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.

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.

On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

Commutative semigroups whose endomorphisms are power functions

For any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

Computing Interval Discrete Logarithm Problem with Restricted Jump Method

The interval discrete logarithm problem(IDLP) is to find a solution n such that gn = h in a finite cyclic group G = 〈g〉, where h ∈ G and n belongs to a given interval. To accelerate solving IDLP, a restricted jump method is given to speed up Pollard’s kangaroo algorithm in this paper. Since the Pollard’ kangaroo-like method need to compute the intermediate value during every iteration, the restricted jump method gives another way to reuse the intermediate value so that each iteration is speeded up at least 10 times. Actually, there are some variants of kangaroo method pre-compute the intermediate value and reuse the pre-computed value in each iteration. Different from the pre-compute method that reuse the pre-computed value, the restricted jump method reuse the value naturally arised in pervious iteration, so that the improved algorithm not only avoids precomputation, but also speeds up the efficiency of each iteration. So only two or three large integer multiplications are needed in each iteration of the restricted jump method. And the average large integer multiplication times is (1:633 + o(1)) N in restricted jump method, which is verified in the experiment.

DETECTING STEINER AND LINEAR ISOMETRIES OPERADS

We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When G is a finite abelian group, we prove that a G-indexing system is realized by a Steiner operad if and only if it is generated by cyclic G-orbits. When G is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than 3, we prove that a G-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill’s horn-filling condition. We also repackage the data in an indexing system as a certain kind of partial order. We call these posets transfer systems, and develop basic tools for computing with them.

Erratum: Vertex connectivity of the power graph of a finite cyclic group II

We retract [1, Lemma 3.4] as the statement is incorrect. In consequence, we correct the statements of Theorems 1.2 and 4.5 and their proofs.

Extended periodic links and HOMFLYPT polynomial

Extended strongly periodic links have been introduced by Przytycki and Sokolov as a symmetric surgery presentation of three-manifolds on which the finite cyclic group acts without fixed points. The purpose of this paper is to prove that the symmetry of these links is reflected by the first coefficients of the HOMFLYPT polynomial.

Hereditary Coreflective Subcategories in Certain Categories of Abelian Semitopological Groups

Let A be an epireflective subcategory of the category of all semitopological groups that consists only of abelian groups. We describe maximal hereditary coreflective subcategories of A that are not bicoreflective in A in the case that the A -reflection of the discrete group of integers is a finite cyclic group, the group of integers with a topology that is not T 0 , or the group of integers with the topology generated by its subgroups of the form p n , where n ∈ N , p ∈ P and P is a given set of prime numbers.