• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.569-608
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• 1994

The subject matter of this survey has to do with holomorphic maps from a compact Riemann surface to projective space, which are also called algebrac curves; the theory we survey lies at the crossroads of function theory, projective geometry, and commutative algebra (although we should mention that the present survey de-emphasizes the algebraic aspect). Algebraic curves have been vigorously and continuously investigated since the time of Riemann. The reasons for the preoccupation with algebraic curves amongst mathematicians perhaps have to do with-other than the usual usual reason, namely, the herd mentality prompting us to follow the leads of a few great pioneering methematicians in the field-the fact that algebraic curves possess a certain simple unity together with a rich and complex structure. From a differential-topological standpoint algebraic curves are quite simple as they are neatly parameterized by a single discrete invariant, the genus. Even the possible complex structures of a fixed genus curve afford a fairly complete description. Yet there are a multitude of diverse perspectives (algebraic, function theoretic, and geometric) often coalescing to yield a spectacular result.

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.565-570
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• 1994

In this article we prove a version of the Perron-Frobenius Theorem in linear algebra using the Brouwer's Fixed Point Theorem in topology. We will mostly concentrate on he qualitative aspect of the Perron-Frobenius Theorem rather than quantitative formulas, which would be enough for theoretical investigations in ergodic theory. By the nature of the method of the proof, we do not expect to obtain a numerical estimate. But we may regard it worthwhile to see why a certain type of result should be true from a topological and geometrical viewpoint. However, a geometric argument alone would give us a sharp numerical bounds on the size of the eigenvalue as shown in Section 2. Eigenvectors of a matrix A will be fixed points of a certain mapping defined in terms of A. We shall modify an existing proof of Frobenius Theorem and that will do the trick for Perron-Frobenius Theorem.

• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.953-967
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• 2011

We show that the characteristic function $1S_{\underline{\alpha}}$ of any Harder-Narasimhan strata $S{\underline{\alpha}}\;{\subset}\;Coh_X^{\alpha}$ belongs to the spherical Hall algebra $H_X^{sph}$ of a smooth projective curve X (defined over a finite field $\mathbb{F}_q$). We prove a similar result in the geometric setting: the intersection cohomology complex IC(${\underline{S}_{\underline{\alpha}}$) of any Harder-Narasimhan strata ${\underline{S}}{\underline{\alpha}}\;{\subset}\;{\underline{Coh}}_X^{\underline{\alpha}}$ belongs to the category $Q_X$ of spherical Eisenstein sheaves of X. We show by a simple example how a complete description of all spherical Eisenstein sheaves would necessarily involve the Brill-Noether stratas of ${\underline{Coh}}_X^{\underline{\alpha}}$.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.1-57
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• 2003

Vector fields in three-space admit bundles of internal variables such as a Heisenberg algebra bundle. Information transmission along field lines of vector fields is described by a wave linked to the Schrodinger representation in the realm of time-frequency analysis. The preservation of local information causes geometric optics and a quantization scheme. A natural circle bundle models quantum information visualized by holographic methods. Features of this setting are applied to magnetic resonance imaging.

• International Journal of Computer Science & Network Security
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• v.22 no.5
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• pp.255-265
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• 2022

Educational programs for students with intellectual disabilities have undergone drastic changes in pursuit of the general curriculum. Accordingly, teachers in various fields, including mathematics, strive to find effective methods that enhance learning. The objective of this systematic review is to examine the field of teaching mathematics to students with intellectual disabilities to investigate relevant effective teaching strategies and required teaching skills. To achieve this goal, studies published during the period 2018-2021 were reviewed. Findings indicate the inclusion of nine studies that met the inclusion criteria out of 55 studies. The included studies found that the system of least prompts (SLP) in conjunction with feedback and error correction, and schema-based instruction are generally the most effective strategies in teaching mathematical skills to students with intellectual disabilities. Addition is the most targeted skill, followed by subtraction and algebra problem solving. The least targeted skills are multiplication, recognition of geometric shapes, calculating price after discount, rapid recognition of numbers, and rapid problem solving. The paper provides recommendations and suggests venues of future research.

• Journal for History of Mathematics
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• v.2 no.1
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• pp.9-27
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• 1985

Islam toot a great interest in the utility sciences such as mathematics and astronomy as it needed them for the religious reasons. It needeed geometry to determine the direction toward Mecca, its holiest place: arithmetic and algebra to settle the dates of the festivals and to calculate the accounts lot the inheritance; astronomy to settle the dates of Ramadan and other festivals. Islam expanded and developed mathematics and sciences which it needed at first for the religious reasons to the benefit of all mankind. This thesis focuses upon the golden age of Islamic culture between 7th to 13th century, the age in which Islam came to possess the spirit of discovery and learning that opened the Islamic Renaissance and provided, in turn, Europeans with the setting for the Renaissance in 14th century. While Europe was still in the midst of the dark age of the feudal society based upon the agricultural economy and its mathematics was barey alive with the efforts of a few scholars in churches, the. Arabs played the important role of bridge between civilizations of the ancient and modern times. In the history of mathematics, the Arabian mathematics formed the orthodox, not collateral, school uniting into one the Indo-Arab and the Greco-Arab mathematics. The Islam scholars made a great contribution toward the development of civilization with their advanced the development of civilization with their advanced knowledge of algebra, arithmetic and trigonometry. the Islam mathematicians demonstrated the value of numerals by using arithmetic in the every day life. They replaced the cumbersome Roman numerals with the convenient Arabic numerals. They used Algebraic methods to solve the geometric problems and vice versa. They proved the correlation between these two branches of mathematics and established the foundation of analytic geometry. This thesis examines the historical background against which Islam united and developed the Indian and Greek mathematics; the reason why the Arabic numerals replaced the Roman numerals in the whole world: and the influence of the Arabic mathematics upon the development of the modern mathematics.

• The Transactions of the Korea Information Processing Society
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• v.6 no.1
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• pp.11-20
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• 1999

Because conventional spatial databases process the spatial information that is valid at current time, it is difficult to manage historical information efficiently which has been changed from the past to current. Recently, there are rapid increasing of interest to solve this problem so that makes databases to support historical information as well as spatial management at the same time. It can be eventually used in a various application areas. The formal semantics in a database is used to represent database structures and operations in order to prove the correctiveness of them in terms or mathematics. It also plays an important role in database to design a database and database management system. So in this paper, we suggest spatiotemporal domain, object, data, and spatiotemporal geometric/topological operations. And we not only formalize relational algebra/calculus using formal semantics for a spatiotemporal data model, but also show the example of real orld with them.

• Journal of the Korean School Mathematics Society
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• v.17 no.4
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• pp.507-539
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• 2014

The algebra is an important tool influencing on a mathematics in general. To make good use of the algebra, it is necessary to transfer from a given situation to a proper algebraic representation. But some research in related to algebraic word problems have reported the difficulty changing to a proper algebraic representation. Our study have focused on transformation and elaboration of algebraic representation. We investigated in detail the responses and perceptions of 29 Grade 7 students while transforming to algebraic representation, only concentrating on the literature expression form the problematic situations given. Most of students showed difficulties in transforming both descriptive and geometric problems to algebraic representation. 10% of them responded wrong answers except only a problem. Four of them were interviewed individually to show their thinking and find the factor influencing on a positive elaboration. As results, we could find some characteristics of their thinking including the misconception that regard the problem finding a functional formula because there are the variables x and y in the problematic situation. In addition, we could find the their fixation which student have to set up the equation. Furthermore we could check that making student explain own algebraic representation was able to become the factor influencing on a positive elaboration. From these, we also discussed about several didactical implications.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.36 no.10C
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• pp.614-620
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• 2011

RSA, the most commonly used public-key cryptosystem, is based on the difficulty of factoring very large integers. The fastest known factoring algorithm is the Number Field Sieve(NFS). NFS first chooses two polynomials having common root modulo N and consists of the following four major steps; 1. Polynomial Selection 2. Sieving 3. Matrix 4. Square Root, of which the most time consuming step is the Sieving step. However, in recent years, the importance of the Polynomial Selection step has been studied widely, because one can save a lot of time and memory in sieving and matrix step if one chooses optimal polynomial for NFS. One of the ideal ways of choosing sieving polynomial is to choose two polynomials with same degree. Montgomery proposed the method of selecting two (nonlinear) quadratic sieving polynomials. We proposed two cubic polynomials using 5-term geometric progression.

• Journal of Korea Spatial Information System Society
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• v.4 no.1 s.7
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• pp.5-14
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• 2002

The generalization of geospatial data is an important way in deriving a new database from an original one. The generalization of a geospatial object changes not only its geometric and aspatial attributes but also results in propagation to other objects along their relationship. We call it generalization propagation of geospatial databases. Without proper handling of the propagation, it brings about an inconsistent database or loss of semantics. Nevertheless, previous studies in the generalization have focused on the derivation of an object by isolating it from others. And they have proposed a set of generalization operators, which were intended to change the geometric and aspatial attributes of an object. In this paper we extend the definition of generalization operators to cover the propagation from an object to others. In order to capture the propagation, we discover a set of rules or constraints that must be taken into account during generalization procedure. Each generalization operator with constraints is expressed in relational algebra and it can be converted to SQL statements with ease. A prototype system was developed to verify the correctness of extended operators.