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Consider a simple symmetric random walk $S$ and another random walk $S'$ whose $k$th increments are the $k$-fold product of the first $k$ increments of $S$.
The random walks $S$ and $S'$ are strongly dependent. Still the 2-dimensional walk $(S, S')$, properly rescaled, converges to a two dimensional Brownian motion. The goal of this talk is to present the proof of this fact, and its generalizations. Based on joint works with K. Hamza and S. Meng.
Originated from applications in signal processing, random evolution,
telecommunications, risk management, financial engineering,
and manufacturing systems, two-time-scale Markovian systems have
drawn much attention. This talk discusses asymptotic
expansions of solutions to the forward equations, scaled and unscaled
occupation measures, approximation error bounds, and associated
switching diffusion processes. Controlled dynamic systems will also
Many problems in control and optimization require the treatment
of systems in which continuous dynamics and discrete events
coexist. This talk presents a survey on some of our recent work on such
systems. In the setup, the discrete event is given by a random
process with a finite state space, and the continuous component is the
solution of a stochastic differential equation. Seemingly similar
to diffusions, the processes have a number of salient features
distinctly different from diffusion processes. After providing
motivational examples arising from wireless communications,
identification, finance, singular perturbed Markovian systems,
manufacturing, and consensus controls, we present necessary and
sufficient conditions for the existence of unique invariant
measure, stability, stabilization, and numerical solutions of
control and game problems.
재미로 풀어보는 퀴즈에나 등장할 법한 추상적인 수학적 개념이 기계공학(예, 응용역학) 연구에 도움을 줄 수 있을까? 수학과 역학 사이의 간극이 가장 좁았던 때는 언제였고, 수학과 역학이 만나는 지점에서 두 학문을 두루 섭렵했던 수리과학자는 누구였을까? 이와 같은 질문에 대한 답변의 일환으로, 본 발표의 전반부에서는 수학과 역학(유체역학, 고체역학, 열역학, 파동학)의 역사가 공존했던 시절을 인물 중심으로 살펴보고자 한다. 본 발표의 후반부에서는, 역학적 파동과 메타물질에 관한 발표자의 연구주제(음향 투명망토, 음향 블랙홀, 생물음향학 등)를 간략하게 소개한다.
We review two examples of interesting interactions between number theory and string compactification and raise some new questions and issues in context of these examples based on review article by Gregory W. Moore (arXiv:hep-th/0401049). The first example concerns the role of the Rademacher expansion of coefficients of modular forms in the AdS/CFT correspondence. The second example concerns the role of the "attractor mechanism" of supergravity in selecting certain arithmetic Calabi-Yau's as distinguished compactifications.
There is a curious relationship between two sets of apparently unrelated numbers. On one side one has the coefficients of the cube root of the classical modular function J, and on the other one has the dimensions of irreducible representations of the exceptional Lie algebra E8. In this talk I will describe how this mysterious connection has been explained in terms of certain infinite dimensional algebras called vertex algebras.
The curious observation discussed in the first lecture is just a prelude to an even more mysterious one, this time between the coefficients of the modular J function itself, and the dimensions of irreducible representations of the largest of the sporadic finite simple groups: the monster group. In this talk I will describe how vertex algebras have been used to explain this connection, as well as to derive remarkable identities satisfied by the J function.