arXiv subjects
Publications and source records attributed to Nguyen Thu Hang.
Let $H$ be a unimodular hypergraph with cover ideal $J(H)$. We prove that the local $v$-numbers of $J(H)^t$ are linear in $t$ for all $t\ge1$. We further show that the global $v$-number of $J(H)^t$ is linear in $t$ for all $t\ge n-1$. Finally, we prove that the global $v$-number of the powers of the cover ideal of any tree is linear in $t$ for all $t\ge1$.
Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S$ in $|V(G)|$ variables. For a matching $M$ of $G$, we denote by $\ell(M)$ the length of the longest $M$-alternating path in $G$. We define $α_t(G)$ to be the maximum size of an ordered matching $M$ of $G$ such that $\ell(M) \le 2t-1$. We then prove that $$\operatorname{depth}(S/J(G)^{(t)}) \le |V(G)| - 1 - α_t(G)$$ for all $t \ge 1$, where $J(G)^{(t)}$ denotes the $t$-th symbolic power of $J(G)$, and that equality holds when $G$ is a forest.
Let $G$ be a simple graph. We demonstrate a method for using $t$-admissible subgraphs of $G$ to determine the regularity of the $t$-th symbolic power of the cover ideal of $G$. As an application, we compute the regularity of powers of cover ideals of bipartite unicyclic graphs.
Let $G$ be a simple graph. We introduce the notion of $t$-admissible subgraphs of $G$ and show how to use them to compute the depth of the $t$-th symbolic powers of the cover ideal of $G$. As an application, we prove that \[ \depth\big(S/J(C_n)^{(t)}\big) = n - 1 - \left\lfloor \frac{tn}{2t+1} \right\rfloor \] for all $t \ge 2$ and $n \ge 3$, where $S = K[x_1,\ldots,x_n]$ and $J(C_n)$ is the cover ideal of the cycle on $n$ vertices.
Let $G=P_n$ be a path graph with cover ideal $J(P_n)$. By using Hochster's depth formula, we prove the explicit formulae to compute the depth functions of powers of cover ideals of paths.
Let $T$ be a perfect binary tree and $I$ be its edge ideal in the polynomial ring $S$. We determine the vertex cover number, independent number, and establish the recursive formula to compute the number of minimal vertex covers. As a consequence, we compute the depth and projective dimension of $S/I$ and show that the total Betti number of $S/I$ at the highest homological degree always equals one.
Let $I$ be a monomial ideal in a polynomial ring. In this paper, we study the asymptotic behavior of the set of associated radical ideals of the (symbolic) powers of $I$. We show that both $\asr(I^s)$ and $\asr(I^{(s)})$ need not stabilize for large value of $s$. In the case $I$ is a square-free monomial ideal, we prove that $\asr(I^{(s)})$ is constant for $s$ large enough. Finally, if $I$ is the cover ideal of a balanced hypergraph, then $\asr(I^s)$ monotonically increases in $s$.
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.
In this paper, we first establish general bounds on the Fisher information distance to the class of normal distributions of Malliavin differentiable random variables. We then study the rate of Fisher information convergence in the central limit theorem for the solution of small noise stochastic differential equations and its additive functionals. We also show that the convergence rate is of optimal order.
We compute the projective dimension and regularity of $3$-path ideals of arbitrary graphs with at most one cycle.
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 $t \ge 1$, $$\operatorname{depth} (S/I^t) = \operatorname{max} \{d -t + 1, 1 \}.$$
Let R = K[x1,...,xr] be a polynomial ring over a field K. Let G be a graph with vertex set {1,...,r} and let J be the cover ideal of G. We give a sharp bound for the stability index of symbolic depth function sdstab(J). In the case G is bipartite, it yields a sharp bound for the stability index of depth function dstab(J) and this bound is exact if G is a forest.
In this note, we investigate the density of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin's calculus, we provide a log-normal upper bound for the density.
We prove that the depth functions of cover ideals of balanced hyper- gragh have the non-increasing property. Furthermore, we also give a bound for the index of depth stability of these ideals.
Let $\H$ be a unimodular hypergraph over the vertex set $[n]$ and let $J(\H)$ be the cover ideal of $\H$ in the polynomial ring $R=K[x_1,\ldots,x_n]$. We show that $\reg J(\H)^s$ is a linear function in $s$ for all $s\geqslant r\left\lceil \frac{n}{2}\right\rceil+1$ where $r$ is the rank of $\H$. Moreover for every $i$, $a_i(R/J(\H)^s)$ is also a linear function in $s$ for $s \geqslant n^2$.