Several characterizations of Bessel functions and their applications

The present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach.

Kemampuan Koneksi Matematik Siswa dalam Menyelesaikan Soal Investigasi Matematika

This study aims to determine and describe the level of mathematical connection ability of students in solving mathematical investigation questions in grade VII students at SMPN 2 Labuapi in the academic year 2020/2021. This type of research is descriptive qualitative research. The subjects in this study were 10 students of VII A class at SMPN 2 Labuapi. The data was collected in the form of test results of students' mathematical connection skills in solving mathematical investigation questions at triangles and quadrangle matter and the results of interviews. The test results were analyzed according to the indicators of students' mathematical connection ability, which is being able to connect the mathematical concepts that have been studied and those being studied and to connect the mathematical concepts to everyday life. Based on the results of data analysis, that the mathematical connection ability of students in solving mathematical investigation questions was still in the sufficient category. Overall there were 3 students (30%) in very poor category, 2 students (20%) in very good category, 2 students (20%) in good category, 2 students (20%) in good enough category, and 1 student (10%) in poor category.

Didactic Sequence for Teaching Exponential Function

This paper presents a methodological proposal for the teaching of exponential function, resulting from the application of a didactic sequence involving exponential function, where evidence of learning and the consolidation and application of mathematical concepts in problem solving were identified and analyzed. The Didactic Engineering of Michèle Artigue (1988) was used as a research methodology. As theoretical contributions that guided and enabled the development of the research, we chose the use of Mathematical Investigation in the classroom; Didactic Sequence in the conception of Zabala (1999); the Articulated Units of Conceptual Reconstruction proposed by Cabral (2017) and assumptions of Vygotsky's theory. A didactic sequence composed of five UARC's was elaborated to work the exponential function, with a view to minimizing the difficulties naturally imposed by the content to be explained. Microgenetic analysis of verbal interactions between teacher and students was used to analyze the results of the application. The results show that the students participating in the experiment showed evidence of learning, recorded during the process, and began to have a good understanding of the concepts and properties related to the topic, in addition to a good performance in carrying out the activities, facts that corroborate the potential of the didactic sequence proposed herein.

Mathematical Investigation of Functions

Generally, when the independent variable of a given exponential function is used as an exponent, the function is considered an exponential. Thus, the following can be examples of exponential functions: $f(x) = ab^x + c$, $f(x) = ae^bx + c$, or $f(x) = e^{a^2+bx+c}$. However, deriving functions of these types given the set of ordered pairs is difficult. This study was conducted to derive formulas for the arbitrary constants a ,b, and $c$ of the exponential function $f(x) = ab^x + c$. It applied the inductive method by using definitions of functions to derive the arbitrary constants from the patterns produced. The findings of the study were: a) For linear, given the table of ordered pairs, equal differences in $x$ produce equal first differences in $y$; b) for quadratic, given the table of ordered pairs, equal differences in $x$ produce equal second differences in $y$; and c) for an exponential function, given a table of ordered pairs, equal differences in $x$ produce a common ratio in the first differences in y. The study obtained the following forms: $b=\sqrt[d]{r}$, $a=\frac{q}{b^n {(b^d-1)}}$, $c=p-ab^n$. Since most models developed used the concept of linear and multiple regressions, it is recommended that pattern analysis be used specifically when data are expressed in terms of ordered pairs.

Mathematical Investigation for Flow Characteristics of Laminar Ferro-Nanofluid Incorporating Cobalt Ferrite Nanoparticles

The effect of a magnetic source of variable strength has been studied on ferro-nanofluid incorporating nanoparticles of Cobalt ferrite (CoFe2O4) with water as a base fluid. Group method has been used to remodel the governing system to a system of ordinary differential equations. The recent study was motivated by inspecting the effect of four parameters including nanoparticles volume fraction, , Prandtl number, , magnetic field strength of the source,, and temperature difference ratio with respect to ambient temperature, . The results showed that the nanofluid velocity and shear stress increased as long as and increase. On the other hand, both are inversely related to the increment in Pr and Temperature distribution inside the boundary layer was noticed to increase due to the increment in Pr values and decrease due to the increment ratios. Contrarily, the heat flux throughout the boundary layer decreased and increased due to increasing respectively. Key words: Ferro-hydrodynamic; Nanofluids; Group Method

Szennyezési jogok hatása a vállalati termelési stratégiára = Effect of pollution rights on production strategy of firm

A cikk a szennyezési jogok bevezetését vizsgálja a mikroökonómia standard vállalatára. A komparatív statika módszerével vizsgáljuk a termelő vállalat lehetséges reakcióit a szennyezési jog bevezetésére. A vállalatok stratégiája a szennyezési jog bevezetésére a technológia változatlanul hagyása, vagy megváltoztatása lehet. Az egyes esetekben arra keresünk választ, hogy hogyan változik a vállalat nyeresége a bevezetés előtti állapothoz képest, és ez milyen szennyezési stratégiával jár. Szennyezési jogot elad-e, vagy vesz a vállalat a szennyezési jogok piacán? = The paper deals with the effect of an introduction of tradable permits on the production strategy of a firm. It is assumed that the firm will maximize its profit. After introducing the emission trading the profit function will contain the linear emission procurement/selling costs. We will compare the optimal production strategy before tradable permits and after that. The mathematical investigation is based on the nonlinear mathematical programming.