The Canonical Module of GT-Varieties and the Normal Bundle of RL-Varieties

In this paper, we study the geometry of GT-varieties $$X_{d}$$ X d with group a finite cyclic group $$\Gamma \subset {{\,\mathrm{GL}\,}}(n+1,\mathbb {K})$$ Γ ⊂ GL ( n + 1 , K ) of order d. We prove that the homogeneous ideal $${{\,\mathrm{I}\,}}(X_{d})$$ I ( X d ) of $$X_{d}$$ X d is generated by binomials of degree at most 3 and we provide examples reaching this bound. We give a combinatorial description of the canonical module of the homogeneous coordinate ring of $$X_{d}$$ X d and we show that it is generated by monomial invariants of $$\Gamma $$ Γ of degree d and 2d. This allows us to characterize the Castelnuovo–Mumford regularity of the homogeneous coordinate ring of $$X_d$$ X d . Finally, we compute the cohomology table of the normal bundle of the so-called RL-varieties. They are projections of the Veronese variety $$\nu _{d}(\mathbb {P}^{n}) \subset \mathbb {P}^{\left( {\begin{array}{c}n+d\\ n\end{array}}\right) -1}$$ ν d ( P n ) ⊂ P n + d n - 1 which naturally arise from level GT-varieties.

ON SOME EQUIVALENCE RELATION ON NON-ABELIAN $\CA$-GROUPS

A non-abelian group $G$ is called a $\CA$-group ($\CC$-group) if $C_G(x)$ is abelian(cyclic) for all $x\in G\setminus Z(G)$. We say $x\sim y$ if and only if $C_G(x)=C_G(y)$.We denote the equivalence class including $x$ by$[x]_{\sim}$. In this paper, we prove thatif $G$ is a $\CA$-group and $[x]_{\sim}=xZ(G)$, for all $x\in G$, then $2^{r-1}\leq|G'|\leq 2^{r\choose 2}$.where $\frac {|G|}{|Z(G)|}=2^{r}, 2\leq r$ and characterize all groups whose $[x]_{\sim}=xZ(G)$for all $x\in G$ and $|G|\leq 100$. Also, we will show that if $G$ is a $\CC$-group and $[x]_{\sim}=xZ(G)$,for all $x \in G$, then $G\cong C_m\times Q_8$ where $C_m$ is a cyclic group of odd order $m$ andif $G$ is a $\CC$-group and $[x]_{\sim}=x^G$, for all $x\in G\setminus Z(G)$, then $G\cong Q_8$.

Dihedral group and classification of G-circuits of length 10

In this paper, we classify G-circuits of length 10 with the help of the location of the reduced numbers lying on G-circuit. The reduced numbers play an important role in the study of modular group action on P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -subset of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ . For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of real quadratic fields. In particular, we classify P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ = ⋃ k ∈ N Q * k 2 m $={\bigcup }_{k\in N}{Q}^{{\ast}}\left(\sqrt{{k}^{2}m}\right)$ containing G-circuits of length 10 and determine that the number of equivalence classes of G-circuits of length 10 is 41 in number. We also use dihedral group to explore cyclically equivalence classes of circuits and use cyclic group to explore similar G-circuits of length 10 corresponding to 10 of these circuits. By using cyclically equivalent classes of circuits and similar circuits, we obtain the exact number of G-orbits and the structure of G-circuits corresponding to cyclically equivalent classes. This study also helps us in classifying the reduced numbers lying in the P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits.

Interactions between Homotopy and Topological Groups in Covering (C, R) Space Embeddings

The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological (C, R) spaces. The covering spaces form a Noetherian structure under topological injective embeddings. The locally path-connected components of covering spaces establish a set of finite topological groups, maintaining group homomorphism. The homeomorphic topological embedding of covering spaces and base space into a fibered non-compact topological (C, R) space generates two classes of fibers based on the location of identity elements of homomorphic groups. A compact general fiber gives rise to the discrete variety of fundamental groups in the embedded covering subspace. The path-homotopy equivalence is admitted by multiple identity fibers if, and only if, the group homomorphism is preserved in homeomorphic topological embeddings. A single identity fiber maintains the path-homotopy equivalence in the discrete fundamental group. If the fiber is an identity-rigid variety, then the fiber-restricted finite and symmetric translations within the embedded covering space successfully admits path-homotopy equivalence involving kernel. The topological projections on a component and formation of 2-simplex in fibered compact covering space embeddings generate a prime order cyclic group. Interestingly, the finite translations of the 2-simplexes in a dense covering subspace assist in determining the simple connectedness of the covering space components, and preserves cyclic group structure.

Good Groups

Good groups are defined in terms of whether capped gropes of height 1.5 contain certain types of immersed discs. The disc embedding theorem holds for 4-manifolds with good fundamental group. It is proven that the infinite cyclic group and finite groups are good, and that extensions and colimits of good groups are good. This shows that all elementary amenable groups are good. The proofs use grope height raising and contraction, together with an analysis of how fundamental group elements behave under these operations. A central open problem in the study of topological 4-manifolds is to determine precisely which groups are good.

Quercetin affects uterine smooth muscle contractile activity in gilts

Quercetin is a polyphenolic flavonoid occurring in leaves, stems, flowers and fruits of many plants. In traditional Chinese medicine, it is used as a natural therapeutic agent with a broad spectrum of activities (antioxidant, neuroprotective, anti-inflammatory, anticancer, antibacterial and antiviral). Moreover, quercetin affects function of the reproductive tract, however the knowledge of this activity is still fragmentary. Therefore, this study aimed to determine the influence of quercetin on the contractile activity of the porcine myometrium collected from immature (n = 6), cyclic (n = 6) and early pregnant (n = 6) gilts. Strips of the myometrium (comprising longitudinal and circular layer) were resected from the middle part of the uterine horns and the isometric contractions were recorded. After 60–90 min of preincubation, the strips were stimulated with quercetin in increasing (10−13–10−1 M) concentrations and the changes in the tension amplitude and frequency of contractions were measured. Quercetin decreased (P<0.01–0.001) the amplitude of contractions at concentrations 10−11–10−1 M and 10−10–10−1 M in cyclic and early pregnant groups, respectively. The frequency of contractions decreased in all groups but was the highest (at concentrations 10−11–10−1 M; P<0.05–0.001) in the cyclic group and the lowest (at concentrations 10−5–10−1 M; P<0.01) in the immature group. The tension decreased only in the cyclic group after quercetin administration in high concentrations (10−6–10−1 M; P<0.05–0.01). The results indicate that quercetin causes relaxation of the porcine uterine smooth muscle but this activity is strongly related to the physiological status of the gilts.

Identity Graph of Finite Cyclic Groups

This paper discusses how to express a finite group as a graph, specifically about the identity graph of a cyclic group. The term chosen for the graph is an identity graph, because it is the identity element of the group that holds the key in forming the identity graph. Through the identity graph, it can be seen which elements are inverse of themselves and other properties of the group. We will look for the characteristics of identity graph of the finite cyclic group, for both cases of odd and even order.

Sums of element orders in groups of odd order

For a finite group [Formula: see text], let [Formula: see text] denote the sum of element orders of [Formula: see text]. If [Formula: see text] is a positive integer let [Formula: see text] be the cyclic group of order [Formula: see text]. It is known that [Formula: see text] is the maximum value of [Formula: see text] on the set of groups of order [Formula: see text], and [Formula: see text] if and only if [Formula: see text] is cyclic of order [Formula: see text]. In this paper, we investigate the second largest value of [Formula: see text] on the set of groups of order [Formula: see text] and the structure of groups [Formula: see text] of order [Formula: see text] with this value of [Formula: see text] when [Formula: see text] is odd.

A generalization of a result on the sum of element orders of a finite group

Let G be a finite group and let ψ(G) denote the sum of element orders of G. It is well-known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group Cn . For nilpotent groups, we prove a natural generalization of this result, obtained by replacing the element orders of G with the element orders relative to a certain subgroup H of G.