陽明交通大學應用數學系

A 2-edge-colouring of a graph G is called locally complete if for each vertex v, the vertices adjacent to v through edges of the same colour induce a complete subgraph in G. Contreras-Balbuena et al characterized locally complete 2-edge-coloured graphs which have alternating Hamiltonian cycles. Chvátal and Sbihi proved that graphs which are locally complete 2-edge-colourable are one of two types of claw-free perfect graphs indecomposable via clique-cutsets. Maffray and Reed gave a forbidden subgraph characterization of these graphs.
We compare locally complete 2-edge-colourable graphs with proper interval graphs and proper circular-arc graphs. We characterize proper interval graphs and proper circular-arc graphs which are locally complete 2-edge-colourable by forbidden subgraphs.

Graphs can be viewed as (non-archimedean) analogs of Riemann surfaces. For example, there is a notion of Jacobians for graphs. More classically, graphs can be viewed as electrical networks. I will explain the interplay between these points of view and some recent applications in algebraic and arithmetic geometry.

Abstract
Fluid-fluid interfaces laden with discrete particles behave like continuous elastic sheets, leading to their applications in emulsion and foam stabilization. Although current continuum models can qualitatively describe the elastic buckling of granular particle-laden interfaces—often referred to as granular rafts—under compression, the connection between their macroscopic collective properties and the microscopic behavior of individual particles remains unclear. Here, by combining systematic experiments with first-principle modeling, we aim to reveal how the macroscopic mechanical properties of particle rafts emerge from particle-scale interactions and predict the failure modes of a granular raft under compression. Our study highlights the dual nature of particle rafts and exemplifies how collective dynamics can arise from discrete components with simple interactions.

Abstract. The use of large language models has changed the world we live in, from communicating in different languages to student use in homework. Recently it has proved to be effective on research level mathematics problems. In this talk, I will discuss the role of the researcher/professor in this newly emerging world.

In living cells, biochemical reactions form intricate networks, such as metabolic pathways, that give rise to essential cellular functions. However, the complexity of these network structures, together with our limited knowledge of their detailed kinetics, makes it challenging to understand how these functions emerge from network dynamics.
In this presentation, we introduce a method that enables us to determine system responses based solely on network structure. Using this method, we show that the effects of parameter perturbations on the steady state are confined to subnetworks satisfying specific topological conditions. Furthermore, we show that these subnetworks govern the steady-state bifurcations of reaction systems, including which parameters control the onset of bifurcations and which chemical species exhibit bifurcating behavior. Our findings demonstrate that network topology can underlie biological robustness and plasticity, two fundamental characteristics of living systems.

摘要
根據基督教的傳統: 在(舊約)先知瑪拉基之後，「上帝的聲音」靜默了四百多年。沉寂被曠野的呼聲所打破，這個人就是出現在舊約和新約中間在耶穌基督誕生之前的關鍵人物。他被上帝揀選呼召成為先知，他是耶穌的先鋒，為將要來臨的耶穌鋪下救恩之路，這人就是 --- 施洗約翰。類比於此我把格林函數稱之為微分方程的施洗約翰，因為格林函數的目的是得到微分方程的解。
這個演講我將從歷史的觀點特別是 J. Green 的思想介紹格林函數(Green function)。基本上這是一個Poisson方程的問題，需要處理Dirac-delta 函數、基本解(fundamental solution)、單層位勢(single layer potential)、雙層位勢(double layer potential)與牛頓位(Newtonian potential)。除了物理的詮釋之外我還從量綱分析(Dimensional Analysis)的角度引導聽眾如何簡單且直觀地猜出這些量。另外這些量也牽涉到數學分析的核心問題: 連續性(continuity)、可微性(differentiability)與可積性(integrability)。最後我也是透過量綱分析來解釋　Holder 空間為何會出現在(橢圓)偏微分方程的研究。

Abstract. Transformer-based machine learning models have achieved state-of-the-art performance in many areas. However, the quadratic complexity of the self-attention mechanism in Transformer models with respect to the input length hinders the applicability of Transformer-based models to long sequences or large images. To address this, we present Fast Multipole Attention (FMA), a new attention mechanism that uses a divide-and-conquer strategy to reduce the time and memory complexity of attention from $O(n^2)$ to $O(n \log n)$ or $O(n)$, while retaining a global receptive field. The hierarchical approach groups queries, keys, and values into $O(\log n)$ levels of resolution, where groups at greater distances are increasingly larger in size and the weights to compute group quantities are learned. As such, the interaction between tokens far from each other is considered in lower resolution in an efficient hierarchical manner. This multi-level divide-and-conquer strategy is inspired by fast summation methods from n-body physics and the Fast Multipole Method. We perform evaluation on language modeling and image processing tasks and compare our FMA model with other efficient attention variants on medium-size datasets. We find empirically that the Fast Multipole Transformer outperforms other efficient transformers in terms of memory size and accuracy. For large language models, the FMA mechanism has the potential to enable greater sequence lengths, taking the full context into account in an efficient, naturally hierarchical manner during training and when generating long sequences.