\;\leq\;k\;\leq\;n,\;n\;\geq\;1$} be an array of random varianbles and $\{a_n$\mid$n\;\geq\;1\}\;and\;\{b_n$\mid$n\;\geq\;1} be a sequence of constants with $a_n\;>\;0,\;b_n\;>\;0,\;n\;\geq\;1. In this paper, for array of row negatively associated(NA) random variables, we establish a general weak law of large numbers (WLLA) of the form (${\sum_{\kappa=1}}^n\;a_{\kappa}X_{n\kappa}\;-\;\nu_{n\kappa})\;/b_n$ converges in probability to zero, as $n\;\rightarrow\;\infty$, where {$\nu_{n\kappa}$\mid$1\;\leq\;\kappa\;\leq\;n,\;n\;\geq\;1$} is a suitable array of constants.

• Bulletin of the Korean Mathematical Society
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• 제54권1호
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• pp.199-214
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• 2017

We consider fixed-point iterations constructed by simple transforming from a quadratic matrix equation to equivalent fixed-point equations and assume that the iterations are well-defined at some solutions. In that case, we suggest real valued functions. These functions provide radii at the solution, which guarantee the local convergence and the uniqueness of the solutions. Moreover, these radii obtained by simple calculations of some constants. We get the constants by arbitrary matrix norm for coefficient matrices and solution. In numerical experiments, the examples show that the functions give suitable boundaries which guarantee the local convergence and the uniqueness of the solutions for the given equations.

• Journal of the Korean Mathematical Society
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• 제54권6호
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• pp.1853-1878
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• 2017

Let $\{U(t,s)\}_{t{\geq}s}$ be a periodic evolutionary process with period ${\tau}$ > 0 on a Banach space X. Also, let L be the generator of the evolution semigroup associated with $\{U(t,s)\}_{t{\geq}s}$ on the phase space $P_{\tau}(X)$ of all ${\tau}$-periodic continuous X-valued functions. Some kind of variation-of-constants formula for the solution u of the equation $({\alpha}I-L)^nu=f$ will be given together with the conditions on $f{\in}P_{\tau}(X)$ for the existence of coefficients in the formula involving the monodromy operator $U(0,-{\tau})$. Also, examples of ODEs and PDEs are presented as its application.

• Journal of the Korean Mathematical Society
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• 제54권5호
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• pp.1521-1536
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• 2017

This paper introduces a representation style of variable binding using dependent types when formalizing meta-theoretic properties. The style we present is a variation of the Coquand-McKinna-Pollack's locally-named representation. The main characteristic is the use of dependent families in defining expressions such as terms and formulas. In this manner, we can handle many syntactic elements, among which wellformedness, provability, soundness, and completeness are critical, in a compact manner. Another point of our paper is to investigate the roles of free variables and constants. Our idea is that fresh constants can entirely play the role of free variables in formalizing meta-theories of first-order predicate logic. In order to show the feasibility of our idea, we formalized the soundness and completeness of LJT with respect to Kripke semantics using the proof assistant Coq, where LJT is the intuitionistic first-order predicate calculus. The proof assistant Coq supports all the functionalities we need: intentional type theory, dependent types, inductive families, and simultaneous substitution.

• Communications of the Korean Mathematical Society
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• 제37권2호
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• pp.551-566
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• 2022

We apply Fourier-Legendre-based integration methods that had been given by Campbell in 2021, to evaluate new rational double hypergeometric sums involving ${\frac{{1}}{\pi}}$. Closed-form evaluations for dilogarithmic expressions are key to our proofs of these results. The single sums obtained from our double series are either inevaluable $_2F_1({\frac{4}{5}})$- or $_2F_1({\frac{1}{2}})$-series, or Ramanujan's ₃F₂(1)-series for the moments of the complete elliptic integral K. Furthermore, we make use of Ramanujan's finite sum identity for the aforementioned ₃F₂(1)-family to construct creative new proofs of Landau's asymptotic formula for the Landau constants.