## Standing

Graduate (PhD)

## Type of Proposal

Oral Research Presentation

## Faculty Sponsor

Dr. Mehdi S. Monfared

## Proposal

In our lives, there are plenty of periodic motions, which repeat in equal intervals of time. For instance, the recurrences of days and nights and the regular changes of seasons. However, a linear combination of two or more periodic motions need not be periodic any longer. Almost periodic functions are more general than periodic functions. Therefore, the class of almost periodic functions forms a more suitable object of study from a structural point of view. As we know sequence is a special case of function. With this knowledge, one part of the main idea of the research was generalizing the existing concept of equidistributed sequences to equidistributed functions by using the property of the invariant mean on almost periodic functions. In the proposed presentation, first, the classic notions of almost periodic functions and equidistributed sequences will be shown with examples. Following this, the definition of almost periodic equidistributed functions on general topological groups will be introduced with the comparison with the classic concepts of equidistributed sequences on compact groups. Furthermore, the Weyl’s criterion, which describes an equivalent condition of equidistributed sequences, will be discussed in the generalized version on almost periodic equidistributed functions as a new result. This presentation is based on part of the results of my Ph.D. thesis, supervised by Dr. Mehdi S. Monfared.

Some References: [1] H. Weyl, Uber die gleichverteilung von zahlen mod. eins, Math. Ann. 77 (1916), 313–352. [2] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Dover Publications, New York, 2006.

## Availability

March 31-April 1 12-3pm

Almost periodic functions and almost periodic equidistributed functions

In our lives, there are plenty of periodic motions, which repeat in equal intervals of time. For instance, the recurrences of days and nights and the regular changes of seasons. However, a linear combination of two or more periodic motions need not be periodic any longer. Almost periodic functions are more general than periodic functions. Therefore, the class of almost periodic functions forms a more suitable object of study from a structural point of view. As we know sequence is a special case of function. With this knowledge, one part of the main idea of the research was generalizing the existing concept of equidistributed sequences to equidistributed functions by using the property of the invariant mean on almost periodic functions. In the proposed presentation, first, the classic notions of almost periodic functions and equidistributed sequences will be shown with examples. Following this, the definition of almost periodic equidistributed functions on general topological groups will be introduced with the comparison with the classic concepts of equidistributed sequences on compact groups. Furthermore, the Weyl’s criterion, which describes an equivalent condition of equidistributed sequences, will be discussed in the generalized version on almost periodic equidistributed functions as a new result. This presentation is based on part of the results of my Ph.D. thesis, supervised by Dr. Mehdi S. Monfared.

Some References: [1] H. Weyl, Uber die gleichverteilung von zahlen mod. eins, Math. Ann. 77 (1916), 313–352. [2] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Dover Publications, New York, 2006.