• Title/Summary/Keyword: The Proving of Theorem

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Some Notes on the Temporal Single-System Interpretation of Marxian Value Theory (마르크스 가치론의 이시적 단일체계 해석에 대한 몇 가지 비판적 논점)

  • Park, Hyun Woong
    • 사회경제평론
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    • v.31 no.2
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    • pp.105-126
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    • 2018
  • In this paper, I makes some critical comments on the temporal single-system interpretation (TSSI) of Marxian value theory. The first concerns the claim that the Fundamental Marxian Theorem (FMT) holds within the TSSI under completely general conditions. Based on the idea that the nominal profit does not well fit the FMT, the TSSI proponents suggest that more relevant for proving the TSSI FMT is the real profit, defined by deflating the output prices. In contrast, I propose a more general approach where three possible concepts of profit are all considered, in which case the result is that whether the FMT holds within the TSSI is indeterminate. Second, the refutation of the Okishio theorem presented in Kliman (1996) is critically examined, focusing on the criticism raised in the literature that the Kliman model ignores the cost-reduction criterion as for the technical change and therefore cannot be considered as an internal refutation of the Okishio theorem. Drawing upon the criticism, I explicitly incorporate the cost-reduction criterion into the Kliman model and show that the continuous labor-saving technical change of the Kliman model is not necessarily cost-reducing and under certain conditions is cost-neutral or cost-raising.

Lebesgue-Stieltjes Measures and Differentiation of Measures

  • Jeon, Won-Kee
    • Honam Mathematical Journal
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    • v.8 no.1
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    • pp.51-74
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    • 1986
  • The thery of measure is significant in that we extend from it to the theory of integration. AS specific metric outer measures we can take Hausdorff outer measure and Lebesgue-Stieltjes outer measure connecting measure with monotone functions.([12]) The purpose of this paper is to find some properties of Lebesgue-Stieltjes measure by extending it from $R^1$ to $R^n(n{\geq}1)$ $({\S}3)$ and differentiation of the integral defined by Borel measure $({\S}4)$. If in detail, as follows. We proved that if $_n{\lambda}_{f}^{\ast}$ is Lebesgue-Stieltjes outer measure defined on a finite monotone increasing function $f:R{\rightarrow}R$ with the right continuity, then $$_n{\lambda}_{f}^{\ast}(I)=\prod_{j=1}^{n}(f(b_j)-f(a_j))$$, where $I={(x_1,...,x_n){\mid}a_j$<$x_j{\leq}b_j,\;j=1,...,n}$. (Theorem 3.6). We've reached the conclusion of an extension of Lebesgue Differentiation Theorem in the course of proving that the class of continuous function on $R^n$ with compact support is dense in $L^p(d{\mu})$ ($1{\leq$}p<$\infty$) (Proposition 2.4). That is, if f is locally $\mu$-integrable on $R^n$, then $\lim_{h\to\0}\left(\frac{1}{{\mu}(Q_x(h))}\right)\int_{Qx(h)}f\;d{\mu}=f(x)\;a.e.(\mu)$.

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MARCINKIEWICZ-TYPE LAW OF LARGE NUMBERS FOR DOUBLE ARRAYS

  • Hong, Dug-Hun;Volodin, Andrei I.
    • Journal of the Korean Mathematical Society
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    • v.36 no.6
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    • pp.1133-1143
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    • 1999
  • Chaterji strengthened version of a theorem for martin-gales which is a generalization of a theorem of Marcinkiewicz proving that if $X_n$ is a sequence of independent, identically distributed random variables with $E{\mid}X_n{\mid}^p\;<\;{\infty}$, 0 < P < 2 and $EX_1\;=\;1{\leq}\;p\;<\;2$ then $n^{-1/p}{\sum^n}_{i=1}X_i\;\rightarrow\;0$ a,s, and in $L^p$. In this paper, we probe a version of law of large numbers for double arrays. If ${X_{ij}}$ is a double sequence of random variables with $E{\mid}X_{11}\mid^log^+\mid X_{11}\mid^p\;<\infty$, 0 < P <2, then $lim_{m{\vee}n{\rightarrow}\infty}\frac{{\sum^m}_{i=1}{\sum^n}_{j=1}(X_{ij-a_{ij}}}{(mn)^\frac{1}{p}}\;=0$ a.s. and in $L^p$, where $a_{ij}$ = 0 if 0 < p < 1, and $a_{ij}\;=\;E[X_{ij}\midF_[ij}]$ if $1{\leq}p{\leq}2$, which is a generalization of Etemadi's marcinkiewicz-type SLLN for double arrays. this also generalize earlier results of Smythe, and Gut for double arrays of i.i.d. r.v's.

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A Study on the Representativeness of Proofs in the Geometry (기하 증명에서의 대표성에 관한 연구)

  • Chung, Young Woo;Kim, Boo Yoon
    • Journal of Educational Research in Mathematics
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    • v.25 no.2
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    • pp.225-240
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    • 2015
  • In this study, we investigated the representativeness of proofs in school mathematics, based on the extension of the midpoint connector theorem for the quadrilateral. To this end, we considered a variety of quadrilateral and proved their extensions of the midpoint connector theorem, and identified the relationships between them, therefore seemed that the proof in school mathematics has a representativeness. On the other hand, in the survey based on this information, students were found only some types of quadrilateral and completed easily the proofs for each quadrilateral they found, but students tended to use other proof or mathematical concepts, if the target figures changes in despite of proving the same mathematical fact. Thus, students were more difficult to figure out the relationship between the proofs. From these facts, we know that students are poorly understood the representativeness of proofs to understand the relationship between concrete proofs and to generalize it, though they are able to proof to the specific figures. Therefore it can be seen that the proof activity needs to be done with organic and semantic.

A Note on Treatment of Axioms for Proof in Middle School Mathematics (중학교 수학에서 증명을 위한 공리 취급에 관한 연구)

  • 김흥기
    • The Mathematical Education
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    • v.40 no.2
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    • pp.291-315
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    • 2001
  • There are some problems in the introduction of proof in middle school mathematics. Among the problems, one is the use of postulates and the another is the methods of proof how to connect a statement with others. The first case has been treated mainly in this note. Since proof means to state the reason logically why the statement is true on the basis of others which have already been known as true and basic properties, in order to prove logically, it is necessary to take the basic properties and the statement known already as true. But the students don't know well what are the basic properties and the statement known already as true for proving. No use of the term postulation(or axiom) cause the confusion to distinguish postulation and theorem. So they don't know which statements are accepted without proof or not accepted without proof, To solve this problems, it is necessary to use the term postulate in middle school mathematics. In middle school mathematics, we present same model of the introduction of proof which are used the postulates needed for the proof.

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A new equilibrium existence via connectedness

  • Rim, Dong-Il;Im, Sung-Mo;Kim, Won-Kyu
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.587-592
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    • 1996
  • In 1950, Nash [5] first proved the existence of equilibrium for games where the player's preferences are representable by continuous quasiconcave utilities and the strategy sets are simplexes. Next Debreu [3] proved the existence of equilibrium for abstract economies. Recently, the existence of Nash equilibrium can be further generalized in more general settings by several athors, e.g. Shafer-Sonnenschein [6], Borglin-Keiding [2], Yannelis-Prabhaker [8]. In the above results, the convexity assumption is very essential and the main proving tools are the continuous selection technique and the existence of maximal elements. Still there have been a number of generalizations and applications of equilibrium existence theorem in generalized games.

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GEOMETRIC CHARACTERISTICS OF GENERIC LIGHTLIKE SUBMANIFOLDS

  • Jha, Nand Kishor;Pruthi, Megha;Kumar, Sangeet;Kaur, Jatinder
    • Honam Mathematical Journal
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    • v.44 no.2
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    • pp.179-194
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    • 2022
  • In the present study, we investigate generic lightlike submanifolds of indefinite nearly Kaehler manifolds. After proving the existence of generic lightlike submanifolds in an indefinite generalized complex space form, a non-trivial example of this class of submanifolds is discussed. Then, we find a characterization theorem enabling the induced connection on a generic lightlike submanifold to be a metric connection. We also derive some conditions for the integrability of distributions defined on generic lightlike submanifolds. Further, we discuss the non-existence of mixed geodesic generic lightlike submanifolds in a generalized complex space form. Finally, we investigate totally umbilical generic lightlike submanifolds and minimal generic lightlike submanifolds of an indefinite nearly Kaehler manifold.

EFFICINET GENERATION OF MAXIMAL IDEALS IN POLYNOMIAL RINGS

  • Kim, Sunah
    • Bulletin of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.137-143
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    • 1992
  • The purpose of this paper is to provide the affirmative solution of the following conjecture due to Davis and Geramita. Conjecture; Let A=R[T] be a polynomial ring in one variable, where R is a regular local ring of dimension d. Then maximal ideals in A are complete intersection. Geramita has proved that the conjecture is true when R is a regular local ring of dimension 2. Whatwadekar has rpoved that conjecture is true when R is a formal power series ring over a field and also when R is a localization of an affine algebra over an infinite perfect field. Nashier also proved that conjecture is true when R is a local ring of D[ $X_{1}$,.., $X_{d-1}$] at the maximal ideal (.pi., $X_{1}$,.., $X_{d-1}$) where (D,(.pi.)) is a discrete valuation ring with infinite residue field. The methods to establish our results are following from Nashier's method. We divide this paper into three sections. In section 1 we state Theorems without proofs which are used in section 2 and 3. In section 2 we prove some lemmas and propositions which are used in proving our results. In section 3 we prove our main theorem.eorem.rem.

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GEOMETRY OF CONTACT STRONGLY PSEUDO-CONVEX CR-MANIFOLDS

  • Cho, Jong-Taek
    • Journal of the Korean Mathematical Society
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    • v.43 no.5
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    • pp.1019-1045
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    • 2006
  • As a natural generalization of a Sasakian space form, we define a contact strongly pseudo-convex CR-space form (of constant pseudo-holomorphic sectional curvature) by using the Tanaka-Webster connection, which is a canonical affine connection on a contact strongly pseudo-convex CR-manifold. In particular, we classify a contact strongly pseudo-convex CR-space form $(M,\;\eta,\;\varphi)$ with the pseudo-parallel structure operator $h(=1/2L\xi\varphi)$, and then we obtain the nice form of their curvature tensors in proving Schurtype theorem, where $L\xi$ denote the Lie derivative in the characteristic direction $\xi$.

Ancient Greece Mathematics and Oriental Mathematics (고대 그리스 수학과 동양 수학)

  • Kim, Jong-Myung
    • Journal for History of Mathematics
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    • v.20 no.2
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    • pp.47-58
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    • 2007
  • In this paper, we shall try to give a comparative study of mathematics developments in ancient Greece and ancient Oriental mathematics. We have found that the Oriental Mathematics. is quantitative, computational and algorithmetic, but the ancient Greece is axiomatic and deductive mathematics in character. The two region mathematics should be unified to give impetus to further development of mathematics in future times.

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