arXiv subjects

Zhiyu Liu

Publications and source records attributed to Zhiyu Liu.

At least 37 records · Page 2Linked to original sources

arXiv2024

Castelnuovo bound for curves in projective 3-folds

The Castelnuovo bound conjecture, which is proposed by physicists, predicts an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau 3-folds of Picard number one. Previously, it is only known for a few cases and all the proofs rely on the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda. In this paper, we prove the Castelnuovo bound conjecture for any Calabi-Yau 3-folds of Picard number one, up to a linear term and finitely many degree, without assuming the conjecture of Bayer-Macrì-Toda. Furthermore, we prove an effective vanishing theorem for surface-counting invariants of Calabi-Yau 4-folds of Picard number one. We also apply our techniques to study low-degree curves on some explicit Calabi-Yau 3-folds. Our approach is based on a general iterative method to obtain upper bounds for the genus of one-dimensional closed subschemes in a fixed 3-fold, which is a combination of classical techniques and the wall-crossing of weak stability conditions on derived categories, and works for any projective 3-fold with at worst isolated singularities over any algebraically closed field.

math.AG↗

arXiv2024

Categorical Torelli theorems for Gushel-Mukai threefolds

We show that a general ordinary Gushel-Mukai(GM) threefold $X$ is reconstructed from the Kuznetsov component $\mathcal{K}u(X)$ together with an extra data coming from tautological sub-bundle of Grassmannian $\mathrm{Gr}(2,5)$. We also prove that $\mathcal{K}u(X)$ determines birational isomorphism class of $X$, while $\mathcal{K}u(X')$ determines the isomorphism class of a general special GM threefold $X'$. As an application, we prove a conjecture of Kuznetsov-Perry in dimension three under a mild assumption. Finally, we use $\mathcal{K}u(X)$ to restate a conjecture of Debarre-Iliev-Manivel regarding fibers of the period map for ordinary GM threefolds.

math.AG↗

arXiv2023

Stability manifolds of Kuznetsov components of prime Fano threefolds

Let $X$ be a cubic threefold, quartic double solid or Gushel--Mukai threefold, and $\mathcal{K}u(X)\subset \mathrm{D}^b(X)$ be its Kuznetsov component. We show that a stability condition $σ$ on $\mathcal{K}u(X)$ is Serre-invariant if and only if its homological dimension is at most $2$. As a corollary, we prove that all Serre-invariant stability conditions on $\mathcal{K}u(X)$ form a contractible connected component of the stability manifold.

math.AG↗

arXiv2023

New perspectives on categorical Torelli theorems for del Pezzo threefolds

Let $Y_d$ be a del Pezzo threefold of Picard rank one and degree $d\geq 2$. In this paper, we apply two different viewpoints to study $Y_d$ via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component: (i) Brill-Noether reconstruction. We show that $Y_d$ can be uniquely recovered as a Brill-Noether locus of Bridgeland stable objects in its Kuznetsov component. (ii) Exact equivalences. We prove that, up to composing with an explicit auto-equivalence, any Fourier-Mukai type equivalence of Kuznetsov components of two del Pezzo threefolds of degree $2\leq d\leq 4$ can be lifted to an equivalence of their bounded derived categories. As a result, we obtain a complete description of the group of Fourier-Mukai type auto-equivalences of the Kuznetsov component of $Y_d$. We also describe the group of Fourier-Mukai type auto-equivalences of Kuznetsov components of index one prime Fano threefolds $X_{2g-2}$ of genus $g=6$ and $8$. As an application, first we identify the group of automorphisms of $X_{14}$ and its associated $Y_3$. Then we give a new disproof of Kuznetsov's Fano threefold conjecture by assuming Gushel-Mukai threefolds are general. In an appendix, we classify instanton sheaves on quartic double solids, generalizing a result of Druel.

math.AG↗

arXiv2023

Conics on Gushel-Mukai fourfolds, EPW sextics and Bridgeland moduli spaces

We identify the double dual EPW sextic $\widetilde{Y}_{A^{\perp}}$ and the double EPW sextic $\widetilde{Y}_A$, associated with a very general Gushel-Mukai fourfold $X$, with the Bridgeland moduli spaces of stable objects of character $Λ_1$ and $Λ_2$ in the Kuznetsov component $\mathcal{K}u(X)$. This provides an affirmative answer to a question of Perry-Pertusi-Zhao. As an application, we prove a conjecture of Kuznetsov-Perry for very general Gushel-Mukai fourfolds.

math.AG↗

arXiv2023

Infinitesimal categorical Torelli theorems for Fano threefolds

Let $X$ be a smooth Fano variety and $\mathcal{K}u(X)$ the Kuznetsov component. Torelli theorems for $\mathcal{K}u(X)$ says that it is uniquely determined by a polarized abelian variety attached to it. An infinitesimal Torelli theorem for $X$ says that the differential of the period map is injective. A categorical variant of infinitesimal Torelli theorem for $X$ says that the morphism $H^1(X,T_X)\xrightarrowη HH^2(\mathcal{K}u(X))$ is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we first prove infinitesimal categorical Torelli theorem for a class of prime Fano threefolds. Then we prove a restatement of the Debarre-Iliev-Manivel conjecture infinitesimally.

math.AG↗

arXiv2023

Flexoelectricity-stabilized ferroelectric phase with enhanced reliability in ultrathin La:HfO2 films

Doped HfO2 thin films exhibit robust ferroelectric properties even for nanometric thicknesses, are compatible with current Si technology and thus have great potential for the revival of integrated ferroelectrics. Phase control and reliability are core issues for their applications. Here we show that, in (111)-oriented 5%La:HfO2 (HLO) epitaxial thin films deposited on (La0.3Sr0.7)(Al0.65Ta0.35)O3 substrates, the flexoelectric effect, arising from the strain gradient along the films normal, induces a rhombohedral distortion in the otherwise Pca21 orthorhombic structure. Density functional calculations reveal that the distorted structure is indeed more stable than the pure Pca21 structure, when applying an electric field mimicking the flexoelectric field. This rhombohedral distortion greatly improves the fatigue endurance of HLO thin films by further stabilizing the metastable ferroelectric phase against the transition to the thermodynamically stable non-polar monoclinic phase during repetitive cycling. Our results demonstrate that the flexoelectric effect, though negligibly weak in bulk, is crucial to optimize the structure and properties of doped HfO2 thin films with nanometric thicknesses for integrated ferroelectric applications.

cond-mat.mtrl-sci↗

arXiv2022

Castelnuovo bound and higher genus Gromov-Witten invariants of quintic 3-folds

We prove a conjectural vanishing result for Gopakumar--Vafa invariants of quintic 3-folds, referred to as Castelnuovo bound in the literature. Furthermore, we calculate Gopakumar--Vafa invariants at Castelnuovo bound $g=\frac{d^2+5d+10}{10}$. As physicists showed, these two properties allow us to compute all Gromov--Witten invariants of quintic 3-folds up to genus $53$, provided that the conifold gap condition holds. We also give a bound for the genus of any one-dimensional closed subscheme in a smooth hypersurface of degree $\leq 5$, which may be of independent interest.

math.AG↗

arXiv2022

Cutting sequence and Sturmian sequence in billiard

The winning rule of billiards is to drive the billiard ball on the table into the designated holes. We try to study the trajectory of the billiard ball, so that we can predict the direction of the ball. For rational slopes, we got cutting sequence by setting up the square torus. We simplified cutting sequence using shearing and flipping and we obtain the transformation between trajectory slope and cutting sequence. For irrational slopes, we look at some properties of Sturmian sequence, which help us distinguish between cutting sequence and Sturmian sequence. In conclusion, in the case of different slopes, we use different sequences to do research.

math.DS↗

arXiv2022

Efficient textual explanations for complex road and traffic scenarios based on semantic segmentation

The complex driving environment brings great challenges to the visual perception of autonomous vehicles. It's essential to extract clear and explainable information from the complex road and traffic scenarios and offer clues to decision and control. However, the previous scene explanation had been implemented as a separate model. The black box model makes it difficult to interpret the driving environment. It cannot detect comprehensive textual information and requires a high computational load and time consumption. Thus, this study proposed a comprehensive and efficient textual explanation model. From 336k video frames of the driving environment, critical images of complex road and traffic scenarios were selected into a dataset. Through transfer learning, this study established an accurate and efficient segmentation model to obtain the critical traffic elements in the environment. Based on the XGBoost algorithm, a comprehensive model was developed. The model provided textual information about states of traffic elements, the motion of conflict objects, and scenario complexity. The approach was verified on the real-world road. It improved the perception accuracy of critical traffic elements to 78.8%. The time consumption reached 13 minutes for each epoch, which was 11.5 times more efficient than the pre-trained network. The textual information analyzed from the model was also accordant with reality. The findings offer clear and explainable information about the complex driving environment, which lays a foundation for subsequent decision and control. It can improve the visual perception ability and enrich the prior knowledge and judgments of complex traffic situations.

cs.CV↗

arXiv2022

A moduli theoretic approach to Lagrangian subvarieties of hyperkähler varieties: Examples

We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperkähler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the conjectures in several cases, recovering classical examples. As a corollary, we confirm a conjecture of O'Grady in several instances on the existence of Lagrangian covering families for hyperkähler varieties.

math.AG↗

arXiv2021

A note on Bridgeland moduli spaces and moduli spaces of sheaves on $X_{14}$ and $Y_3$

We study Bridgeland moduli spaces of semistable objects of $(-1)$-classes and $(-4)$-classes in the Kuznetsov components on index one prime Fano threefold $X_{4d+2}$ of degree $4d+2$ and index two prime Fano threefold $Y_d$ of degree $d$ for $d=3,4,5$. For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of $(-1)$-classes on $X_{4d+2}$ and $Y_d$ are isomorphic. We show that moduli spaces of stable objects of $(-1)$-classes on $X_{14}$ are realized by Fano surface $\mathcal{C}(X)$ of conics, moduli spaces of semistable sheaves $M_X(2,1,6)$ and $M_X(2,-1,6)$ and the correspondent moduli spaces on cubic threefold $Y_3$ are realized by moduli spaces of stable vector bundles $M^b_Y(2,1,2)$ and $M^b_Y(2,-1,2)$. We show that moduli spaces of semistable objects of $(-4)$-classes on $Y_{d}$ are isomorphic to the moduli spaces of instanton sheaves $M^{inst}_Y$ when $d\neq 1,2$, and show that there're open immersions of $M^{inst}_Y$ into moduli spaces of semistable objects of $(-4)$-classes when $d=1,2$. Finally, when $d=3,4,5$ we show that these moduli spaces are all isomorphic to $M^{ss}_X(2,0,4)$.

math.AG↗

arXiv2020

Specification mining and automated task planning for autonomous robots based on a graph-based spatial temporal logic

We aim to enable an autonomous robot to learn new skills from demo videos and use these newly learned skills to accomplish non-trivial high-level tasks. The goal of developing such autonomous robot involves knowledge representation, specification mining, and automated task planning. For knowledge representation, we use a graph-based spatial temporal logic (GSTL) to capture spatial and temporal information of related skills demonstrated by demo videos. We design a specification mining algorithm to generate a set of parametric GSTL formulas from demo videos by inductively constructing spatial terms and temporal formulas. The resulting parametric GSTL formulas from specification mining serve as a domain theory, which is used in automated task planning for autonomous robots. We propose an automatic task planning based on GSTL where a proposer is used to generate ordered actions, and a verifier is used to generate executable task plans. A table setting example is used throughout the paper to illustrate the main ideas.

cs.AI↗

arXiv2020

A graph-based spatial temporal logic for knowledge representation and automated reasoning in cognitive robots

We propose a new graph-based spatial temporal logic for knowledge representation and automated reasoning in this paper. The proposed logic achieves a balance between expressiveness and tractability in applications such as cognitive robots. The satisfiability of the proposed logic is decidable. We apply a Hilbert style axiomatization for the proposed graph-based spatial temporal logic, in which Modus ponens and IRR are the inference rules. We show that the corresponding deduction system is sound and complete and can be implemented through SAT.

cs.LO↗

arXiv2019

Respiratory Motion Correction in Abdominal MRI using a Densely Connected U-Net with GAN-guided Training

Abdominal magnetic resonance imaging (MRI) provides a straightforward way of characterizing tissue and locating lesions of patients as in standard diagnosis. However, abdominal MRI often suffers from respiratory motion artifacts, which leads to blurring and ghosting that significantly deteriorate the imaging quality. Conventional methods to reduce or eliminate these motion artifacts include breath holding, patient sedation, respiratory gating, and image post-processing, but these strategies inevitably involve extra scanning time and patient discomfort. In this paper, we propose a novel deep-learning-based model to recover MR images from respiratory motion artifacts. The proposed model comprises a densely connected U-net with generative adversarial network (GAN)-guided training and a perceptual loss function. We validate the model using a diverse collection of MRI data that are adversely affected by both synthetic and authentic respiration artifacts. Effective outcomes of motion removal are demonstrated. Our experimental results show the great potential of utilizing deep-learning-based methods in respiratory motion correction for abdominal MRI.

eess.IV↗

arXiv2018

Distributed Communication-aware Motion Planning for Networked Mobile Robots under Formal Specifications

Control and communication are often tightly coupled in motion planning of networked mobile robots, due to the fact that robotic motions will affect the overall communication quality, and the quality of service (QoS) of the communication among the robots will in turn affect their coordination performance. In this paper, we propose a control theoretical motion planning framework for a team of networked mobile robots in order to accomplish high-level spatial and temporal motion objectives while optimizing communication QoS. Desired motion specifications are formulated as Signal Temporal Logic (STL), whereas the communication performances to be optimized are captured by recently proposed Spatial Temporal Reach and Escape Logic (STREL) formulas. Both the STL and STREL specifications are encoded as mixed integer linear constraints posed on the system and/or environment state variables of the mobile robot network, where satisfactory control strategies can be computed by exploiting a distributed model predictive control (MPC) approach. To the best of the authors' knowledge, we are the first to study controller synthesis for STREL specifications. A two-layer hierarchical MPC procedure is proposed to efficiently solve the problem, whose soundness and completeness are formally ensured. The effectiveness of the proposed framework is validated by simulation examples.

eess.SY↗

arXiv2018

Linearizable Iterators for Concurrent Sets

This paper proposes a general framework for adding linearizable iterators to a class of data structures that implement set operations. We introduce a condition on set operations, called local consistency, which informally states that set operations never make elements unreachable to a sequential iterator's traversal. We show that sets with locally consistent operations can be augmented with a linearizable iterator via the framework. Our technique is broadly applicable to a variety of data structures, including hash tables and binary search trees. We apply the technique to sets taken from existing literature, prove their operations are locally consistent, and demonstrate that iterators do not significantly affect the performance of concurrent set operations.

cs.DS↗

arXiv2018

Parameter and Insertion Function Co-synthesis for Opacity Enhancement in Parametric Stochastic Discrete Event Systems

Opacity is a property that characterizes the system's capability to keep its "secret" from being inferred by an intruder that partially observes the system's behavior. In this paper, we are concerned with enhancing the opacity using insertion functions, while at the same time, enforcing the task specification in a parametric stochastic discrete event system. We first obtain the parametric Markov decision process that encodes all the possible insertions. Based on which, we convert this parameter and insertion function co-synthesis problem into a nonlinear program. We prove that if the output of this program satisfies all the constraints, it will be a valid solution to our problem. Therefore, the security and the capability of enforcing the task specification can be simultaneously guaranteed.

cs.CR↗