• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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• pp.711-720
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• 2010

In this note, we will establish the integration formulae for functionals such as $F(x)=\prod_{j=1}^{n}\;x(s_j)^2$ and $G(x)=\exp\{{\lambda}{\int}_{0}^{t}\;x(s)^2dm_L(s)\}$ in the analogue of Wiener measure space and using our formulae, we will derive some formulae for series.

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.351-361
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• 2010

For a bounded linear operator T on a separable complex infinite dimensional Hilbert space $\mathcal{H}$, we say that T is a quasi-class (A, k) operator if $T^{*k}|T^2|T^k\;{\geq}\;T^{*k}|T|^2T^k$. In this paper we prove that if T is a quasi-class (A, k) operator and f is an analytic function on an open neighborhood of the spectrum of T, then f(T) satisfies Weyl's theorem. Also, we consider the tensor product for quasi-class (A, k) operators.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.771-780
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• 2008

Let $M_C=\(\array{A&C\\0&B}\)$ be a $2{\times}2$ upper triangular operator matrix acting on the Hilbert space $H{\bigoplus}K\;and\;let\;{\sigma}_w(\cdot)$ denote the Weyl spectrum. We give the necessary and sufficient conditions for operators A and B which ${\sigma}_w\(\array{A&C\\0&B}\)={\sigma}_w\(\array{A&C\\0&B}\)\;or\;{\sigma}_w\(\array{A&C\\0&B}\)={\sigma}_w(A){\cup}{\sigma}_w(B)$ holds for every $C{\in}B(K,\;H)$. We also study the Weyl's theorem for operator matrices.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.743-748
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• 2009

In 1970, Fernique proved that there is a positive real number $\alpha$ such that $\int_{\mathbb{B}}\exp\{\alpha{\parallel}x{\parallel}^{2}\}dP(x)$ is finite where ($\mathbb{B},\;P$) is an abstract Wiener measure space and ${\parallel}\;{\cdot}\;{\parallel}$ is a measurable norm on ($\mathbb{B},\;P$) in [2, 3]. In this article, we investigate the existence of the integral $\int_{c}\exp\{\alpha(sup_t{\mid}x(t){\mid})^p\}dm_{\varphi}(x)$ where ($\mathcal{C}$, $m_{\varphi}$) is the analogue of Wiener measure space and p and $\alpha$ are both positive real numbers.

• Communications of the Korean Mathematical Society
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• v.32 no.4
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• pp.865-874
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• 2017

In the theory of hypergeometric and generalized hypergeometric series, classical summation theorems have been found interesting applications in obtaining various series identities for ${\Pi}$, ${\Pi}^2$ and ${\frac{1}{\Pi}}$. The aim of this research paper is to provide twelve general formulas for ${\frac{1}{\Pi}}$. On specializing the parameters, a large number of very interesting series identities for ${\frac{1}{\Pi}}$ not previously appeared in the literature have been obtained. Also, several other results for multiples of ${\Pi}$, ${\Pi}^2$, ${\frac{1}{{\Pi}^2}}$, ${\frac{1}{{\Pi}^3}}$ and ${\frac{1}{\sqrt{\Pi}}}$ have been obtained. The results are established with the help of the extensions of classical Gauss's summation theorem available in the literature.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.251-274
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• 2020

We study a dynamical system that was originally defined by Romik in 2008 using an old theorem of Berggren concerning Pythagorean triples. Romik's system is closely related to the Farey map on the unit interval which generates an additive continued fraction algorithm. We explore some number theoretical properties of the Romik system. In particular, we prove an analogue of Lagrange's theorem in the case of the Romik system on the unit quarter circle, which states that a point possesses an eventually periodic digit expansion if and only if the point is defined over a real quadratic extension field of rationals.

• Journal of IKEEE
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• v.18 no.2
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• pp.255-262
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• 2014

The uncertain integral linear system is the integral-augmented uncertain system to improve the resultant performance. In this note, for a MI(Multi Input) uncertain integral linear case, Utkin's theorem is proved clearly and comparatively. With respect to the two transformations(diagonalizations), the equation of the sliding mode is invariant. By using the results of this note, in the SMC for MIMO uncertain integral linear systems, the existence condition of the sliding mode on the predetermined sliding surface is easily proved. The effectiveness of the main results is verified through an illustrative example and simulation study.

• The Pure and Applied Mathematics
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• v.18 no.1
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• pp.31-39
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• 2011

Stone's theorem states that "A bounded linear operator A is infinitesimal generator of a $C_0$-group of unitary operators on a Hilbert space H if and only if iA is self adjoint". In this paper we establish a generalization of Stone's theorem in the framework of Hilbert $C^*$-modules.

• The Mathematical Education
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• v.29 no.1
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• pp.55-61
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• 1990

There are three objectives of this paper. First, we present an elegant and simple generalization of Abel's theorem (i .e. tile Abel summability (on the unit disk of the euclidean plane) is regular). Second, we consider the definition of Abel summability through lim (equation omitted) which immediately has clear connexctions with CeSARO summability and Cesaro sums (equation omitted). This approach examplifies some simple aspects of so-called Tauberian theorems of divergent series. Third, we present the applications of the previous results to find the limits of transition probabilities of homogeneous Marker chain. Finally, we explain why the original Abel's theorem which looks obvious is difficult to be proved, and can not be proved analytically.

• Journal of Educational Research in Mathematics
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• v.19 no.4
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• pp.479-491
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• 2009

This paper researches the possibility of introducing Descartes' theorem to mathematically gifted students. Not only is Descartes' theorem logically equivalent to Euler's theorem but is hierarchically connected with Gauss-Bonnet theorem which is the core concept on differential geometry. It is possible to teach mathematically gifted students Descartes' theorem by generalizing mathematical property in solid geometry through analogical reasoning, that is, so in a polyhedrons the sum of the deficient angles is $720^\circ$ as in an polygon the sum of the exterior angles is $360^\circ$. This study introduces an alternative method of instruction that we enable mathematically gifted students to reinvent Descartes' theorem through analogical reasoning instead of deductive reasoning.