arXiv subjects
Publications and source records attributed to Qikai Wang.
We give a counterexample to completeness descent from ideal cotorsion pairs to their objects. A radical-square-zero algebra on the two-cycle gives a finite-dimensional example. The construction is intrinsically non-weakly-idempotent-complete.
Skill selection is a key stage in LLM-agent workflows, determining which installed skill should handle a user request. Existing attacks on this stage primarily rely on explicit prompt injection or instruction-level steering, which can expose recognizable manipulation signals. In this work, we identify a new implicit attack surface for skill selection: even when the user prompt and skill description appear benign in isolation, their semantic relationship can still be strategically shaped to favor an attacker-chosen skill. Based on this observation, we present Implicit Skill-Selection Manipulation via Semantic Matching (ISM), which jointly shapes target-skill metadata and reusable prompts to manipulate skill selection without explicit selection instructions. Specifically, we develop a three-stage strategy to broaden semantic coverage, strengthen target distinctiveness, and preserve natural prompt wording. Across four task domains and eight selector models, ISM increases the average target-selection rate (TSR) from 15.2% to 63.5%. In a matched comparison, ISM achieves a 73.5% TSR, only 9.8 percentage points below Explicit Steering. Human reviewers block ISM in only 2.9% of judgments, versus 91.4% for Explicit Steering, while five LLM-based inspectors pass ISM at an average rate of 82.9%, versus 37.4% for Explicit Steering. Moreover, ISM remains effective against PPL-W, Llama Prompt Guard 2, and PIGuard.
Let $(\mathcal{T}',\mathcal{T},\mathcal{T}'')$ be a recollement of triangulated categories.A complete ideal cotorsion pair in $\mathcal{T}$ induces complete ideal cotorsion pairs in $\mathcal{T}'$ and $\mathcal{T}''$. In addition, if $(\mathcal{I}, \mathcal{I}^\perp )$ and $(\mathcal{J},\mathcal{J}^\perp)$ are two complete ideal cotorsion pairs in a triangulated category, then $(\mathcal{I}\cap\mathcal{J}, \langle\mathcal{I}^\perp,\mathcal{J}^\perp\rangle)$ is also a complete ideal cotorsion pair. By this method, starting from two complete ideal cotorsion pairs in $\mathcal{T}'$ and $\mathcal{T}''$, one can induce a family of complete ideal cotorsion pairs in $\mathcal{T}$.
Given two (hereditary) complete cotorsion pairs $(\mathcal{X}_1,\mathcal{Y}_1)$ and $(\mathcal{X}_2,\mathcal{Y}_2)$ in an exact category with $\mathcal{X}_1\subseteq \mathcal{Y}_2$, we prove that $\left({\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle,\mathcal{Y}_1\cap \mathcal{Y}_2\right)$ is also a (hereditary) complete cotorsion pair, where ${\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle$ is the class of direct summands of extension of $\mathcal{X}_1$ and $\mathcal{X}_2$. As an application, we construct complete cotorsion pairs, such as $(^\perp\mathcal{GI}^{\leqslant n},\mathcal{GI}^{\leqslant n})$, where $\mathcal{GI}^{\leqslant n}$ is the class of modules of Gorenstein injective dimension at most $n$. And we also characterize the left orthogonal class of exact complexes of injective modules and the classes of modules with finite Gorenstein projective, Gorenstein flat, and PGF dimensions.
This paper investigates the conditions under which the centralizer algebra $S_n(c,R)$ of a matrix $ c\in M_n(R)$ is a (separable) Frobenius extension of the base algebra $R$. For an algebra $R$ over an integral domain $\mathbb{k}$, we provide necessary and sufficient conditions for $S_n(c,R)/R$ to be a (separable) Frobenius extension when $c$ is in Jordan canonical form with eigenvalues in $\mathbb{k}$. We extend this analysis to arbitrary matrices over a field and derive conditions for matrix diagonalizability through Frobenius extensions.
Let $\mathcal{A}$ be a Frobenius category and $ω$ the full subcategory consisting of projective objects. The relations between special precovering (resp., precovering) ideals in $\mathcal{A}$ and special precovering (resp., preenveloping) ideals in the stable category $\mathcal{A}/ω$ are explored. In combination with a result due to Breaz and Modoi, we conclude that every precovering or preenveloping ideal $\mathcal{I}$ in $\mathcal{A}$ with $1_{X}\in{\mathcal{I}}$ for any $X\inω$ is special. As a consequence, it is proved that an ideal cotorsion pair $(\mathcal{I},\mathcal{J})$ in $\mathcal{A}$ is complete if and only if $\mathcal{I}$ is precovering if and only if $\mathcal{J}$ is preenveloping. This leads to an ideal version of the Bongartz-Eklof-Trlifaj Lemma in $\mathcal{A}/ω$, which states that an ideal cotorsion pair in $\mathcal{A}/ω$ generated by a set of morphisms is complete. As another consequence, we provide some partial answers to the question about the completeness of cotorsion pairs posed by Fu, Guil Asensio, Herzog and Torrecillas.
Let $\mathcal{I}$ and $\mathcal{J}$ be object ideals in an exact category $(\mathcal{A}; \mathcal{E})$. It is proved that $(\mathcal{I},\mathcal{J})$ is a perfect ideal cotorsion pair if and only if $({\rm Ob}(\mathcal{I}),{\rm Ob}(\mathcal{J}))$ is a perfect cotorsion pair, where ${\rm Ob}(\mathcal{I})$ and ${\rm Ob}(\mathcal{J})$ is the objects of $\mathcal{I}$ and $\mathcal{J}$, respectively. If in addition $(\mathcal{A}; \mathcal{E})$ has enough projective objects and injective objects, and $\mathcal{J}$ is enveloping, then $(\mathcal{I},\mathcal{J})$ is a complete ideal cotorsion pair if and only if $({\rm Ob}(\mathcal{I}),{\rm Ob}(\mathcal{J}))$ is a complete cotorsion pair. This gives a partial answer to the question posed by Fu, Guil Asensio, Herzog and Torrecillas. Moreover, for any object ideal $\mathcal{I}$ in the category of left $R$-modules, it is proved that $\mathcal{I}$ satisfies Enochs Conjecture if and only if ${\rm Ob}(\mathcal{I})$ satisfies Enochs Conjecture. Applications are given to projective morphisms and ideal cotorsion pairs $(\mathcal{I},\mathcal{J})$ of object ideals under certain conditions.
Semi-supervised learning can significantly boost model performance by leveraging unlabeled data, particularly when labeled data is scarce. However, real-world unlabeled data often contain unseen-class samples, which can hinder the classification of seen classes. To address this issue, mainstream safe SSL methods suggest detecting and discarding unseen-class samples from unlabeled data. Nevertheless, these methods typically employ a single-model strategy to simultaneously tackle both the classification of seen classes and the detection of unseen classes. Our research indicates that such an approach may lead to conflicts during training, resulting in suboptimal model optimization. Inspired by this, we introduce a novel framework named Diverse Teacher-Students (\textbf{DTS}), which uniquely utilizes dual teacher-student models to individually and effectively handle these two tasks. DTS employs a novel uncertainty score to softly separate unseen-class and seen-class data from the unlabeled set, and intelligently creates an additional ($K$+1)-th class supervisory signal for training. By training both teacher-student models with all unlabeled samples, DTS can enhance the classification of seen classes while simultaneously improving the detection of unseen classes. Comprehensive experiments demonstrate that DTS surpasses baseline methods across a variety of datasets and configurations. Our code and models can be publicly accessible on the link https://github.com/Zhanlo/DTS.