- Date: 25 Jun. (Thu.)
- Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
- Time: 16:50-19:00
- Speaker 1: Junichiro Matsuda (Ritsumeikan University)
- Time:16:50-17:50
- Title: Recent approaches to expander quantum graphs
- Abstract:
The notion of quantum graph was introduced in the early 2010s motivated by quantum information theory. Quantum graphs are a non-commutative analogue of classical graphs, replacing the function algebra over the vertices with a non-commutative algebra and considering the quantum adjacency matrix acting on it. As in the classical case, we can characterize several properties of quantum graphs in terms of the spectrum of the related operators, e.g., connectedness, bipartiteness, and expanders. Expanders are families of $d$-regular graphs with a common lower bound of the spectral gap, on which the random walk expands rapidly.
In this talk, I will overview recent approaches to expander quantum graphs, and discuss possible definitions of quantum walks on quantum graphs.
- Speaker 2: Vu Thi Huong (University of Transport and Communications)
- Time:18:00-19:00
- Title: Some results on Caputo stochastic fractional delay differential equations
- Abstract:
This talk consists of two parts.
Part I. A brief overview of recent developments on Caputo stochastic fractional delay differential equations. The main topics include
• Well-posedness and regularity for solutions of Caputo stochastic fractional
delay differential equations.
• Variation-of-constants formulas
• Stability analysis.
• Several estimates for integrals involving singular kernels of the form (t −
s)
1−α, as well as matrix Mittag–Leffler functions Eα,α
(t − s)
αB
and
Eα,α
(t − s)
αB
. A key ingredient in these analyses is the exploitation of
the structural properties of the matrix Mittag–Leffler function Eα,α
(t −
s)
αB
in combination with the Jordan canonical form and a Djrbashiantype summation formula.
• Several numerical approximation schemes for this class of equations that
have been developed in the existing literature.
Part II. I will present some recent results from a joint work with my coauthors, which has been submitted to the Journal of Computational and Applied
Mathematics under the title Stability and Infinite-Time Convergence of the θMittag–Leffler Euler–Maruyama Approximation for Stochastic Fractional Delay
Differential Equations. This work is joint work with Prof. Ngo Hoang Long,
Hanoi National University of Education and Dr. Phan Thi Huong, Le Quy Don
Technical University.
This paper establishes, for the first time, the strong convergence of numerical
approximations over infinite time intervals for stochastic fractional delay differential equations. In contrast to existing results on finite horizons, where error
bounds typically depend on Mittag–Leffler functions that grow with the terminal
time, we establish a time-uniform error estimate. The analysis is based on a suitable weighted norm combined with stability properties of the linear part, which
allows us to control the singular kernels and the memory terms. Under suitable assumptions, we establish the uniform moment boundedness, stability, and
global-in-time strong convergence rate of the θ-Mittag–Leffler Euler–Maruyama
scheme. Numerical experiments illustrate the theoretical results and confirmthe
effectiveness of the method over long time intervals.
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