• Title/Summary/Keyword: Borel distribution

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THE ZEROS DISTRIBUTION OF SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS IN AN ANGULAR DOMAIN

  • Huang, Zhibo;Chen, Zongxuan
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.443-454
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    • 2010
  • In this paper, we investigate the zeros distribution and Borel direction for the solutions of linear homogeneous differential equation $f^{(n)}+A_{n-2}(z)f^{(n-2)}+{\cdots}+A_1(z)f'+A_0(z)f=0(n{\geq}2)$ in an angular domain. Especially, we establish a relation between a cluster ray of zeros and Borel direction.

BI-UNIVALENT FUNCTIONS CONNECTED WITH THE MITTAG-LEFFLER-TYPE BOREL DISTRIBUTION BASED UPON THE LEGENDRE POLYNOMIALS

  • El-Deeb, Sheza M.;Murugusundaramoorthy, Gangadharan;Alburaikan, Alhanouf
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.2
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    • pp.331-347
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    • 2022
  • In this paper, we introduce new subclasses of analytic and bi-univalent functions associated with the Mittag-Leffler-type Borel distribution by using the Legendre polynomials. Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3| for functions in these subclasses and obtain Fekete-Szegő problem for these subclasses. We also state certain new subclasses of Σ and initial coefficient estimates and Fekete-Szegő inequalities.

ANGULAR DISTRIBUTION OF SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

  • Wu, Zhaojun;Sun, Daochun
    • Journal of the Korean Mathematical Society
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    • v.44 no.6
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    • pp.1329-1338
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    • 2007
  • In this paper, we study the location of zeros and Borel direction for the solutions of linear homogeneous differential equations $$f^{(n)}+A_{n-1}(z)f^{(n-1)}+{\cdots}+A_1(z)f#+A_0(z)f=0$$ with entire coefficients. Results are obtained concerning the rays near which the exponent of convergence of zeros of the solutions attains its Borel direction. This paper extends previous results due to S. J. Wu and other authors.

MISCLASSIFICATION IN SIZE-BIASED MODIFIED POWER SERIES DISTRIBUTION AND ITS APPLICATIONS

  • Hassan, Anwar;Ahmad, Peer Bilal
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.13 no.1
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    • pp.55-72
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    • 2009
  • A misclassified size-biased modified power series distribution (MSBMPSD) where some of the observations corresponding to x = c + 1 are misclassified as x = c with probability $\alpha$, is defined. We obtain its recurrence relations among the raw moments, the central moments and the factorial moments. Discussion of the effect of the misclassification on the variance is considered. To illustrate the situation under consideration some of its particular cases like the size-biased generalized negative binomial (SBGNB), the size-biased generalized Poisson (SBGP) and sizebiased Borel distributions are included. Finally, an example is presented for the size-biased generalized Poisson distribution to illustrate the results.

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ON RADIAL OSCILLATION OF ENTIRE SOLUTIONS TO NONHOMOGENEOUS ALGEBRAIC DIFFERENTIAL EQUATIONS

  • Zhang, Guowei
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.545-559
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    • 2018
  • In this paper we mainly investigate the properties of the solutions to a type of nonhomogeneous algebraic differential equation in an angular domain. It includes the Borel directions of the solutions, the width of angular domains in which the solutions take its order and the measure of radial distributions of Julia sets of the solutions.

COMPLEX DELAY-DIFFERENTIAL EQUATIONS OF MALMQUIST TYPE

  • NAGASWARA, P.;RAJESHWARI, S.
    • Journal of applied mathematics & informatics
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    • v.40 no.3_4
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    • pp.507-513
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    • 2022
  • In this paper, we investigate some results on complex delay-differential equations of the classical Malmquist theorem. A classic illustrations of their results states us that if a complex delay equation w(t + 1) + w(t - 1) = R(t, w) with R(t, w) rational in both arguments admits (concede) a transcendental meromorphic solution of finite order, then degwR(t, w) ≤ 2. Development and upgrade of such results are presented in this paper. In addition, Borel exceptional zeros and poles seem to appear in special situations.

UNIFORM DISTRIBUTIONS ON CURVES AND QUANTIZATION

  • Joseph Rosenblatt;Mrinal Kanti Roychowdhury
    • Communications of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.431-450
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    • 2023
  • The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of n-means and the nth quantization errors for different values of n with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise [2], which says that for a Borel probability measure P with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number.

SOME NECESSARY CONDITIONS FOR ERGODICITY OF NONLINEAR FIRST ORDER AUTOREGRESSIVE MODELS

  • Lee, Chan-Ho
    • Journal of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.227-234
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    • 1996
  • Consider nonlinear autoregressive processes of order 1 defined by the random iteration $$ (1) X_{n + 1} = f(X_n) + \epsilon_{n + 1} (n \geq 0) $$ where f is real-valued Borel measurable functin on $R^1, {\epsilon_n : n \geq 1}$ is an i.i.d.sequence whose common distribution F has a non-zero absolutely continuous component with a positive density, $E$\mid$\epsilon_n$\mid$ < \infty$, and the initial $X_0$ is independent of ${\epsilon_n : n > \geq 1}$.

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RADIAL OSCILLATION OF LINEAR DIFFERENTIAL EQUATION

  • Wu, Zhaojun
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.5
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    • pp.911-921
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    • 2012
  • In this paper, the radial oscillation of the solutions of higher order homogeneous linear differential equation $$f^{(k)}+A_{n-2}(z)f^{(k-2)}+{\cdots}+A_1(z)f^{\prime}+A_0(z)f=0$$ with transcendental entire function coefficients is studied. Results are obtained to extend some results in [Z. Wu and D. Sun, Angular distribution of solutions of higher order linear differential equations, J. Korean Math. Soc. 44 (2007), no. 6, 1329-1338].

Large Deviations for random walks with time stationary random distribution function

  • Hong, Dug-Hun
    • Journal of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.279-287
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    • 1995
  • Let $F$ be a set of distributions on R with the topology of weak convergence, and let $A$ be the $\sigma$-field generated by the open sets. We denote by $F_1^\infty$ the space consisting of all infinite sequence $(F_1, F_2, \cdots), F_n \in F and R_1^\infty$ the space consisting of all infinite sequences $(x_1, x_2, \cdots)$ of real numbers. Take the $\sigma$-field $F_1^\infty$ to be the smallest $\sigma$-field of subsets of $F_1^\infty$ containing all finite-dimensional rectangles and take $B_1^\infty$ to be the Borel $\sigma$-field $R_1^\infty$.

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