On the two sided barrier problem

Upper and lower bounds are given for the probability that a separable random process X(t) will take values outside the interval (— λ 1, λ 2) for 0 ≦ t ≦ T, where λ 1 and λ 2 are positive constants. The random process needs to be neither stationary, Gaussian nor purely random (white noise). In engineering applications, X(t) is usually a random process decaying with time at least in the long run such as the structural response to the acceleration of ground motion due to earthquake. Numerical examples show that the present method estimates the probability between the upper and lower bounds which are sufficiently close to be useful when the random processes decay with time.

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On the two sided barrier problem

Journal of Applied Probability ◽

10.2307/3211875 ◽

1965 ◽

Vol 2 (1) ◽

pp. 79-87 ◽

Cited By ~ 15

Author(s):

Masanobu Shinozuka

Keyword(s):

Random Process ◽

Ground Motion ◽

Lower Bounds ◽

Present Method ◽

Structural Response ◽

Random Processes ◽

Upper And Lower Bounds ◽

Numerical Examples ◽

Long Run ◽

Barrier Problem

Upper and lower bounds are given for the probability that a separable random process X(t) will take values outside the interval (— λ1, λ2) for 0 ≦ t ≦ T, where λ1 and λ2 are positive constants.The random process needs to be neither stationary, Gaussian nor purely random (white noise).In engineering applications, X(t) is usually a random process decaying with time at least in the long run such as the structural response to the acceleration of ground motion due to earthquake.Numerical examples show that the present method estimates the probability between the upper and lower bounds which are sufficiently close to be useful when the random processes decay with time.

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The accuracy of Rayleigh’s method of calculating the natural frequencies of vibrating systems

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1952.0034 ◽

1952 ◽

Vol 211 (1105) ◽

pp. 204-224 ◽

Cited By ~ 29

Keyword(s):

Lower Bounds ◽

Natural Frequencies ◽

Hermitian Operator ◽

Normal Modes ◽

Comparison Theorems ◽

Upper And Lower Bounds ◽

Subsequent Paper ◽

Numerical Examples ◽

Modes Of Vibration ◽

Vibrating Systems

In this paper a theorem of Kato (1949) which provides upper and lower bounds for the eigenvalues of a Hermitian operator is modified and generalized so as to give upper and lower bounds for the normal frequencies of oscillation of a conservative dynamical system. The method given here is directly applicable to a system specified by generalized co-ordinates with both elastic and inertial couplings. It can be applied to any one of the normal modes of vibration of the system. The bounds obtained are much closer than those given by Rayleigh’s comparison theorems in which the inertia or elasticity of the system is changed, and they are in fact the ‘best possible’ bounds. The principles of the computation of upper and lower bounds is explained in this paper and will be illustrated by some numerical examples in a subsequent paper.

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Bounds for spectral radius of nonnegative tensors using matrix-digragh-based approach

Journal of Industrial and Management Optimization ◽

10.3934/jimo.2021176 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Guimin Liu ◽

Hongbin Lv

Keyword(s):

Spectral Radius ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Nonnegative Tensor ◽

Related Literature ◽

Numerical Examples ◽

Nonnegative Tensors

We obtain the improved results of the upper and lower bounds for the spectral radius of a nonnegative tensor by its majorization matrix's digraph. Numerical examples are also given to show that our results are significantly superior to the results of related literature.

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The two-barrier problem for continuously differentiable processes

Advances in Applied Probability ◽

10.1017/s0001867800024174 ◽

1992 ◽

Vol 24 (01) ◽

pp. 71-94

Author(s):

Igor Rychlik

Keyword(s):

Vector Field ◽

First Passage Time ◽

Passage Time ◽

Upper And Lower Bounds ◽

Gaussian Vector ◽

First Passage ◽

Zero Crossing ◽

Numerical Examples ◽

Cycle Amplitude ◽

Barrier Problem

An efficient algorithm to compute upper and lower bounds for the first-passage time in the presence of a second absorbing barrier by means of a continuously differentiable decomposable process, e.g. a smooth function of a continuously differentiable Gaussian vector field, is given. The method is used to obtain accurate approximations for the joint density of the zero-crossing wavelength and amplitude and the distribution of the rainflow cycle amplitude. Numerical examples illustrating the results are also given.

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Some Refinements and Generalizations of I. Schur Type Inequalities

The Scientific World JOURNAL ◽

10.1155/2014/709358 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Xian-Ming Gu ◽

Ting-Zhu Huang ◽

Wei-Ru Xu ◽

Hou-Biao Li ◽

Liang Li ◽

...

Keyword(s):

Lower Bounds ◽

Recent Literature ◽

Structural Characteristics ◽

Variable Parameter ◽

Upper Bounds ◽

Upper And Lower Bounds ◽

Numerical Examples ◽

Schur Inequality

Recently, extensive researches on estimating the value ofehave been studied. In this paper, the structural characteristics of I. Schur type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help ofMapleand automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation of these new upper and lower bounds, which improve some known results in the recent literature.

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Bounds for eigenvalues of nonsingular H-tensor

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3116 ◽

2015 ◽

Vol 29 ◽

pp. 3-16 ◽

Cited By ~ 6

Author(s):

Xue-Zhong Wang ◽

Yimin Wei

Keyword(s):

Spectral Radius ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Numerical Examples

The bounds for the Z-spectral radius of nonsingular H -tensor, the upper and lower bounds for the minimum H-eigenvalue of nonsingular (strong) M -tensor are studied in this paper. The sharper bounds are obtained. Numerical examples illustrate that our bounds give tighter bounds.

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Numerical Generation of Compound Random Processes with an Arbitrary Autocorrelation Function

Fluctuation and Noise Letters ◽

10.1142/s0219477518500013 ◽

2018 ◽

Vol 17 (01) ◽

pp. 1850001 ◽

Cited By ~ 2

Author(s):

Dima Bykhovsky ◽

Tom Trigano

Keyword(s):

Random Process ◽

Autocorrelation Function ◽

Analytical Study ◽

Random Processes ◽

The Other ◽

Gaussian Random Process ◽

Numerical Examples ◽

Compound Process ◽

Numerical Generation ◽

Non Gaussian

The generation of non-Gaussian random processes with a given autocorrelation function (ACF) is addressed. The generation is based on a compound process with two components. Both components are solutions of appropriate stochastic differential equations (SDEs). One of the components is a Gaussian process and the other one is non-Gaussian with an exponential ACF. The analytical study shows that a compound combination of these processes may be used for the generation of a non-Gaussian random process with a required ACF. The results are verified by two numerical examples.

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The two-barrier problem for continuously differentiable processes

Advances in Applied Probability ◽

10.2307/1427730 ◽

1992 ◽

Vol 24 (1) ◽

pp. 71-94 ◽

Cited By ~ 6

Author(s):

Igor Rychlik

Keyword(s):

Vector Field ◽

First Passage Time ◽

Passage Time ◽

Upper And Lower Bounds ◽

Gaussian Vector ◽

First Passage ◽

Zero Crossing ◽

Numerical Examples ◽

Cycle Amplitude ◽

Barrier Problem

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Sharp bounds on the minimum $M$-eigenvalue and strong ellipticity condition of elasticity $Z$-tensors-tensors

Journal of Industrial and Management Optimization ◽

10.3934/jimo.2021205 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Chong Wang ◽

Gang Wang ◽

Lixia Liu

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Necessary Conditions ◽

Upper And Lower Bounds ◽

Symmetric Matrices ◽

Strong Ellipticity ◽

Ellipticity Condition ◽

Numerical Examples ◽

Extreme Eigenvalue ◽

Strong Ellipticity Condition

In this paper, we establish sharp upper and lower bounds on the minimum M-eigenvalue via the extreme eigenvalue of the symmetric matrices extracted from elasticity Z-tensors without irreducible conditions. Based on the lower bound estimations for the minimum M-eigenvalue, we provide some checkable sufficient or necessary conditions for the strong ellipticity condition. Numerical examples are given to demonstrate the proposed results.

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Iterative Bounds on the Equilibrium Distribution of a Finite Markov Chain

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800000334 ◽

1987 ◽

Vol 1 (1) ◽

pp. 117-131 ◽

Cited By ~ 6

Author(s):

Jan Van Der Wal ◽

Paul J. Schweitzer

Keyword(s):

Markov Chain ◽

Iterative Method ◽

Lower Bounds ◽

Equilibrium Distribution ◽

Upper And Lower Bounds ◽

Finite Markov Chain ◽

Expected Number ◽

Numerical Examples ◽

Number Of Visits ◽

New Iterative Method

This article presents a new iterative method for computing the equilibrium distribution of a finite Markov chain, which has the significant advantage of providing good upper and lower bounds for the equilibrium probabilities. The method approximates the expected number of visits to each state between two successive visits to a given reference state. Numerical examples indicate that the performance of this method is quite good.

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