Automorphism Of Order Research Articles

Binary Sequences With a Low Correlation via Cyclotomic Function Fields

Due to wide applications of binary sequences with a low correlation to communications, various constructions of such sequences have been proposed in the literature. Many efforts have been made to construct good binary sequences with various lengths. However, most of the known constructions make use of the multiplicative cyclic group structure of finite field <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2^{n}}$ </tex-math></inline-formula> for a positive integer <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. It is often overlooked in this community that all <inline-formula> <tex-math notation="LaTeX">$2^{n}+1$ </tex-math></inline-formula> rational places (including &#x201C;the place at infinity&#x201D;) of the rational function field over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2^{n}}$ </tex-math></inline-formula> form a cyclic structure under an automorphism of order <inline-formula> <tex-math notation="LaTeX">$2^{n}+1$ </tex-math></inline-formula>. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length <inline-formula> <tex-math notation="LaTeX">$2^{n}+1$ </tex-math></inline-formula> via cyclotomic function fields over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2^{n}}$ </tex-math></inline-formula>. Each family of our sequences has size <inline-formula> <tex-math notation="LaTeX">$2^{n}-1$ </tex-math></inline-formula> and its correlation is upper bounded by <inline-formula> <tex-math notation="LaTeX">$\lfloor 2^{(n+2)/2}\rfloor $ </tex-math></inline-formula>. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length <inline-formula> <tex-math notation="LaTeX">$2^{n}+1$ </tex-math></inline-formula>. Moreover, our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length <inline-formula> <tex-math notation="LaTeX">$2^{n}-1$ </tex-math></inline-formula> for even <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller.

Nonsymplectic automorphisms of prime order on O’Grady’s sixfolds

We classify nonsymplectic automorphisms of prime order on irreducible holomorphic symplectic manifolds of O’Grady’s 6-dimensional deformation type. More precisely, we give a classification of the invariant and coinvariant sublattices of the second integral cohomology group.

Extremal Binary and Ternary Codes of Length 60 with an Automorphism of Order 29 and a Generalization

In this paper, all extremal Type I and Type III codes of length 60 with an automorphism of order 29 are classified up to equivalence. In both cases, it has been proven that there are three inequivalent codes. In addition, a new family of self-dual codes over non-binary fields is presented.

Classification of Cartan embeddings which are austere submanifolds

We classify automorphisms of finite order on compact connected simple Lie groups by which the induced Cartan embedding is an austere embedding. Furthermore, we consider the problem whether an austere Cartan embedding is weakly reflective or not.

On binary LCD codes possessing an automorphism of order 13

In this work we apply the method for constructing binary LCD codes via an automorphism of prime order described in [3] and [4]. Thus we obtain all optimal LCD codes of lengths 26, 27 and 28 possessing an automorphism of order 13 with two cycles.

The integral monodromy of the cycle type singularities

The middle homology of the Milnor fiber of a quasihomogeneous polynomial with an isolated singularity is a ${\mathbb Z}$-lattice and comes equipped with an automorphism of finite order, the integral monodromy. Orlik (1972) made a precise conjecture, which would determine this monodromy in terms of the weights of the polynomial. Here we prove this conjecture for the cycle type singularities. A paper of Cooper (1982) with the same aim contained two mistakes. Still it is very useful. We build on it and correct the mistakes. We give additional algebraic and combinatorial results.

Spectral order isomorphisms and AW$AW$*‐factors

AbstractThe paper deals with spectral order isomorphisms in the framework of *‐algebras. We establish that every spectral order isomorphism between sets of all self‐adjoint elements (or between sets of all effects, or between sets of all positive elements) in *‐factors of Type I has a canonical form induced by a continuous function calculus and an isomorphism between projection lattices. In particular, this solves an open question about spectral order automorphisms of the set of all (bounded) self‐adjoint operators on an infinite‐dimensional Hilbert space. We also discuss spectral order isomorphisms preserving, in addition, orthogonality in both directions.

CCEPTED Automorphisms on complex simple Lie algebras of order 3

For complex simple Lie algebras, the article provides classification of all automorphisms of order 3. The method is an extension of Dynkin diagrams, so that the classification is a listing of diagrams which represent automorphisms of order 3. This work extends an earlier result on automorphisms of order 2. As an application, it shows that for automorphisms of orders 2 and 3 only, the invariant subalgebra determines the automorphism.

Non-purely non-symplectic automorphisms of order 6 on $K3$ surfaces

In this paper we study non-symplectic automorphisms of order 6 on $K3$ surfaces which are not purely. In particular we shall describe their fixed loci.

Sigma involutions associated with parafermion vertex operator algebra

An irreducible module for the parafermion vertex operator algebra is said to be of σ-type if an automorphism of the fusion algebra of of order k is trivial on it. For any integer k ≥ 3, we show that there exists an automorphism of order 2 of the subalgebra of the fusion algebra of spanned by the irreducible direct summands of σ-type irreducible -modules, where θ is an involution of . We discuss some examples of such an automorphism as well.

A special configuration of 12 conics and generalized Kummer surfaces

A generalized Kummer surface X obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $$9\mathbf{A}_{2}$$ -configuration of $$(-2)$$ -curves. Such a configuration plays the role of the $$16\mathbf{A}_{1}$$ -configurations for usual Kummer surfaces. In this paper we construct 9 other such $$9\mathbf{A}_{2}$$ -configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $$9\mathbf{A}_{2}$$ -configurations are obtained by taking the pullback of a certain configuration of 12 conics which are in special position with respect to the branch curve, plus some singular quartic curves. We then construct some automorphisms of the K3 surface sending one configuration to another. We also give various models of X and of the generic fiber of its natural elliptic pencil.

Noninner automorphisms of order p for finite p-groups of restricted coclass

Noninner automorphisms of order p for finite p-groups of restricted coclass

On Periodic Groups with a Regular Automorphism of Order 4

On Periodic Groups with a Regular Automorphism of Order 4

On automorphism groups of a biplane (121,16,2)

The existence of a biplane with parameters (121,16,2) is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a possible biplane D of order 14 divides 27⋅32⋅5⋅7⋅11⋅13. In this paper we show that such a biplane does not have an automorphism of order 11 or 13, and thereby establish that |Aut(D)| divides 27⋅32⋅5⋅7. Further, we study a possible action of an automorphism group of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters (121,16,2).

Finite p-groups with non-cyclic center have a non-inner automorphism of order p

Finite p-groups with non-cyclic center have a non-inner automorphism of order p

Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

A right Engel sink of an element g of a group G is a set {{mathscr {R}}}(g) such that for every xin G all sufficiently long commutators [...[[g,x],x],dots ,x] belong to {mathscr {R}}(g). (Thus, g is a right Engel element precisely when we can choose {{mathscr {R}}}(g)={ 1}.) We prove that if a profinite group G admits a coprime automorphism varphi of prime order such that every fixed point of varphi has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set {{mathscr {E}}}(g) such that for every xin G all sufficiently long commutators [...[[x,g],g],dots ,g] belong to {{mathscr {E}}}(g). (Thus, g is a left Engel element precisely when we can choose {mathscr {E}}(g)={ 1}.) We prove that if a profinite group G admits a coprime automorphism varphi of prime order such that every fixed point of varphi has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

Twisted rational r-matrices and algebraic Bethe ansatz: Application to generalized Gaudin and Richardson models

In the present paper we develop the algebraic Bethe ansatz approach to the case of non-skew-symmetric gl(2)⊗gl(2)-valued Cartan-non-invariant classical r-matrices with spectral parameters. We consider the two families of these r-matrices, namely, the two non-standard rational r-matrices twisted with the help of second order automorphisms and realize the algebraic Bethe ansatz method for them. We study physically important examples of the Gaudin-type and BCS-type systems associated with these r-matrices and obtain explicitly the Bethe vectors and the spectrum for the corresponding quantum hamiltonians in terms of solutions of Bethe equations.

An inverse Jacobian algorithm for Picard curves

We study the inverse Jacobian problem for the case of Picard curves over {mathbb {C}}. More precisely, we elaborate on an algorithm that, given a small period matrix varOmega in {mathbb {C}}^{3times 3} corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most 4. In particular, we obtain the complete list of maximal CM Picard curves defined over {mathbb {Q}}. In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].

Automorphisms of Finite Order of Soluble Groups of Finite Rank

We study the effect on sections of a soluble-by-finite group G of finite rank of an almost fixed-point-free automorphism φ of G of finite order. We also elucidate the structure of G if φ has order 4 and if G is also (torsion-free)-by-finite. The latter extends recent work of Xu, Zhou and Liu.

PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Let \U0001d52e be a finite-dimensional Lie algebra. The symmetric algebra (\U0001d52e) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (\U0001d52e) to any finite order automorphism ϑ of \U0001d52e. We study related Poisson-commutative subalgebras (\U0001d52e; ϑ) of \U0001d4ae(\U0001d52e) and associated Lie algebra contractions of \U0001d52e. To obtain substantial results, we have to assume that \U0001d52e = \U0001d524 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If \U0001d524 = \U0001d525⊕⋯⊕\U0001d525 (sum of k copies), where \U0001d525 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (\U0001d52e; ϑ) is polynomial and maximal. Furthermore, we quantise this (\U0001d524; ϑ) using a Gaudin subalgebra in the enveloping algebra \U0001d4b0(\U0001d524).