• Journal for History of Mathematics
• /
• v.16 no.3
• /
• pp.45-56
• /
• 2003

In this paper, we investigate the origin of Euclid's fifth postulate. For this we analyze the Euclid's proof of the Pythagorean theorem, so form a hypothesis "The Euclid's fifth postulate originated from the Pythagorean theorem." And we test our hypothesis by some historical evidences.evidences.

• Bulletin of the Korean Mathematical Society
• /
• v.29 no.1
• /
• pp.81-87
• /
• 1992

Wellknown invariance of domain theorems are Brower's invariance of domain theorem for continuous mappings defined on a finite dimensional space and Schauder-Leray's invariance of domain theorem for the class of mappings I+C defined on a infinite dimensional Banach space with I the identity and C compact. The two classical invariance of domain theorems were proved by applying the homotopy invariance of Brower's degree and Leray-Schauder's degree respectively. Degree theory for some class of mappings is a useful tool for mapping theorems. And mapping theorems (or surjectivity theorems of mappings) are closely related with invariance of domain theorems for mappings. In[4, 5], Browder and Petryshyn constructed a multi-valued degree theory for A-proper mappings. From this degree Petryshyn [9] obtained some invariance of domain theorems for locally A-proper mappings. Recently Browder [6] has developed a degree theory for demicontinuous mapings of type ( $S_{+}$) from a reflexive Banach space X to its dual $X^{*}$. By applying this degree we obtain some invariance of domain theorems for demicontinuous mappings of type ( $S_{+}$). ( $S_{+}$).

• Journal of the Earthquake Engineering Society of Korea
• /
• v.7 no.4
• /
• pp.81-87
• /
• 2003

In this paper, the feasibility of using Shannon's sampling theorem to reconstruct exact mode shapes of a structural system from a limited number of sensor points and localizing damage in that structure with reconstructed mode shapes is investigated. Shannon's sampling theorem for the time domain is reviewed. The theorem is then extended to the spatial domain. To verify the usefulness of extended theorem, mode shapes of a simple beam are reconstructed from a limited amount of data and the reconstructed mode shapes are compared to the exact mode shapes. On the basis of the results, a simple rule is proposed for the optimal placement of accelerometers in modal parameter extraction experiments. Practicality of the proposed rule and the extended Shannon's theorem is demonstrated by detecting damage in laboratory beam structure with two-span via applying to mode shapes of pre and post damage states.

• Research in Mathematical Education
• /
• v.14 no.3
• /
• pp.211-218
• /
• 2010

It is a truism that mathematics is about relations (cf. [Halford, G. S. (1999). The properties of representations used in higher cognitive processes: Developmental implications. In: Sigel, I. E. (Ed.), The Development of Mental Representation: Theories and Applications (pp. 147-168). Mahwah, New Jersey: Erlbaum]). In this article we are considering few problems related to the Viviani's and Routh's Theorems. All Problems are connected by the relation which exists between the distances of the point inside the triangle to it sides. We show how reasoning about the relations could lead the student's problem solving process and give easy to understand solutions of the problems. Among the problems being considered are the proof of the Converse to Viviani's Theorem, the formulas for areas of all figures formed by the sides of triangle and its cevians.

• Communications of the Korean Mathematical Society
• /
• v.38 no.4
• /
• pp.1127-1139
• /
• 2023

We investigate an extension of Schauder's theorem by studying the relationship between various s-numbers of an operator T and its adjoint T^*. We have three main results. First, we present a new proof that the approximation number of T and T^* are equal for compact operators. Second, for non-compact, bounded linear operators from X to Y, we obtain a relationship between certain s-numbers of T and T^* under natural conditions on X and Y . Lastly, for non-compact operators that are compact with respect to certain approximation schemes, we prove results for comparing the degree of compactness of T with that of its adjoint T^*.

• Journal for History of Mathematics
• /
• v.30 no.1
• /
• pp.1-15
• /
• 2017

The original proof-argument of Riemann in 1851 for the Riemann mapping theorem, one of the most central theorems in Complex analysis, was found faulty and essentially buried underneath the proof by $Carath{\acute{e}}odory$ of 1929, now accepted as the "textbook" proof. On the other hand, the original Riemann's "proof" was rediscovered and made correct by R.E. Greene and the author of this article in 2016. In this article, we try to shed lights onto the history related to the Riemann mapping theorem and the surrounding developments of 1850-1930 by reflecting upon the main flow of ideas and methods of the proof by R. E. Greene and K.-T. Kim.

• Communications of the Korean Mathematical Society
• /
• v.37 no.2
• /
• pp.503-520
• /
• 2022

We introduce a new type of fractional derivative, which we call as the right local general truncated M-fractional derivative for α-differentiable functions that generalizes the fractional derivative type introduced by Anastassiou. This newly defined operator generalizes the standard properties and results of the integer order calculus viz. the Rolle's theorem, the mean value theorem and its extension, inverse property, the fundamental theorem of calculus and the theorem of integration by parts. Then we represent a relation of the newly defined fractional derivative with known fractional derivative and in context with this derivative a physical problem, Kirchoff's voltage law, is generalized. Also, the importance of this newly defined operator with respect to the flexibility in the parametric values is described via the comparison of the solutions in the graphs using MATLAB software.

• Journal of applied mathematics & informatics
• /
• v.42 no.2
• /
• pp.433-444
• /
• 2024

Our study explores the connection between the Pythagorean theorem and the Fixed-point theorem in metric spaces. Both of which center around the concepts of distance transformations and point relationships. The Pythagorean theorem deals with right triangles in Euclidean space, emphasizing distances between points. In contrast, fixed-point theorems pertain to the points that remain unchanged under specific transformations thereby preserving distances. The article delves into the intrinsic correlation between these concepts and presents a novel study in Euclidean metric spaces, examining the relationship between contraction mapping and Pythagorean Right Triangles. Practical applications are also discussed particularly in the context of image compression. Here, the integration of the Pythagorean right triangle paradigm with contraction mappings results in efficient data representation and the preservation of visual data relation-ships. This illustrates the practical utility of seemingly abstract theories in addressing real-world challenges.

• Journal of the Chungcheong Mathematical Society
• /
• v.25 no.3
• /
• pp.527-529
• /
• 2012

It is proved that the fractional Sobolev spaces $W^s_p(\mathbb{R}^n)$ 0 < $s$ < $n$, are compactly embedded into Lebesgue spaces $L^q(\Omega)$ for some bounded set $\Omega$­.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.1
• /
• pp.163-168
• /
• 2010

In this paper it is shown that if [$T^*$,T] is of rank one and ker [$T^*$,T] is invariant for T, then T is quasinormal. Thus, we can know that the hyponormal condition is superfluous in the Morrel's theorem.