Almost Periodic Functions on Topological Groups

We study the definition and properties of almost periodic functions on topological groups. We show the equivalence between Bochner’s and Bohr’s definitions of almost periodicity. We discuss Weil’s construction of Bohr compactification b(G) and study its properties. Using Peter-Weyl’s density theorem we show that a function f in Cb(G) is almost periodic if and only if it is the uniform limit of linear combinations of coefficients of the finite-dimensional irreducible unitary representations of G. We show the existence of a unique invariant mean on the space of almost periodic functions. We investigate the Fourier series of almost periodic functions, and show that it extends the classical Fourier series of 2-periodic functions

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

Floating binary planets from ejections during close stellar encounters

The discovery of planetary systems beyond our solar system has posed challenges to established theories of planetary formation. Planetary orbits display a variety of architectures not predicted by first principles, and free-floating planets appear ubiquitous. The recent discovery of candidate Jupiter Mass Binary Objects (JuMBOs) by the James Webb Space Telescope (JWST) further expanded this enigma. Here, by means of high-accuracy, direct $N$-body simulations, we evaluate the possibility that JuMBOs may form as a result of ejection after a close stellar flyby. We consider a system of two Jupiter-like planets moving in circular orbits with velocities $v_1$ and $v_2$ at distances $a_1$ and $a_2$ around a Sun-like star. The interloper is another Sun-like star approaching with asymptotic velocity $v_\infty$. We find that JuMBOs can indeed be formed upon ejection if the two planets are nearly aligned as the interloper reaches the closest approach. The ratio of the cross section of JuMBOs production to that of single ejected free-floating planets can approach $\sim 20\%$ for $v_\infty/v_2 \sim 0.1 - 0.2$ and $a_1/a_2\sim 0.75-0.8$. JuMBOs formed via this channel are expected to have an average semi-major axis comparable to $\Delta a = (a_2-a_1)$ and high eccentricity, with a distinctive superthermal distribution which can help to observationally identify this formation channel and distinguish it from primordial formation. If the ejection channel is confirmed for these or future JWST observations, these JuMBOs will directly inform us of the conditions where these giant planets formed in protoplanetary disks, putting stringent constraints on the giant planet formation theory.Comment: 18 pages, 5 figures. Videos are available at https://yihanwangastro.github.io/#posts

Stellar/BH Population in AGN Disks: Direct Binary Formation from Capture Objects in Nuclei Clusters

The Active Galatic Nuclei(AGN) disk has been proposed as a potential channel for the merger of binary black holes. The population of massive stars and black holes in AGN disks captured from the nuclei cluster plays a crucial role in determining the efficiency of binary formation and final merger rate within the AGN disks. In this paper, we investigate the capture process using analytical and numerical approaches. We discover a new constant integral of motion for one object's capture process. {Applying this result to the whole population of the nuclei cluster captured by the AGN disk, we find that the population of captured objects} depends on the angular density and eccentricity distribution of the nuclei clusters and is effectively independent of the radial density profile of the nuclei cluster and disk models. An isotropic nuclei cluster with thermal eccentricity distribution predicts a captured profile \dd N/\dd r\propto r^{-1/4}. The captured objects are found to be dynamically crowded within the disk. Direct binary formation right after the capture would be promising, especially for stars. The conventional migration traps that help pile up single objects in AGN disks for black hole mergers might not be required.Comment: Comments are welcom

Effective two-body scatterings around a massive object

Two-body scatterings under the potential of a massive object are very common in astrophysics. If the massive body is far enough away that the two small bodies are in their own gravitational sphere of influence, the gravity of the massive body can be temporarily ignored. However, this requires the scattering process to be fast enough that the small objects do not spend too much time at distances near the surface of the sphere of influence. In this paper, we derive the validation criteria for effective two-body scattering and establish a simple analytical solution for this process, which we verify through numerical scattering experiments. We use this solution to study star-black hole scatterings in the disks of Active Galactic Nuclei and planet-planet scatterings in planetary systems, and calculate their one-dimensional cross-section analytically. Our solution will be valuable in reducing computational time when treating two-body scatterings under the potential of a much more massive third body, provided that the problem settings are in the valid parameter space region identified by our study.Comment: Comments are Welcom

Enhancing career-preparation equity for international students

In this session, we will explore the unique challenges of international students with respect to support for career-related skill development and experiences with career services on campus. We will present findings from quantitative and qualitative responses of more than 4,500 senior international students over the past five years at hundreds of four-year colleges and universities. Through discussion, participants will learn about international students’ confidence in using skills and abilities essential to their future careers and how institutions can create new career services supports for this large group of minoritized student

Fast synchronization of complex dynamical networks with time-varying delay via periodically intermittent control

The fast synchronization problem for a class of complex dynamical networks with time varying delay by means of periodically intermittent control is studied. Based on the finite-time stability theory and periodically intermittent control technique, some sufficient synchronization criteria are obtained to guarantee the fast synchronization. Furthermore, the essential condition for guaranteeing periodically intermittent control realized in finite time is given in this paper. Finally, two examples are illustrated to verify the proposed theoretical results.http://www.elsevier.com/locate/neucom2017-09-30hb2016Electrical, Electronic and Computer Engineerin