scholarly journals Polynomials, numerical triangles and sequences associated with the probability distribution of the hyperbolic cosine type for odd values of the parameter

2021 ◽  
Vol 2052 (1) ◽  
pp. 012045
Author(s):  
M S Tokmachev
Keyword(s):  
Probability Distribution ◽  
Density Function ◽  
Distribution Density ◽  
Stirling Numbers ◽  
Practical Significance ◽  
New Class ◽  
Distribution Density Function ◽  
Physical Problems ◽  
Hyperbolic Cosine ◽  
And Algebra

Abstract The article introduces a new class of polynomials that first appeared in the probability distribution density function of the hyperbolic cosine type. With an integer change in one of the parameters of this distribution, polynomials in the form of a product of positive factors are written out with an increasing degree. Earlier, the author found a connection between the distribution of the hyperbolic cosine type and numerical sets, in particular, in the simplest cases with the triangle of coefficients of Bessel polynomials, the triangle of Stirling numbers, sequences of coefficients in the expansion of various functions, etc. Also from the distribution formed numerous numerical sequences, both new and widely known. Consideration of polynomials separately from the density function made it possible to reconstruct numerical sets of coefficients, ordered in the form of numerical triangles and numerical sequences. The connections between the elements of the sets are established. Among the sequences obtained, in the simplest cases, there are those known from others, for example, physical problems. However, the overwhelming majority of the found number sets have not been encountered earlier in the literature. The obvious applications of this research are number theory and algebra. And the interdisciplinarity of the results indicates the possibility of applications and enhances their practical significance in other areas of knowledge.

scholarly journals Polynomials and number sets associated with the probability distribution of the hyperbolic cosine type for even values of the parameter

2021 ◽  
Vol 2052 (1) ◽  
pp. 012044
Author(s):  
M S Tokmachev
Keyword(s):  
Probability Theory ◽  
Probability Distribution ◽  
Distribution Density ◽  
Stirling Numbers ◽  
Probability Distribution Density ◽  
Natural Parameter ◽  
New Class ◽  
Distribution Density Function ◽  
Real Distribution ◽  
Hyperbolic Cosine

Abstract The polynomials used in the formation of the probability distribution density function of the hyperbolic cosine type are investigated. Earlier, on the basis of a hyperbolic cosine distribution, the author obtained numerical sets, among which not only new ones, but also, for example, the triangle of Stirling numbers, the triangle of the coefficients of Bessel polynomials, sequences of coefficients in the expansion of various functions, etc. In this paper, depending on the natural parameter m and the real distribution parameter β , a new class of polynomials is obtained. For even and odd m , the polynomials are constructed using similar, but different formulas. The article presents polynomials for even values m . Structurally, polynomials consist of quadratic factors. The coefficients of the polynomials, ordered by m , form numerical triangles depending on β . Some relations are found between the coefficients. From the numerical triangles, a set of numerical sequences is obtained, which for integers β are integers. Also, polynomials with respect to x turn out to be polynomials with respect to β . With this interpretation the variable acts as a parameter. New numerical triangles and sequences for different x were found. The overwhelming majority of the obtained numerical sequences are new. The class of polynomials arising from problems of probability theory indicates the possibility of applying the results.

scholarly journals Diagnostics of impregnation defects of reinforcing filaments of polymer composite with built-in fibre-optic sensor with distributed Bragg grating

2020 ◽  
pp. 60-72
Author(s):  
A. A Pan’kov
Keyword(s):  
Composite Material ◽  
Optical Fiber ◽  
Density Function ◽  
Strain Distribution ◽  
Distribution Density ◽  
Optical Fiber Sensor ◽  
Fiber Sensor ◽  
Distribution Density Function ◽  
Optic Sensor ◽  
Probability Of Presence

Mathematical model of unidirectional fibrous polymer composite material with optical fiber sensor built into reinforcing fiber (filament of elementary fibers) with distributed Bragg grating is developed in order to diagnoste defects of filament impregnation - finding probability of impregnation defect as relative length of local sections of filament without impregnation, i.e. without filling binder of space between its elementary fibers. The technique of digital processing of reflection spectrum according to the solution of the integral Fredholm equation of the 1st kind is used in order to find the desired informative function of density of distribution of axial strains along the length of the sensitive section of the fibre-optic sensor. The approach assumes that the optical fiber sensor is embedded in the composite material at the stage of its manufacture, wherein the low-reflective nature of the sensitive portion of the optical fiber allows linear summation of reflection coefficients from its various local portions regardless of their mutual positions. Algorithm of numerical processing of strain distribution density function is developed for finding of sought probability of presence of impregnation defects along filament length. It has been revealed that the distribution density function has pronounced informative pulses, from the location and value of which the sought-after values of probability of presence of impregnation defects along the length of the filament can be found. The results of diagnostics of different values of the sought probability of the filament impregnation defect are presented based on the results of numerical simulation of the measured reflection spectra and the sought function of strain distribution density along the length of the sensitive section of the optical fiber sensor at different values of the volume fraction of the filaments, combinations of transverse and longitudinal loads of the representative domain of the unidirectional fibrous composite material in comparison with graphs for the case without load.

scholarly journals Some identities and formulas derived from analysis of distribution functions including Bernoulli polynomials and Stirling numbers

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 521-527 ◽  
Author(s):  
Burcin Simsek
Keyword(s):  
Probability Distribution ◽  
Characteristic Function ◽  
Density Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Distribution Functions ◽  
Uniform Density ◽  
Probability Distribution Functions ◽  
Negative Order

In this paper, we investigate applications of characteristic function of the uniform density function. Using this characteristic function, some identities and formulas associated with Bernoulli polynomials of negative order, Stirling numbers and probability distribution functions were derived.

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