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Truong Thi Hien

Publications and source records attributed to Truong Thi Hien.

11 recordsLinked to original sources
arXiv2026

Regularity of symbolic powers of complementary edge ideals

Let \(G\) be a finite simple graph on \(n\) vertices, and let \(I_c(G)\) be its complementary edge ideal. We determine \(\reg(I_c(G)^{(t)})\) for every \(t\geq1\) in terms of the number \(c(G)\) of nontrivial connected components of \(G\) and the presence of a \(K_2\)-component. Consequently, \(\reg(I_c(G)^{(t)})\leq \reg(I_c(G)^t) \) for all \(t\geq1\), with equality for every \(t\) if and only if either \(c(G)=1\), or \(c(G)=2\) and \(G\) has a \(K_2\)-component. We also characterize Serre's condition \((S_2)\) for \(R/I_c(G)\), and classify the graphs for which \(R/I_c(G)^{(t)}\) is Cohen-Macaulay for every \(t\geq1\).

math.AC↗
arXiv2025

Microwave power and chamber pressure studies for single-crystalline diamond film growth using microwave plasma CVD

Single-crystalline diamond (SCD) films possess exceptional thermal, chemical, and optical properties, making them ideal for advanced applications. However, achieving uniform film quality via microwave plasma chemical vapor deposition (MPCVD) remains challenging due to spatial variations in plasma characteristics. This study systematically examines the influence of microwave power and chamber pressure on the growth of SCD films using CH4/H2 gas mixtures. Under optimized conditions (3,900 W, 120 Torr), the films exhibit low surface roughness (~2.0 nm), a sharp sp3 Raman peak at 1,332.2 cm-1, and no detectable C-H related features, indicating high crystalline purity. Cross-sectional TEM analysis confirms a uniform (100)-oriented single-crystal structure across the entire sample. These findings advance the understanding of the interplay between deposition parameters and film quality, and establish a more robust foundation for optimizing MPCVD processes in large-area, high-purity diamond fabrication.

cond-mat.mtrl-sci↗
arXiv2025

Cohen-Macaulayness of Powers of Edge Ideals of Weighted Oriented Graphs

For the edge ideal $I(\D)$ of a weighted oriented graph $\D$, we prove that its symbolic powers $I(\D)^{(t)}$ are Cohen-Macaulay for all $t\geqslant 1$ if and only if the underlying graph $G$ is composed of a disjoint union of some complete graphs. We also completely characterize the Cohen-Macaulayness of the ordinary powers $I(\D)^t$ for all $t\geqslant 2$. Furthermore, we provide a criterion for determining whether $I(\D)^t=I(\D)^{(t)}$.

math.AC↗
arXiv2025

Depth and regularity of powers of edge ideals of edge-weighted trees

For an increasing weighted tree $G_ω$, we obtain an asymptotic value and a sharp bound on the index stability of the depth function of its edge ideal $I(G_ω)$. Moreover, if $G_ω$ is a strictly increasing weighted tree, we provide the minimal free resolution of $I(G_ω)$ and an exact formula for the regularity of all powers of $I(G_ω)$.

math.AC↗
arXiv2024

On the set of associated radicals of powers of monomial ideals

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 $s$.

math.AC↗
arXiv2023

Depth of powers of edge ideals of Cohen-Macaulay trees

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 $t \ge 1$, $$\operatorname{depth} (S/I^t) = \operatorname{max} \{d -t + 1, 1 \}.$$

math.AC↗
arXiv2023

Quantum diamond microscopy with optimized magnetic field sensitivity and sub-ms temporal resolution

Quantum diamond magnetometers using lock-in detection have successfully detected weak bio-magnetic fields from neurons, a live mammalian muscle, and a live mouse heart. This opens up the possibility of quantum diamond magnetometers visualizing microscopic distributions of the bio-magnetic fields. Here, we demonstrate a lock-in-based wide-field quantum diamond microscopy, achieving a mean volume-normalized per pixel sensitivity of 43.9 $\mathrm{nTμm^{1.5}/Hz^{0.5}}$. We optimize the sensitivity by implementing a double resonance with hyperfine driving and magnetic field alignment along the $<$001$>$ orientation of the diamond. Additionally, we show that sub-ms temporal resolution ($\sim$ 0.4 ms) can be achieved while keeping the per-pixel sensitivity at a few tens of nanotesla per second using quantum diamond microscopy. This lock-in-based diamond quantum microscopy could be a step forward in mapping functional activity in neuronal networks in micrometer spatial resolution.

quant-ph↗
arXiv2022

Depth stability of cover ideals

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.

math.AC↗
arXiv2021

Regularity of symbolic powers of square-free monomial ideals

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_Δ$ is the Stanley-Reisner ideal of a simplicial complex $Δ$, then $\reg(I^{(n)}) \leqslant δ(n-1) +b$ for all $n\geqslant 1$, where $δ= \lim\limits_{n\to\infty} \reg(I^{(n)})/n$, and $b = \max\{\reg(I_Γ) \mid Γ\text{ is a subcomplex of } Δ\text{ with } \F(Γ) \subseteq \F(Δ)\}$. This bound is sharp for any $n$. When $I = I(G)$ is the edge ideal of a simple graph $G$, we obtain a general linear upper bound $\reg(I^{(n)}) \leqslant 2n + \ordmatch(G)-1$, where $\ordmatch(G)$ is the ordered matching number of $G$.

math.AC↗
arXiv2021

Regularity and Koszul property of symbolic powers of monomial ideals

Let $I$ be a homogeneous ideal in a polynomial ring over a field. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. Motivated by results about ordinary powers of $I$, we study the asymptotic behavior of the regularity function $\text{reg}~ (I^{(n)})$ and the maximal generating degree function $ω(I^{(n)})$, when $I$ is a monomial ideal. It is known that both functions are eventually quasi-linear. We show that, in addition, the sequences $\{\text{reg}~ I^{(n)}/n\}_n$ and $\{ω(I^{(n)})/n\}_n$ converge to the same limit, which can be described combinatorially. We construct an example of an equidimensional, height two squarefree monomial ideal $I$ for which $ω(I^{(n)})$ and $\text{reg}~ (I^{(n)})$ are not eventually linear functions. For the last goal, we introduce a new method for establishing the componentwise linearity of ideals. This method allows us to identify a new class of monomial ideals whose symbolic powers are componentwise linear.

math.AC↗