scholarly journals An investigation of PT -symmetry breaking in tight-binding chains

2022 ◽  
Vol 2022 (1) ◽  
pp. 013302
Author(s):  
Jean-Marc Luck
Keyword(s):  
Chain Length ◽  
Tight Binding ◽  
Random Variable ◽  
Mean Value ◽  
Finite Variance ◽  
Exact Enumeration ◽  
Logarithmic Corrections ◽  
Optical Potentials ◽  
Gain Loss ◽  
Symmetric Phase

Abstract We consider non-Hermitian PT -symmetric tight-binding chains where gain/loss optical potentials of equal magnitudes ±iγ are arbitrarily distributed over all sites. The main focus is on the threshold γ c beyond which PT -symmetry is broken. This threshold generically falls off as a power of the chain length, whose exponent depends on the configuration of optical potentials, ranging between 1 (for balanced periodic chains) and 2 (for unbalanced periodic chains, where each half of the chain experiences a non-zero mean potential). For random sequences of optical potentials with zero average and finite variance, the threshold is itself a random variable, whose mean value decays with exponent 3/2 and whose fluctuations have a universal distribution. The chains yielding the most robust PT -symmetric phase, i.e. the highest threshold at fixed chain length, are obtained by exact enumeration up to 48 sites. This optimal threshold exhibits an irregular dependence on the chain length, presumably decaying asymptotically with exponent 1, up to logarithmic corrections.

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.

Design Sensitivity Method for Sampling-Based RBDO With Varying Standard Deviation

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
Ikjin Lee ◽  
David Lamb
Keyword(s):  
Standard Deviation ◽  
Design Variable ◽  
Score Function ◽  
Random Variable ◽  
Probability Of Failure ◽  
Design Sensitivity ◽  
Mean Value ◽  
Reliability Based Design ◽  
The Mean ◽  
Sensitivity Method

Conventional reliability-based design optimization (RBDO) uses the mean of input random variable as its design variable; and the standard deviation (STD) of the random variable is a fixed constant. However, the constant STD may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is present as a percentage of the mean value. For this kind of design problem, the STD of the input random variable should vary as the corresponding design variable changes. In this paper, a method to calculate the design sensitivity of the probability of failure for RBDO with varying STD is developed. For sampling-based RBDO, which uses Monte Carlo simulation (MCS) for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STD in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with varying STD. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed design sensitivity method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.

scholarly journals The general theory of canonical correlation and its relation to functional analysis

1961 ◽  
Vol 2 (2) ◽  
pp. 229-242 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Linear Combination ◽  
Canonical Correlation ◽  
Random Variables ◽  
Random Variable ◽  
Finite Variance ◽  
Finite Sample ◽  
Standard Description ◽  
Joint Distributions ◽  
Gaussian Case ◽  
The Relationship

The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variablesxs, and any linear combination ofqrandom variablesytinsofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, forp=q= 1, to include a description of the correlation of any function of a random variablexand any function of a random variabley(both functions having finite variance) for a class of joint distributions ofxandywhich is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations wherepandqare not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.

scholarly journals On the Accuracy of Error Propagation Calculations by Analytic Formulas Obtained for the Inverse Transformation

2019 ◽  
Vol 64 (3) ◽  
pp. 217
Author(s):  
V. I. Romanenko ◽  
N. V. Kornilovska
Keyword(s):  
Error Propagation ◽  
Random Variable ◽  
Mean Value ◽  
Inverse Transformation ◽  
First Order ◽  
Order Of Magnitude ◽  
The Mean ◽  
Normally Distributed

The accuracy of error propagation calculations is estimated for the transformation x → y = f(x) of the normally distributed random variable x. The estimation is based on the formulas for the error propagation obtained for the inverse transformation y → x of the normally distributed random variable y. In the general case, the calculation accuracy for the mean value and the variance of the random variable y is shown to be of the first order of magnitude in the variance of the random variable x.

scholarly journals Modelling the Efficiency of TGARCH Model in Nigeria Inflation Rate

2021 ◽  
Vol 3 (2) ◽  
pp. 20-35
Author(s):  
Michael Sunday Olayemi ◽  
Adenike Oluwafunmilola Olubiyi ◽  
Oluwamayowa Opeyimika Olajide ◽  
Omolola Felicia Ajayi
Keyword(s):  
Inflation Rate ◽  
Random Variable ◽  
Mean Value ◽  
Volatility Models ◽  
Downward Spiral ◽  
The Garch Model ◽  
Autoregressive Conditional Heteroscedasticity ◽  
Best Fit ◽  
Lower Production ◽  
Better Than

In general, volatility is known and referred to as variance and it is a degree of spread of a random variable from its mean value. Two volatility models were considered in this paperwork. Nigeria's inflation rate was modeled by applying the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) and Threshold GARCH models. Symmetric and asymmetric models captured the most commonly stylized facts about the rate of inflation in Nigeria like leverage effects and irregularities in clustering and were studied. These models are GARCH (1,1) and TGARCH (1,1). This work estimated the comparison of volatility models in term of best fit and forecasting. The result showed that TGARCH (1,1) model outperformed GARCH (1,1) models in term of best fit, because it has the least AIC of 2.590438. We forecasted to see the level of volatility using Theils Inequality Coefficient and the result shows that TGARCH has the highest Theils Inequality Coefficient of 0.065075 which makes the TGARCH model better than the GARCH model in this research. From the initial and modified sample static forecast, it was discovered that the return on inflation is stable and shows that volatility slows towards the end of the month, we can see a downward spiral, which means price reaction to economic crisis led to lower production, lower wages, decreased demand, and still lower prices.

Study on Minimum Wall Thickness Requirement of Reactor Vessel of Fast Reactor for Seismic Buckling by System Based Code

2013 ◽  
Author(s):  
Shigeru Takaya ◽  
Daigo Watanabe ◽  
Shinobu Yokoi ◽  
Yoshio Kamishima ◽  
Kenichi Kurisaka ◽  
...  
Keyword(s):  
Wall Thickness ◽  
Fast Reactor ◽  
Design Method ◽  
Random Variable ◽  
Mean Value ◽  
Reactor Vessel ◽  
Seismic Load ◽  
Reliability Design ◽  
Plant Safety ◽  
Distribution Type

In this paper, minimum wall thickness requirement of reactor vessel of fast reactor for seismic buckling is discussed on the basis of the System Based Code (SBC) concept. One of key concepts of SBC is the margin optimization. To implement this concept, reliability design method is employed, and the target reliability for seismic buckling of reactor vessel is derived from nuclear plant safety goals. Input data for reliability evaluation such as distribution type, mean value and standard deviation of random variable are prepared. Seismic hazard is considered to evaluate uncertainty of seismic load. Wall thickness needed to achieve the target reliability is evaluated, and as a result, it is shown that the minimum wall thickness can be reduced from that required by a deterministic design method.

scholarly journals Asymptotic results for multiplexing subexponential on-off processes

1999 ◽  
Vol 31 (02) ◽  
pp. 394-421 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Queueing System ◽  
Random Variable ◽  
Mean Value ◽  
Activity Period ◽  
Arrival Process ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Arrival Processes ◽  
Image Position

Consider an aggregate arrival process A N obtained by multiplexing N on-off processes with exponential off periods of rate λ and subexponential on periods τon. As N goes to infinity, with λN → Λ, A N approaches an M/G/∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process A t ∞ and capacity c. When on periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable Q t P observed at the beginning of the arrival process activity periods where ρ = 𝔼A t ∞ &lt; c; r (c ≤ r) is the rate at which the fluid is arriving during an on period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions regarding on periods than regular variation. In addition, we analyse a queueing system in which one on-off process, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate 𝔼e t . This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value 𝔼e t .

Probabilistic Design and Analysis of Foundation Forces for a Class of Unbalanced Rotating Machines

1984 ◽  
Vol 106 (1) ◽  
pp. 90-99
Author(s):  
L. Boyce ◽  
T. J. Kozik ◽  
E. Parzen
Keyword(s):  
Degrees Of Freedom ◽  
Random Variable ◽  
Simulation Method ◽  
Quantile Function ◽  
Mean Value ◽  
Housing System ◽  
Mass Eccentricity ◽  
Force Data ◽  
Rotor Mass

Rotors of rotating machinery inherently have mass eccentricities that transfer forces to the bearings, housing, and foundation of the machine. This paper considers, from a probabilistic viewpoint, ways to determine the foundation forces and their probabilities. A rotor-housing system is modeled with three degrees-of-freedom, a translation in the direction of the machine supports, a roll, and a pitch. Equations are presented for the motion of the model and the expression for maximum foundation force is developed. All parameters are constant except for rotor mass eccentricity, which is introduced into the problem as a random variable. The probabilistic analysis includes the calculation of the mean value of the foundation force and some measure for its error or variance. An approximate calculation for variance yields values too large for meaningful interpretation. However, a simulation method using a sufficient number of individual rotors leads to a quantile function that displays the variability of the foundation force data in useful form, and allows determination of foundation force probabilities. A numerical example illustrates how the equations can be applied to a sample taken from a population of mass-produced rotors.

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