Research Article
About Absolute Convergence of Fourier Series of Almost Periodic Functions
Farkhodzhon Makhmadshevich Talbakov*
Issue:
Volume 13, Issue 3, June 2024
Pages:
36-43
Received:
18 February 2024
Accepted:
12 March 2024
Published:
2 July 2024
Abstract: The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x1, x2,...xp), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x1=a1 t, x2=a2 t,..., xp=apt, where a1, a2, ..., ap - some real, non-zero numbers. Considering the value of the function f(x1, x2,...xp) on this line, he obtains a function of one variable φ(t) = f(a1 t, a2 t,...ap t) and proves that this function is almost periodic. With some choice of numbers a1, a2, ..., ap - it may happen that this function is periodic. However, if the numbers a1, a2, ..., ap are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a1, a2, ..., ap associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λk→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.
Abstract: The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issu...
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Research Article
Improving Primary School Pupils’ Academic Achievement and Retention in Mathematics Using Jigsaw Teaching Method
Obilor Wisdom Enyinnaya*
,
InekweIsrael Onwuegbu,
Ogbonna Cecilia Chinyere
Issue:
Volume 13, Issue 3, June 2024
Pages:
44-50
Received:
27 May 2024
Accepted:
24 June 2024
Published:
31 July 2024
DOI:
10.11648/j.pamj.20241303.12
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Views:
Abstract: This study focused on improving primary school pupils’ academic achievement and retention in mathematics using jigsaw teaching method. It was guided by two research questions and two hypotheses. The pretest, posttest quasi experimental research design involving two groups was used for the study. The population of the study comprised of all the primary five (5) pupils of 2022/2023 academic session in Umuahia Education Zone of Abia state. The education zone has 201 primary schools with 4,447 primary five (5) pupils. The Two (2) primary schools with class teachers in primary five (5) that have mathematics background at NCE and Degree levels from Umuahia Educational Zone of Abia State were purposively sampled for the research.Simple random sampling technique was used in determining which school becomes the experimental and control group, 44 pupils participated as a sample for the study (19 males and 25 females). A Mathematics Achievement Test (MAT) and Mathematics Retention Test (MRT) were used for data collection. Research questions were answered using mean and standard deviation while the hypotheses were tested using Analysis of covariance at 0.05 alpha level, with the aid of SPSS version 21. The results showed a significant difference between the academic achievement and retention of the experimental group and control group in favour of the experimental group. It also revealed that there was no significant difference in the academic achievement and retention of male and female pupils taught geometry in the experimental group which implies that the use of the jigsaw teaching method is independent of gender. Based on the findings recommendations were made such as teachers should employ the use of cooperative method of teaching such as jigsaw teaching method in the teaching of geometrical concepts at the primary level of education to enhance pupils’ academic achievement and retention. Finally, workshops and seminars on the use of cooperative method of teaching such as jigsaw Teaching Method should be organized by Ministry of Education for in-service primary school mathematics teachers.
Abstract: This study focused on improving primary school pupils’ academic achievement and retention in mathematics using jigsaw teaching method. It was guided by two research questions and two hypotheses. The pretest, posttest quasi experimental research design involving two groups was used for the study. The population of the study comprised of all the prim...
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