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In the context of hyperbolic systems of balance laws with dissipative source manifesting relaxation, recent pr"Ogress will be reported in the program of passing to the limit, in 1he BV setting, as the relaxation lime tends to zero.
Event date: 16/2/2023
Speaker: Prof. Constantine Dafermos (Brown University)
Hosted by: Department of Applied Mathematics
- Subjects:
- Mathematics and Statistics
- Keywords:
- Equilibrium -- Mathematical models Relaxation Differential equations Hyperbolic
- Resource Type:
- Video
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Video
Models arising in biology are often written in terms of Ordinary Differential Equations. The celebrated paper of Kermack-McKendrick (19271, founding mathematical epidemiology, showed the necessity to include parameters in order to describe the state of the individuals, as time elapsed after infection. During the 70s, many mathematical studies were developed when equations are structured by age, size, more generally a physiological trait. The renewal, growth-fragmentation are the more standard equations. The talk will present structured equations, show that a universal generalized relative entropy structure is available in the linear case, which imposes relaxation to a steady state under non-degeneracy conditions. In the nonlinear cases, it might be that periodic solutions occur, which can be interpreted in biological terms, e.g., as network activity in the neuroscience. When the equations are conservation laws, a variant of the Monge-Kantorovich distance (called Fortet-Mourier distance) also gives a general non-expansion property of solutions.
Event date: 19/1/2023
Speaker: Prof. Benoît Perthame (Sorbonne University)
Hosted by: Department of Applied Mathematics
- Subjects:
- Biology and Mathematics and Statistics
- Keywords:
- Biomathematics Equations
- Resource Type:
- Video
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Video
Universities conduct research for three reasons: to educate students, to contribute to society, and to understand the world. While society often holds a view of the scholar as a solitary and singular genius, in reality scholars today participate in a highly collaborative, worldwide search for shared understandings that stand the test of time and the scrutiny of others. The problems in the 21st century often demand effort by teams of researchers with resources at scale: laboratories and equipment, compute resources, and expert staffing. Working with faculty, students, and other stakeholders to identify the greatest opportunities and the resources needed to address them is both a privilege and a challenge for modern academic administrators. In this talk, I will share three examples: fostering collaborative proposal-writing; planning for shared capabilities in experimental facilities, data, and computation; and transforming academic structures.
Event date: 12/4/2023
Speaker: Prof. Kathryn Ann Moler
Hosted by: PolyU Academy for Interdisciplinary Research
- Subjects:
- Statistics and Research Methods
- Keywords:
- Research Science
- Resource Type:
- Video
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Video
More than one hundred years ago, Albert Einstein published his Theory of General Relativity (GR). One year later, Karl Schwarzschild solved the GR equations for a non-rotating, spherical mass distribution; if this mass is sufficiently compact, even light cannot escape from within the so-called event horizon, and there is a mass singularity at the center. The theoretical concept of a 'black hole' was born, and was refined in the next decades by work of Penrose, Wheeler, Kerr, Hawking and many others. First indirect evidence for the existence of such black holes in our Universe came from observations of compact X-ray binaries and distant luminous quasars. I will discuss the forty-year journey, which my colleagues and I have been undertaking to study the mass distribution in the Center of our Milky Way from ever more precise, long-term studies of the motions of gas and stars as test particles of the space time. These studies show the existence of a four million solar mass object, which must be a single massive black hole, beyond any reasonable doubt.
Event date: 09/02/2023
Speaker: Prof. Reinhard GENZEL
Hosted by: PolyU Academy for Interdisciplinary Research
- Subjects:
- Cosmology and Astronomy and Physics
- Keywords:
- Astrophysics Astronomy Deep space -- Milky Way Nobel Prize winners General relativity (Physics) Black holes (Astronomy)
- Resource Type:
- Video
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Video
In this lesson, we'll be looking at the cell cycle. This is the lifespan of a eukaryotic somatic cell. A somatic cell is any cell in the body of an organism, except for sex cells such as sperm and egg cells. The cell cycle describes the sequence of cell growth and division. A cell spends most of its life a state called interphase. Interphase has three phases, the G1, S, and G2 phases. Interphase is followed by cell division, which has one phase, the M phase. Together these four phases make up the entire cell cycle. G1 of interphase is sometimes called growth 1 or gap phase 1. In G1, a cell is busy growing and carrying out whatever function it's supposed to do. Note that some cells, such as muscle and nerve cells, exit the cell cycle after G1 because they do not divide again. A cell enters the S phase after it grows to the point where it's no longer able to function well and needs to divide. The S stands for synthesis, which means to make, because a copy of DNA is being made during this phase. Once DNA replication is complete, the cell enters the shortest and the last part of interphase called G2, also known as growth 2 or gap phase 2. Right now, it's enough to know that further preparations for cell division take place in the G2 phase. Now that interphase is over, the cell is ready for cell division, which happens in the M phase. The M phase has two events. The main one is mitosis, which is division of the cell's nucleus, followed by cytokinesis, a division of the cytoplasm. So, at the end of M phase, you have two daughter cells identical to each other and identical to the original cell. Let's review. The cell cycle describes the life cycle of an individual cell. It has four phases, three in interphase and one for cell division. Most cell growth and function happen during G1. The cell enters the S phase when it needs to divide. In this phase the cell replicates its DNA. Replication just means the cell makes a copy of its DNA. In G2, the cell undergoes further preparations for cell division. Finally, we have cell division in the M phase. The M phase consists of mitosis, which is nuclear division, and cytokinesis, or division of the cytoplasm. We'll explore the details of mitosis and cytokinesis separately
- Subjects:
- Biology
- Keywords:
- Cell cycle
- Resource Type:
- Video
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Video
Stanford Electrical Engineering Course on Convex Optimization.
- Course related:
- AMA4850 Optimization Methods
- Subjects:
- Mathematics and Statistics
- Keywords:
- Mathematical optimization Convex functions
- Resource Type:
- Video
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Video
With calculus well behind us, it's time to enter the next major topic in any study of mathematics. Linear Algebra! The name doesn't sound very intimidating, but there are some pretty abstract concepts in this subject. Let's start nice and easy simply by learning about what this subject covers and some basic terminology.
- Course related:
- COMP4434 Big Data Analytics
- Subjects:
- Mathematics and Statistics
- Keywords:
- Algebras Linear
- Resource Type:
- Video
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Video
Lecture videos from Gilbert Strang's course on Linear Algebra at MIT.
- Course related:
- AMA1120 Basic Mathematics II - Calculus and Linear Algebra
- Subjects:
- Mathematics and Statistics
- Keywords:
- Algebras Linear
- Resource Type:
- Video
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Video
This course is an introduction to game theory and strategic thinking. Ideas such as dominance, backward induction, Nash equilibrium, evolutionary stability, commitment, credibility, asymmetric information, adverse selection, and signaling are discussed and applied to games played in class and to examples drawn from economics, politics, the movies, and elsewhere.
- Subjects:
- Mathematics and Statistics and Economics
- Keywords:
- Game theory
- Resource Type:
- Video
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Video
Before the advent of computers around 1950, optimization centered either on small-dimensional problems solved by looking at zeroes of first derivatives and signs of second derivatives, or on infinite-dimensional problems about curves and surfaces. In both cases, "variations" were employed to understand how a local solution might be characterized. Computers changed the picture by opening the possibility of solving large-scale problems involving inequalities, instead of only equations. Inequalities had to be recognized as important because the decisions to be optimized were constrained by the need to respect many upper or lower bounds on their feasibility. A new kind of mathematical analysis, beyond traditional calculus, had to be developed to address these needs. It built first on appealing to the convexity of sets and functions, but went on to amazingly broad and successful concepts of variational geometry, subgradients, subderivatives, and variational convergence beyond just that. This talk will explain these revolutionary developments and why there were essential.
Event date: 1/11/2022
Speaker: Prof. Terry Rockafellar (University of Washington)
Hosted by: Department of Applied Mathematics
- Subjects:
- Mathematics and Statistics
- Keywords:
- Convex functions Convex sets Mathematical optimization Computer science -- Mathematics
- Resource Type:
- Video