Linear Automorphisms Research Articles

On Linear Automorphisms Over ℝ and a Note of Hajja

We derive a description of the linear automorphisms of rational function fields over the field ℝ of real numbers. We use this description to establish that the field of invariants for linear ℝ-automorphisms is rational over ℝ.

The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes

The main aim of the classification of linear codes is the evaluation of complete lists of representatives of the isometry classes. These classes are mostly defined with respect to linear isometry, but it is well known that there is also the more general definition of semilinear isometry taking the field automorphisms into account. This notion leads to bigger classes so the data becomes smaller. Hence we describe an algorithm that gives canonical representatives of these bigger classes by calculating a unique generator matrix to a given linear code, in a well defined manner.   &nbsp The algorithm is based on the partitioning and refinement idea which is also used to calculate the canonical labeling of a graph [12] and it similarly returns the automorphism group of the given linear code. The time needed by the implementation of the algorithm is comparable to Leon's program [10] for the calculation of the linear automorphism group of a linear code, but it additionally provides a unique representative and the automorphism group with respect to the more general notion of semilinear equivalence. The program can be used online under http://www.algorithm.uni-bayreuth.de/en/research/Coding_Theory/CanonicalForm/index.html.

EXISTENCE OF SOLUTIONS OF SEMILINEAR SYSTEMS IN <img width=32 height=32 src="http:/fbpe/img/proy/v27n2/img02.JPG">

Let Q : ℓ² → ℓ² be a symmetric and positive semi-definite linear operator and fj : R → R (j = 1, 2, ...) be real functions so that, fj(0) = 0 and, for every x = (x1, x2, ....) ∈ ℓ², it holds that f(x) := (f1(x1), f2(x2), ...) ∈ ℓ². Sufficient conditions for the existence of non-trivial solutions to the semilinear problem Qx = f(x) are provided. Moreover, if G is a group of orthogonal linear automorphisms of ℓ² which commute with Q, then such sufficient conditions ensure the existence of non-trivial solutions which are invariant under G. As a consequence, sufficient conditions to ensure solutions of nonlinear partial difference equations on finite degree graphs with vertex set being either finite or infinitely countable are obtained. We consider adaptations to graphs of both Matukuma type equations and Helmholtz equations and study the existence of their solutions.

$C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus

We consider two $C^2$ Anosov diffeomorphisms in a $C^1$ neighborhood of a linear hyperbolic automorphism ofthree dimensional torus with real spectrum. We prove that they are $C^{1+\nu}$ conjugate if and only if thedifferentials of the return maps at corresponding periodic points have the same eigenvalues.

Relative negligibility of linear automorphisms

Let U and V be finite-dimensional vector spaces over a field k, α ∈ GL ( U ) , β ∈ GL ( V ) and I be the identity transformation on V. Denote by α * β and α * I the induced linear automorphisms on U ⊕ V ; α * β and α * I can also be regarded as k-automorphisms on the function field k ( U ⊕ V ) . It is elementary to check whether α * β and α * I are conjugate within GL ( U ⊕ V ) by examining their rational canonical forms. In this paper we shall give necessary and sufficient conditions for α * β and α * I to be conjugate within Aut k ( k ( U ⊕ V ) ) . For this characterization, we introduce the concept of the generalized order. Through this invariant we also settle the question of when two different polynomials are minimal polynomials of the same linear automorphism of a rational function field.

Toric Legendrian Subvarieties

We give the full classification of smooth toric Legendrian subvarieties in P2n-1. We also prove that under some minor assumptions the group of linear automorphisms preserving a given Legendrian subvariety preserves the contact structure of the ambient projective space.

Linear dynamical entropy and free-independence for quantized linear maps on the torus

We study the relations between the averaged linear entropy production in periodically measured quantum systems and ergodic properties of their classical counterparts. Quantized linear automorphisms of the torus, both classically chaotic and regular ones, are used as examples. Numerical calculations show different entropy production regimes depending on the relation between the Kolmogorov–Sinai entropy and the measurement entropy. The hypothesis of free independence relations between the dynamics and measurement proposed to explain the initial constant and maximal entropy production is tested numerically for those models.

Projective subschemes with many linear automorphisms

Here we study projective subschemes of P with many automorphisms, pointing out some peculiarities of the positive characteristic case. Mathematics Subject Classification: 14H60

Closed subschemes of quadrics with large groups of linear automorphisms

Let Qn ⊂ P n+1 , n ≥ 2, be an n-dimensional smooth quadric hypersurface. Fix P ∈ Qn. Set Gn := Aut(Qn) and Gn(P ): ={f ∈: f (P )= P }. We first study low dimensional linear subspaces of H 0 (P n+1 , OPn+1(k)). Then we study the zero-dimensional schemes Z ⊂ Qn such that f (Z )= Z for all f ∈ Gn(P ) and length(Z) is small.

The algebraic entropy of the special linear character automorphisms of a free group on two generators

In this note, we establish a connection between the dynamical degree, or algebraic entropy of a certain class of polynomial automorphisms of R 3 \mathbb {R}^3 , and the maximum topological entropy of the action when restricted to compact invariant subvarieties. Indeed, when there is no cancellation of leading terms in the successive iterates of the polynomial automorphism, the two quantities are equal. In general, however, the algebraic entropy overestimates the topological entropy. These polynomial automorphisms arise as extensions of mapping class actions of a punctured torus S S on the relative SU ⁡ ( 2 ) \operatorname {SU}(2) -character varieties of S S embedded in R 3 \mathbb {R}^3 . It is known that the topological entropy of these mapping class actions is maximized on the relative character variety comprised of reducible characters (those whose boundary holonomy is 2 2 ). Here we calculate the algebraic entropy of the induced polynomial automorphisms on the character varieties and show that it too solely depends on the topology of S S .

Equivalence of polynomials under automorphisms of [formula omitted

Let K [ x , y ] be the algebra of polynomials in two variables over an arbitrary field K . We show that if the maximum of the x - and y -degrees of a given polynomial p ( x , y ) cannot be decreased by a single triangular or linear automorphism of K [ x , y ] , then it cannot be decreased by any automorphism of K [ x , y ] . If K is an algebraically closed constructible field, this result yields an algorithm for deciding whether or not two polynomials p , q ∈ K [ x , y ] are equivalent under an automorphism of K [ x , y ] . We also show that if there is an automorphism of K [ x , y ] taking p to q , then it is “almost” unique. More precisely: if an automorphism α of K [ x , y ] is not conjugate to a triangular or linear automorphism, then any polynomial invariant (or even semiinvariant) under α is a constant.

JORDAN AUTOMORPHIC GENERATORS OF EUCLIDEAN JORDAN ALGEBRAS

In this paper we show that the Koecher's Jordan automorphic generators of one variable on an irreducible symmetric cone are enough to determine the elements of scalar multiple of the Jordan identity on the attached simple Euclidean Jordan algebra. Its various geometric, Jordan and Lie theoretic interpretations associated to the Cartan-Hadamard metric and Cartan decomposition of the linear automorphisms group of a symmetric cone are given with validity on infinite-dimensional spin factors

Extensions of isomorphisms for affine Grassmannians over

Abstract In Blok [R. J. Blok, Extensions of isomorphisms for affine dual polar spaces and strong parapolar spaces. Adv. Geom. 5 (2005), 509–532.] affinely rigid classes of geometries were studied. These are classes B of geometries with the following property: Given any two geometries Γ1, Γ2 ∈ B with subspaces 𝒮1 and 𝒮2 respectively, then any isomorphism Γ1 − 𝒮1 → Γ2 − 𝒮2 uniquely extends to an isomorphism Γ1 → Γ2. Suppose Γ belongs to an affinely rigid class. Then for any subspace 𝒮 we have Aut(Γ − 𝒮) ≤ Aut(Γ). Suppose that, in addition, Γ is embedded into the projective space ℙ(V) for some vector space V. Then one may think of V as a “natural” embedding if every automorphism of Γ is induced by some (semi-) linear automorphism of V. This is for instance true of the projective geometry Γ = ℙ(V) itself by the fundamental theorem of projective geometry. Clearly since Γ belongs to an affinely rigid class and has a natural embedding into ℙ(V), also the embedding Γ − 𝒮 into ℙ(V) is natural. In Blok [R. J. Blok, Extensions of isomorphisms for affine dual polar spaces and strong parapolar spaces. Adv. Geom. 5 (2005), 509–532.] the notion of a layer-extendable class was introduced and it was shown that layer-extendable classes are affinely rigid. As an application, it was shown that the union of most projective geometries, (dual) polar spaces, and strong parapolar spaces forms an affinely rigid class. However, the geometries motivating that study, the Grassmannians defined over 𝔽2, were not included in this class because they do not form a layer-extendable class. Since affine projective geometries (1-Grassmannians, if you will) are simply complete graphs, clearly they are not affinely rigid at all. In the present note we show that also the class of 2-Grassmannians over 𝔽2 fails to form an affinely rigid class, although in a less dramatic way, whereas the class of k-Grassmannians of projective spaces of dimension n over 𝔽2 where 3 ≤ k ≤ n − 2 is in fact affinely rigid.

Conjugacy classes of affine automorphisms of and linear automorphisms of ℙn in the Cremona groups

We describe the conjugacy classes of affine automorphisms in the group $Aut(n,\K)$ (respectively $Bir(\K^n)$) of automorphisms (respectively of birational maps) of $\K^n$. From this we deduce also the classification of conjugacy classes of automorphisms of $\Pn$ in the Cremona group $Bir(\K^n)$.

Quartic K3 surfaces without nontrivial automorphisms

For any field k we fix an algebraic closure of k, denoted by k. For a variety X ⊂ P k we set X = X×k k, we say that X is smooth if X is regular, and we let AutX denote the group of k-automorphisms of X, while LinX denotes the group of linear automorphisms of X, i.e., automorphisms induced by a linear transformation of the coordinates of P. The following theorem was proved by Poonen, see [Po], Thm. 1.6.

A construction of non-regularly orbicular modules for Galois coverings

For a given finite dimensional k -algebra A which admits a presentation in the form R / G , where G is an infinite group of k -linear automorphisms of a locally bounded k -category R , a class of modules lying out of the image of the push-down functor associated with the Galois covering R → R / G , is studied. Namely, the problem of existence and construction of the so called non-regularly orbicular indecomposable R / G -modules is discussed. For a G -atom B (with a stabilizer G B ), whose endomorphism algebra has a suitable structure,a representation embedding Φ B ( f , s ) | : I n - s p r l ( s ) ( k G B ) → m o d ( R / G ) , which yields large families of non-regularly orbicular indecomposable R / G -modules,is constructed (Theorem 2.2). An important role in consideration is played by a result interpreting some class of R / G -modules in terms of Cohen-Macaulay modules over certain skew grup algebra (Theorem 3.3). Also, Theorems 4.5 and 5.4, adapting the generalized tensor product construction and Galois covering scheme, respectively, for Cohen-Macaulay modules context, are proved and intensively used.

Diffeomorfismi birazionali del piano proiettivo reale

We study real birational transformations of the real projective plane which are diffeomorphisms. It turns out that their degree must be congruent to 1 mod 4, and that they are generated by linear automorphisms and transformations of degree 5 centred at 3 pairs of conjugated imaginary points. Our approach is inspired by recent proofs of the classical theorem of Noether and Castelnuovo that use the Sarkisov program.

Dual MacWilliams Pair

A pair of posets (P,Q ) on [n] is called a weak dual MacWilliams pair (wdMp) if the P-weight enumerator of a linear code uniquely determines the Q-weight enumerator of the dual of that code for every linear code of length n over a finite field. First, we show that (P, Pbreve) is a wdMp if and only if the group of all P-weight preserving linear automorphisms of the ambient n-dimensional space over the finite field acts transitively on every P-sphere centered at 0. Here Pbreve is the dual poset of P. Also, we show some equivalent conditions which say that P being weak order poset with Q = Pbreve is essentially the only possible case for (P,Q) to be a wdMp

Stable ergodicity of certain linear automorphisms of the torus

We find a class of ergodic linear automorphisms of TN that are stably ergodic. This class includes all non-Anosov ergodic automorphisms when N = 4. As a corollary, we obtain the fact that all ergodic linear automorphism of TN are stably ergodic when N iU 5.

Automorphisms of Tensor Completions of Algebras

In the classical representation of different groups, frequent use is made of a linear automorphism group of various algebras. Since the linear automorphism group is only part of a full automorphism group, such an approach might seem to be too restrictive. In this connection, we point out a natural, wide class of algebras whose automorphisms are standard and are reducible to linear. Thus, for algebras in this class, studying the full automorphism group reduces to treating the linear, a traditional approach in the class of such algebras being quite general.