Irreducible Maps Research Articles

DYNAMICAL SYNTHESIS OF POINCARÉ MAPS

We present a theory of constructive Poincaré maps. The basis of our theory is the concept of irreducible nonlinear maps closely associated to concepts from Lie groups. Irreducible nonlinear maps are, heuristically, nonlinear maps which cannot be made simpler without removing the nonlinearity. A single irreducible map cannot produce chaos or any complex nonlinear effect. It can be implemented in an electronic circuit, and there are only a finite number of families of irreducible maps in any n-dimensional space. The composition of two or more irreducible maps can produce chaos and most of the maps studied today that produce chaos are compositions of two or more irreducible maps. The composition of a finite number of irreducible maps is called a completely reducible map and a map which can be approximated pointwise by completely reducible maps is called a reducible map. Poincaré maps from sinusoidally forced oscillators are the most familiar examples of reducible maps. This theoretical framework provides an approach to the construction of "closed form" Poincaré maps having the properties of Poincaré maps of systems for which the Poincaré map cannot be obtained in closed form. In particular, we derive a three-dimensional ODE for which the Hénon map is the Poincaré map and show that there is no two-dimensional ODE which can be written down in closed form for which the Hénon map is the Poincaré map. We also show that the Chirikov (standard) map is a Poincaré map for a two-dimensional closed form ODE. As a result of our theory, these differential equations can be mapped into electronic circuits, thereby associating them with real world physical systems. In order to clarify our results with respect to the abstract mathematical concept of suspension, which says that every C1 invertible map is a Poincaré map, we introduce the concept of a constructable Poincaré map. Not every map is a constructable Poincaré map and this is an important distinction between dynamical synthesis and abstract nonlinear dynamics. We also show how to use any one-dimensional map to induce a two-dimensional Poincaré map which is a completely reducible map and hence for a very broad class of maps that includes the logistic map we derive closed form ODEs for which these one-dimensional maps are "embedded" in a Poincaré map. This provides an avenue for the study of one-dimensional maps, such as the logistic map, as two-dimensional Poincaré maps that arise from square-wave forced electronic circuits.

Contravariantly finite subcategories and irreducible maps

Let Λ \Lambda be an Artin algebra. This paper presents a condition that is necessary for a subcategory of mod Λ \bmod \Lambda to be contravariantly finite. Several examples are explored in the case that Λ \Lambda is the group algebra of a finite group over a field of finite characteristic.

Contravariantly Finite Subcategories and Irreducible Maps

Contravariantly Finite Subcategories and Irreducible Maps

Irreducible maps of commutative rings

We give necessary conditions for a map to be irreducible (in the category of finitely generated, torsion free modules) over a non-local, commutative ring and sufficient conditions when the ring is Bass. In particular, we show that an irreducible map of ZG, where G is a square free abelian group, must be a monomorphism with a simple cokernel. We also show that local endomorphism rings are necessary and sufficient for the existence of almost split sequences over a commutative Bass ring and we explicitly describe the modules and the maps in those sequences. The results in this paper enable us to describe the Auslander-Reiten quiver of a non-local Bass ring in [8].

Irreducible maps of commutative rings

Irreducible maps of commutative rings

Quasi θ-spaces and pairwise θ-perfect irreducible mappings

AbstractIn this paper we extend the notion of perfect, θ-continuous, irreducible and θ-perfect mappings to bitopological spaces. The main result is the following: the (small) image of an (i, j)-canonical open sets is an (i, j)-canonical open set under a pairwise θ-closed irreducible surjective mapping. Also we extend the notion of θ-proximity spaces to quasi θ-proximity spaces and point out the interrelation between it and separated quasi-proximity spaces by means of a pairwise θ-perfect irreducible mappings.

Fuzzy θ-perfect irreducible mappings and fuzzy θ-proximity spaces

In this paper we introduce and study the concepts of fuzzy perfect, fuzzy θ-continuous, fuzzy irreducible and fuzzy θ-perfect mappings. It is shown that the small image of a canonical open fuzzy set remains canonical open under fuzzy θ-closed irreducible mapping. Also, we introduce and study the concept of fuzzy θ-proximity spaces and point out their interrelation with separated fuzzy proximity spaces by means of fuzzy θ-perfect irreducible mappings.

𝑑-final continua

The Hilbert cube is known to be an irreducible quotient of every perfect metric space. Its irreducible quotients are identified: all nondegenerate Peano continua no open set in which is homeomorphic with R R . In compact metric spaces, every irreducible surjection X → Y X \to Y embeds a dense G δ {G_\delta } subset of X X in Y Y . The Peano continua I which are strongly initial, every irreducible map from a Peano continuum to I being a homeomorphism, are identified: the dendrites the closure of whose end points is zero-dimensional.

Absolutes of almost realcompactifications

AbstractGiven Hausdorff spacesXandYand a perfect irreducible and θ-continuous mapffromXontoY, technique that carries open (ultra) filters onXto open (ultra) filters onYback and forth in a natural way is introduced. It is proved that iffis a perfect irreducible and θ-continuous map fromXontoY, thenXis almost realcompact if and only ifYis almost realcompact. Several commutativity relations between the ‘absolutes of almost realcompactifications’ and the ‘almost realcompactifications of absólutes’ of a spaceXare discussed.

Remainders of H-closed extensions

The remainders of compactifications of Tychonoff spaces have been studied extensively during the last five decades, but, the remainders of H-closed extensions of Hausdorff spaces have not been studied in a concerted manner so far. In this paper, we show that two Hausdorff spaces X and Y have the same remainders in their H-closed extensions if there exists a perfect, irreducible and θ-continuous mapping from X onto Y. Further, a Hausdorff space Z is shown to be the remainder of some H-closed extension of a given space X if and only if there exists a perfect (not necessarily continuous) mapping from σX⧹ X onto Z. We also show that if X is locally H-closed and |σX⧹X|⩾ℵ 0 , then the set of remainders of H-closed extensions of X contains all separable H-closed spaces. We give examples to illustrate the differences between the sets of remainders of compactifications and remainders of H-closed extensions of a fixed Tychonoff space.

A characterization of continuous images of compact ordered spaces and the Hahn-Mazurkiewicz problem

In the present paper we give the internal characterization of the following three classes of spaces: continuous images of compact ordered spaces, images of Hausdorff arcs under irreducible maps and continuous images of Hausdorff arcs.

Representation-Finite Algebras whose Auslander-Reiten Quiver is Planar

Throughout the paper A will denote a finite-dimensional connected and basic /c-algebra of finite representation type. We assume the ground field k to be algebraically closed. By FA we denote the Auslander-Reiten quiver of A, which has as vertices the isomorphism classes of indecomposable left A-modules and in which we put an arrow from the class of M to the class of N whenever there exists an irreducible map from M to N. Algebras whose associated Auslander-Reiten quivers have no oriented cycles have played an important role in representation theory: hereditary algebras, tree algebras, simply connected algebras, etc. In this paper we study some conditions which enable us, in some cases, to decide, in terms of the quiver with relations of A, whether FA has oriented cycles or not. Our conditions give, a constructive characterization of all finite representation type algebras A whose Auslander-Reiten quiver FA is planar, that is, FA has no oriented cycles and every Auslander-Reiten sequence in FA has at most two indecomposable summands in its middle term. We thank Prof. F. Larrion for some suggestions about the presentation of this paper.

The existence of covering functors for orders and consequences for stable components with oriented cycles

The starting point for the covering theory for algebras of finite type over algebraically closed fields was Riedtmann’s paper, “Algebren, Darstellungskocher, uberlagerungen und zurtick” [ 14 1. Her main result, from today’s point of view, is the existence of covering functors, i.e., the existence of a bijection between the morphisms between indecomposable modules and purely combinatorial defined vector spaces generated by paths in a translation quiver (cf. (1.1) [ 141). This bijection has recently turned out to be more and more important in the representation theory of algebras. It was the first step leading to the use of Auslander-Reiten quivers as a tool for the classification of algebras of linite type [ 15, 81. The main purpose of this paper is to initiate an analog for classical orders over complete Dedekind domains. We shall formulate our results and proofs only for orders, but it is always obvious that everything carries immediately over to the algebra situation. Let R be a complete Dedekind domain with quotient field K, and let n be an R-order in a separable K-algebra A [ 161: n is a subring of A with the same identity, and moreover A is a full R-sublattice, which means that /i is an R-lattice and contains a K-basis of A. We denote by r the Auslander-Reiten quiver of/i, i.e., vertices of r are the isomorphism classes of indecomposable n-lattices, and two vertices are joined by an arrow provided there exists an irreducible /i-map between the corresponding A-lattices. Moreover, we denote by f the universal cover of I[8], in particular, the vertices of r” are homotopy classes of walks in I-, with covering morphism F: F-P r [8]. For each arrow x + y in F we define $,, as the space of irreducible maps Irr(Fx, Fy) [4, 171 from Fx to Fy. Moreover, for indecomposable /i-lattices IV, N let r’(M, N) c Hom,(M, N) be the ith functorial radical, i.e., r’(M, N) = ri(-, N)(M), where ri(-, N) is the ith power of the radical of the presentable fun&or (-, N) = Hom,((, N) [ 21. 292 0021-8693/85 $3.00

Artin algebras with irreducible maps between projective and injective modules

Artin algebras with irreducible maps between projective and injective modules

Two-sided spectral bounds of two irreducible linear maps

The difference of the spectral radii of two irreducible linear operators is estimated both from above and from below in terms of values of a suitable positive operator on appropriate elements. This generalizes a well-known result from matrix theory [6].

Two-sided spectral bounds of two irreducible linear maps

Two-sided spectral bounds of two irreducible linear maps

The Preprojective Partition for Hereditary Artin Algebras

The purpose of this paper is to study the preprojective partition of a hereditary artin algebra. For a hereditary algebra of finite representation type, we give some numerical invariants in terms of the length of chains of irreducible maps, also in terms of the length of the maximal indecomposable module, and the orientation of the quiver of the algebra. Similar results are given for algebras stably equivalent to hereditary artin algebras.

A Note on Preprojective Partitions Over Hereditary Artin Algebras

If $\Lambda$ is an artin algebra there is a partition of $\operatorname {ind} \Lambda$, the category of indecomposable finitely generated $\Lambda$-modules, $\operatorname {ind} \Lambda = { \cup _{i \geqslant 0}}{\underline {\underline {P}}_i}$, called the preprojective partition. We show that $\underline {\underline {P}}_i$ can be easily constructed for hereditary artin algebras, if $\underline {\underline {P}}_{i - 1}$ is known: $A$ is in $\underline {\underline {P}}_i$ if and only if $A$ is not in $\underline {\underline {P}}_{i - 1}$ and there is an irreducible map $B \to A$, where $B$ is in $\underline {\underline {P}}_{i - 1}$.

A note on preprojective partitions over hereditary Artin algebras

If Λ \Lambda is an artin algebra there is a partition of ind ⁡ Λ \operatorname {ind} \Lambda , the category of indecomposable finitely generated Λ \Lambda -modules, ind ⁡ Λ = ∪ i ⩾ 0 P _ _ i \operatorname {ind} \Lambda = { \cup _{i \geqslant 0}}{\underline {\underline {P}}_i} , called the preprojective partition. We show that P _ _ i \underline {\underline {P}}_i can be easily constructed for hereditary artin algebras, if P _ _ i − 1 \underline {\underline {P}}_{i - 1} is known: A A is in P _ _ i \underline {\underline {P}}_i if and only if A A is not in P _ _ i − 1 \underline {\underline {P}}_{i - 1} and there is an irreducible map B → A B \to A , where B B is in P _ _ i − 1 \underline {\underline {P}}_{i - 1} .

The preprojective partition for hereditary Artin algebras

The purpose of this paper is to study the preprojective partition of a hereditary artin algebra. For a hereditary algebra of finite representation type, we give some numerical invariants in terms of the length of chains of irreducible maps, also in terms of the length of the maximal indecomposable module, and the orientation of the quiver of the algebra. Similar results are given for algebras stably equivalent to hereditary artin algebras.