Inverse problems and rigidity questions in billiard dynamics

2021 ◽  
pp. 1-34
Author(s):  
VADIM KALOSHIN ◽  
ALFONSO SORRENTINO

Abstract A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.

Author(s):  
A.G. CORBALAN ◽  
M. MAZON ◽  
T. RECIO

In this paper we study some unexpected geometric properties of the family of bisector lines for a convex distance d, showing that bisectors do not always have an asymptotic line (Section 2). Moreover, although bisectors are homeomorphic to lines, pairs of them can exist intersecting infinitely many times (Section 3). This leads to the conclusion that convex distances are not always nice in the sense of Klein and Wood.7 On the other hand, we prove that distances d, having d-balls whose boundary is given by finitely many algebraic conditions, produce nice distances (Section 3).


2020 ◽  
Vol 1664 (1) ◽  
pp. 012038
Author(s):  
Saied A. Jhonny ◽  
Buthainah A. A. Ahmed

Abstract In this paper, we ⊥ B J C ϵ -orthogonality and explore ⊥ B J C ϵ -symmetricity such as a ⊥ B J C ϵ -left-symmetric ( ⊥ B J C ϵ -right-symmetric) of a vector x in a real Banach space (𝕏, ‖·‖𝕩) and study the relation between a ⊥ B J C ϵ -right-symmetric ( ⊥ B J C ϵ -left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.


2020 ◽  
Author(s):  
Tomohiro Nishiyama

The divergence minimization problem plays an important role in various fields. In this note, we focus on differentiable and strictly convex divergences. For some minimization problems, we show the minimizer conditions and the uniqueness of the minimizer without assuming a specific form of divergences. Furthermore, we show geometric properties related to the minimization problems.


2020 ◽  
Vol 29 (2) ◽  
pp. 290-299
Author(s):  
Julie G. Arenberg ◽  
Ray H. Hull ◽  
Lisa Hunter

Purpose From the Audiology Education Summit held in 2017, several working groups were formed to explore ideas about improving the quality and consistency in graduate education in audiology and externship training. The results are described here from one of the working groups formed to examine postgraduate specialization fellowships. Method Over the course of a year, the committee designed and implemented two surveys: one directed toward faculty and one toward students. The rationale for the survey and the results are presented. Comparisons between faculty and student responses are made for similar questions. Results Overall, the results demonstrate that the majority of both students and faculty believe that postgraduation specialization fellowships are needed for either 1 year or a flexible length. There was a consensus of opinion that the fellowship should be paid, as these would be designed for licensed audiologists. Most believed that the fellowships should be “governed by a professional organization (e.g., American Speech-Language-Hearing Association, American Academy of Audiology, American Doctors of Audiology, etc.),” or less so, a “separate body for this specific purpose.” Potential topics for specialization identified were the following: tinnitus, vestibular, cochlear implants, pediatrics, and intraoperative monitoring. The highest priority attributes for a specialization site were “abundant access to patient populations,” “staff of clinical experts,” and “active research.” The weight put toward these attributes differed between faculty and students with faculty prioritizing “university/academic centers,” and “access to academic coursework in the fellowship area.” The faculty rated “caseload diversity,” “minimum hours,” “research,” and “academic affiliation” as requirements for a fellowship site, with less weight for “coursework” and “other.” Finally, the students valued “improved personal ability to provide exceptional patient care,” “the potential for increased job opportunities,” and the “potential for a higher salary” as benefits most important to them, with lower ratings for “recognition as a subject matter expert” or “potential pathway to Ph.D. program.” Conclusions As a result of the survey, further exploration of a postgraduate specialization fellowship is warranted, especially to determine funding opportunities to offset cost for the sites and to ensure that fellows are paid adequately.


2011 ◽  
Author(s):  
Misty M. Glover ◽  
Michelle E. Ronayne ◽  
Johnny P. Nguyen
Keyword(s):  

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