Applications of Continuous Fractions in Orthogonal Polynomials

Several applications of continuous fractions are restricted to theoretical studies, such as problems associated with the approximation of functions, determination of rational and irrational numbers, applications in physics in determining the resistance of electric circuits and integral equations and in several other areas of mathematics. This work aimed to study the results that open the way for the connection of continuous fractions with the orthogonal polynomials. As support, we will study the general case, where the applications of the Wallis formulas in a monolithic orthogonal polynomial, which generates a continuous fraction of the Jacobi type. It will be allowed applications with relations of recurrence of three terms in the polynomials of Tchebyshev and Legendre, through the results found, establishing connection between them with the continuous fractions. And finally, will be presented the "Number of gold", that is an application of this theory.

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Comparison of the Orthogonal Polynomial Solutions for Fractional Integral Equations

Mathematics ◽

10.3390/math7010059 ◽

2019 ◽

Vol 7 (1) ◽

pp. 59

Author(s):

Ayşegül Daşcıoğlu ◽

Serpil Salınan

Keyword(s):

Orthogonal Polynomial ◽

Orthogonal Polynomials ◽

Integral Equations ◽

Collocation Method ◽

Fractional Integral ◽

The Other ◽

Numerical Examples ◽

Polynomial Solutions

In this paper, a collocation method based on the orthogonal polynomials is presented to solve the fractional integral equations. Six numerical examples are given to illustrate the method. The results are compared with the other methods in the literature, and the results obtained by different kinds of polynomials are compared.

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Universal Synthesizer of Mueller Matrices Based on the Symmetry Properties of the Enpolarizing Ellipsoid

Symmetry ◽

10.3390/sym13060983 ◽

2021 ◽

Vol 13 (6) ◽

pp. 983

Author(s):

José J. Gil ◽

Ignacio San José

Keyword(s):

Mathematical Structure ◽

Mueller Matrix ◽

Morphological Properties ◽

Nondestructive Analysis ◽

Structural Decomposition ◽

Symmetry Properties ◽

Mueller Matrices ◽

Scattering Measurement ◽

The Way

Polarimetry is today a widely used and powerful tool for nondestructive analysis of the structural and morphological properties of a great variety of material samples, including aerosols and hydrosols, among many others. For each given scattering measurement configuration, absolute Mueller polarimeters provide the most complete polarimetric information, intricately encoded in the 16 parameters of the corresponding Mueller matrix. Thus, the determination of the mathematical structure of the polarimetric information contained in a Mueller matrix constitutes a topic of great interest. In this work, besides a structural decomposition that makes explicit the role played by the diattenuation-polarizance of a general depolarizing medium, a universal synthesizer of Muller matrices is developed. This is based on the concept of an enpolarizing ellipsoid, whose symmetry features are directly linked to the way in which the polarimetric information is organized.

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III. Further experiments on the vapour densities of potassium and sodium

Proceedings of the Royal Society of London ◽

10.1098/rspl.1879.0073 ◽

1879 ◽

Vol 29 (196-199) ◽

pp. 490-493 ◽

Cited By ~ 2

Keyword(s):

Considerable Error ◽

Exact Determination ◽

The Way

In one former communication “On the Vapour Densities of Potassium and Sodium,” we pointed out the chief obstacles which lay in the way of an exact determination of these constants. Having overcome the chief manipulative difficulties in connexion with the method we described, there still remained the problem for solution as to how far the use of iron bottles in our experiments might affect the results. If the iron retained the metals or allowed their vapours to diffuse with rapidity through it, a considerable error might be produced without its being easily detected.

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Calculation of Technical Characteristics of the Technical Fleet Vessels Involved in Ensuring Safety of the Offshore Facilities of Oil and Gas Complex

Occupational Safety in Industry ◽

10.24000/0409-2961-2021-1-7-13 ◽

2021 ◽

pp. 7-13

Author(s):

S.V. Matsenko ◽

◽

V.M. Minko ◽

A.A. Koshelev ◽

V.Yu. Piven ◽

...

Keyword(s):

Oil And Gas ◽

Oil Pollution ◽

Raw Materials ◽

Experimental Studies ◽

Industrial Safety ◽

Theoretical Studies ◽

Physical Processes ◽

Oil Spill Response ◽

Sea Area

Violation of industrial safety rules during the operation of offshore facilities for the production, storage and transportation of the hydrocarbon raw materials leads in the most cases to pollution of the marine environment with oil and its components. The works on localization and elimination of such pollution are carried out with the help of vessels of the technical support fleet and booms. When developing oil spill response plans at such facilities, a calculated determination of the technical characteristics of vessels and booms is required that are sufficient to carry out the planned activities. The basic design principles for determining the towing capacity of the technical fleet vessels involved in the localization and elimination of oil and oil product spills by trawling methods are given in the article. The calculation is based on theoretical studies performed by the authors of the physical processes occurring during the movement of objects of a mobile trawling order in the sea area. The results obtained during the course of theoretical studies were confirmed by the experimental studies carried out by the authors personally using the real pieces of equipment in the actual development of tasks for training spill containment by trawling. As a result, the empirical dependencies were obtained and experimentally confirmed, which can be used to calculate technical characteristics of the ships as part of the mobile orders and anchor systems as part of stationary orders intended for the localization and elimination of oil pollution. These results can be used, among other things, for the calculated substantiation of the technical characteristics of the technical fleet vessels designed to ensure safety of the offshore facilities for production, storage, and transportation of the hydrocarbon raw materials.

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On the Inadequacy of existing Data for determining the Rate of Mortality among Select Lives

The Assurance Magazine and Journal of the Institute of Actuaries ◽

10.1017/s2046165800022279 ◽

1854 ◽

Vol 4 (1) ◽

pp. 1-9

Author(s):

William Spens

Keyword(s):

First Year ◽

Sufficient Data ◽

Existing Data ◽

Selection Of ◽

The Way

I. While so much improvement has recently taken place in the arrangement and construction of various tables for facilitating calculations founded on existing data, very little has been done in the way of investigating and correcting the data themselves; and it is feared that the question of the rate of mortality among select lives is still involved in the greatest doubt and obscurity.II. It is not proposed in the present paper to go farther than to show that the rate of mortality, during the first year of selection, of select assured lives is so materially different from what it has hitherto been represented, as to lead to the inference that the data from which the erroneous deduction has been made cannot be true data for the ascertainment of the value of selection. To investigate the rate of mortality of select lives at separate ages, I conceive to be of the utmost importance for the elucidation of truth, and the proper direction of sanatory inquiries; but I do not consider that sufficient data at present exist for the determination of this, and these can only be obtained by a united inquiry. I shall be very happy if the present observations have any effect in hastening such an investigation, which sooner or later must be entered upon.

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Study on Calculation of Rock Pressure and Determination of Depth for Shallow Asymmetric Tunnel

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.261-263.1034 ◽

2011 ◽

Vol 261-263 ◽

pp. 1034-1038 ◽

Cited By ~ 1

Author(s):

Xiao Jun Zhou

Keyword(s):

Rock Pressure ◽

Calculation Method ◽

Tunnel Structure ◽

The Way

The method to calculate rock pressure for shallow asymmetric tunnel is analyzed by means of taking a mountainous tunnel with semicircular crown and straight sidewall as the object in this paper. The calculation method of tunnel rock pressure has been presented with consideration of both tunnel structure size and its overburden depth. Finally the way to determine the shallow or profound depth of asymmetric tunnel is also obtained.

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Statistical error in counting experiments

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

10.1098/rspa.1935.0076 ◽

1935 ◽

Vol 149 (868) ◽

pp. 467-486 ◽

Cited By ~ 83

Keyword(s):

Finite Number ◽

Statistical Error ◽

Single Particles ◽

Exact Determination ◽

Statistical Laws ◽

The Way

Experiments in which single particles are studied with the aid of counters would, in principle, lead to an exact determination of the statistical laws governing the behaviour of these particles if the number of counted particles were infinitely large. With a finite number of counts, however, a finite statistical error will always remain. This error depends upon the number of counts and upon the way in which one makes use of the counter readings to calculate the parameters entering into the statistical laws. The purpose of the following investigation is to show for some typical cases which way of calculating has to be adopted in order to make the error a minimum.

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