We study Borel equivalence relations equipped with a uniformly Borel family of Polish topologies on each equivalence class, and more generally, standard Borel groupoids equipped with such a family of topologies on each connected component. Such "componentwise Polish topologies" capture precisely the topological information determined by the Borel structure of a Polish group action, by the Becker--Kechris theorem. We prove that conversely, every abstract such Borel componentwise Polish groupoid obeying suitable axioms admits a Borel equivalence of groupoids to a global open Polish groupoid. Together with known results, this implies that every such groupoid is Borel equivalent to an action groupoid of a Polish group action; in particular, the induced equivalence relations are Borel bireducible. Our results are also valid for Borel groupoids with componentwise quasi-Polish topologies; and under stronger uniformity assumptions, we show that such groupoids in fact themselves admit global quasi-Polish topologies. As a byproduct, we also generalize several standard tools for Polish groups and their actions to the setting of componentwise quasi-Polish groupoids, including Vaught transforms
In this paper, we investigate Polish semigroup topologies on the endomorphism monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$. We introduce a new structural condition, property $\mathbb{XX}$, which yields automatic continuity of Borel measurable homomorphisms between certain topological semigroups. This provides a new method for analyzing Polish semigroup topologies on monoids with small groups of units. We show that for all monoids considered, the semigroup Zariski topology coincides with the pointwise topology and is therefore the coarsest Hausdorff semigroup topology. We prove that the submonoid $\operatorname{End}^{\infty}(\mathbb{N},\leq)$ of $\operatorname{End}(\mathbb{N},\leq)$ consisting of all endomorphisms with infinite image admits a unique Polish semigroup topology, namely the pointwise topology. On the other hand, despite possessing a finest Polish semigroup topology, the monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$, admit infinitely many distinct Polish semigroup topologies. Also, we show that the monoid $\operatorname{End}(\mathbb{N},<)$ admits exactly $2^{\aleph_0}$ Polish semigrou
This data report introduces PolSeT (Polish Semantic Timbre), a dataset designed to facilitate research in psychoacoustics and Music Information Retrieval (MIR) in Polish and cross-cultural contexts. The dataset contains data from two sequential experiments. Experiment 1 (N=60) was a free-verbalization task aimed at creating a lexicon of Polish semantic descriptors. Using 11 stimuli, a total of 1901 descriptors (701 unique) were gathered. Experiment 2 (N=105) utilized this lexicon to conduct a semantic differential study, where participants rated 18 instrument sounds on 8 bipolar scales, with repeated trials for reliability analysis. The released dataset includes raw listener responses, comprehensive demographics (experience, gender, age), audio stimuli, and extracted acoustic features with Python extraction code. This dataset addresses a gap in open timbre research data, providing both the qualitative linguistic groundwork and the quantitative ratings necessary for psychoacoustic research and the training of multilingual semantic embedding models.
In this paper, we initiate the study of pro-Lie Polish abelian groups from the perspective of homological algebra. We extend to this context the type-decomposition of locally compact Polish abelian groups of Hoffmann and Spitzweck, and prove that the category $\mathbf{proLiePAb}$ of pro-Lie Polish abelian groups is a thick subcategory of the category of Polish abelian groups. We completely characterize injective and projective objects in $\mathbf{proLiePAb}$. We conclude that $\mathbf{proLiePAb}$ has enough projectives but not enough injectives and homological dimension $1$. We also completely characterize injective and projective objects in the category of non-Archimedean Polish abelian groups, concluding that it has enough injectives and projectives and homological dimension $1$. Injective objects are also characterized for the categories of topological torsion Polish abelian groups and for Polish abelian topological $p$-groups, showing that these categories have enough injectives and homological dimension $1$.
This article introduces the first comprehensive benchmark for the Polish language at this scale: LLMzSzŁ (LLMs Behind the School Desk). It is based on a coherent collection of Polish national exams, including both academic and professional tests extracted from the archives of the Polish Central Examination Board. It covers 4 types of exams, coming from 154 domains. Altogether, it consists of almost 19k closed-ended questions. We investigate the performance of open-source multilingual, English, and Polish LLMs to verify LLMs' abilities to transfer knowledge between languages. Also, the correlation between LLMs and humans at model accuracy and exam pass rate levels is examined. We show that multilingual LLMs can obtain superior results over monolingual ones; however, monolingual models may be beneficial when model size matters. Our analysis highlights the potential of LLMs in assisting with exam validation, particularly in identifying anomalies or errors in examination tasks.
In recent years, much work has been done to measure and compare the complexity of orbit equivalence relations, especially for certain classes of Polish groups. We start by introducing some language to organize this previous work, namely the notion of \textbf{classification strength} of Polish groups. Broadly speaking, a Polish group $G$ has stronger classification strength than $H$ if every orbit equivalence relation induced by a continuous action of $H$ on a Polish space can be ``emulated" by such an action of $G$ in the sense of Borel reduction. Among the non-Archimedean Polish groups, the groups with the highest classification strength are those that involve $S_\infty$, the Polish group of permutations of a countably-infinite set. We prove that several properties, including a weakening of the disjoint amalgamation in Fraïssé theory, a weakening of the existence of an absolute set of generating indiscernibles, and not having ordinal rank in a particular coanalytic rank function, are all equivalent to a non-Archimedean Polish group involving $S_\infty$. Furthermore, we show the equivalence relation $=^+$, which is a relatively simple benchmark equivalence relation in the theory of
For each ordinal $α<ω_1$, we introduce the class of $α$-balanced Polish groups. These classes form a hierarchy that completely stratifies the space between the class of Polish groups admitting a two-side-invariant metric (TSI) and the class of Polish groups admitting a complete left-invariant metric (CLI). We establish various closure properties, provide connections to model theory, and we develop a boundedness principle for CLI groups by showing that $α$-balancedness is an initial segment of a regular coanalytic rank. In the spirit of Hjorth's turbulence theory we also introduce "generic $α$-unbalancedness": a new dynamical condition for Polish $G$-spaces which serves as an obstruction to classification by actions of $α$-balanced Polish groups. We use this to provide, for each $α<ω_1$, an action of an $α$-balanced Polish group whose orbit equivalence relation is strongly generically ergodic against actions of any $β$-balanced Polish group with $β<α$.
We solve a long-standing open problem, formulated by Krasner in the 1950's, in the context of Polish (i.e. separable complete) ultrametric spaces by providing a characterization of their isometry groups using suitable forms of generalized wreath products of full permutation groups. Since our solution is developed in the finer context of topological (Polish) groups, it also solves a problem of Gao and Kechris from 2003. Furthermore, we provide an exact correspondence between the isometry groups of Polish ultrametric spaces belonging to some natural subclasses and various kinds of generalized wreath products proposed in the literature by Hall, Holland, and Malicki.
The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^ω/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. Let $G$ be a Polish group. (1) $G$ is a discrete countable group containing at least two elements iff $E(G)\sim_BE_0$; (2) if $G$ is TSI uncountable non-archimedean, then $E(G)\sim_BE_0^ω$; (3) $G$ is non-archimedean iff $E(G)\le_B=^+$; (4) if $H$ is a CLI Polish group but $G$ is not, then $E(G) ot\le_BE(H)$; (5) if $H$ is a non-archimedean Polish group but $G$ is not, then $E(G) ot\le_BE(H)$. The notion of $α$-l.m.-unbalanced Polish group for $α<ω_1$ is introduced. Let $G,H$ be Polish groups, $0<α<ω_1$. If $G$ is $α$-l.m.-unbalanced but $H$ is not, then $E(G) ot\le_B E(H)$. For TSI Polish groups, the existence of Borel reduction is transformed into the existence of a well-behaved continuous mapping between topological groups. As its applications, for any Polish group $G$, let $G_0$ be the connected component of the identity element
The development of bare fiber or air-gapped microlens-fiber coupled Integral Field Units (IFUs) for astronomical applications requires careful treatment of the fiber end-faces (terminations). Previous studies suggest that minimization of fiber end face irregularity leads to better optical performance in terms of the diminishing effect of focal ratio degradation. Polishing has typically been performed using commercial rotary polishers with multiple gradually decreasing grit sizes. These polishers generally lack the ability to carefully adjust angular position and polishing force. Control of these parameters vastly help in getting a repeatable and controllable polish over a variety of glass/epoxy/metal matrices that make up integral filed units and fiber slits. A polishing arm is developed to polish the fiber terminations (IFU, mini-bundles and v-grooves) of the NIR Fiber System for the RSS spectrograph at SALT. The polishing arm angular adjustments ensure the correct position and orientation of each termination on the polishing surface during the polish. Various studies have indicated that the fiber focal ratio also degrades if the fiber end face comes under excessive stress. The po
We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is level Π_2 in the Borel hierarchy. Quasi-Polish spaces can be characterized within the framework of Type-2 Theory of Effectivity as precisely the countably based spaces that have an admissible representation with a Polish domain. They can also be characterized domain theoretically as precisely the spaces that are homeomorphic to the subspace of all non-compact elements of an ω-continuous domain. Every countably based locally compact sober space is quasi-Polish, hence every ω-continuous domain is quasi-Polish. A metrizable space is quasi-Polish if and only if it is Polish. We show that the Borel hierarchy on an uncountable quasi-Polish space does not collapse, and that the Hausdorff-Kuratowski theorem generalizes to all quasi-Polish spaces.
We extend some results of Carderi and Le Maître on full groups in the probability context to the infinite measure one: there exists at most one Polish group topology (refining the weak topology and coarser than the uniform topology) on an ergodic full group, and the orbit full group of a locally compact group acting in a Borel manner can be endowed with a Polish group topology. Moreover, orbit full groups are complete invariants for orbit equivalence. We then generalize a result from Le Maître on non-Polishability of the group of finitely supported bijections: the finitely supported elements of an ergodic full group carries a Polish group topology if and only if the associated full group comes from a countable equivalence relation. We finish with algebraic and topological results on ergodic (orbit) full groups concerning normal subgroups, contractibility and genericity of aperiodic elements.
Aspect-Sentiment Triplet Extraction (ASTE) is one of the most challenging and complex tasks in sentiment analysis. It concerns the construction of triplets that contain an aspect, its associated sentiment polarity, and an opinion phrase that serves as a rationale for the assigned polarity. Despite the growing popularity of the task and the many machine learning methods being proposed to address it, the number of datasets for ASTE is very limited. In particular, no dataset is available for any of the Slavic languages. In this paper, we present two new datasets for ASTE containing customer opinions about hotels and purchased products expressed in Polish. We also perform experiments with two ASTE techniques combined with two large language models for Polish to investigate their performance and the difficulty of the assembled datasets. The new datasets are available under a permissive licence and have the same file format as the English datasets, facilitating their use in future research.
In this paper we further develop the theory of canonical approximations of Polishable subgroups of Polish groups, building on previous work of Solecki and Farah--Solecki. In particular, we obtain a characterization of such canonical approximations in terms of their Borel complexity class. As an application we provide a complete list of all the possible Borel complexity classes of Polishable subgroups of Polish groups or, equivalently, of the ranges of continuous group homomorphisms between Polish groups. We also provide a complete list of all the possible Borel complexity classes of the ranges of: continuous group homomorphisms between non-Archimedean Polish groups; continuous linear maps between separable Fréchet spaces; continuous linear maps between separable Banach spaces.
We define and study Poissonian actions of Polish groups as a framework to Poisson suspensions, characterize them spectrally, and provide a complete characterization of their ergodicity. We further construct 'spatial' Poissonian actions, answering partially a question of Glasner, Tsirelson & Weiss about Lévy groups. We also construct for every diffeomorphism group an ergodic free spatial probability preserving actions. This constitutes a new class of Polish groups admitting non-essentially countable orbit equivalence relations, obtaining progress on a problem of Kechris.
We prove that every Polish group with more than two elements admits a non-trivial topological group automorphism. As a consequence, a hypothetical uniquely homogeneous Polish space with more than two points cannot be a semitopological group.
An attempt is made to evaluate progress of the Polish astrophysical research of stars and of the inter-stellar medium (ISM) on the basis of scientific paper citations in the ADS database. Rather modest citation levels were observed in the years before the mid-1950's. In the years 1958 - 1973, thanks to the partly opened foreign contacts and to strong support from astronomers of the older generation, work of a number of young, energetic enthusiasts reached the world science levels and formed a strong basis for the well recognized, international successes of the next generations.
To enhance the multi-step reasoning capabilities of large language models, researchers have extensively explored prompting methods, notably the Chain-of-Thought (CoT) method which explicitly elicits human-like rationales. However, they have inadvertently overlooked the potential of enhancing model reasoning performance by formulating higher-quality problems. In this work, we start from the problem side and propose Self-Polish (SP), a novel method that facilitates the model's reasoning by guiding it to progressively refine the given problems to be more comprehensible and solvable. We also explore several automatic prompting varients and propose the Self-Polish prompt bank for the community. SP is orthogonal to all other prompting methods of answer/reasoning side like CoT, allowing for seamless integration with state-of-the-art techniques for further improvement. Thorough experiments show that the proposed method attains notable and consistent effectiveness on five reasoning benchmarks across different models. Furthermore, our method also showcases impressive performance on robustness evaluation. Codes and prompts are available at https://github.com/WooooDyy/Self-Polish.
We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup have the Pettis property. We present several examples of Polish inverse subsemigroup of the symmetric inverse semigroup I(N) of all partial bijections between subsets of N. We also study whether our examples satisfy automatic continuity.
We introduce a new benchmark for assessing the quality of text-to-text models for Polish. The benchmark consists of diverse tasks and datasets: KLEJ benchmark adapted for text-to-text, en-pl translation, summarization, and question answering. In particular, since summarization and question answering lack benchmark datasets for the Polish language, we describe their construction and make them publicly available. Additionally, we present plT5 - a general-purpose text-to-text model for Polish that can be fine-tuned on various Natural Language Processing (NLP) tasks with a single training objective. Unsupervised denoising pre-training is performed efficiently by initializing the model weights with a multi-lingual T5 (mT5) counterpart. We evaluate the performance of plT5, mT5, Polish BART (plBART), and Polish GPT-2 (papuGaPT2). The plT5 scores top on all of these tasks except summarization, where plBART is best. In general (except for summarization), the larger the model, the better the results. The encoder-decoder architectures prove to be better than the decoder-only equivalent.