arXiv subjects

Shijie Zhu

Publications and source records attributed to Shijie Zhu.

18 recordsLinked to original sources

arXiv2026

Accessing both electrochemical SEIRA and SERS with ultrasensitive metamaterials for enhanced molecular identification

Surface-enhanced IR absorption (SEIRA) and surface-enhanced Raman spectroscopy (SERS) are complementary techniques that allow for ultrasensitive chemical fingerprinting. Non-invasive optical sensing would be significantly improved by a robust implementation of a reusable substrate that combines these techniques. Here, we present an electrochemically-cleanable metamaterial that enables combined real-time SEIRA and SERS in flow. This metamaterial facilitates the study of surface-adsorbed species and diffusion layers, elicits spectral shifts from changes in nanogap refractive index of 1400 nm/RIU, and delivers ultrasensitive analyte detection. Combining SERS and SEIRA clarifies molecular (electro)chemical transformations and tracks changes in selection rules and symmetry breaking at the analyte-electrode interface. This development in enhanced multimodal spectro-electrochemistry is suited for multiple domains, including understanding charge transport mechanisms and interfacial dynamics at electrodes, and is capable of real-time flow monitoring for a wide range of molecular processes.

physics.optics↗

arXiv2025

Weakly Gorensteinness of tensor algebras and Morita algebras

An algebra $A$ is left weakly Gorenstein if any semi-Gorenstein-projective left $A$-modules is Gorenstein-projective. The weakly Gorensteinness of two kinds of algebras are answered. Using the method of the monomorphism category, it is proved that the tensor algebra $A\otimes B$ with ${\rm gl.dim} B< \infty$ is left weakly Gorenstein if and only if so is $A$. For a class of Morita algebras $Λ=\begin{pmatrix}\begin{smallmatrix} A & N \\ M & B \\ \end{smallmatrix}\end{pmatrix}$, the (semi-)Gorenstein-projective left $Λ$-modules are computed and described; and then it is proved that $Λ$ is left weakly Gorenstein if and only if so are $A$ and $B$. As an application, the upper triangular matrix algebra $T_n(A)$ is left weakly Gorenstein if and only if so is $A$.

math.RT↗

arXiv2025

Auslander-Reiten translations in the monomorphism categories of exact categories

Let $Λ$ be a finite dimensional algebra. Let $\mathcal C$ be a functorially finite exact subcategory of $Λ$-mod with enough projective and injective objects and $\mathcal S (\mathcal C)$ be its monomorphism category. It turns out that the category $\mathcal S (\mathcal C)$ has almost split sequences. We show an explicit formula for the Auslander-Reiten translation in $\mathcal S (\mathcal C)$. Furthermore, if $\mathcal C$ is a stably $d$-Calabi-Yau Frobenius category, we calculate objects under powers of Auslander-Reiten translation in the triangulated category $\overline{\mathcal S(\mathcal C)}$.

math.RT↗

arXiv2025

Convergence Theory of Flexible ALADIN for Distributed Optimization

The Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method is a cutting-edge distributed optimization algorithm known for its superior numerical performance. It relies on each agent transmitting information to a central coordinator for data exchange. However, in practical network optimization and federated learning, unreliable information transmission often leads to packet loss, posing challenges for the convergence analysis of ALADIN. To address this issue, this paper proposes Flexible ALADIN, a random polling variant of ALADIN, and presents a rigorous convergence analysis, including global convergence for convex problems and local convergence for non-convex problems.

eess.SY↗

arXiv2025

Gradient Deconfliction via Orthogonal Projections onto Subspaces For Multi-task Learning

Although multi-task learning (MTL) has been a preferred approach and successfully applied in many real-world scenarios, MTL models are not guaranteed to outperform single-task models on all tasks mainly due to the negative effects of conflicting gradients among the tasks. In this paper, we fully examine the influence of conflicting gradients and further emphasize the importance and advantages of achieving non-conflicting gradients which allows simple but effective trade-off strategies among the tasks with stable performance. Based on our findings, we propose the Gradient Deconfliction via Orthogonal Projections onto Subspaces (GradOPS) spanned by other task-specific gradients. Our method not only solves all conflicts among the tasks, but can also effectively search for diverse solutions towards different trade-off preferences among the tasks. Theoretical analysis on convergence is provided, and performance of our algorithm is fully testified on multiple benchmarks in various domains. Results demonstrate that our method can effectively find multiple state-of-the-art solutions with different trade-off strategies among the tasks on multiple datasets.

cs.LG↗

arXiv2024

Defect Invariant Nakayama Algebras

We show that for a given Nakayama algebra $Θ$, there exist countably many cyclic Nakayama algebras $Λ_i$, where $i \in \mathbb{N}$, such that the syzygy filtered algebra of $Λ_i$ is isomorphic to $Θ$ and we describe those algebras $Λ_i$. We show, among these algebras, there exists a unique algebra $Λ$ where the defects, representing the number of indecomposable injective but not projective modules, remain invariant for both $Θ$ and $Λ$. As an application, we achieve the classification of cyclic Nakayama algebras that are minimal Auslander-Gorenstein and dominant Auslander-regular algebras of global dimension three. Specifically, by using the Auslander-Iyama correspondence, we obtain cluster-tilting objects for certain Nakayama algebras. Additionally, we introduce cosyzygy filtered algebras and show that it is dual of syzygy filtered algebra.

math.RT↗

arXiv2023

Optimal Demand Shut-offs of AC Microgrid using AO-SBQP Method

Microgrids are increasingly being utilized to improve the resilience and operational flexibility of power grids, and act as a backup power source during grid outages. However, it necessitates that the microgrid itself could provide power to the critical loads. This paper presents an algorithm named alternating optimization based sequential boolean quadratic programming tailored for solving optimal demand shut-offs problems arising in microgrids. Moreover, we establish local superlinear convergence of the proposed approximate Boolean quadratic programming method over nonconvex problems. In the end, the performance of the proposed method is illustrated on the modified IEEE 30-bus case study.

math.OC↗

arXiv2022

Separated monic correspondence of cotorsion pairs and semi-Gorenstein-projective modules

Given a finite dimensional algebra $A$ over a field $k$, and a finite acyclic quiver $Q$, let $Λ= A\otimes_k kQ/I$, where $kQ$ is the path algebra of $Q$ over $k$ and $I$ is a monomial ideal. We show that $(\mathcal X,\mathcal Y)$ is a (complete) hereditary cotorsion pair in $A$-mod if and only if $({\rm smon}(Q,I,\mathcal X), {\rm rep}(Q,I,\mathcal Y))$ is a (complete) hereditary cotorsion pair in $Λ$-mod. We also show that $A$ is left weakly Gorenstein if and only if so is $Λ$. Provided that $kQ/I$ is non-semisimple, the category $^{\perp}Λ$ of semi-Gorenstein-projective $Λ$-modules coincides with the category of separated monic representations ${\rm smon}(Q,I,^{\perp}A)$ if and only if $A$ is left weakly Gorenstein.

math.RT↗

arXiv2022

Pointed Hopf Algebras of Discrete Corepresentation Type

We classify pointed Hopf algebras of discrete corepresentation type over an algebraically closed field K with characteristic zero. For such algebras $H$, we explicitly determine the algebra structure up to isomorphism for the link indecomposable component $B$ containing the unit. It turns out that $H$ is a crossed product of $B$ and a certain group algebra.

math.RT↗

arXiv2022

Alternating Direction Based Sequential Boolean Quadratic Programming Method for Transmit Antenna Selection

The wireless mobile communication system is updated and iterated on the whole almost every decade. It is now in the development period of the application scenarios of the fifth generation mobile communication system (5G). Unfortunately, 5G relies on plenty of small base stations with a large number of antennas that consume a lot of energy. In this paper, a novel Boolean variable quadratic programming algorithm is designed for the antenna selection optimization problem to reduce power consumption. Experiments show that the proposed algorithm achieves high complementarity satisfaction accuracy with only a few steps.

eess.SY↗

arXiv2022

Continuous Nakayama Representations

We introduce continuous analogues of Nakayama algebras. In particular, we introduce the notion of (pre-)Kupisch functions, which play a role as Kupisch series of Nakayama algebras, and view continuous Nakayama representations as a special type of representation of $\mathbb{R}$ or $\mathbb{S}^1$. We investigate equivalences and connectedness of the categories of Nakayama representations. Specifically, we prove that orientation-preserving homeomorphisms on $\mathbb{R}$ and on $\mathbb{S}^1$ induce equivalences between these categories. Connectedness is characterized by a special type of points called separation points determined by (pre-)Kupisch functions. We also construct an exact embedding from the category of finite-dimensional representations for any finite-dimensional Nakayama algebra, to a category of continuous Nakayama representaitons.

math.RT↗

arXiv2021

Adversarial Directed Graph Embedding

Node representation learning for directed graphs is critically important to facilitate many graph mining tasks. To capture the directed edges between nodes, existing methods mostly learn two embedding vectors for each node, source vector and target vector. However, these methods learn the source and target vectors separately. For the node with very low indegree or outdegree, the corresponding target vector or source vector cannot be effectively learned. In this paper, we propose a novel Directed Graph embedding framework based on Generative Adversarial Network, called DGGAN. The main idea is to use adversarial mechanisms to deploy a discriminator and two generators that jointly learn each node's source and target vectors. For a given node, the two generators are trained to generate its fake target and source neighbor nodes from the same underlying distribution, and the discriminator aims to distinguish whether a neighbor node is real or fake. The two generators are formulated into a unified framework and could mutually reinforce each other to learn more robust source and target vectors. Extensive experiments show that DGGAN consistently and significantly outperforms existing state-of-the-art methods across multiple graph mining tasks on directed graphs.

cs.SI↗

arXiv2021

A Robust and Generalized Framework for Adversarial Graph Embedding

Graph embedding is essential for graph mining tasks. With the prevalence of graph data in real-world applications, many methods have been proposed in recent years to learn high-quality graph embedding vectors various types of graphs. However, most existing methods usually randomly select the negative samples from the original graph to enhance the training data without considering the noise. In addition, most of these methods only focus on the explicit graph structures and cannot fully capture complex semantics of edges such as various relationships or asymmetry. In order to address these issues, we propose a robust and generalized framework for adversarial graph embedding based on generative adversarial networks. Inspired by generative adversarial network, we propose a robust and generalized framework for adversarial graph embedding, named AGE. AGE generates the fake neighbor nodes as the enhanced negative samples from the implicit distribution, and enables the discriminator and generator to jointly learn each node's robust and generalized representation. Based on this framework, we propose three models to handle three types of graph data and derive the corresponding optimization algorithms, i.e., UG-AGE and DG-AGE for undirected and directed homogeneous graphs, respectively, and HIN-AGE for heterogeneous information networks. Extensive experiments show that our methods consistently and significantly outperform existing state-of-the-art methods across multiple graph mining tasks, including link prediction, node classification, and graph reconstruction.

cs.LG↗

arXiv2020

Dynamical Combinatorics and Torsion Classes

For finite semidistributive lattices the map $κ$ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the $κ$-map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the $κ$-map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend $κ$ to a map on torsion classes which have canonical join representations given by the special torsion classes associated to the minimal extending modules introduced by the first and third authors and A. Carroll. For hereditary algebras, we show that the extended $κ$-map on torsion classes is essentially the same as Ringel's $ε$-map on wide subcategories. Also in hereditary case, we relate the square of $κ$ to the Auslander-Reiten translation.

math.RT↗

arXiv2017

Minimal inclusions of torsion classes

Let $Λ$ be a finite-dimensional associative algebra. The torsion classes of $mod\, Λ$ form a lattice under containment, denoted by $tors\, Λ$. In this paper, we characterize the cover relations in $tors\, Λ$ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection with bricks. Second, we characterize faces of the canonical join complex of $tors\, Λ$ in terms of representation theory. Finally, we show that, in general, the algebra $Λ$ is not characterized by its lattice $tors\, Λ$. In particular, we study the torsion theory of a quotient of the preprojective algebra of type $A_n$. We show that its torsion class lattice is isomorphic to the weak order on $A_n$.

math.RT↗

arXiv2017

Functors and morphisms determined by subcategories

We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a Hom-finite hereditary abelian category with enough projectives, we prove that the Auslander-Reiten-Smalø-Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers.

math.RT↗

arXiv2017

Dominant dimension and tilting modules

We study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules. Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension $2$, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least $2$, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting-cotilting module generated-cogenerated by projective-injective modules are precisely $1$-Auslander-Gorenstein algebras. When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

math.RT↗

arXiv2017

Preprojective algebras of tree-type quivers

Let $Q$ be a tree-type quiver, $\mathbf{k} Q$ its path algebra, and $λ$ a nonzero element in the field $\mathbf{k}$. We construct irreducible morphisms in the Auslander-Reiten quiver of the transjective component of the bounded derived category of $\mathbf{k} Q$ that satisfy what we call the $λ$-relations. When $λ=1$, the relations are known as mesh relations. When $λ=-1$, they are known as commutativity relations. Using this technique together with the results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, we show that for any tree-type quiver, several descriptions of its preprojective algebra are equivalent.

math.RA↗