arXiv subjects

Chencheng Zhang

Publications and source records attributed to Chencheng Zhang.

15 recordsLinked to original sources

arXiv2026

Gabriel--Zisman Localizations, Products, Coproducts, and Product Categories

This paper studies (co)products and additive structures under Gabriel--Zisman localization $Q:\mathcal A\longrightarrow\mathcal A[S^{-1}]$. Examples show that localization may destroy existing (co)products or fail to preserve (co)products that remain, even when $Q$ is additive. For a set-indexed family $(\mathcal A_i,S_i)_{i\in I}$, necessary and sufficient conditions are given for the canonical functor $\big(\prod_{i\in I}\mathcal A_i\big)\big[\big(\prod_{i\in I}S_i\big)^{-1}\big]\longrightarrow\prod_{i\in I}\mathcal A_i[S_i^{-1}]$ to be full or faithful. These criteria yield compatibility with $I$-indexed (co)products for localizations admitting a calculus of fractions or arising from model categories, provided the relevant (co)products exist and the designated classes of morphisms are closed under them.

math.CT↗

arXiv2026

Exceptional model structures and the induced extriangulated categories

A Hovey triple $(\mathcal{C}, \mathcal{F}, \mathcal{W})$ is {\it exceptional}, if $(\mathcal{C} \cap \mathcal{F}, \ \mathcal{C} \cap \mathcal{F} \cap \mathcal{W})$ is not a Frobenius pair. One has a disjoint union $\{\text{Hovey triple}\} = \{\text{Hereditary Hovey triple}\} \ \dot\bigcup \ \{\text{Exceptional Hovey triple}\}$ \ $\dot\bigcup \ \{\text{non-hereditary and non-exceptional Hovey triple}\}.$ Exceptional Hovey triples appear widely. For a selfinjective Nakayama algebra $A= kC_n/J^t$, $A\mbox{-}{\rm mod}$ admits exceptional Hovey triples if and only if $\gcd (n, t) \ge 2$ and $t \geq 3$. Their homotopy categories reveal new phenomena. Nakaoka-Palu's Theorem implies that there is a triangulation on $\frac{\mathcal C\cap\mathcal F}{\mathcal C\cap\mathcal F\cap\mathcal W}.$ It is proved that a Hovey triple in a weakly idempotent complete extriangulated category $(\mathcal A, \mathbb E, \mathfrak s)$ is exceptional if and only if the induced extriangulated structure $\bigl(\frac{\mathcal C\cap\mathcal F}{\mathcal C\cap\mathcal F\cap\mathcal W}, \ \overline{\mathbb E}, \ \overline{\mathfrak s}\bigr)$ is not {\it canonically triangulated}. Between this extriangulated structure and the one arising from the triangulation, the intermediate extriangulated structures are in one-to-one correspondence with Serre subcategories of $\mathrm{fp}((\frac{\mathcal P(\mathcal C\cap\mathcal F)}{\mathcal C\cap\mathcal F\cap\mathcal W})^{\mathrm{op}}, \mathrm{Ab})$.

math.RT↗

arXiv2026

Two-sided coherent algebras over any field with $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$

For a ring $R$, let $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, and $\mathcal{PGF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We answer negatively the question whether $\mathcal{GP}(R)=\mathcal{PGF}(R)$ for every ring. More precisely, over every field $k$ we construct a left and right coherent central $k$-algebra $T$ and a strongly Gorenstein projective left $T$-module which is not Gorenstein flat; hence $\mathcal{PGF}(T)\subsetneq\mathcal{GP}(T)$.

math.RA↗

arXiv2026

A strongly compact cardinal yields a left and right coherent ring with $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$

For a ring $R$, let $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, and $\mathcal{PGF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We isolate the local ultrafilter hypothesis $\textsf{LUH}$: the existence of a strongly compact cardinal implies $\textsf{LUH}$, while $\textsf{LUH}$ implies the existence of a measurable cardinal. Assuming $\textsf{LUH}$, we construct a left and right coherent ring $R$ and a strongly Gorenstein projective left $R$-module $G$ which is not Gorenstein flat; hence $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$.

math.RA↗

arXiv2026

A pre-triangulated category which is not triangulated

In this article, we construct an explicit pre-triangulated category which is not a triangulated category. Its underlying additive category is the category of finitely generated projective modules of the type-$A_5$ preprojective algebra over $\mathbb F_2$, and the suspension is induced by the graph-reflection automorphism.

math.CT↗

arXiv2026

Rigidity of the multi-bubble solutions to the energy critical wave equation in dimension five

We study the asymptotic dynamics of multi-bubble solutions to the focusing energy-critical wave equation in five dimensions. Assuming that the solution asymptotically decomposes into a finite superposition of spatially separated bubbles with comparable scales, we prove a rigidity result that describes the precise long-time behavior of these scales. More precisely, we show that all scaling parameters are necessarily of order $t^{-2}$, and that the corresponding renormalized modulation vector converges to a connected component of a finite-dimensional algebraic set determined by the limiting spatial configuration of the bubbles. This algebraic system encodes the strong interactions between the polynomial tails of the bubbles and governs the effective asymptotic dynamics of the multi-bubble regime.

math.AP↗

arXiv2026

Homotopic morphisms and diagram theorems in extriangulated categories

Homotopic morphisms of $\mathbb E$-triangles in extriangulated categories are introduced. Any morphism of $\mathbb E$-triangles is a composition of homotopic morphisms. Any morphism $(α_1, α_2, α_3)$ of $\mathbb E$-triangles can be modified to be homotopic, by changing one of $α_i$; moreover, all the 15 cases where $α_i$ is an $\mathbb E$-inflation ($\mathbb E$-deflation) are analyzed. Some diagram theorems, especially $4\times 4$ Lemma and its $14$ variants, including $3\times 3$ diagram and Horseshoe Lemma, are investigated. A relation between homotopic morphisms and (middling) good morphisms in triangulated categories are given. Weakly idempotent complete extriangulated categories are characterized.

math.CT↗

arXiv2026

Stability and Robustness of Tensor-Coupled Flow-Conservation Dynamical Systems on Hypergraphs

This paper develops an entropy-based stability and robustness framework for nonlinear hypergraph dynamics with conservation and flow balance. We consider generator-form systems on the simplex whose state-dependent transition rates capture higher-order (tensor) interactions among nodes. Under a tensor generalized detailed-balance (TGDB) condition, we show that the system admits a unique equilibrium and an entropy Lyapunov function ensuring global asymptotic stability. The Jacobian restricted to the tangent subspace of the simplex is Hurwitz, and its spectral gap determines the exponential convergence rate. Building on this structure, we derive first-order sensitivity bounds of the equilibrium under perturbations of the coupling tensor and establish a local input-to-state stability (ISS) estimate with respect to external inputs. The results reveal a quantitative link between the spectral gap and the system's robustness margin: larger spectral gaps imply smaller equilibrium shifts and faster recovery under structural or parametric perturbations. Numerical experiments on tensor-coupled flow models confirm the theoretical predictions and illustrate how the proposed entropy-dissipating framework unifies stability and robustness analysis for conservative higher-order network systems.

eess.SY↗

arXiv2026

Predicting Neuromodulation Outcome for Parkinson's Disease with Generative Virtual Brain Model

Parkinson's disease (PD) affects over ten million people worldwide. Although temporal interference (TI) and deep brain stimulation (DBS) are promising therapies, inter-individual variability limits empirical treatment selection, increasing non-negligible surgical risk and cost. Previous explorations either resort to limited statistical biomarkers that are insufficient to characterize variability, or employ AI-driven methods which is prone to overfitting and opacity. We bridge this gap with a pretraining-finetuning framework to predict outcomes directly from resting-state fMRI. Critically, a generative virtual brain foundation model, pretrained on a collective dataset (2707 subjects, 5621 sessions) to capture universal disorder patterns, was finetuned on PD cohorts receiving TI (n=51) or DBS (n=55) to yield individualized virtual brains with high fidelity to empirical functional connectivity (r=0.935). By constructing counterfactual estimations between pathological and healthy neural states within these personalized models, we predicted clinical responses (TI: AUPR=0.853; DBS: AUPR=0.915), substantially outperforming baselines. External and prospective validations (n=14, n=11) highlight the feasibility of clinical translation. Moreover, our framework provides state-dependent regional patterns linked to response, offering hypothesis-generating mechanistic insights.

q-bio.NC↗

arXiv2025

Global and local observability of hypergraphs

This paper studies observability for non-uniform hypergraphs with inputs and outputs. To capture higher-order interactions, we define a canonical non-homogeneous dynamical system with nonlinear outputs on hypergraphs. We then construct algebraic necessary and sufficient conditions based on polynomial ideals and varieties for global observability at an initial state of hypergraphs. An example is given to illustrate the proposed criteria for observability. Further, necessary and sufficient conditions for local observability are derived based on rank conditions of observability matrices, which provide a framework to study local observability for non-uniform hypergraphs. Finally, the similarity of observability for hypergraphs is proposed using similarity of tensors, which reveals the relation of observability between two hypergraphs and helps to check the observability intuitively.

eess.SY↗

arXiv2025

Higher-order Laplacian dynamics on hypergraphs with cooperative and antagonistic interactions

Laplacian dynamics on a signless graph characterize a class of linear interactions, where pairwise cooperative interactions between all agents lead to the convergence to a common state. On a structurally balanced signed graph, the agents converge to values of the same magnitude but opposite signs (bipartite consensus), as illustrated by the well-known Altafini model. These interactions have been modeled using traditional graphs, where the relationships between agents are always pairwise. In comparison, higher-order networks (such as hypergraphs), offer the possibility to capture more complex, group-wise interactions among agents. This raises a natural question: can collective behavior be analyzed by using hypergraphs? The answer is affirmative. In this paper, higher-order Laplacian dynamics on signless hypergraphs are first introduced and various collective convergence behaviors are investigated, in the framework of homogeneous and non-homogeneous polynomial systems. Furthermore, by employing gauge transformations and leveraging tensor similarities, we extend these dynamics to signed hypergraphs, drawing parallels to the Altafini model. Moreover, we explore non-polynomial interaction functions within this framework. The theoretical results are demonstrated through several numerical examples.

eess.SY↗

arXiv2024

On melting for the 3D radial Stefan problem

We consider the three-dimensional radial Stefan problem which describes the evolution of a radial symmetric ice ball with free boundary \begin{equation*} \left\{\begin{aligned} &\partial_{t}u-\partial_{rr}u-\frac{2}{r}\partial_{r}u=0 \quad in\ r\geqλ(t),\\ &\partial_{r}u(t,λ(t))=-\dotλ(t),\\ &u(t,λ(t))=0,\\ &u(0,\cdot)=u_{0},\quad λ(0)=λ_{0}. \end{aligned}\right. \end{equation*} We prove the existence in the radial class of finite time melting with rates \begin{equation*} λ(t)=\left\{\begin{aligned} &4\sqrtπ\frac{\sqrt{T-t}}{|\log (T-t)|}(1+o_{t\rightarrow T}(1)),\\ &c(u_{0},k)(1+o_{t\rightarrow T}(1))(T-t)^{\frac{k+1}{2}},\quad k\in{\mathbb{N}}^{*}, \end{aligned}\right. \end{equation*} which respectively correspond to the fundamental stable melting rate and a sequence of codimension $k$ unstable rates. Our analysis mainly depend on the methods developed in [17] which deals with the similar problems in two dimensions and also the construction of both stable and unstable finite time blow-up solutions for the harmonic heat flow in [49],[50].

math.AP↗

arXiv2023

Flocking control against the malicious agent

This paper investigates the flocking control of a swarm with a malicious agent that falsifies its controller parameters to cause collision, division, and escape of agents in the swarm. A novel geometric flocking condition is established by designing the configuration of the malicious agent and its neighbors, under which we propose a hierarchal geometric configuration-based flocking control method. To help detect the malicious agent, a parameter estimate mechanism is also provided. The proposed method can achieve the flocking control goal and meanwhile contain the malicious agent in the swarm without removing it. Experimental result shows the effectiveness of the theoretical result.

eess.SY↗

arXiv2022

A Long Short-term Memory Based Recurrent Neural Network for Interventional MRI Reconstruction

Interventional magnetic resonance imaging (i-MRI) for surgical guidance could help visualize the interventional process such as deep brain stimulation (DBS), improving the surgery performance and patient outcome. Different from retrospective reconstruction in conventional dynamic imaging, i-MRI for DBS has to acquire and reconstruct the interventional images sequentially online. Here we proposed a convolutional long short-term memory (Conv-LSTM) based recurrent neural network (RNN), or ConvLR, to reconstruct interventional images with golden-angle radial sampling. By using an initializer and Conv-LSTM blocks, the priors from the pre-operative reference image and intra-operative frames were exploited for reconstructing the current frame. Data consistency for radial sampling was implemented by a soft-projection method. To improve the reconstruction accuracy, an adversarial learning strategy was adopted. A set of interventional images based on the pre-operative and post-operative MR images were simulated for algorithm validation. Results showed with only 10 radial spokes, ConvLR provided the best performance compared with state-of-the-art methods, giving an acceleration up to 40 folds. The proposed algorithm has the potential to achieve real-time i-MRI for DBS and can be used for general purpose MR-guided intervention.

cs.CV↗

arXiv2020

Predicting the Impact of Electric Field Stimulation in a Detailed Computational Model of Cortical Tissue

Neurostimulation using weak electric fields has generated excitement in recent years due to its potential as a medical intervention. However, study of this stimulation modality has been hampered by inconsistent results and large variability within and between studies. In order to begin addressing this variability we need to properly characterise the impact of the current on the underlying neuron populations. To develop and test a computational model capable of capturing the impact of electric field stimulation on networks of neurons. We construct a cortical tissue model with distinct layers and explicit neuron morphologies. We then apply a model of electrical stimulation and carry out multiple test case simulations. The cortical slice model is compared to experimental literature and shown to capture the main features of the electrophysiological response to stimulation. Namely, the model showed 1) a similar level of depolarisation in individual pyramidal neurons, 2) acceleration of intrinsic oscillations, and 3) retention of the spatial profile of oscillations in different layers. We then apply alternative electric fields to demonstrate how the model can capture differences in neuronal responses to the electric field. We demonstrate that the tissue response is dependent on layer depth, the angle of the apical dendrite relative to the field, and stimulation strength. We present publicly available computational modelling software that predicts the neuron network population response to electric field stimulation.

q-bio.NC↗