Venue: College of Science Room SC4009-1, National Sun Yat-sen University Venue: [ trasportation campus map ]
Announcements:
The conference's primary objective is to create a conducive environment where researchers can exchange findings, insights, and challenges within the fields of graph theory, combinatorics, and their practical applications. It encompasses a comprehensive range of topics in discrete mathematics and their real-world implementations, with a particular focus on the intersection with artificial intelligence. This conference represents a continuation of previous gatherings at NSYSU, paying tribute to the individuals who have made substantial contributions to our shared endeavors.
Each talk is 40 minutes for presentation and 10 minuntes for questions.
The original talk scheduled from 10:00 to 11:00 has been canceled, so the talk of Jephian Lin is moved to this time instead.
Neural networks, a cornerstone of machine learning, have profoundly impacted the modern world. Their foundation lies in a blend of mathematical disciplines, including matrix theory, gradient descent optimization, matrix derivatives, randomized algorithms, and more. This harmonious integration of fields has contributed to the remarkable success of neural networks in various applications. In this talk, we will explore the details of a neural network prototype and demonstrate how to construct one from scratch.
A conflict-avoiding code (CAC) $\mathcal{C}$ of length $L$ with weight $w$ is a collection of $w$-subsets of $\mathbb{Z}_L$ such that $d^*(S_1)\cap d^*(S_2)=\emptyset$ for any two distinct $w$-subsets $S_1, S_2\in\mathcal{C}$, where $d^*(S)=\{a-b\ (\bmod\, L):\,a,b\in S, a\neq b\}$. A CAC is a deterministic transmission scheme for asynchronous multiple-access without feedback. When the number of simultaneously active users is less than or equal to $w$, a CAC of length $L$ with weight $w$ can provide a hard guarantee that each active user has at least one successful transmission within every consecutive $L$ time slots. The design goal of CACs is to determine the maximum code size, denoted by $K(L,w)$, for given $L$ and $w$. A CAC is called optimal if its code size achieves the value $K(L,w)$. In this talk, we will provide a series of optimal CACs by the help of Kneser’s Theorem and some other techniques in Additive Combinatorics. Mixed-weight CACs and multichannel CACs, two class of natural generalization of CACs, will be discussed as well.
The spectral radius of a square matrix is the maximum of the magnitude of its eigenvalues. The spectral radius of a graph, which is the spectral radius of its adjacency matrix, is related to many properties of the graph and is an important and extensively studied topic in spectral graph theory. This talk will introduce the extremal values of spectral radius under various constraints, and the tools used to address these problems.
Spanning trees play an important role in graph theory, providing valuable insights into connectivity and enumeration. In this talk, I will provide the methods for counting spanning trees in various graphs, such as the fan graphs, the wheel graphs, and the ladder graphs, using both algebraic and combinatorial approaches. For algebraic approach, we use the spectrum of Laplacian matrix to acquire the number of spanning trees, as for the combinatorial approach, we derive the quantity by finding a recursive relation. We also establish connections between counting spanning trees and tiling problems, which easily give a simple recursive formula.
The talk will also present an alternative proof of Cayley’s formula using the idea of labeled Dyck path. Additionally, we adopt the concept of extended Prüfer code to find out a deeper structural properties of trees. We also enumerate the number of spanning trees that have the same order of deleted leaves during the construction of Prüfer code.
There is no default arrangement for dinner. You are welcome to stay, and we may have an informal gathering at our own cost.