Characteristic Function and Moments of the Distribution of Storage of Finite Dam with Exponential Input and of the Outflows

1968 ◽  
Vol 17 (1) ◽  
pp. 42-48
Author(s):  
K. C. Majumdar
Keyword(s):  
Characteristic Function ◽  
Exact Solutions ◽  
Stationary State ◽  
Characteristic Functions ◽  
Stationary States ◽  
Mean And Variance ◽  
Stationary Conditions ◽  
Finite Dam ◽  
Explicit Expressions

Though exact solutions of the problems of the distribution of storage and emptiness of dam are available for simple release scheme with exponential, Gamma and Poisson inputs under stationary state conditions and for Holdaway release scheme with exponential input under stationary as well as non­stationary conditions, the exact expressions for the characteristic functions and moments of these distributions have not been given. In this paper, simple and explicit expressions for the characteristic functions and moments of the distribution of storage have been derived under non­stationary as well as stationary conditions with exponential input for simple release rules. The distribution of the outflows below the dam, its characteristic function and moments have also been derived in this paper. The results obtained herein thus provide for the exact values of the useful measures like mean and variance of the storage and outflows below the dam at any point of time as well as for the stationary states. The method developed here can be extended to the more general cases where telease rules obey Holdaway scheme.

SOME EXTENSIONS OF A LEMMA OF KOTLARSKI

2012 ◽  
Vol 28 (4) ◽  
pp. 925-932 ◽  
Author(s):  
Kirill Evdokimov ◽  
Halbert White
Keyword(s):  
Characteristic Function ◽  
Characteristic Functions ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Weaker Conditions

This note demonstrates that the conditions of Kotlarski’s (1967, Pacific Journal of Mathematics 20(1), 69–76) lemma can be substantially relaxed. In particular, the condition that the characteristic functions of M, U1, and U2 are nonvanishing can be replaced with much weaker conditions: The characteristic function of U1 can be allowed to have real zeros, as long as the derivative of its characteristic function at those points is not also zero; that of U2 can have an isolated number of zeros; and that of M need satisfy no restrictions on its zeros. We also show that Kotlarski’s lemma holds when the tails of U1 are no thicker than exponential, regardless of the zeros of the characteristic functions of U1, U2, or M.

scholarly journals STATIONARY STATE ENTANGLEMENT AND TOTAL CORRELATION OF TWO QUBITS OR QUTRITS

2005 ◽  
Vol 19 (30) ◽  
pp. 1803-1811
Author(s):  
SHANG-BIN LI ◽  
JING-BO XU
Keyword(s):  
Mutual Information ◽  
Stationary State ◽  
Sufficient Condition ◽  
Stationary States ◽  
Driving Field ◽  
Total Correlation ◽  
Action Time ◽  
Damped Oscillation ◽  
External Driving

We investigate the mutual information and entanglement of stationary states of two locally driven qubits under the influence of collective dephasing. It is shown that both the mutual information and the entanglement of two qubits in the stationary state exhibit damped oscillation with the scaled action time γT of the local external driving field. It means that we can control both the entanglement and total correlation of the stationary state of two qubits by adjusting the action time of the driving field. We also consider the influence of collective dephasing on the entanglement of two qutrits and obtain the sufficient condition that the stationary state is entangled.

Certain Fourier Transforms of Distributions: II

1954 ◽  
Vol 6 ◽  
pp. 186-189 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szász
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Necessary Conditions ◽  
Laplace Transforms ◽  
Characteristic Functions ◽  
Necessary Condition ◽  
Multiple Roots ◽  
Not Given ◽  
Class Of Functions

In an earlier paper (1), published in this journal, a necessary condition was given which the reciprocal of a polynomial without multiple roots must satisfy in order to be a characteristic function. This condition is, however, valid for a wider class of functions since it can be shown (2, theorem 2 and corollary to theorem 3) that it holds for all analytic characteristic functions. The proof given in (1) is elementary and has some methodological interest since it avoids the use of theorems on singularities of Laplace transforms. Moreover the method used in (1) yields some additional necessary conditions which were not given in (1) and which do not seem to follow easily from the properties of analytic characteristic functions.

A model for a system subject to random shocks

1993 ◽  
Vol 30 (4) ◽  
pp. 979-984 ◽  
Author(s):  
Eui Yong Lee ◽  
Jiyeon Lee
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Stochastic Model ◽  
Poisson Process ◽  
The State ◽  
Stationary Case ◽  
Random Shocks ◽  
Random Amount ◽  
Explicit Expressions

A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold α. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if . The stationary case is also discussed.

John Stuart Mill on China’s Stationary State

2021 ◽  
pp. 833-856
Author(s):  
Yue Xiao
Keyword(s):  
Stationary State ◽  
Capital Accumulation ◽  
Negative Impact ◽  
Mutual Effect ◽  
John Stuart Mill ◽  
East India Company ◽  
Stationary States ◽  
Individual Liberty ◽  
The West ◽  
Economic Stagnation

Current literature on John Stuart Mill’s writings about Asia have focused mainly on his relationship with India because of Mill’s thirty-five year career in the East India Company. Scholars in both China and the West have not paid attention to Mill’s views on China. This paper delves into Mill’s notion of China’s stationary state from two perspectives: (1) a stationary state of capital accumulation and (2) a stationary state of human improvement. In Principles, Mill explained his conception of stationary state. He linked China’s economic stagnation to the low desire for capital accumulation. In On Liberty and Considerations, Mill explored the reasons for China’s stagnancy in human improvement. He discussed the negative impact of the “despotism of custom” on individual liberty and the defects of a bureaucratic government in nineteenth-century China. Mill thought that a stationary state of capital accumulation does not necessarily imply a stationary state of human improvement. However, he seemed to argue that, in China, these two types of stationary states have a mutual effect upon each other.

A new weighting scheme for arc based circle cone-beam CT reconstruction

2021 ◽  
pp. 1-19
Author(s):  
Wei Wang ◽  
Xiang-Gen Xia ◽  
Chuanjiang He ◽  
Zemin Ren ◽  
Jian Lu
Keyword(s):  
Characteristic Function ◽  
Weighting Function ◽  
Cone Beam Ct ◽  
Reconstruction Algorithm ◽  
Cone Beam ◽  
Characteristic Functions ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Scanning Angle ◽  
Beam Geometry

In this paper, we present an arc based fan-beam computed tomography (CT) reconstruction algorithm by applying Katsevich’s helical CT image reconstruction formula to 2D fan-beam CT scanning data. Specifically, we propose a new weighting function to deal with the redundant data. Our weighting function ϖ ( x _ , λ ) is an average of two characteristic functions, where each characteristic function indicates whether the projection data of the scanning angle contributes to the intensity of the pixel x _ . In fact, for every pixel x _ , our method uses the projection data of two scanning angle intervals to reconstruct its intensity, where one interval contains the starting angle and another contains the end angle. Each interval corresponds to a characteristic function. By extending the fan-beam algorithm to the circle cone-beam geometry, we also obtain a new circle cone-beam CT reconstruction algorithm. To verify the effectiveness of our method, the simulated experiments are performed for 2D fan-beam geometry with straight line detectors and 3D circle cone-beam geometry with flat-plan detectors, where the simulated sinograms are generated by the open-source software “ASTRA toolbox.” We compare our method with the other existing algorithms. Our experimental results show that our new method yields the lowest root-mean-square-error (RMSE) and the highest structural-similarity (SSIM) for both reconstructed 2D and 3D fan-beam CT images.

Integral Limit Laws for Additive Functions

1973 ◽  
Vol 25 (1) ◽  
pp. 194-203
Author(s):  
J. Galambos
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Mean Value ◽  
Characteristic Functions ◽  
Asymptotic Formulas ◽  
Mean Value Theorem ◽  
Multiplicative Functions ◽  
Additive Functions ◽  
Limit Laws ◽  
Integral Limit

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

Analytic solution of a finite dam governed by a general input

1974 ◽  
Vol 11 (01) ◽  
pp. 134-144 ◽  
Author(s):  
S. K. Srinivasan
Keyword(s):  
Total Duration ◽  
Random Variables ◽  
Rational Functions ◽  
Characteristic Functions ◽  
Dry Period ◽  
Laplace Transform Technique ◽  
Independent Families ◽  
Set Up ◽  
Input Form ◽  
Finite Dam

A stochastic model of a finite dam in which the epochs of input form a renewal process is considered. It is assumed that the quantities of input at different epochs and the inter-input times are two independent families of random variables whose characteristic functions are rational functions. The release rate is equal to unity. An imbedding equation is set up for the probability frequency governing the water level in the first wet period and the resulting equation is solved by Laplace transform technique. Explicit expressions relating to the moments of the random variables representing the number of occasions in which the dam becomes empty as well as the total duration of the dry period in any arbitrary interval of time are indicated for negative exponentially distributed inter-input times.

The Characteristic Function, a Method-Specific Alternative to the Horwitz Function

2012 ◽  
Vol 95 (6) ◽  
pp. 1803-1806 ◽  
Author(s):  
Michael Thompson
Keyword(s):  
Detection Limit ◽  
Characteristic Function ◽  
Characteristic Functions ◽  
Mathematical Form ◽  
Essential Difference ◽  
Food Sector ◽  
Wide Range ◽  
Specific Alternative ◽  
Additional Role

Abstract The Horwitz function is compared with the characteristic function as a descriptor of the precision of individual analytical methods. The Horwitz function describes the trend of reproducibility SDs observed in collaborative trials in the food sector over a wide range of concentrations of the analyte. However, it is imperfectly adaptable for describing the precision of individual methods, which is the role of the characteristic function. An essential difference between the two functions is that the characteristic function can accommodate a detection limit. This makes it a useful alternative when the precision of a method down to a detection limit is of interest. Many characteristic functions have a simple mathematical form, the parameters of which can be estimated with the usual resources. The Horwitz function serves an additional role as a fitness-for-purpose criterion in the form of the Horwitz ratio (HorRat). This use also has some shortcomings. The functional form of the characteristic function (with suitable prescribed parameters) is better adapted to this task.

The optical imagery of curved surfaces

1957 ◽  
Vol 240 (1223) ◽  
pp. 458-461
Keyword(s):  
Characteristic Function ◽  
Functional Equations ◽  
Individual Characteristic ◽  
Optical Systems ◽  
Characteristic Functions ◽  
Curved Surfaces ◽  
Angular Characteristic ◽  
Optical Imagery ◽  
The Individual ◽  
Angular Characteristic Function

The form of Hamilton’s angular characteristic function for the aberrationless imagery of one surface of rotation on another, and the connexions between the coefficients of the surface and functional equations, are found. When several optical systems of the type considered are arranged in succession the relations between the coefficients of the individual characteristic functions and those of the combination are obtained. These connexions enable all aberrations to be computed without resorting to ray tracing.

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