On the minimum of gaps generated by one-dimensional random packing

1980 ◽  
Vol 17 (01) ◽  
pp. 134-144 ◽  
Author(s):  
Yoshiaki Itoh
Keyword(s):  
Asymptotic Behaviour ◽  
Random Variable ◽  
Random Packing ◽  
One Dimensional ◽  
Image Position

Let L(t) be the random variable which represents the minimum of length of gaps generated by random packing of unit intervals into [0, t]. We have with Using this equation the asymptotic behaviour of P(L(x)≧h) is discussed.

On the minimum of gaps generated by one-dimensional random packing

1980 ◽  
Vol 17 (1) ◽  
pp. 134-144 ◽  
Author(s):  
Yoshiaki Itoh
Keyword(s):  
Asymptotic Behaviour ◽  
Random Variable ◽  
Random Packing ◽  
One Dimensional ◽  
Image Position

Let L(t) be the random variable which represents the minimum of length of gaps generated by random packing of unit intervals into [0, t]. We have with Using this equation the asymptotic behaviour of P(L(x)≧h) is discussed.

Characterizations of certain multivariate distributions

1974 ◽  
Vol 75 (2) ◽  
pp. 219-234 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Random Variable ◽  
Sample Variance ◽  
Sufficient Condition ◽  
Multivariate Distributions ◽  
Sample Mean ◽  
One Dimensional ◽  
Independent Observations ◽  
Image Position

Let X1, X2, …, Xn, be n (n ≥ 2) independent observations on a one-dimensional random variable X with distribution function F. Letbe the sample mean andbe the sample variance. In 1925, Fisher (2) showed that if the distribution function F is normal then and S2 are stochastically independent. This property was used to derive the student's t-distribution which has played a very important role in statistics. In 1936, Geary(3) proved that the independence of and S2 is a sufficient condition for F to be a normal distribution under the assumption that F has moments of all order. Later, Lukacs (14) proved this result assuming only the existence of the second moment of F. The assumption of the existence of moments of F was subsequently dropped in the proofs given by Kawata and Sakamoto (7) and by Zinger (27). Thus the independence of and S2 is a characterizing property of the normal distribution.

scholarly journals Remarks on the Asymptotic Behaviour of Perturbed Linear Systems

1970 ◽  
Vol 11 (1) ◽  
pp. 84-84 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Asymptotic Behaviour ◽  
Linear Systems ◽  
Image Position ◽  
Perturbed Linear Systems

Remarks 1, 3 and 5 are incorrect as stated. They should be supplemented by the following observations:(i) In case the perturbing term is linear in y, i.e. f(t, y) = B(t)y, the conclusion of Theorem 1 will follow from Lemma 1 when applied to equation (15) if we assume, instead of (6),The hypothesis given in Trench's theorem is sufficient to imply (*) but not (6). A similar comment applies to Remark 5.

scholarly journals Homogenization of the spectral problem on the Riemannian manifold consisting of two domains connected by many tubes

2013 ◽  
Vol 143 (6) ◽  
pp. 1255-1289 ◽  
Author(s):  
Andrii Khrabustovskyi
Keyword(s):  
Riemannian Manifold ◽  
Small Parameter ◽  
Asymptotic Behaviour ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Beltrami Operator ◽  
Perforated Domain ◽  
Perforated Domains ◽  
Accumulation Points ◽  
Image Position

The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limitsare positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

Random Packing of Particles: Simulation With a One Dimensional Parking Model

Technometrics ◽  
1974 ◽  
Vol 16 (2) ◽  
pp. 301-309 ◽  
Author(s):  
Aaron S. Goldman ◽  
Homer D. Lewis ◽  
Willliam M. Visscher
Keyword(s):  
Random Packing ◽  
One Dimensional

scholarly journals A Remark on the Principle of Zero Utility

1982 ◽  
Vol 13 (2) ◽  
pp. 133-134 ◽  
Author(s):  
Hans U. Gerber
Keyword(s):  
Risk Aversion ◽  
Utility Function ◽  
Affine Transformation ◽  
Random Variable ◽  
Affine Transformations ◽  
Premium Calculation ◽  
Image Position

Let u(x) be a utility function, i.e., a function with u′(x)>0, u″(x)<0 for all x. If S is a risk to be insured (a random variable), the premium P = P(x) is obtained as the solution of the equationwhich is the condition that the premium is fair in terms of utility. It is clear that an affine transformation of u generates the same principle of premium calculation. To avoid this ambiguity, one can standardize the utility function in the sense thatfor an arbitrarily chosen point y. Alternatively, one can consider the risk aversionwhich is the same for all affine transformations of a utility function.Given the risk aversion r(x), the standardized utility function can be retrieved from the formulaIt is easily verified that this expression satisfies (2) and (3).The following lemma states that the greater the risk aversion the greater the premium, a result that does not surprise.

scholarly journals Some Remarks on a Recent Paper by Borch

1961 ◽  
Vol 1 (5) ◽  
pp. 265-272 ◽  
Author(s):  
Paul Markham Kahn
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Fixed Number ◽  
Point Of View ◽  
Optimum Amount ◽  
Measurable Transformation ◽  
The Difference ◽  
Image Position ◽  
Stop Loss ◽  
Condition B

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such thatwhere c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media

2006 ◽  
Vol 136 (6) ◽  
pp. 1131-1155 ◽  
Author(s):  
B. Amaziane ◽  
L. Pankratov ◽  
A. Piatnitski
Keyword(s):  
Elliptic Equation ◽  
Asymptotic Behaviour ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Quasilinear Equation ◽  
Quasilinear Elliptic Equations ◽  
High Contrast ◽  
Small Measure ◽  
Image Position ◽  
Quasilinear Elliptic

The aim of the paper is to study the asymptotic behaviour of the solution of a quasilinear elliptic equation of the form with a high-contrast discontinuous coefficient aε(x), where ε is the parameter characterizing the scale of the microstucture. The coefficient aε(x) is assumed to degenerate everywhere in the domain Ω except in a thin connected microstructure of asymptotically small measure. It is shown that the asymptotical behaviour of the solution uε as ε → 0 is described by a homogenized quasilinear equation with the coefficients calculated by local energetic characteristics of the domain Ω.

Numerical integration by systematic sampling

1950 ◽  
Vol 46 (1) ◽  
pp. 111-115 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Numerical Integration ◽  
Bounded Variation ◽  
Mathematical Problem ◽  
Random Variable ◽  
Mean Value ◽  
Lattice Points ◽  
Systematic Sampling ◽  
Image Position ◽  
Infinite Sum

Recent investigations by F. Yates (1) in agricultural statistics suggest a mathematical problem which may be formulated as follows. A function f(x) is known to be of bounded variation and Lebesgue integrable on the range −∞ < x < ∞, and its integral over this range is to be determined. In default of any knowledge of the position of the non-negligible values of the function the best that can be done is to calculate the infinite sumfor some suitable δ and an arbitrary origin t, where s ranges over all possible positive and negative integers including zero. S is evidently of period δ in t and ranges over all its values as t varies from 0 to δ. Previous writers (Aitken (2), p. 45, and Kendall (3)) have examined the resulting errors for fixed t. (They considered only symmetrical functions, and supposed one of the lattice points to be located at the centre.) Here we do not restrict ourselves to symmetrical functions and consider the likely departure of S(t) from J (the required integral) when t is a random variable uniformly distributed in (0, δ). It will be shown that S(t) is distributed about J as mean value, with a variance which will be evaluated as a function of δ, the scale of subdivision.

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