• Title/Summary/Keyword: theoretical generalization

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The Theoretical Generalization Appling the Strategy(WIOS) finding an Intrinsic Attribute (본질적 속성 찾기 전략(WIOS)을 통한 이론적 일반화)

  • Roh, Eun-Hwan;Jun, Young-Bae;Kang, Jeong-Gi
    • Communications of Mathematical Education
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    • v.26 no.1
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    • pp.51-69
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    • 2012
  • The cognition of an intrinsic attribute play an important role in the process of theoretical generalization. It is the aim of this paper to study how the theoretical generalization is made. First of all, we suggest the What-if-only-strategy(WIOS) which is the strategy helping the cognition of an intrinsic attribute. And we propose the process of the theoretical generalization that go on the cognitive stage, WIOS stage, conjecture stage, justification stage and insight into an intrinsic attribute in order. We propose the process of generalization adding the concrete process cognizing an intrinsic attribute to the existing process of generalization. And we applied the proposed process of generalization to two mathematical theorem which is being managed in middle school. We got a conclusion that the what-if-only strategy is an useful method of generalization for the proposition. We hope that the what-if-only strategy is helpful for both teaching and learning the mathematical generalization.

MARGIN-BASED GENERALIZATION FOR CLASSIFICATIONS WITH INPUT NOISE

  • Choe, Hi Jun;Koh, Hayeong;Lee, Jimin
    • Journal of the Korean Mathematical Society
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    • v.59 no.2
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    • pp.217-233
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    • 2022
  • Although machine learning shows state-of-the-art performance in a variety of fields, it is short a theoretical understanding of how machine learning works. Recently, theoretical approaches are actively being studied, and there are results for one of them, margin and its distribution. In this paper, especially we focused on the role of margin in the perturbations of inputs and parameters. We show a generalization bound for two cases, a linear model for binary classification and neural networks for multi-classification, when the inputs have normal distributed random noises. The additional generalization term caused by random noises is related to margin and exponentially inversely proportional to the noise level for binary classification. And in neural networks, the additional generalization term depends on (input dimension) × (norms of input and weights). For these results, we used the PAC-Bayesian framework. This paper is considering random noises and margin together, and it will be helpful to a better understanding of model sensitivity and the construction of robust generalization.

On the design of a teaching unit for the exploration of number patterns in Pascal graphs and triangles applying theoretical generalization. (이론적 일반화를 적용한 파스칼 그래프와 삼각형에 내재된 수의 패턴 탐구를 위한 교수단원의 설계)

  • Kim, Jin Hwan
    • East Asian mathematical journal
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    • v.40 no.2
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    • pp.209-229
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    • 2024
  • In this study, we design a teaching unit that constructs Pascal graphs and extended Pascal triangles to explore number patterns inherent in them. This teaching unit is designed to consider the diachronic process of teaching-learning by combining Dörfler's theoretical generalization model with Wittmann's design science ideas, which are applied to the didactical practice of mathematization. In the teaching unit, considering the teaching-learning level of prospective teachers who studied discrete mathematics, we generalize the well-known Pascal triangle and its number patterns to extended Pascal triangles which have directed graphs(called Pascal graphs) as geometric models. In this process, the use of symbols and the introduction of variables are exhibited as important means of generalization. It provides practical experiences of mathematization to prospective teachers by going through various steps of the generalization process targeting symbols. This study reflects Wittmann's intention in that well-understood mathematics and the context of the first type of empirical research as structure-genetic didactical analysis are considered in the design of the learning environment.

Linguistic Description and Theory

  • Nakajima, Heizo
    • Korean Journal of English Language and Linguistics
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    • v.1 no.3
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    • pp.349-368
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    • 2001
  • We have brought up several distinct types of English clausal constructions, and have been lead to the descriptive generalization in (14),repeated here as (33): (33) Reduced clauses cannot occur in non-complement positions. The generalization in (33) refers to two theory-internal notions, reduced clauses and non-complement positions. Both notions are concerned with the composition of syntactic structures to be defined by X-bar theory. Without these theoretical notions, it would be difficult to describe in a general form the fact that certain types of complement clauses-namely, null-that clauses, if-clauses, Acc-ing gerund, ECM complement clauses, and Raising complement clauses-cannot occur in particular syntactic positions. Instead, one would have to describe this fact for each clause type, in such a way that null-that clauses cannot occur in such and such positions, and if-clauses cannot occur in such and such positions, and Acc-ing gerund cannot occur in such and such positions, and so on, although the positions in which they cannot occur are totally the same. Given the terminology of X-bar theory, however, it has turned out that these types of complement clauses are all reduced clauses, and the positions where they cannot occur are all non-complement positions. Then, the generalization has obtained that reduced clauses cannot occur in non-complement positions. It is a theoretical issue, and differs depending upon theories, how to explain why such a descriptive generalization holds at all. Hopefully, the demonstration here provides a piece of evidence showing that a theory or a particular theoretical nation plays an important role in the description of linguistic facts. Moreover, I have made a crucial prediction on the basis of the well-accepted theoretical assumption the ECM complement clauses and Raising complement clauses are reduced clauses; namely, the prediction that these types of clauses cannot occur in non-complement position. The prediction based upon the theoretical assumption is actually borne out, as illustrated earlier. The illustration of the prediction, I hope, shows that a theory or a particular theoretical assumption, coupled with another theoretical assumption, allows us to make some interesting predictions. Predictions serve to widen a range of linguistic facts to be described. A theory plays a crucial part in finding out interesting facts as well as in describing them in some general forms. Finally, let me state a few words as to the recent generative theory in connection with linguistic description. The recent generative theory is getting more and more abstract. I think it is moving toward a good direction as cognitive science. It will contribute, among others, to the inquiry into what is knowledge that is very specific to language faculty, and into how it interacts with other cognitive faculties. However, I am suspicious about how much the abstract generative theory will contribute to the description of linguistic facts in a particular language. While generative theory is claimed to aim both for descriptive adequacy and for explanatory adequacy, the recent generative theory is likely to put much more weight on explanatory adequacy. In my view, a less abstract theory is enough, or even more useful, for the purpose of linguistic description. Of course, how abstract theory one should adopt as a framework differs depending upon what aspect of language one attempts to describe. What I would like to emphasize here is that linguistic theory does not conflicts with linguistic description, and a linguistic theory with an appropriate degree of abstractness serves as a tool for finding out new interesting facts, as well as for describing them in some general, elegant forms.

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Pain and Muscle Elasticity for Deficiency-Excessiveness Discussed by the View of Oriental and Western Medicine (경근(頸筋)의 동통(疼痛) 및 근(筋) 탄력상태(彈力狀態)에 대한 허실(虛實)의 동서의학적 고찰)

  • Lee Dong-Kyu;Seo Hyung-Joo;Na Chang-Su
    • Korean Journal of Acupuncture
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    • v.17 no.1
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    • pp.141-156
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    • 2000
  • Prognosis in oriental medicine gathers information by four examination methods. It provides important information to understand the degree of deficiency - excessiveness of a patient to treat properly. To generalize the degree of deficiency - excessiveness can be found by seeing the patient's muscle response and pain perception to the palpitations.The theoretical basis to generalize deficiency - excessiveness, oriental and western medical understanding of pain perception and the elasticity of muscle were discussed.The usual symptoms for the excessiveness could include Pain (dislikeness to the palpitation), Stiffness of nape and limbs, Contracture of the limbs, Clonic convulsion and Fast pain. The symptoms for the deficiency could include Pain (likeness to the palpitation, Gastrocnemius muscle spasm, Flaccid paralysis of limbs and Slow pain. More theoretical bases for generalization of deficiency - excessiveness are needed along with the simplifying the complex clinical symptoms. In this way, we can discuss about deficiency - excessiveness with the regard to western medicine to help its generalization.

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NEW GENERALIZATION OF THE WRIGHT SERIES IN TWO VARIABLES AND ITS PROPERTIES

  • Belafhal, Abdelmajid;Chib, Salma;Usman, Talha
    • Communications of the Korean Mathematical Society
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    • v.37 no.1
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    • pp.177-193
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    • 2022
  • The main aim of this paper is to introduce a new generalization of the Wright series in two variables, which is expressed in terms of Hermite polynomials. The properties of the freshly defined function involving its auxiliary functions and the integral representations are established. Furthermore, a Gauss-Hermite quadrature and Gaussian quadrature formulas have been established to evaluate some integral representations of our main results and compare them with our theoretical evaluations using graphical simulations.

A phonological study and historical view on IC clusters in English (영어 lC 자음군에 관한 역사적 조명과 음운적 고찰)

  • Oh, Kwanyoung
    • English Language & Literature Teaching
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    • v.16 no.4
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    • pp.201-222
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    • 2010
  • The purpose of this study is to investigate /l/-deletion in lC clusters which are composed of a lateral followed by consonants at syllable-final position in English. For this, I have analyzed /l/-deletion in words depending on conditions and theoretical analyses such as Sonority Sequencing Generalization, Cluster Simplification, Complex sounds and merger, and Feature Geometry, but they didn't offer a very satisfactory explanation to the phenomenon. Therefore, I adopted a historical approach in order to determine the cause and origin of /l/-deletion in lC clusters, and then as a phonological analysis tool, I relied on the constraints and their ranking in Optimal Theory framework for explaining /l/-deletion in the clusters more consistently. As a result, I can explain the phenomenon more explicitly than from the above mentioned analyses.

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ON A MULTI-PARAMETRIC GENERALIZATION OF THE UNIFORM ZERO-TWO LAW IN L1-SPACES

  • MUKHAMEDOV, FARRUKH
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.6
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    • pp.1819-1826
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    • 2015
  • Following an idea of Ornstein and Sucheston, Foguel proved the so-called uniform "zero-two" law: let T : $L^1$(X, $\mathcal{F}$, ${\mu}$) ${\rightarrow}$ $L^1$(X, $\mathcal{F}$, ${\mu}$) be a positive contraction. If for some $m{\in}{\mathbb{N}}{\cup}\{0\}$ one has ${\parallel}T^{m+1}-T^m{\parallel}$ < 2, then $\lim_{n{\rightarrow}{\infty}}{\parallel}T^{m+1}-T^m{\parallel}=0$. There are many papers devoted to generalizations of this law. In the present paper we provide a multi-parametric generalization of the uniform zero-two law for $L^1$-contractions.

A Study On The Realization Of Multi-Threshold Function By Partition Of Switching Functions (스윗칭함수 분할에 의한 다역치함수 실현에 관한 연구)

  • Chae Tak Lim
    • 전기의세계
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    • v.23 no.4
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    • pp.53-59
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    • 1974
  • This paper investigates the theoretical properties of a logic element called the multithreshold threshold element, which is a generalization of the single-threshold threshold element. The primary partition os a systematic method of obtaining the multi-threshold realization of a switching function by the index numbers. The concept of comparable vertices of the same index numbers introduced in this paper is very promising for testing the multi-threshold partition by the initial condition to be defined by the minterms of the same index numbers.

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