Automorphisms Of Type Research Articles

On Generic Automorphisms

In this article we investigate generic automorphisms of countable models. Hodges et al. 1993 introduces the notion of SI (small index) generic automorphisms. They used the existence of small index generics to show the small index property of the model. Truss 1989 defines the notion of Truss generic automorphisms. An automorphism f ofM is called Truss generic if its conjugacy class is comeagre in the automorphism group ofM. We study the relationship between these two types of generic automorphisms. We show that either the countable random graph or a countable arithmetically saturated model of True Arithmetic have both SI generic and Truss generic automorphisms. We prove that the dense linear order has the small index property and Truss-generic automorphisms but it does not have SI generic automorphisms. We also construct an example of a countable structure which has SI generics but it does not have Truss generics.

Об автоморфизмах и дифференцированиях редуцированных алгебр и коалгебр инцидентности

Incidence algebras of partially ordered sets over commutative rings are an important and characteristic example of function rings. From a partially ordered set, one can obtain an incidence coalgebra. Using certain equivalence relations on the set of all intervals of a locally finite poset, reduced incidence algebras and reduced incidence coalgebras are defined. These objects have a much more complex structure compared to incidence algebras and incidence coalgebras. This article introduces two types of automorphisms of the reduced incidence algebra – multiplicative and order, as well as one type of derivations – additive derivation. As for incidence coalgebras, there are no works devoted to their automorphisms or derivations. The article discusses a possible approach to the study of automorphisms and derivations of incidence coalgebras.

Multi-dimensional almost automorphic type sequences and applications

Abstract In this paper, we investigate several new classes of multi-dimensional almost automorphic type sequences and focus on their applications to various difference equations involving Volterra difference equations. We provide many structural results, illustrative examples and open problems about the notion under consideration.

Polynomial Automorphisms of Characteristic Order and Their Invariant Rings

Let k be a field of characteristic p > 0. We discuss the automorphisms of the polynomial ring k[x1,…,xn] of order p, or equivalently the Z/pZ-actions on the affine space ${\textbf {A}}_{k}^{n}$ . When n = 2, such an automorphism is known to be a conjugate of an automorphism fixing a variable. It is an open question whether the same holds when n ≥ 3. In this paper, (1) we give a negative answer to this question when n = 3. In fact, we show that every Ga-action on ${\textbf {A}}_{k}^{3}$ of rank three yields an automorphism of k[x1,x2,x3] of order p which is not a conjugate of an automorphism fixing a variable. We give a family of such automorphisms of k[x1,x2,x3] by constructing a family of rank three Ga-actions on ${\textbf {A}}_{k}^{3}$ . (2) For the automorphisms of k[x1,x2,x3] induced by this family of Ga-actions, we show that the invariant ring is isomorphic to k[x1,x2,x3] if and only if the plinth ideal is principal, under some mild assumptions. (3) We study the Nagata type automorphism of R[x1,x2], where R is a UFD of characteristic p > 0. This type of automorphism is of order p. We give a necessary and sufficient condition for the invariant ring to be isomorphic to R[x1,x2]. This condition is equivalent to the condition that the plinth ideal is principal.

Linear invariance and extension operators of Pfaltzgraff-Suffridge type

"We consider the image of a linear-invariant family $\mathcal{F}$ of normalized locally biholomorphic mappings defined in the Euclidean unit ball $\B_n$ of $\C^n$ under the extension operator $$\Phi_{n,m,\beta}[f](z,w) = \mleft( f(z), [Jf(z)]^\beta w\mright), \quad (z,w) \in \B_{n+m} \subseteq \C^n \times \C^m,$$ where $\beta \in \C$, $Jf$ denotes the Jacobian determinant of $f$, and the branch of the power function taking $0$ to $1$ is used. When $\beta=1/(n+1)$ and $m=1$, this is the Pfaltzgraff--Suffridge extension operator. In particular, we determine the order of the linear-invariant family on $\B_{n+m}$ generated by the image in terms of the order of $\mathcal{F}$, taking note that the resulting family has minimum order if and only if either $\beta \in (-1/m,1/(n+1)]$ and the family $\mathcal{F}$ has minimum order or $\beta=-1/m$. We will also see that order is preserved when generating a linear-invariant family from the family obtained by composing $\mathcal{F}$ with a certain type of automorphism of $\C^n$, leading to consequences for various extension operators including the modified Roper--Suffridge extension operator introduced by the author."

Automorphism groups and normal forms in Normaliz

We discuss the computation of automorphism groups and normal forms of cones and polyhedra in Normaliz and indicate its implementation via nauty. The types of automorphisms include integral, rational, Euclidean and combinatorial, as well as algebraic for polytopes defined over real algebraic number fields. Examples treated in detail are the icosahedron and linear-ordering polytopes whose Euclidean automorphism groups are determined.

Multi-dimensional Almost Automorphic Type Functions and Applications

Multi-dimensional Almost Automorphic Type Functions and Applications

Ore Extensions for the Sweedler’s Hopf Algebra H4

The aim of this paper is to classify all Hopf algebra structures on the quotient of Ore extensions H4[z;σ] of automorphism type for the Sweedler′s 4-dimensional Hopf algebra H4. Firstly, we calculate all equivalent classes of twisted homomorphisms (σ,J) for H4. Then we give the classification of all bialgebra (Hopf algebra) structures on the quotients of H4[z;σ] up to isomorphism.

On almost automorphic type solutions of abstract integral equations, a Bohr-Neugebauer type property and some applications

In the present work we give some sufficient conditions to obtain a unique almost automorphic solution to abstract nonlinear integral equations which are simultaneously of advanced and delayed type and also a unique asymptotically almost automorphic mild solution to abstract integro-differential equations with nonlocal initial conditions, both situations are posed on Banach spaces. Also, we develop a Bohr-Neugebauer type result for the abstract integral equations. Before that, we introduce the notion of λ-bounded functions, develop the appropriate abstract theory and discuss the almost periodic situation. As applications, we study the existence of an asymptotically almost automorphic solution to integro-differential equations modeling heat conduction in materials with memory and also the existence of the almost automorphic solution to semilinear parabolic evolution equations with finite delay.

Some finiteness results for groups of automorphisms of manifolds

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of $R^n$ and various types of automorphisms of 2-connected manifolds.

Ore Extensions of Automorphism Type for Hopf Algebras

The aim of this paper is to study bialgebras(Hopf algebras) on the skew polynomial algebras $$R[z;\sigma ]$$ of automorphism type, which comultiplication defined on the variable z is extended by a Drinfeld twist J for a Hopf algebra R. Firstly, we establish some necessary and sufficient conditions that are of Hopf algebra structures on the localization of skew polynomial ring $$R[z;\sigma ]$$ and on the quotients of $$R[z;\sigma ]/I$$, for a certain Hopf ideal I of $$R[z;\sigma ]$$. Then, we give some examples for these results. Specially, the non-semisimple and non-pointed 16 dimension self-dual Hopf algebra is also constructed by the Ore extensions of automorphism type.

Classification of Binary Self-Dual [76, 38, 14] Codes With an Automorphism of Order 9

Using the method for constructing binary self-dual codes with an automorphism of order square of a prime number we have classified all binary self-dual codes with length 76 having minimum distance $d=14$ and automorphism of order 9. Up to equivalence, there are six self-dual $[76, 38, 14]$ codes with an automorphism of type $9$-$(8,0,4)$. All codes obtained have new values of the parameter in their weight enumerator thus more than doubling the number of known values.

Self-dual codes with an automorphism of order 7 and s-extremal codes of length 68

This paper studies and classifies optimal binary self-dual codes having an automorphism of order 7 with 9 cycles. This classification is done by applying a method for constructing binary self-dual codes with an automorphism of odd prime order p. There are exactly 69781 inequivalent binary self-dual [64,32,12] codes with an automorphism of type 7−(9,1). As for binary [66,33,12] self-dual codes with an automorphism of type 7−(9,3) there are 1652432 such codes. We also construct more than 4 million new optimal codes of length 68 among which are the first known examples of the very elusive s-extremal self-dual codes. We prove the nonexistence of [70,35,14] codes with an automorphism of type 7−(9,7). Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.

An alternative proof of Miyashita's Theorem in a skew polynomial ring II

Y. Miyashita gave characterizations of a separable polynomial and a Hirata separable polynomial in skew polynomial rings. In the previous paper, the author and S. Ikehata gave direct and elementary proofs of Miyashita's theorems in skew polynomial rings of Automorphism type B[X;ρ] and Derivation type B[X;D], respectively. The purpose of this paper is to give proofs for them in the general skew polynomial ring B[X;ρ,D].

Boundaries and automorphisms of hierarchically hyperbolic spaces

Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichm\"uller space, right-angled Artin groups, and many other cubical groups. Given such a space $\mathcal X$, we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathcal X$ is proper, we get a compactification of $\mathcal X$; we also prove that our construction generalizes the Gromov boundary of a hyperbolic space. In our first main set of applications, we introduce a notion of geometrical finiteness for hierarchically hyperbolic subgroups of hierarchically hyperbolic groups in terms of boundary embeddings. As primary examples of geometrical finiteness, we prove that the natural inclusions of finitely generated Veech groups and the Leininger-Reid combination subgroups extend to continuous embeddings of their Gromov boundaries into the boundary of the mapping class group, both of which fail to happen with the Thurston compactification of Teichm\"uller space. Our second main set of applications are dynamical and structural, built upon our classification of automorphisms of hierarchically hyperbolic spaces and analysis of how the various types of automorphisms act on the boundary. We prove a generalization of the Handel-Mosher "omnibus subgroup theorem" for mapping class groups to all hierarchically hyperbolic groups, obtain a new proof of the Caprace-Sageev rank-rigidity theorem for many CAT(0) cube complexes, and identify the boundary of a hierarchically hyperbolic group as its Poisson boundary; these results rely on a theorem detecting \emph{irreducible axial} elements of a group acting on a hierarchically hyperbolic space (which generalize pseudo-Anosov elements of the mapping class group and rank-one isometries of a cube complex not virtually stabilizing a hyperplane).

Asymptotically almost automorphic solutions of differential equations with piecewise constant argument

Abstract This paper is concerned with the existence and uniqueness of asymptotically almost automorphic solutions to differential equations with piecewise constant argument. To study that, we first introduce several notions about asymptotically almost automorphic type functions and obtain some properties of such functions. Then, on the basis of a systematic study on the associated difference system, the existence and uniqueness theorem is established. Compared with some earlier results, we do not assume directly that the Green’s function is a Bi-almost automorphic type function.

Langlands reciprocity for certain Galois extensions

In this note, we study Artin's conjecture via group theory and derive Langlands reciprocity for certain solvable Galois extensions of number fields, which extends the previous work of Arthur and Clozel. In particular, we show that all nearly nilpotent groups and all groups of order less than 60 are of automorphic type.

On Some Automorphisms of Order 3 of the Extremal Binary Codes

We prove that if an extremal binary self-dual code of length n ≥ 72 has an automorphism of type 3-(c, f) then c ≥ f.

Self-dual codes with an automorphism of order 13

Using a method for constructing binary self-dual codes having an automorphism of odd prime order \begin{document}$p$\end{document} we classify, up to equivalence, all singly-even self-dual \begin{document}$[78,39,14]$\end{document} , \begin{document}$[80,40,14]$\end{document} , \begin{document}$[82,41,14],$\end{document} and \begin{document}$[84,42,14]$\end{document} codes as well as all doubly-even \begin{document}$[80,40,16]$\end{document} codes for \begin{document}$p=13$\end{document} . The results show that there are exactly 1592 inequivalent binary self-dual \begin{document}$[78,39,14]$\end{document} codes with an automorphism of type \begin{document}$13-(6,0)$\end{document} and we found 6 new values of the parameter in the weight function thus increasing more than twice the number of known values. As for binary \begin{document}$[80,40]$\end{document} self-dual codes with an automorphism of type \begin{document}$13-(6,2)$\end{document} there are 162696 singly-even self-dual codes with minimum distance 14 and 195 doubly-even such codes with minimum distance 16. We also construct many new codes of lengths 82 and 84 with minimum distance 14. Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.

Group actions on metric spaces: fixed points and free subgroups

We look at group actions on graphs and other metric spaces, e. g., at group actions on geodesic hyperbolic spaces. We classify the types of automorphisms on these spaces and prove several results about the density of the hyperbolic limit set of the group in the whole limit set of the group.