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We prove that $Aut({\mathbb S}^1)$ coincides with the automorphism group of the \emph{circle graph} $\mathcal{C}$, i.e. the intersection graph of the family of chords of ${\mathbb S}^1$. We prove that the countable subgraph of $\mathcal{C}$ induced by the rational chords is a strongly universal element of the family of circle graphs, and that it is invariant under local complementation. The only other known connected graphs that have the latter property are $K_2$ and the Rado graph.

Motivation: With the rapid expansion of large-scale biological datasets, DNA and protein sequence alignments have become essential for comparative genomics and proteomics. These alignments facilitate the exploration of sequence similarity patterns, providing valuable insights into sequence conservation, evolutionary relationships and for functional analyses. Typically, sequence alignments are stored in formats such as the Multiple Alignment Format (MAF). Counting k-mer occurrences is a crucial task in many computational biology applications, but currently, there is no algorithm designed for k-mer counting in alignment files. Results: We have developed MAFcounter, the first k-mer counter dedicated to alignment files. MAFcounter is multithreaded, fast, and memory efficient, enabling k-mer counting in DNA and protein sequence alignment files. Availability: The MAFcounter package and its Python bindings are released under GPL license as a multi-platform application and are available at: https://github.com/Georgakopoulos-Soares-lab/MAFcounter

The accurate diagnosis of machine breakdowns is crucial for maintaining operational safety in smart manufacturing. Despite the promise shown by deep learning in automating fault identification, the scarcity of labeled training data, particularly for equipment failure instances, poses a significant challenge. This limitation hampers the development of robust classification models. Existing methods like model-agnostic meta-learning (MAML) do not adequately address variable working conditions, affecting knowledge transfer. To address these challenges, a Related Task Aware Curriculum Meta-learning (RT-ACM) enhanced fault diagnosis framework is proposed in this paper, inspired by human cognitive learning processes. RT-ACM improves training by considering the relevance of auxiliary sensor working conditions, adhering to the principle of ``paying more attention to more relevant knowledge", and focusing on ``easier first, harder later" curriculum sampling. This approach aids the meta-learner in achieving a superior convergence state. Extensive experiments on two real-world datasets demonstrate the superiority of RT-ACM framework.

Zimin words are words that have the same prefix and suffix. They are unavoidable patterns, with all sufficiently large strings encompassing them. Here, we examine for the first time the presence of k-mers not containing any Zimin patterns, defined hereafter as Zimin avoidmers, in the human genome. We report that in the reference human genome all k-mers above 104 base-pairs contain Zimin words. We find that Zimin avoidmers are most enriched in coding and Human Satellite 1 regions in the human genome. Zimin avoidmers display a depletion of germline insertions and deletions relative to surrounding genomic areas. We also apply our methodology in the genomes of another eight model organisms from all three domains of life, finding large differences in their Zimin avoidmer frequencies and their genomic localization preferences. We observe that Zimin avoidmers exhibit the highest genomic density in prokaryotic organisms, with E. coli showing particularly high levels, while the lowest density is found in eukaryotic organisms, with D. rerio having the lowest. Among the studied genomes the longest k-mer length at which Zimin avoidmers are observed is that of S. cerevisiae at k-mer length of 115 base-pairs. We conclude that Zimin avoidmers display inhomogeneous distributions in organismal genomes, have intricate properties including lower insertion and deletion rates, and disappear faster than the theoretical expected k-mer length, across the organismal genomes studied.

Motivation: Genome and Proteome Alignments, represented by the Multiple Alignment File (MAF) format, have become a standard approach in the field of comparative genomics and proteomics. However, current approaches lack a direct method for motif detection within MAF files. To address this gap, we present MAFin, a novel tool that enables efficient motif detection and conservation analysis in MAF files, streamlining genomic and proteomic research. Results: We developed MAFin, the first motif detection tool for Multiple Alignment Format files. MAFin enables the multithreaded search of conserved motifs using three approaches: 1) by using user-specified k-mers to search the sequences. 2) with regular expressions, in which case one or more patterns are searched, and 3) with predefined Position Weight Matrices. Once the motif has been found, MAFin detects the motif instances and calculates the conservation across the aligned sequences. MAFin also calculates a conservation percentage, which provides information about the conservation levels of each motif across the aligned sequences, based on the number of matches relative to the length of the motif. A set of statistics enable the interpretation of each motif's conservation level, and the detected motifs are exported in JSON and CSV files for downstream analyses. Availability: MAFin is released as a Python package under the GPL license as a multi-platform application and is available at: https://github.com/Georgakopoulos-Soares-lab/MAFin. Contact: izg5139@psu.edu

We study a higher-dimensional analogue of the {Random Travelling Salesman Problem}: let the complete $d$-dimensional simplicial complex $K_n^{d}$ on $n$ vertices be equipped with i.i.d.\ volumes on its facets, uniformly random in $[0,1]$. What is the minimum volume $M_{n,d}$ of a sub-complex homeomorphic to the $d$-dimensional sphere $\mathbb{S}^d$, containing all vertices? We determine the growth rate of $M_{n,2}$, and prove that it is well-concentrated. For $d>2$ we prove such results to the extent that current knowledge about the number of triangulations of $\mathbb{S}^d$ allows. We remark that this can be thought of as a model of random geometry in the spirit of Angel \& Schramm's UIPT, and provide a generalised framework that interpolates between our model and the uniform random triangulation of $\mathbb{S}^d$., Comment: 25 pages, 1 figure

We study the potentially undecidable problem of whether a given 2-dimensional CW complex can be embedded into $\mathbb{R}^4$. We provide operations that preserve embeddability, including joining and cloning of 2-cells, as well as $\Delta\mathrm Y$-transformations. We also construct a CW complex for which $\mathrm Y\Delta$-transformations do not preserve embeddability. We use these results to study 4-flat graphs, i.e., graphs that embed in $\mathbb{R}^4$ after attaching any number of 2-cells to their cycles; a graph class that naturally generalizes planarity and linklessness. We verify several conjectures of van der Holst; in particular, we prove that each of the 78 graphs of the Heawood family is an excluded minor for the class of 4-flat graphs.

With the increasing use of Artificial Intelligence in Natural Language Processing, concerns have been raised regarding the detection of AI-generated text in various domains. This study aims to investigate this issue by proposing a methodology to accurately distinguish AI-generated and human-written book reviews. Our approach utilizes transfer learning, enabling the model to identify generated text across different topics while improving its ability to detect variations in writing style and vocabulary. To evaluate the effectiveness of the proposed methodology, we developed a dataset consisting of real book reviews and AI-generated reviews using the recently proposed Vicuna open-source language model. The experimental results demonstrate that it is feasible to detect the original source of text, achieving an accuracy rate of 96.86%. Our efforts are oriented toward the exploration of the capabilities and limitations of Large Language Models in the context of text identification. Expanding our knowledge in these aspects will be valuable for effectively navigating similar models in the future and ensuring the integrity and authenticity of human-generated content.

We introduce notions of dimension of an infinite group, or more generally, a metric space, defined using percolation. Roughly speaking, the percolation dimension $pdim(G)$ of a group $G$ is the fastest rate of decay of a symmetric probability measure $\mu$ on $G$, such that Bernoulli percolation on $G$ with connection probabilities proportional to $\mu$ behaves like a Poisson branching process with parameter 1 in a sense made precise below. We show that $pdim(G)$ has several natural properties: it is monotone decreasing with respect to subgroups and quotients, and coincides with the growth rate exponent for several classes of groups.

Internet of Things (IoT) sensor data or readings evince variations in timestamp range, sampling frequency, geographical location, unit of measurement, etc. Such presented sequence data heterogeneity makes it difficult for traditional time series classification algorithms to perform well. Therefore, addressing the heterogeneity challenge demands learning not only the sub-patterns (local features) but also the overall pattern (global feature). To address the challenge of classifying heterogeneous IoT sensor data (e.g., categorizing sensor data types like temperature and humidity), we propose a novel deep learning model that incorporates both Convolutional Neural Network and Bi-directional Gated Recurrent Unit to learn local and global features respectively, in an end-to-end manner. Through rigorous experimentation on heterogeneous IoT sensor datasets, we validate the effectiveness of our proposed model, which outperforms recent state-of-the-art classification methods as well as several machine learning and deep learning baselines. In particular, the model achieves an average absolute improvement of 3.37% in Accuracy and 2.85% in F1-Score across datasets, Comment: Accepted for Publication and Presented in EAI MobiQuitous 2023 - 20th EAI International Conference on Mobile and Ubiquitous Systems: Computing, Networking and Services

Mohar recently adapted the classical game of Cops and Robber from graphs to metric spaces, thereby unifying previously studied pursuit-evasion games. He conjectured that finitely many cops can win on any compact geodesic metric space, and that their number can be upper-bounded in terms of the ranks of the homology groups when the space is a simplicial pseudo-manifold. We disprove these conjectures by constructing a metric on $\mathbb{S}^3$ with infinite cop number. More problems are raised than settled.

We present problems and results that combine graph-minors and coarse geometry. For example, we ask whether every geodesic metric space (or graph) without a fat $H$ minor is quasi-isometric to a graph with no $H$ minor, for an arbitrary finite graph $H$. We answer this affirmatively for a few small $H$. We also present a metric analogue of Menger's theorem and Konig's ray theorem. We conjecture metric analogues of the Erdos--Posa Theorem and Halin's grid theorem.

We prove that every recurrent graph $G$ quasi-isometric to $\mathbb{R}$ admits an essentially unique Lipschitz harmonic function $h$. If $G$ is vertex-transitive, then the action of $Aut(G)$ preserves $\partial h$ up to a sign, a fact that we exploit to prove various combinatorial results about $G$. As a consequence, we prove the 2-ended case of the conjecture of Grimmett & Li that the connective constant of a non-degenerate vertex-transitive graph is at least the golden mean. Moreover, answering a question of Watkins from 1990, we construct a cubic, 2-ended, vertex-transitive graph which is not a Cayley graph.

The widespread adoption of Large Language Models and publicly available ChatGPT has marked a significant turning point in the integration of Artificial Intelligence into people's everyday lives. The academic community has taken notice of these technological advancements and has expressed concerns regarding the difficulty of discriminating between what is real and what is artificially generated. Thus, researchers have been working on developing effective systems to identify machine-generated text. In this study, we utilize the GPT-3 model to generate scientific paper abstracts through Artificial Intelligence and explore various text representation methods when combined with Machine Learning models with the aim of identifying machine-written text. We analyze the models' performance and address several research questions that rise during the analysis of the results. By conducting this research, we shed light on the capabilities and limitations of Artificial Intelligence generated text.

We determine the excluded minors characterising the class of countable graphs that embed into some compact surface., Comment: Final version for Combinatorica

A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in \mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or $K_\infty$ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that (do and) do not have a universal element. Some of our results and questions may be of interest to the finite graph theorist. In particular, one of our side-results is that every $K_5$-minor-free graph is a minor of a $K_5$-minor-free graph of maximum degree 22.

The joint optimization of the reconstruction and classification error is a hard non convex problem, especially when a non linear mapping is utilized. In order to overcome this obstacle, a novel optimization strategy is proposed, in which a Convolutional Autoencoder for dimensionality reduction and a classifier composed by a Fully Connected Network, are combined to simultaneously produce supervised dimensionality reduction and predictions. It turned out that this methodology can also be greatly beneficial in enforcing explainability of deep learning architectures. Additionally, the resulting Latent Space, optimized for the classification task, can be utilized to improve traditional, interpretable classification algorithms. The experimental results, showed that the proposed methodology achieved competitive results against the state of the art deep learning methods, while being much more efficient in terms of parameter count. Finally, it was empirically justified that the proposed methodology introduces advanced explainability regarding, not only the data structure through the produced latent space, but also about the classification behaviour., Comment: to be submitted

We prove that any two countable, compact, subsets of $\mathbb{S}^n, n\geq 2$ that are homeomorphic also have homeomorphic complements. Thus any wild subspace like the classical construction of Antoine must contain a Cantor set.

We prove that a group $\Gamma$ admits a discrete topological (equivalently, smooth) action on some simply-connected 3-manifold if and only if $\Gamma$ has a Cayley complex embeddable -- with certain natural restrictions -- in one of the following four 3-manifolds: (i) $\mathbb{S}^3$, (ii) $\mathbb{R}^3$, (iii) $\mathbb{S}^2 \times \mathbb{R}$, (iv) the complement of a tame Cantor set in $\mathbb{S}^3$.

Consistency in product quality is of critical importance in manufacturing. However, achieving a target product quality typically involves balancing a large number of manufacturing attributes. Existing manufacturing practices for dealing with such complexity are driven largely based on human knowledge and experience. The prevalence of manual intervention makes it difficult to perfect manufacturing practices, underscoring the need for a data-driven solution. In this paper, we present an Industrial Internet of Things (IIoT) machine model which enables effective monitoring and control of plant machinery so as to achieve consistency in product quality. We present algorithms that can provide product quality prediction during production, and provide recommendations for machine control. Subsequently, we perform an experimental evaluation of the proposed solution using real data captured from a food processing plant. We show that the proposed algorithms can be used to predict product quality with a high degree of accuracy, thereby enabling effective production monitoring and control., Comment: 10 pages, 8 figures

A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected $2$-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of as a generalisation of the 3-dimensional Schoenflies theorem.

We introduce a method for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate $a(G)$ of lattice animals and vice-versa. We exploit this in both directions. We improve on the best known asymptotic lower and upper bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. We use percolation as a tool to obtain the latter, and conversely we use the former to obtain lower bounds on $p_c(\mathbb{Z}^d)$. We obtain the rigorous lower bound $\dot{p}_c(\mathbb{Z}^3)>0.2522$ for 3-dimensional site percolation.

In today's wireless communication systems, the exponentially growing needs of mobile users require the combination of new and existing techniques to meet the demands for reliable and high-quality service provision. This may not always be possible, as the resources in wireless telecommunication systems are limited and a significant number of users, usually located at the cell edges, can suffer from severe interference. For this purpose, a new Joint Transmission Coordinated Multipoint (JT-CoMP) scheme, in which the transmission points' clustering is based on a coalition formation game, is deployed alongside with Non-Orthogonal Multiple Access (NOMA) in a Cloud Radio Access Network (C-RAN) consisting of small cells. To further enhance the network's performance, Multiuser Multiple-Input Multiple-Output (MU-MIMO) with Zero-Forcing (ZF) beamforming is applied. The proposed scheme's performance, in terms of user throughput, is then compared to that of a scheme where JT-CoMP with static clustering is selected, a JT-CoMP scheme with clustering based on a greedy algorithm and a scenario where JT-CoMP is not deployed. Simulation results confirm that the proposed scheme reliably improves the throughput of users with poor wireless channels while guaranteeing that the performance of the rest is not severely undermined., Comment: This paper has been submitted for publication in IEEE Access

It is known that for every $\alpha \geq 1$ there is a planar triangulation in which every ball of radius $r$ has size $\Theta(r^\alpha)$. We prove that for $\alpha <2$ every such triangulation is quasi-isometric to a tree. The result extends to Riemannian 2-manifolds of finite genus, and to large-scale-simply-connected graphs. We also prove that every planar triangulation of asymptotic dimension 1 is quasi-isometric to a tree., Comment: Revised version submitted to Annales Henri Lebesgue

The unprecedented growth of Internet of Things (IoT) and its applications in areas such as Smart Agriculture compels the need to devise newer ways for evaluating the quality of such applications. While existing models for application quality focus on the quality experienced by the end-user (captured using likert scale), IoT applications have minimal human involvement and rely on machine to machine communication and analytics to drive decision via actuations. In this paper, we first present a conceptual framework for the evaluation of IoT application quality. Subsequently, we propose, develop and validate via empirical evaluations a novel model for evaluating sensor data quality that is a key component in assessing IoT application quality. We present an implementation of the sensor data quality model and demonstrate how the IoT sensor data quality can be integrated with a Smart Agriculture application. Results of experimental evaluations conducted using data from a real-world testbed concludes the paper., Comment: Technical Report under review with IEEE micro

We give minimal presentations for the $RO(C_2)$-graded Bredon cohomology of the equivariant classifying spaces $B_{C_2}U(n), B_{C_2}SO(n)$ and $B_{C_2}Sp(n)$ with coefficients in the rational Burnside Green functor $A_{\mathbf Q}$. This results in an efficient description of rational $C_2$ equivariant Chern, Pontryagin and symplectic characteristic classes. These classes are then related to each other using the inclusions of maximal tori., Comment: Comments are most welcome! 34 pages. Code is available on https://github.com/NickG-Math/Symmetric_Polynomials

In this short note, we compute the rational $C_{2^n}$-equivariant stable stems and give minimal presentations for the $RO(C_{2^n})$-graded Bredon cohomology of the equivariant classifying spaces $B_{C_{2^n}}S^1$ and $B_{C_{2^n}}\Sigma_2$ over the rational Burnside functor $A_{\mathbf Q}$. We also examine for which compact Lie groups $L$ the maximal torus inclusion $T\to L$ induces an isomorphism from $H^*_{C_{2^n}}(B_{C_{2^n}}L;A_{\mathbf Q})$ onto the fixed points of $H^*_{C_{2^n}}(B_{C_{2^n}}T;A_{\mathbf Q})$ under the Weyl group action. We prove that this holds for $L=U(m)$ and any $n,m\ge 1$ but does not hold for $L=SU(2)$ and $n>1$., Comment: Comments are most welcome! 9 pages

Following the Hu-Kriz method of computing the $C_2$ genuine dual Steenrod algebra $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$, we calculate the $C_4$ equivariant Bredon cohomology of the classifying space $\mathbf R P^{\infty \rho}=B_{C_4}\Sigma_{2}$ as an $RO(C_4)$ graded Green-functor. We prove that as a module over the homology of a point (which we also compute), this cohomology is not flat. As a result, it can't be used as a test module for obtaining generators in $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$ as Hu-Kriz do in the $C_2$ case., Comment: Comments are most welcome! 41 pages, 5 figures. Code available on https://github.com/NickG-Math/Mackey

In this very short note, we expand the Hu-Kriz computation of the $G$-equivariant Borel dual Steenrod algebra in characteristic $2$, from the group $G=C_2$ to all power-$2$ cyclic groups $G=C_{2^n}$., Comment: Comments are most welcome! 5 pages

Fog computing is a promising computing paradigm for time-sensitive Internet of Things (IoT) applications. It helps to process data close to the users, in order to deliver faster processing outcomes than the Cloud; it also helps to reduce network traffic. The computation environment in the Fog computing is highly dynamic and most of the Fog devices are battery powered hence the chances of application failure is high which leads to delaying the application outcome. On the other hand, if we rerun the application in other devices after the failure it will not comply with time-sensitiveness. To solve this problem, we need to run applications in an energy-efficient manner which is a challenging task due to the dynamic nature of Fog computing environment. It is required to schedule application in such a way that the application should not fail due to the unavailability of energy. In this paper, we propose a multiple linear, regression-based resource allocation mechanism to run applications in an energy-aware manner in the Fog computing environment to minimise failures due to energy constraint. Prior works lack of energy-aware application execution considering dynamism of Fog environment. Hence, we propose A multiple linear regression-based approach which can achieve such objectives. We present a sustainable energy-aware framework and algorithm which execute applications in Fog environment in an energy-aware manner. The trade-off between energy-efficient allocation and application execution time has been investigated and shown to have a minimum negative impact on the system for energy-aware allocation. We compared our proposed method with existing approaches. Our proposed approach minimises the delay and processing by 20%, and 17% compared with the existing one. Furthermore, SLA violation decrease by 57% for the proposed energy-aware allocation., Comment: 8 Pages, 9 Figures