陽明交通大學應用數學系

Abstract. In this talk, we consider some nonlinear hyperbolic systems of balance laws near resonance. We focus on the classical solutions near or away from vacuum state (resonance ) to the compressible Euler equations for either isentropic or non-isentropic flows through non-uniform ducts. We establish the global existence of classical solutions for both cases. By the normalization of systems, maximum principle, generalized Lax transformation and the detail study of Riccati equations, we are able to obtain uniformly bounded C^1 norms of those solutions . The asymptotic behavior of solutions is also studied.

This series of projects are the joint works with Hsin-Yi Lee, Jay Chu, Ying-Chieh Lin, Shih-Wei Chou and Shih-Ming Wang.

Abstract. It is well known that the 2D Keller-Segel system has finite time blowup solutions if the initial density has total mass greater than and finite second moment. We have several constructive examples showing that the solution blows up with the only -mass concentration. We will exhibit a new blowup mechanism formed by a collision of two sub-collapses, resulting in a -mass concentrating solution. A similar phenomenon appears in the 2D mass critical NLS by the work of Martel-Raphael in 2018, where the construction relies on two specific features: a conformal invariance and a minimal mass blowup constraint. These properties are not available for parabolic equations such as the Keller-Segel system. We bring here for the first time a directly rigorous construction of such a multiple-collapsing blowup solution.

演講網址：meet.google.com/bfa-aykg-mjt

Abstract. West Nile virus (WNv) is a mosquito-borne disease caused by Flavivirus. To investigate the combined effects of vertical transmission, temperature-dependent incubation periods and environmental heterogeneity on the transmission of WNv, we develop a delay differential system with stage-structure and time-varying delays. We then derive the mosquito reproduction number and WNv reproduction number, respectively, and further show that these two numbers serve as threshold parameters that determine whether WNv will spread or not.

Numerical simulations support our analytic results and suggest that time-periodic delays and environmental heterogeneity can play important roles in the WNv dynamics

Abstract. We study diverse solutions to optimization problems and design fair scheduling of demands with different priorities on wireless networks The problem of finding several sufficiently-diverse, yet approximately-optimal solutions to an optimization problem can be described as follows: given an integer k, an approximation factor $\alpha$, and a diversity measure $\sigma$ on a set of solutions to an optimization problem P, find a set of k solutions to P that (a) are all $\alpha$ approximately-optimal for P and (b) maximize the diversity measure $\sigma$ over all such sets of k solutions. The optimization problems considered here are maximum matching, spanning tree, global min-cut, shortest path, and minimum weight bases of a matroid. Here, a solution is a set of edges, and the number of edges two given solutions differ in as the diversity measure of the pair. The diversity measure $\sigma$ of k solutions is the sum of the pairwise Hamming distance between the bit-vectors representing the k solutions. We propose the first polynomial-time algorithms that (except for the unweighted spanning tree), for these five graph problems, guarantee that their diversity is at least 1/4 times the optimal diversity under different $\alpha$.

This result is applied for minimum weight bases of a matroid, which means it gives us diverse $\alpha$-approximate minimum spanning trees, advancing a step towards achieving diverse $\alpha$-approximate TSP tours.

Abstract
The exploration of submanifolds in spaces of constant sectional curvature is fundamental to Differential Geometry. Since 2015, significant progress has been achieved by Chiu-Huang-Lai and Chiu-Ho, who established the Fundamental Theory of Curves and Surfaces in the Heisenberg group H₁ and the CR sphere S³ respectively. Their research resulted in the identification of a comprehensive set of geometric invariants, specifically curvatures for curves and surfaces. Acting as pseudo-Hermitian manifolds, H₁ and S³ represent special cases of CR manifolds, embodying zero and positive constant sectional curvatures, respectively. Recently, Pak Tung Ho (Tamkang University) and I have extended our research to include the final model case—the constant negative sectional curvature—in the product space consisting of a unit 2-sphere and the real line. This presentation will provide a concise review of our previous findings and will concentrate on developing geometric invariants in this product space. Furthermore, I will explore the proof of the Fundamental Theorem of Curves, supported by a series of intriguing examples.

Abstract. We will present recent numerical methods for solving partial differential equations on surfaces. In the first part of the talk, we will introduce a simple discretization called the Modified Virtual Grid Difference (MVGD), which allows for the numerical approximation of the Laplace-Beltrami operator on manifolds sampled by point clouds. In the second part, we will present a diffeomorphic embedding method that has been recently developed for computing differential equations on implicit surfaces. This embedding equation is based on a push-forward operator, which can extend any tangential flux vectors from the surface to a neighboring level surface. The computation of this operator is straightforward and only requires the level set function and the corresponding Hessian. We will provide examples from scalar hyperbolic conservation laws and eigenproblems of the Laplace-Beltrami operators.

Abstract. We provide a systematic convergence analysis of the Hessian operator estimator from random samples supported on a low dimensional manifold. We show that the impact of the nonuniform sampling and the curvature on the widely applied Hessian operator estimator is asymptotically negligible. This is joint work with Hau-Tieng Wu (Courant Institute of Mathematical Sciences, NYU).

摘要
傳統術數(或稱「命理學」)源遠流長，目前已經發產出龐大的思想體系，多元的論命方法與派別，以及深邃的理論基礎；同時也和東亞文化、精神生活、哲學思想、以及天人合一的世界觀都緊密的結合在一起。相對於這些完善的「質性」理論與實踐基礎，「量化」研究目前正處於發展中的階段。近年來本人致力於傳統術數量化以及學術化的工作，大致分成三個方向在進行：(1)研發「術數量化研究」的方法學，包括量化問卷發展、研究設計、以及資料分析；(2)執行術數量化實證研究；以及(3)推廣術數的教育傳播，包括設計能夠與其他學科結合的大學課程、撰寫科學普及文章、以及研究術數學科化時諸多科技與社會的議題。 在研發方法學方面，我們提出未來術數量化研究的架構與三個重點方向：(1)命運的量化與測量、(2)信度研究、以及(3)效度研究。命運的量化與測量大致可以分成心理學、社會學、以及醫學三個方面，除了使用現有的問卷與資料庫(例如臺灣地區社會變遷基本調查資料庫)，我們也同時重新針對傳統術數可預測且可量化的命運項目發展問卷，建立標準量化指標。信度研究目的是估計命理師之間對於論命結果的共識程度。效度研究目的是估計「命理師預測命主的命運」是否和「命主真實發生的命運」的吻合程度。我們過去提出「所有的命理學的理論體系皆可以視為一組預測模型」的概念，將術數的效度化為可以實證的命題，同時也提出使用機器學習模型建構「推命理論參數」與實際發生事件的預測模型的研究設計方法。我們目前正在使用上述提出的方法學進行實證研究中，使用的資料主要以臺灣社會變遷基本調查計畫資料庫為主，量化紫微斗數論命理論的針對婚姻狀態的結果。未來也預計推展到婚姻狀態以外的其他變項(諸如收入、學歷、以即六親相處情形)，在論命方法將紫微斗數擴展到所有以生辰八字為主的推命數、姓名學、以及各項中華術數領域進行大規模的量化實證研究。

Abstract
This presentation introduces the AI-meteorological applications developed in NTU in recent years.
The first part showcases the usefulness of deep learning (DL) for reconstructing homogenized and trustworthy global tropical cyclone (TC) wind profile datasets since 1981 and thus facilitating an examination of climate trends of TC structure/energy extremes. Understanding past TC trends and variability is critical for projecting future TC impacts on human society considering the changing climate. By training with uniquely labeled data integrating best tracks and numerical model analysis, our model converts multichannel satellite imagery to a 0-750-km wind profile of axisymmetric surface winds. The model performance is verified to be sufficient for climate studies by comparing it to independent satellite-radar surface winds. Based on the new homogenized dataset, the major TC proportion has increased by ~13% in the past four decades. Moreover, the proportion of extremely high-energy TCs has increased by ~25%, along with an increasing trend (> one standard deviation of the 40-y variability) of the mean total energy of high-energy TCs. Although the warming ocean favors TC intensification, the TC track migration to higher latitudes and altered environments further affect TC structure and energy.
In the second half of this talk, we will review a recent and impactful paper published in Nature: the Pangu-Weather. This framework swtich from the deep-learning regression tasks to modeling the entire atmospheric circulation system with data-driven AI. Our preliminary tests of this new approach for weather prediction is also carried out. Subsequently, we discuss new opportunities and plans for integrating the above DL analysis/forecasting techniques into a full data-driven mesoscale weather prediction model.

Abstract. As the most abundant tissue on Earth, xylem is responsible for lateral growth in plants. Typical xylem has a radial system composed of ray parenchyma cells and an axial system of fusiform cells. In most angiosperms, fusiform cells are a combination of vessel elements for water transportation and libriform fibers for mechanical support, while both functions are performed together by tracheids in other vascular plants. However, little is known about the developmental programs and evolutionary relationships of these xylem cell types. Through both single-cell and laser-capture microdissection transcriptomic profiling, here we demonstrate the developmental lineages of ray and fusiform cells in stem-differentiating xylem across four divergent woody angiosperms. Cross-species analyses of single-cell trajectories reveal highly conserved ray, yet variable fusiform, lineages across angiosperms. Core eudicots Populus trichocarpa and Eucalyptus grandis share nearly identical fusiform lineages. The tracheids in the basal eudicot Trochodendron aralioides, an evolutionarily reversed character, exhibit strong transcriptomic similarity to vessel elements but not libriform fibers, suggesting that water transportation, instead of mechanical support, is the major feature. We also found that the more basal angiosperm Liriodendron chinense has a fusiform lineage distinct from that in core eudicots. This evo-developmental framework provides a comprehensive understanding of the formation of xylem cell lineages across multiple plant species spanning over a hundred million years of evolutionary history. Within the past one and half years, four research groups have published their studies in the top 5% journals on elucidating stem-differentiating xylem development. Yet, these four articles led to four conflicting models. In an attempt to sort out this puzzle, we reached out to the other corresponding authors to determine the current most plausible model.

Abstract. In this talk, I will first briefly review the history of sum-product estimates in the setting of finite fields and mention the applications of sum-product type problems. Then I will start introducing the celebrated result of Bourgain about the discretized sum-product estimates and the applications to geometric measure theory problems. Finally I will talk about our recent results about expanding polynomials in continuous setting and discuss their sum-product phenomena.

Abstract. This talk will cover three closely related problems: universal coding for data compression, online portfolio selection for long-term investment, and online learning of quantum states for quantum state tomography. While the application scenarios appear distinct, these problems share similar mathematical structures. I will offer a brief introduction to all three problems and delve into state-of-the-art results for the latter two.

Abstract. Let $X$ be a.compact metric space and $T: X \to X$ be a continuous transformation. A dynamical system $(X, T)$ is called Bohr chaotic if for each weight sequence $(w_n) \in \ell^{\infty}(\mathbb{N}, \mathbb{R})$ there are $f \in \mathcal{C}(X)$ and $x \in X$ such that $(w_n)$ is orthogonal to $\{f \circ T^n(x)\}$. In this talk, we introduce the number-conserving shifts and show that a number-conserving shift is either finite or Bohr chaotic. Furthermore, a number-conserving shift is consisting of periodic points whenever it is finite.

Abstract. In this talk I will present a recent result showing that linear combinations of tensor products of k gradients of harmonic functions, with k at least three, are dense in C_0^infty(\Omega), for any bounded domain \Omega in dimension 3 or higher. This kind of density result has applications to inverse problems for elliptic quasilinear equations/systems in divergence form, where the nonlinear part of the ”conductivity is anisotropic.

摘 要：近年來全世界的太空產業發展蓬勃，自火箭回收、立方衛星崛起到近期的低軌衛星通訊、太空短期旅行等，太空領域產業將來前景無可限量。國家太空中心是臺灣太空發展推動的專責單位，自2023 年一月開始將轉型為行政法人研究機構，除了英文名將由NSPO (National Space Organization)改為 TASA(Taiwan Space Agency)，中心的資源（人、地、物）將隨之擴大，補強過去台灣太空科技研發能量的不足，肩負起臺灣未來太空科技與相關產業發展的重責大任，讓我們一起了解臺灣太空的歷程，面對未來的挑戰，一同建構與實現臺灣的太空夢。