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.

Topological Properties of Braid-Paths Connected 2-Simplices in Covering Spaces under Cyclic Orientations

In general, the braid structures in a topological space can be classified into algebraic forms and geometric forms. This paper investigates the properties of a braid structure involving 2-simplices and a set of directed braid-paths in view of algebraic as well as geometric topology. The 2-simplices are of the cyclically oriented variety embedded within the disjoint topological covering subspaces where the finite braid-paths are twisted as well as directed. It is shown that the generated homotopic simplicial braids form Abelian groups and the twisted braid-paths successfully admit several varieties of twisted discrete path-homotopy equivalence classes, establishing a set of simplicial fibers. Furthermore, a set of discrete-loop fundamental groups are generated in the covering spaces where the appropriate weight assignments generate multiplicative group structures under a variety of homological formal sums. Interestingly, the resulting smallest non-trivial group is not necessarily unique. The proposed variety of homological formal sum exhibits a loop absorption property if the homotopy path-products are non-commutative. It is considered that the topological covering subspaces are simply connected under embeddings with local homeomorphism maintaining generality.

Gastrointestinal manifestations as early symptoms to diagnose COVID-19 paediatric cases

Background: The international situation surrounding the COVID-19 pandemic is seeing multiple countries battle various waves of the SARS-CoV-2 virus infections, with millions of individuals being infected globally. COVID-19 cases initially involved the immuno-compromised and elderly. As the virus has infected millions globally, the demographic profile of cases has shifted with more children being infected; this increase in younger individuals contracting the infection has resulted in new symptoms with altered manifestations and presentations of the disease in the young. In comparison to the severe symptoms of COVID-19 in adults children present with a more trivial group of symptoms. The majority of children develop mild symptoms or remain asymptomatic. This is in stark contrast to adults who have a higher admission rate with severe symptoms. A sign of great importance and now incidence in pediatric cases with COVID-19 is that of the gastrointestinal tract. The virus has a tropism for the GIT due to the presence of ACE2 receptors, which facilitate the entry of the virus into the cell. Conclusion: It is now established that the GIT symptoms form part of a newly recognized multisystem inflammatory syndrome (MIS-C) which occurs as a result and or manifestation of the COVID-19 infection. The innate difficulty in correctly and accurately diagnosing such a case is that the symptoms very close mimic gastroenteritis and acute abdominal pathologies. Therefore, physicians need to be aware of the various manners in which the COVID-19 infection manifests itself in children to diagnose better and isolate the cases.

Locally projective graphs and their densely embedded subgraphs

The article contributes to the classification project of locally projective graphs and their locally projective groups of automorphisms outlined in Chapter 10 of Ivanov (The Mathieu Groups, Cambridge University Press, Cambridge, 2018). We prove that a simply connected locally projective graph $$\Gamma $$ Γ of type (n, 3) for $$n \ge 3$$ n ≥ 3 contains a densely embedded subtree provided (a) it contains a (simply connected) geometric subgraph at level 2 whose stabiliser acts on this subgraph as the universal completion of the Goldschmidt amalgam $$G_3^1\cong \{S_4 \times 2,S_4 \times 2\}$$ G 3 1 ≅ { S 4 × 2 , S 4 × 2 } having $$S_6$$ S 6 as another completion, (b) for a vertex x of $$\Gamma $$ Γ the group $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) which stabilizes every line passing through x induces on the neighbourhood $$\Gamma (x)$$ Γ ( x ) of x the (dual) natural module $$2^n$$ 2 n of $$G(x)/G_{\frac{1}{2}}(x) \cong L_n(2)$$ G ( x ) / G 1 2 ( x ) ≅ L n ( 2 ) , (c) G(x) splits over $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) , (d) the vertex-wise stabilizer $$G_1(x)$$ G 1 ( x ) of the neighbourhood of x is a non-trivial group, and (e) $$n \ne 4$$ n ≠ 4 .

Homologically trivial group actions on elliptic surfaces

In this paper, we study the minimal symplectic elliptic surfaces [Formula: see text] with homologically trivial symplectic finite group actions, and get a rigidity theorem under some restriction.

On the indeterminacy of Milnor’s triple linking number

In the 1950s Milnor defined a family of higher-order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received fruitful study since its inception. In the case that a link [Formula: see text] has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking number is only well-defined modulo their greatest common divisor. In recent work Davis–Nagel–Orson–Powell produce a single invariant called the total triple linking number refining the triple linking number and taking values in an abelian group called the total Milnor quotient. They present examples for which this quotient is nontrivial even though each of the individual triple linking numbers take values in the trivial group, [Formula: see text], and so carry no information. As a consequence, the total triple linking number carries more information than do the classical triple linking numbers. The goal of this paper is to compute this group and show that when [Formula: see text] is a link of at least six components it is nontrivial. Thus, this total triple linking number carries information for every [Formula: see text]-component link, even though the classical triple linking numbers often carry no information.

Quotients of del Pezzo surfaces

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

Characterization of small polygroups by their fundamental groups

In this paper, we consider polygroup [Formula: see text] and prove necessary and sufficient conditions such that [Formula: see text] is non-commutative. Then by using the Maple programming, we obtain all polygroups of order less than five up to isomorphism. In fact, we determine all 115 non-isomorphic polygroups of order less than five and characterize them by their fundamental groups, i.e., polygroups with same fundamental group, say [Formula: see text], classifies in the class [Formula: see text]. Finally, we obtain that the fundamental groups of 94 polygroups are the trivial group. The numbers of polygroups in classes [Formula: see text] and [Formula: see text] are 16 and 3, respectively, and the classes [Formula: see text] and [Formula: see text] are singleton.

Sizes of spaces of triangulations of 4-manifold and balanced presentations of the trivial group

Let [Formula: see text] be any compact four-dimensional PL-manifold with or without boundary (e.g. the four-dimensional sphere or ball). Consider the space [Formula: see text] of all simplicial isomorphism classes of triangulations of [Formula: see text] endowed with the metric defined as follows: the distance between a pair of triangulations is the minimal number of bistellar transformations required to transform one of the triangulations into the other. Our main result is the existence of an absolute constant [Formula: see text] such that for every [Formula: see text] and all sufficiently large [Formula: see text] there exist more than [Formula: see text] triangulations of [Formula: see text] with at most [Formula: see text] simplices such that pairwise distances between them are greater than [Formula: see text] ([Formula: see text] times). This result follows from a similar result for the space of all balanced presentations of the trivial group. (“Balanced” means that the number of generators equals to the number of relations). This space is endowed with the metric defined as the minimal number of Tietze transformations between finite presentations. We prove a similar exponential lower bound for the number of balanced presentations of length [Formula: see text] with four generators that are pairwise [Formula: see text]-far from each other. If one does not fix the number of generators, then we establish a super-exponential lower bound [Formula: see text] for the number of balanced presentations of length [Formula: see text] that are [Formula: see text]-far from each other.