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.

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 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].

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.

EXTRACTION OF COGNITIVE CHARACTERISTIC FUNCTIONS ACCORDING TO VARIOUS PARAMETER CHANGES OF VIBRATORY STIMULATION

2019 ◽  
Vol 19 (08) ◽  
pp. 1940052
Author(s):  
MI-HYUN CHOI ◽  
SOON-CHEOL CHUNG
Keyword(s):  
Characteristic Function ◽  
Subjective Assessment ◽  
Characteristic Functions ◽  
Vibratory Stimulation ◽  
Cognitive Characteristics ◽  
Cognitive Characteristic ◽  
Normal Cognitive Function ◽  
Four Levels ◽  
Vibration Simulation ◽  
The Right

In this study, vibratory stimuli with various intensities (four levels) and frequencies (10, 50, 100, 150, 200, 225, 250 and 300[Formula: see text]Hz) were applied to the right index finger to extract cognitive characteristic functions for the intensity and frequency. For subjective assessment, an experiment was conducted with 30 healthy adults in their twenties who were right-handed and had normal cognitive function. One trial of the experiment was composed of a vibration phase (0.1 s) and a rest phase (10 s). After vibration simulation was applied once for each intensity and frequency, the participants filled out a subjective assessment form. After extracting the score of each word, the cognitive characteristic function was derived through discriminant analysis and regression analysis according to the frequency and intensity. Through the cognitive characteristic function, the level of cognitive characteristics of each word can be investigated according to the influence of the frequency and intensity. The results observed through the cognitive characteristics function of each word showed that the words affected by the frequency and intensity were “light,” “thick,” “heavy,” “blunt,” “vibrating,” “fast” and “weak.” “Itchy” and “slow” were cognitive characteristics affected only by the frequency, and “strong” and “push” were affected only by the intensity. Through the vibratory stimulation presentation parameters, i.e., frequency and intensity, the results capable of predicting the degrees of various cognitive characteristics were presented.

The average number of real zeros of a random trigonometric polynomial

1968 ◽  
Vol 64 (3) ◽  
pp. 721-730 ◽  
Author(s):  
Minaketan Das
Keyword(s):  
Trigonometric Polynomial ◽  
Mathematical Expectation ◽  
Random Variables ◽  
Trigonometric Polynomials ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
Image Position ◽  
Mutually Independent

AbstractLet a1, a2,… be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity; let b1, b2,… be a set of positive constants. In this work, we obtain the average number of zeros in the interval (0, 2π) of trigonometric polynomials of the formfor large n. The case when bk = kσ (σ > − 3/2;) is studied in detail. Here the required average is (2σ + 1/2σ + 3)½.2n + o(n) for σ ≥ − ½ and of order n3/2; + σ in the remaining cases.

THE ASYMPTOTIC BEHAVIOR OF SEMI-INVARIANTS FOR LINEAR STOCHASTIC SYSTEMS

2002 ◽  
Vol 02 (02) ◽  
pp. 281-294
Author(s):  
G. N. MILSTEIN
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Exponent ◽  
Linear System ◽  
Characteristic Function ◽  
Stochastic Systems ◽  
Random Variable ◽  
Characteristic Functions ◽  
Linear Stochastic Systems ◽  
Moment Lyapunov Exponent ◽  
The Moment

The asymptotic behavior of semi-invariants of the random variable ln |X(t,x)|, where X(t,x) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(p). Namely, it is obtained that the nth semi-invariant is asymptotically proportional to the time t with the coefficient of proportionality g(n)(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the analytic characteristic function of ln |X(t,x)| in a neighborhood of the origin of the complex plane is controlled by the extension g(iz) of g(p).

scholarly journals A NUMERICAL EXPERIMENT ON MATHEMATICAL MODEL OF FORECASTING THE RESULTS OF KNOWLEDGE TESTING

2011 ◽  
Vol 17 (1) ◽  
pp. 42-61 ◽  
Author(s):  
Natalja Kosareva ◽  
Aleksandras Krylovas
Keyword(s):  
Mathematical Model ◽  
Characteristic Function ◽  
Goodness Of Fit ◽  
Characteristic Functions ◽  
Model Parameters ◽  
Information Function ◽  
Test Information ◽  
Parametric Functions ◽  
Office Rooms ◽  
Test Information Function

In this paper the new approach to the forecasting the results of knowledge testing, proposed earlier by authors, is extended with four classes of parametric functions, the best fitting one from which is selected to approximate item characteristic function. Mathematical model is visualized by two numerical experiments. The first experiment was performed with the purpose to show the procedure of selecting the most appropriate item characteristic function and adjusting the parameters of the model. Goodness-of-fit statistic for detecting misfit of the selected model is calculated. In the second experiment a test of 10 items is constructed for the population with latent ability having normal distribution. Probability distribution of total test result and test information function are calculated when item characteristic functions are selected from four classes of parametric functions. In the next step it is shown how test information function value could be increased by adjusting parameters of item characteristic functions to the observed population. This model could be used not only for knowledge testing but also when solving diagnostic tasks in various fields of human activities. Other advantage of this method is the reduction of resources of testing process by more precise adjustment of the model parameters and decreasing the standard error of measurement of the estimated examinee ability. In the presented example the methodology is applied for solving the problem of microclimate evaluation in office rooms.

scholarly journals A Note on Characteristic Functions Which Vanish Identically in an Interval

1962 ◽  
Vol 58 (2) ◽  
pp. 430-432 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Characteristic Function ◽  
Random Variable ◽  
Characteristic Functions ◽  
Unpublished Work

Some years ago, in connexion with some unpublished work in the theory of queues, the question arose as to whether the characteristic function of a non-negative random variable could vanish identically in an interval. The purpose of this note is to show that such a thing is impossible.

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