arXiv subjects

Qingguo Li

Publications and source records attributed to Qingguo Li.

At least 19 recordsLinked to original sources

arXiv2026

A continuous $3$-distributive frame that is not $\omega$-distributive

We give a negative answer to the question, posed by Ern\'e, whether every $3$-distributive lattice is $\omega$-distributive. More precisely, we exhibit a continuous frame that is $\kappa$-distributive for every integer $\kappa\geq 2$, but is not a wide coframe. The frame is the open-set lattice of a compact, locally compact, countably based $T_0$ topological meet-semilattice, obtained from Lawson's construction in the logarithmic form described by Goubault-Larrecq. The failure of $\omega$-distributivity is witnessed by an explicit matrix with countably many nonempty finite rows: all row joins are the same nonzero element, whereas every choice of one entry from each row has meet zero. The same space answers negatively Ern\'e's accompanying question whether every $4$-web space is a wide web space. All properties of the construction needed for these conclusions are proved directly.

math.GN↗

arXiv2026

Monad Structures on Topological Spaces Comprising Mislove's Random Variables

Mislove, Goubault and Varacca investigated how to define random variables in Domain theory to form monads over the category of bounded complete domains. They intended to model probabilistic programming languages with their random variables monads. In this paper, we focus on the random variables defined by Mislove from a topological perspective. We provide a topology for $\surd$-max continuous random variables on a $T_0$ space, we construct a new $T_0$ space, where $\surd$-max property is essential for the monad structures. We show that the spaces of normalized $\surd$-max simple random variables form a monad over the category of $T_0$ spaces and that the spaces of normalized $\surd$-max continuous random variables give a monad over the category of d-spaces. In addition, on a sober space, the space of normalized $\surd$-max continuous random variables is the sobrification of the space of normalized $\surd$-max simple random variables.

cs.LO↗

arXiv2026

Chronicle: A Multimodal Foundation Model for Joint Language and Time Series Understanding

Real-world time series come with text: metadata, descriptions, news, reports. Yet time series foundation models process numerical sequences in isolation, and the multimodal text-and-time-series models that attempt to bridge the two all adapt a pretrained language model post hoc, inheriting representations shaped without ever seeing temporal data. These models are also evaluated almost exclusively against other multimodal baselines, not against the strongest unimodal foundation models in either domain, leaving open whether joint training is needed at all. We present Chronicle, a compact 324M-parameter decoder-only transformer trained from scratch on natural language and time series within a single unified architecture. Both modalities share the same transformer blocks, attention mechanism, and residual stream; the bulk of pretraining uses unimodal batches so cross-modal capability emerges purely from shared parameters, with a short alignment stage that interleaves the two. To our knowledge, Chronicle is the first model jointly pretrained on text and time series from scratch, and the first multimodal model evaluated against dedicated foundation models in both domains. It matches Gemma-3-270M-PT on 19 NLU tasks, sets a new bar for frozen-embedding time series classification on 24 UCR/UEA datasets, and produces multimodal forecasts on Time-MMD that beat every supervised fusion baseline, all from a single backbone.

cs.LG↗

arXiv2026

ADAPTive Input Training for Many-to-One Pre-Training on Time-Series Classification

Recent work on time-series models has leveraged self-supervised training to learn meaningful features and patterns in order to improve performance on downstream tasks and generalize to unseen modalities. While these pretraining methods have shown great promise in one-to-many scenarios, where a model is pre-trained on one dataset and fine-tuned on a downstream dataset, they have struggled to generalize to new datasets when more datasets are added during pre-training. This is a fundamental challenge in building foundation models for time-series data, as it limits the ability to develop models that can learn from a large variety of diverse datasets available. To address this challenge, we present a new pre-training paradigm for time-series data called ADAPT, which can efficiently align the physical properties of data in the time-series domain, enabling mixed-batch pre-training despite the extreme discrepancies in the input sizes and channel dimensions of pre-training data. We trained on 162 time-series classification datasets and set new state-of-the-art performance for classification benchmarks. We successfully train a model within the time-series domain on a wide range of datasets simultaneously, which is a major building block for building generalist foundation models in time-series domains.

cs.LG↗

arXiv2026

Chat-TS: Enhancing Multi-Modal Reasoning Over Time-Series and Natural Language Data

Large language models are being rapidly deployed across many fields such as healthcare, finance, transportation, and energy, where time-series data are fundamental components. The current works are still limited in their ability to perform reasoning that involves both time-series and the corresponding textual content. We address this gap by introducing Chat-TS, a large language model (LLM) based framework designed to support reasoning over time series and textual data. Unlike traditional models, Chat-TS integrates time-series tokens into LLMs' vocabulary, enhancing its reasoning ability over both modalities without compromising core natural language capabilities. To support learning and evaluation, we contribute new datasets: the TS Instruct Training Dataset (pairing diverse time-series data with relevant text instructions and responses for instruction tuning), the TS Instruct Question and Answer (QA) Gold Dataset (multiple-choice questions to evaluate multimodal reasoning), and a TS Instruct Quantitative Probing Set (a small subset of TS Instruct QA reasoning tasks alongside math and decision-making questions for LLM evaluation). We design a training strategy to preserve the inherent reasoning capabilities of LLMs while augmenting them for time-series reasoning. Experiments show that Chat-TS achieves state-of-the-art performance in multimodal reasoning tasks by maintaining strong natural language proficiency while improving time-series reasoning.

cs.AI↗

arXiv2026

Co-Noetherian spaces

In non-Hausdorff topology, many spaces exhibit significant separation properties, such as sober spaces, well-filtered spaces and d-spaces. These properties serve to fundamentally classify T0 topological spaces. In this paper, we introduce and study a new class of topological spaces called co-Noetherian spaces, which can refine the classification of T0 spaces. We discuss some basic properties of co-Noetherian spaces and obtain an equivalent characterization of compactness under the strong topology. Additionally, we investigate the connections among KC-spaces, strong R-spaces and co-Noetherian spaces. Moreover, we establish an equivalence between the category of T0 co-Noetherian spaces with continuous mappings and a subcategory of the poset category. Finally, we provide counterexamples to show that the Hoare powerspace of a T0 space may fail to be co-Noetherian, and that the Smyth powerspace of a co-Noetherian space need not be co-Noetherian.

math.GN↗

arXiv2026

S-well-filtered spaces and d-spaces

Recently, Xu proposed a strongly well-filtered space in [24] and systematically investigated some of its properties and characterizations. In this paper, we introduce a new class of T0-spaces called S*-well-filtered spaces, which is strictly larger than the class of strongly well-filtered spaces. First, we establish some connections among S*-well-filtered spaces, d*-spaces and weak well-filtered spaces. Then it is demonstrated that for any dcpo P, the Scott space ΣP is a d*-space if and only if it is S*-well-filtered. Furthermore, some basic properties of S*-well-filtered spaces are discussed. We prove that if Y is an S*-well-filtered space, the function space TOP(X,Y) equipped with the Isbell topology may not be an S*-well-filtered space. Finally, we study the S*-well-filteredness of Smyth power spaces. In addition, Johnstone's non-sober dcpo example is shown to be S*-well-filtered yet it is not strongly well-filtered, thereby establishing an obvious distinction between these two classes of dcpos.

math.GN↗

arXiv2025

Products of two sober dcpo's need not be sober

We construct two dcpo's whose Scott spaces are sober, but the Scott space of their order product is not sober. This answers an open problem on the sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special type of sober complete lattices, then the Scott space of their order product $M\times N$ is sober.

math.GN↗

arXiv2025

Sober topologies on a set

The collection of all topologies on a set X forms a complete lattice with respect to the inclusion order, which have been investigated by many researchers. Sobriety is one of the core and extensively studied properties in non-Hausdorff topology. This property plays a crucial role in characterizing the spectral spaces of commutative rings and topological spaces determined by their lattices of open sets. In this paper, we investigate the statute of sober topologies in the complete lattice of all topologies on a given set. The main results to be proved include: (1) every T1 topology is the join of some sober topologies; (2) every topology is the meet of some sober topologies; (3) the set of all sober topologies is directed complete; (4) every Alexanderoff - discrete topology is the meet of some sober Alexanderoff - discrete topologies; (5) the minimal sober topologies are exactly the Scott topologies of sup-complete chains; (6) an example will be constructed to show that the intersection of a decreasing sequence of Hausdorff topologies need not be sober.

math.GN↗

arXiv2025

Separating domains from algebraic domains

We prove that every domain that fails to be algebraic admits the unit interval $[0, 1]$ as its Scott-continuous retract. As a result, every countable domain is algebraic.

math.GN↗

arXiv2025

Some Consistent Power Constructions

Consistent Hoare, Smyth and Plotkin power domains are introduced and discussed by Yuan and Kou. The consistent algebraic operation $+$ defined by them is a binary partial Scott continuous operation satisfying the requirement: $a+b$ exists whenever there exists a $c$ which is greater than $a$ and $b$. We extend the consistency to be a categorical concept and obtain an approach to generating consistent monads from monads on dcpos whose images equipped with some algebraic operations. Then we provide two new power constructions over domains: the consistent Plotkin index power domain and the consistent probabilistic power domain. Moreover, we verify these power constructions are free.

cs.LO↗

arXiv2025

Posets uniquely determined by its compact saturated subsets

Inspired by Zhao and Xu's study on which a dcpo can be determined by its Scott closed subsets lattice, we further investigate whether a poset (or dcpo) $P$ is able to be determined by the family $\mathcal Q(P)$ of its Scott compact saturated subsets, in the sense that the isomorphism between $(\mathcal Q(P), \supseteq)$ and $(\mathcal Q(M), \supseteq)$ implies the isomorphism between $P$ and $M$ for any poset (or dcpo) $M$, in such case, $P$ is called $\mathcal Q_σ$-unique. Quasicontinuous domains are proved to be $\mathcal Q_σ$-unique posets and draw support from which, we provide a class of $\mathcal Q_σ$-unique dcpos. We also define a new kind of posets called $K_D$ and show that every co-sober $K_D$ poset is $\mathcal Q_σ$-unique. It even yields another kind of $\mathcal Q_σ$-unique dcpos. It is gratifying that weakly well-filtered co-sober posets are also $\mathcal Q_σ$-unique. At last, we distinguish among the conditions which make a poset (or dcpo) $\mathcal Q_σ$-unique from each other by some examples; meanwhile, it is confirmed that none of them except the property of being co-sober are necessary for a poset (or dcpo) to be $\mathcal Q_σ$-unique.

math.GN↗

arXiv2025

Low-Dimensional Representation-Driven TSK Fuzzy System for Feature Selection

Feature selection can select important features to address dimensional curses. Subspace learning, a widely used dimensionality reduction method, can project the original data into a low-dimensional space. However, the low-dimensional representation is often transformed back into the original space, resulting in information loss. Additionally, gate function-based methods in Takagi-Sugeno-Kang fuzzy system (TSK-FS) are commonly less discrimination. To address these issues, this paper proposes a novel feature selection method that integrates subspace learning with TSK-FS. Specifically, a projection matrix is used to fit the intrinsic low-dimensional representation. Subsequently, the low-dimensional representation is fed to TSK-FS to measure its availability. The firing strength is slacked so that TSK-FS is not limited by numerical underflow. Finally, the $\ell _{2,1}$-norm is introduced to select significant features and the connection to related works is discussed. The proposed method is evaluated against six state-of-the-art methods on eighteen datasets, and the results demonstrate the superiority of the proposed method.

cs.LG↗

arXiv2024

The category of well-filtered dcpos is not $Γ$-faithful

The Ho-Zhao problem asks whether any two dcpo's with isomorphic Scott closed set lattices are themselves isomorphic, that is, whether the category $\mathbf{DCPO}$ of dcpo's and Scott-continuous maps is $Γ$-faithful. In 2018, Ho, Goubault-Larrecq, Jung and Xi answered this question in the negative, and they introduced the category $\mathbf{DOMI}$ of dominated dcpo's and proved that it is {$Γ$-faithful}. Dominated dcpo's subsume many familiar families of dcpo's in domain theory, such as the category of bounded-complete dcpo's and that of sober dcpo's, among others. However, it is unknown whether the category of dominated dcpo's dominates all well-filtered dcpo's, a class strictly larger than that of bounded-complete lattices and that of sober dcpo's. In this paper, we address this very natural question and show that the category $\mathbf{WF}$ of well-filtered dcpo's is not $Γ$-faithful, and as a result of it, well-filtered dcpo's need not be dominated in general. Since not all dcpo's are well-filtered, our work refines the results of Ho, Goubault-Larrecq, Jung and Xi. As a second contribution, we confirm that the Lawson's category of $Ω^{*}$-compact dcpo's is $Γ$-faithful. Moreover, we locate a class of dcpo's which we call weakly dominated dcpo's, and show that this class is $Γ$-faithful and strictly larger than $\mathbf{DOMI}$.

cs.LO↗

arXiv2024

$ω$-well-filtered spaces, revisited

We prove that a $T_0$ topological space is $ω$-well-filtered if and only if it does not admit either the natural numbers with the cofinite topology or with the Scott topology as its closed subsets in the strong topology. Based on this, we offer a refined topological characterization for the $ω$-well-filterification of $T_0$-spaces and solve a problem posed by Xiaoquan Xu. In the setting of second countable spaces, we also characterise sobriety by convergences of certain $Π^0_2$-Cauchy subsets of the spaces.

math.GN↗

arXiv2024

Topological representations for frame-valued domains via $L$-sobriety

With a frame $L$ as the truth value table, we study the topological representations for frame-valued domains. We introduce the notions of locally super-compact $L$-topological space and strong locally super-compact $L$-topological space. Using these concepts, continuous $L$-dcpos and algebraic $L$-dcpos are successfully represented via $L$-sobriety. By means of Scott $L$-topology and specialization $L$-order, we establish a categorical isomorphism between the category of the continuous (resp., algebraic) $L$-dcpos with Scott continuous maps and that of the locally super-compact (resp., strong locally super-compact) $L$-sober spaces with continuous maps. As an application, for a continuous $L$-poset $P$, we obtain a categorical isomorphism between the category of directed completions of $P$ with Scott continuous maps and that of the $L$-sobrifications of $(P, σ_{L}(P))$ with continuous maps.

math.GN↗

arXiv2024

Representations of domains via closure spaces in the quantale-valued setting

With a commutative unital quantale $L$ as the truth value table, this study focuses on the representations of $L$-domains by means of $L$-closure spaces. First, the notions of interpolative generalized $L$-closure spaces and directed closed sets are introduced. It is proved that in an interpolative generalized $L$-closure space (resp., $L$-closure space), the collection of directed closed sets with respect to the inclusion $L$-order forms a continuous $L$-dcpo (resp., an algebraic $L$-dcpo). Conversely, it is shown that every continuous $L$-dcpo (resp., algebraic $L$-dcpo) can be reconstructed by an interpolative generalized $L$-closure space (resp., $L$-closure space). Second, when $L$ is integral, the notion of dense subspaces of generalized $L$-closure spaces is introduced. By means of dense subspaces, an alternative representation for algebraic $L$-dcpos is given. Moreover, the concept of $L$-approximable relations between interpolative generalized $L$-closure spaces is introduced. Consequently, a categorical equivalence between the category of interpolative generalized $L$-closure spaces (resp., $L$-closure spaces) with $L$-approximable relations and that of continuous $L$-dcpos (resp., algebraic $L$-dcpos) with Scott continuous mappings is established.

math.GN↗

arXiv2023

Commutation of Smyth and Hoare Power Constructions in Well-filtered Dcpos

Prior work [11] established a commutativity result for the Hoare power construction and a modified version of the Smyth power construction consisting of strongly compact sets, which is defined for Us-admitting dcpos, where Us-admissability is well-filteredness with compact sets replaced by strongly compact sets. In this paper, we consider the Hoare power construction H and the Smyth power construction Q on the category WF of well-filtered dcpos with Scott-continuous maps. Actually, the functors H and Q can be extended to monads. We prove that H and Q commute, that is, HQ(L) is isomorphic to QH(L) for a well-filtered dcpo L, if and only if L satisfies a property similar to consonance that we call (KC) and the Scott topology coincides with the upper Vietoris topology on Q(L). We also investigate the Eilenberg-Moore category of the monad composed by H and Q under a distributive law on WF and characterize it to be a subcategory of the category Frm, which is composed of all frames and all frame homomorphisms.

math.CT↗