G-Hypergroups: Hypergroups with a Group-Isomorphic Heart

Hypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is the trivial group. However, very little is known about hypergroups that are neither 1-hypergroups nor belong to the first class. The goal of this work is to take a first step in classifying G-hypergroups, that is, hypergroups whose heart is a nontrivial group. We introduce their main properties, with an emphasis on G-hypergroups whose the heart is a torsion group. We analyze the main properties of the stabilizers of group actions of the heart, which play an important role in the construction of multiplicative tables of G-hypergroups. Based on these results, we characterize the G-hypergroups that are of type U on the right or cogroups on the right. Finally, we present the hyperproduct tables of all G-hypergroups of size not larger than 5, apart of isomorphisms.

Benefits and Challenges of Online Teaching During the COVID-19 Pandemic at Rundu Campus of the University of Namibia

This paper presents the benefits and challenges of online teaching during the COVID-19 pandemic experienced by educators at the Rundu Campus of the University of Namibia. Researchers used a structured interview guide to collect data from 14 conveniently selected lecturers from a population of 65. Findings of the study indicate that the benefits of using online teaching and learning include flexibility, ability to teach large classes; increased interaction and engagement between lecturers and students; and increased learning opportunities for lecturers. The study further found that some of the significant challenges lecturers experienced with online teaching and learning include lack of information and technology skills, internet connectivity and availability; poor student attendance; and loneliness. The study provided crucial information on lecturers' progress within the framework of online teaching and learning mode. The paper recommends that lecturers receive formal training on online teaching and learning tools to minimise the limitations. The study also suggests increased psychosocial support for lecturers to curb feelings of isolation and loneness during this time.

Sobolev-to-Lipschitz property on $${\mathsf {QCD}}$$-spaces and applications

We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature-dimension condition recently introduced in Milman (Commun Pure Appl Math, to appear). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These results apply in particular to large classes of (ideal) sub-Riemannian manifolds.

Positive answer to the invariant and hyperinvariant subspaces problems for hyponormal operators

The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

Cohen–Macaulay binomial edge ideals and accessible graphs

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen–Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen–Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness.

Machine Learning in Mental Health: Recognizing the Symptoms of Depressive and Anxiety Disorders

Machine learning technologies can be used to extract important information about mental health of individuals from unstructured texts, including social media posts and transcriptions of counselling sessions. So far machines have been trained to detect the presence of mental disorder, but they still need to learn to recognize individual symptoms in order to make a valid diagnosis. This study presents an attempt to train a machine learning model to recognize individual symptoms of anxiety and depressive disorders. We collected 1065 posts about depression and anxiety from online psychological forums; divided messages into 7149 replicas and classified each replica according to the DSM-5 criteria. We found that users mention emotional symptoms far more often than physical ones. An imbalanced dataset did not allow us to recognize the full spectrum of symptoms with sufficient accuracy. A two-stage model was developed: at the first stage the model recognized large classes of depression, anxiety or irritability. At the second stage it recognized sub-classes of symptoms, such as depressed mood, suicidal intent and negative self-talk within the depression class; and excessive worry and social anxiety within the anxiety class. The research has demonstrated the potential possibility of extracting symptoms of mental disorders from unstructured data on a larger dataset.

Support theorems for Funk-type isodistant Radon transforms on constant curvature spaces

A connected maximal submanifold in a constant curvature space is called isodistant if its points are in equal distances from a totally geodesic of codimension 1. The isodistant Radon transform of a suitable real function f on a constant curvature space is the function on the set of the isodistants that gives the integrals of f over the isodistants using the canonical measure. Inverting the isodistant Radon transform is severely overdetermined because the totally geodesic Radon transform, which is a restriction of the isodistant Radon transform, is invertible on some large classes of functions. This raises the admissibility problem that is about finding reasonably small subsets of the set of the isodistants such that the associated restrictions of the isodistant Radon transform are injective on a reasonably large set of functions. One of the main results of this paper is that the Funk-type sets of isodistants are admissible, because the associated restrictions of the isodistant Radon transform, we call them Funk-type isodistant Radon transforms, satisfy appropriate support theorems on a large set of functions. This unifies and sharpens several earlier results for the sphere, and brings to light new results for every constant curvature space.

How Teacher and Grammarly Feedback Complement One Another in Myanmar EFL Students’ Writing

Providing feedback on students’ writing is considered important by both writing teachers and students. However, contextual constraints including excess workloads and large classes pose major and recurrent challenges for teachers. To lighten the feedback burden, teachers can take advantage of a range of automated feedback tools. This paper investigated how automated feedback can be integrated into traditional teacher feedback by analyzing the focus of teacher and Grammarly feedback through a written feedback analysis of language- and content-related issues. This inquiry considered whether and how successfully students exploited feedback from different sources in their revisions and how the feedback provisions helped improve their writing performance. The study sample of texts was made up of 216 argumentative and narrative essays written by 27 low-intermediate level students at a Myanmar university over a 13-week semester. By analyzing data from the feedback analysis, we found that Grammarly provided feedback on surface-level errors, whereas teacher feedback covered both lower- and higher-level writing concerns, suggesting a potential for integration. The results from the revision analysis and pre- and post-tests suggested that students made effective use of the feedback received, and their writing performance improved according to the assessment criteria. The data were triangulated with self-assessment questionnaires regarding students’ emic perspectives on how useful they found the feedback. The pedagogical implications for integrating automated and teacher feedback are presented.