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The structure of automorphism groups of $\kappa$-existentially closed groups are studied by Kaya-Kuzucuo\u{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^\kappa$ whenever $\kappa$ is an inaccessible cardinal and $G$ is the unique $\kappa$-existentially closed group of cardinality $\kappa$. The cardinality of the automorphism group of a $\kappa$-existentially closed group of cardinality $\lambda>\kappa$ is asked in Kourovka Notebook Question 20.40. Here we answer positively the promised case $\kappa=\lambda$ namely: If $G$ is a $\kappa$-existentially closed group of cardinality $\kappa$, then $|Aut(G)|=2^{\kappa}$. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal $\kappa$, there exist universal groups of cardinality $\kappa$.

In this paper, we show that Frucht's theorem holds in Borel setting. More specifically, we prove that any standard Borel group can be realized as the Borel automorphism group of a Borel graph. A slight modification of our construction also yields the following result in topological setting: Any Polish group can be realized as the homeomorphic automorphism group of a $\mathbf{\Delta^0_2}$-graph on a Polish space.

We investigate the automorphisms of some $\kappa$- existentially closed groups. In particular, we prove that $Aut(G)$ is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^{\kappa}$ whenever $\kappa$ is inaccessible and $G$ is the unique $\kappa$-existentially closed group of cardinality $\kappa$. Indeed, the latter result is a byproduct of an argument showing that, for any uncountable $\kappa$ and any group $G$ that is the limit of regular representation of length $\kappa$ with countable base, we have $|Aut(G)|=\beth_{\kappa+1}$, where $\beth$ is the beth function. Such groups are also $\kappa$-existentially closed if $\kappa$ is regular. Both results are obtained by an analysis and classification of level preserving automorphisms of such groups.

In this paper, we contribute to the study of topological partition relations for pairs of countable ordinals and prove that, for all integers $n \geq 3$, \begin{align*} R^{cl}(\omega+n,3) &\geq \omega^2 \cdot n + \omega \cdot (R(n,3)-n)+n\\ R^{cl}(\omega+n,3) &\leq \omega^2 \cdot n + \omega \cdot (R(2n-3,3)+1)+1 \end{align*} where $R^{cl}(\cdot,\cdot)$ and $R(\cdot,\cdot)$ denote the closed Ramsey numbers and the classical Ramsey numbers respectively. We also establish the following asymptotically weaker upper bound \[ R^{cl}(\omega+n,3) \leq \omega^2 \cdot n + \omega \cdot (n^2-4)+1\] eliminating the use of Ramsey numbers. These results improve the previously known upper and lower bounds.

This paper presents an accurate and robust embedded motor-imagery brain-computer interface (MI-BCI). The proposed novel model, based on EEGNet, matches the requirements of memory footprint and computational resources of low-power microcontroller units (MCUs), such as the ARM Cortex-M family. Furthermore, the paper presents a set of methods, including temporal downsampling, channel selection, and narrowing of the classification window, to further scale down the model to relax memory requirements with negligible accuracy degradation. Experimental results on the Physionet EEG Motor Movement/Imagery Dataset show that standard EEGNet achieves 82.43%, 75.07%, and 65.07% classification accuracy on 2-, 3-, and 4-class MI tasks in global validation, outperforming the state-of-the-art (SoA) convolutional neural network (CNN) by 2.05%, 5.25%, and 5.48%. Our novel method further scales down the standard EEGNet at a negligible accuracy loss of 0.31% with 7.6x memory footprint reduction and a small accuracy loss of 2.51% with 15x reduction. The scaled models are deployed on a commercial Cortex-M4F MCU taking 101ms and consuming 4.28mJ per inference for operating the smallest model, and on a Cortex-M7 with 44ms and 18.1mJ per inference for the medium-sized model, enabling a fully autonomous, wearable, and accurate low-power BCI.

In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups, groups of exponential growth and groups with decidable word problem are $\mathbf{\Sigma}^0_2$-complete and that the sets of periodic groups and groups of intermediate growth are $\mathbf{\Pi}^0_2$-complete. We also provide bounds for the descriptive complexity of simplicity, amenability, residually finiteness, Hopficity and co-Hopficity. This paper is intended to serve as a compilation of results on this theme., Comment: Some minor mistakes in the earlier version are fixed. Also some improvements and expansions are made

In this note, we analyze the classification problem for compact metrizable $G$-ambits for a countable discrete group $G$ from the point of view of descriptive set theory. More precisely, we prove that the topological conjugacy relation on the standard Borel space of compact metrizable $G$-ambits is Borel for every countable discrete group $G$.

In this paper, we analyze the Borel complexity of the topological conjugacy relation on Toeplitz subshifts. More specifically, we prove that topological conjugacy of Toeplitz subshifts with separated holes is hyperfinite. Indeed, we show that the topological conjugacy relation is hyperfinite on a larger class of Toeplitz subshifts which we call Toeplitz subshifts with growing blocks. This result provides a partial answer to a question asked by Sabok and Tsankov.

In this paper, we analyze the complexity of topological conjugacy of pointed Cantor minimal systems from the point of view of descriptive set theory. We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation $\Delta_{\mathbb{R}}^+$ on $\mathbb{R}^{\mathbb{N}}$ defined by $x \Delta_{\mathbb{R}}^+ y \Leftrightarrow \{x_i:i \in \mathbb{N}\}=\{y_i:i \in \mathbb{N}\}$. Moreover, we show that $\Delta_{\mathbb{R}}^+$ is a lower bound for the Borel complexity of topological conjugacy of Cantor minimal systems. Finally, we interpret our results in terms of properly ordered Bratteli diagrams and discuss some applications.