arXiv subjects
Publications and source records attributed to Thanh Vu.
We study simple graphs for which the maximal homogeneous ideal is an associated prime of the second power of their closed neighborhood ideals. In particular, we show that such graphs must have diameter at most $6$, and that those with diameter $2$ must be vertex diameter-$2$-critical.
Large language models (LLMs) have shown impressive performance in \emph{code} understanding and generation, making coding tasks a key focus for researchers due to their practical applications and value as a testbed for LLM evaluation. Data synthesis and filtering techniques have been widely adopted and shown to be highly effective in this context. In this paper, we present a focused survey and taxonomy of these techniques, emphasizing recent advancements. We highlight key challenges, explore future research directions, and offer practical guidance for new researchers entering the field.
We prove that the regularity of edge ideals of powers of forests is weakly decreasing. We then compute the regularity of edge ideals of powers of cycles.
We associate a sequence of positive integers, termed the type sequence, with a cochordal graph. Using this type sequence, we compute all graded Betti numbers of its edge ideal. We then classify all positive integer $n$ such that the zero divisor graph of $\mathbb{Z}/n \mathbb{Z}$ is cochordal and determine all the graded Betti numbers of its edge ideal.
Let $I$ be an arbitrary nonzero squarefree monomial ideal of dimension $d$ in a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_n]$. Let $μ$ be the number of associated primes of $S/I$ of dimension $d$. We prove that the multiplicity of powers of $I$ is given by $$e_0(S/I^s) = μ\binom{n-d+s-1}{s-1},$$ for all $s \ge 1$. Consequently, we compute the multiplicity of all powers of path ideals of cycles.
We associate a {\it skew tableau ideal} to each filling of a skew Ferrers diagram with positive integers. We classify all unmixed and sequentially Cohen-Macaulay skew tableau ideals. Consequently, we classify all Cohen-Macaulay, Buchsbaum, and generalized Cohen-Macaulay skew tableau ideals.
We compute the regularity of powers and symbolic powers of edge ideals of all cubic circulant graphs. In particular, we establish Conjecture of Minh for cubic circulant graphs.
We classify all normal edge ideals of edge-weighted graphs.
We prove that the regularity cannot increase when taking the integral closure for edge ideals of arbitrary weighted oriented graphs.
We prove a sharp bound for the regularity of a squarefree monomial ideal with a linear presentation. This result also answers in positive a question on the cohomological dimension of squarefree monomial ideals satisfying Serre's $S_2$-condition proposed by Dao and Takagi.
We classify all graphs for which the Rees algebras of their edge ideals are normal and have regularity equal to their matching numbers.
Let $G$ be a simple graph on $n$ vertices. We introduce the notion of bipartite connectivity of $G$, denoted by $\operatorname{bc}(G)$ and prove that $$\lim_{s \to \infty} \operatorname{depth} (S/I(G)^{(s)}) \le \operatorname{bc}(G),$$ where $I(G)$ denotes the edge ideal of $G$ and $S = \mathrm{k}[x_1, \ldots, x_n]$ is a standard graded polynomial ring over a field $\mathrm{k}$. We further compute the depth of symbolic powers of edge ideals of several classes of graphs, including odd cycles and whisker graphs of complete graphs to illustrate the cases where the above inequality becomes equality.
Let $I_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n+1}x_{n+2}\cdots x_{n+m})$ be the $m$-path ideal of a path of length $n + m-1$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{n+m}]$. We compute all the graded Betti numbers of all powers of $I_{n,m}$.
Let $J_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n}x_1\cdots x_{m-1})$ be the $m$-path ideal of a cycle of length $n \ge 5$ over a polynomial ring $S = k[x_1,\ldots,x_n]$. Let $t\geq 1$ be an integer. We show that $J_{n,m}^t$ has a linear free resolution and give a precise formula for all of its Betti numbers when $m = n-1, n-2$.
We compute the projective dimension and regularity of $3$-path ideals of arbitrary graphs with at most one cycle.
The success of data mixing augmentations in image classification tasks has been well-received. However, these techniques cannot be readily applied to object detection due to challenges such as spatial misalignment, foreground/background distinction, and plurality of instances. To tackle these issues, we first introduce a novel conceptual framework called Supervision Interpolation (SI), which offers a fresh perspective on interpolation-based augmentations by relaxing and generalizing Mixup. Based on SI, we propose LossMix, a simple yet versatile and effective regularization that enhances the performance and robustness of object detectors and more. Our key insight is that we can effectively regularize the training on mixed data by interpolating their loss errors instead of ground truth labels. Empirical results on the PASCAL VOC and MS COCO datasets demonstrate that LossMix can consistently outperform state-of-the-art methods widely adopted for detection. Furthermore, by jointly leveraging LossMix with unsupervised domain adaptation, we successfully improve existing approaches and set a new state of the art for cross-domain object detection.
We compute the depth and regularity of ideals associated with arbitrary fillings of positive integers to a Young diagram, called the tableau ideals.
Let $I$ be the edge ideal of a Cohen-Macaulay tree of dimension $d$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{d},y_1,\ldots,y_d]$. We prove that for all $t \ge 1$, $$\operatorname{depth} (S/I^t) = \operatorname{max} \{d -t + 1, 1 \}.$$