• Title/Summary/Keyword: exponents

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WEAK HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENTS AND APPLICATIONS

  • Souad Ben Seghier
    • Journal of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.33-69
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    • 2023
  • Let α ∈ (0, ∞), p ∈ (0, ∞) and q(·) : ℝn → [1, ∞) satisfy the globally log-Hölder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator M and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley g-function and the Littlewood-Paley $g^*_{\lambda}$-function.

Analysis of Modified Distance-and-Elevation Ratio Method with Different Exponents of Distance and Elevation (거리와 고도의 지수를 구분한 수정거리고도비율법의 분석)

  • Yoo, Ju-Hwan
    • Journal of Korea Water Resources Association
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    • v.48 no.5
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    • pp.357-365
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    • 2015
  • Both exponents of distance and elevation into distance-and-elevation ratio method for estimating missing rainfall data are expressed as squares together but in this study the two exponents are differently separated and analyzed. We used 326 hourly rainfall events of precipitation data during 10 years of 2004 to 2013 observed at a base station of Pyeongchang and the five neighboring index stations-Bangrim, Suju, Cheongoksan, Jinbu, Yeongwol1-in Han River basin for a case study. As a result, exponent values of distance and elevation appropriate for a topography of the site appear as 3.7 and 0.57 respectively. The exponents of distance and elevation difference need to be applied according to topographical characteristics of site where estimating missing data or interpolation are required.

Teachers' understanding of the definition of rational exponent (유리수 지수 정의에 대한 교사 이해 분석)

  • Shin, Bomi
    • The Mathematical Education
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    • v.60 no.1
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    • pp.21-39
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    • 2021
  • The aim of this study was to deduce implications of the growth of mathematics teachers' specialty for effective instruction about the formulae of exponentiation with rational exponents by analyzing teachers' understanding of the definition of rational exponent. In order to accomplish the aim, this study ascertained the nature of the definition of rational exponent through examining previous literature and established specific research questions with reference to the results of the examination. A questionnaire regarding the nature of the definition was developed in order to solve the questions and was taken to 50 in-service high school teachers. By analysing the data from the written responses by the teachers, this study delineated four characteristics of the teachers' understanding with regard to the definition of rational exponent. Firstly, the teachers did not explicitly use the condition of the bases with rational exponents while proving f'(x)=rxr-1. Secondly, few teachers explained the reason why the bases with rational exponents must be positive. Thirdly, there were some teachers who misunderstood the formulae of exponentiation with rational exponents. Lastly, the majority of teachers thought that $(-8)^{\frac{1}{3}}$ equals to -2. Additionally, several issues were discussed related to teacher education for enhancing teachers' knowledge about the definition, features of effective instruction on the formulae of exponentiation and improvement points to explanation methods about the definition and formulae on the current high school textbooks.

The sound quality evaluation of an air-conditioner based on consumer sensory evaluation (소비자 감성 평가를 통한 에어컨 음질 평가)

  • Kim, Hooi-Joong;Lee, Jin-Kyung;Lee, Jea-Won;Cho, Sung-Jin
    • Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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    • 2007.11a
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    • pp.630-633
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    • 2007
  • This paper proposes method to find the subjective satisfaction index of an air-conditioner using consumer's sensory evaluation for an air-conditioner. A satisfaction index of an air-conditioner is presented the annoyance and freshness as ${\alphs}$, ${\beta}$ exponents of 1/f slope and is changed the annoyance and freshness as exponents of 1/f slope are changed. The satisfaction index of an air-conditioner is not improved if SPL is above the proper level although the 1/f slope has optimal ${\alpha}$, ${\beta}$ values. Therefore, this paper proposes the satisfaction index of an air-conditioner considered both SPL and ${\alpha}$, ${\beta}$ exponents of 1/f slope.

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Effects of Strain Hardening Exponents on the Retardation of Fatigue Crack Propagation (가공경화지수가 피로균열 지연거동에 끼치는 영향)

  • 김상철;강동명
    • Transactions of the Korean Society of Mechanical Engineers
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    • v.14 no.5
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    • pp.1193-1199
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    • 1990
  • Effects of strain hardening exponents on the behavior of fatigue crack propagation are experimentally investigated. The retardation effect of fatigue crack propagation after single overloading is investigated in relation to strain hardening exponent and crack closure. A relationship between crack opening ratio and strain hardening exponents is inspected through an examination of the crack closure behavior. An empirical equation relating retardation effect of fatigue crack propagation after single overloading, percent peak load and strain hardening exponent of materials is proposed.

Moment Lyapunov exponents of the Parametrical Hill's equation under the excitation of two correlated wideband noises

  • Janevski, Goran;Kozic, Predrag;Pavlovic, Ivan
    • Structural Engineering and Mechanics
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    • v.52 no.3
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    • pp.525-540
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    • 2014
  • The Lyapunov exponent and moment Lyapunov exponents of Hill's equation with frequency and damping coefficient fluctuated by correlated wideband random processes are studied in this paper. The method of stochastic averaging, both the first-order and the second-order, is applied. The averaged $It\hat{o}$ differential equation governing the pth norm is established and the pth moment Lyapunov exponents and Lyapunov exponent are then obtained. This method is applied to the study of the almost-sure and the moment stability of the stationary solution of the thin simply supported beam subjected to time-varying axial compressions and damping which are small intensity correlated stochastic excitations. The validity of the approximate results is checked by the numerical Monte Carlo simulation method for this stochastic system.