A Characterization of Domination Reducible Graphs

We introduce a hereditary class of domination reducible graphs where the minimum dominating set problem is polynomially solvable, and characterize this class in terms of forbidden induced subgraphs.Acknowledgments The research was supported by DIMACS 2002 and 2003 Awards. The author would like to thank the both referees for their valuable suggestions.Final version received: October 3, 2003     

Reducible polar representations

In this paper the reducible polar representations of the compact connected Lie groups are classified. It turns out that there only exist “interesting” reducible polar representations of Lie groups of the types A ₃, A ₃×T ¹, B ₃, B ₃×T ¹, D ₄, D ₄×T ¹ and D ₄×A ₁. Up to equivalence, there is just one such representation of the first four Lie groups, there are three reducible polar representations of D ₄ and six of D ₄×T ¹ and D ₄×A ₁, respectively. From this follows immediately the classification of the compact connected subgroups of SO(n) which act transitively on products of spheres. Received: 28 April 2000     

Reducible Dupin submanifolds

Pinkall's standard constructions for obtaining a Dupin hypersurface W in ^N from a Dupin hypersurface M in ⁿ , N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ^N with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ⁿ if and only if g=2.This research was supported by NSF Grant No. DMS 87-06015.     

Reducible Runge-Kutta methods

Reducible Runge-Kutta methods are characterized by means of special matrices. Previous definitions of reducibility are incorporated. This characterization may be useful in the study of algebraic stability and in studies of existence and uniqueness.     

Reducible Specht modules

James and Mathas conjecture a criterion for the Specht module S^λ for the symmetric group to be irreducible over a field of prime characteristic. We extend a result of Lyle to prove this conjecture in one direction; our techniques are elementary.     

可约的把柄添加

Let M be a hyperbolic 3-manifold with boundary. Suppose thatαandβare two separating slopes on the same component of (?)M. We shall prove that if both M[α] and M[β] are reducible, then△(α/β)≤8.     

可约的Dehn手术

本文将利用C.Gordon和J.Luecke发展起来的组合拓扑的方法来证明缆式猜想对一类特殊纽结而言是正确的。     

Reducible Ordinary Differential Equations

The class of reducible differential equations under consideration here includes the class of symmetric systems, and examples show that the inclusion is proper. We first discuss reducibility, as well as the stronger concept of complete reducibility, from the viewpoint of Lie algebras of vector fields and their invariants, and find Lie algebra conditions for reducibility which generalize the conditions in the symmetric case. Completely reducible equations are shown to correspond to a special class of abelian Lie algebras. Then we consider the inverse problem of determining all vector fields which are reducible by some given map. We find conditions imposed on the vector fields by the map, and present an algorithmic access for a given polynomial or local analytic map to Next, reducibility of polynomial systems is discussed, with applications to local reducibility near a stationary point. We find necessary conditions for reducibility, including restrictions for possible reduction maps to a one-dimensional equation.     

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.     

Reducible means and reducible inequalities

It is well-known that if a real valued function acting on a convex set satisfies the n-variable Jensen inequality, for some natural number \(n\ge 2\), then, for all \(k\in \{1,\dots , n\}\), it fulfills the k-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility property. Finally, we give a sufficient condition for the reducibility of the (M, N)-convexity property of functions and also for Hölder–Minkowski type inequalities.     

Reducible and finite Dehn fillings

We show that the distance between a finite filling slope anda reducible filling slope on the boundary of a hyperbolic knotmanifold is one.     

Completely Reducible Infinite-Dimensional Skew Linear Groups

 Let V be a left vector space over a division ring D and the group of all D-automorphisms of V. A subgroup G of is completely reducible of V is completely reducible as D–G bimodule. Our aim in this brief note is to point out that in a sense the very useful notion of a local marker extends from V finite-dimensional to V infinite-dimensional. (A local marker of a subgroup G of is any finitely generated subgroup X of G such that row n-space has least composition length as D–X bimodule. A local marker of G controls to a considerable extent the local behaviour of G.) Our main result is the following. Let G be a completely reducible subgroup of and let W be any finite-dimensional D-subspace of V. Then G has a finitely generated subgroup X such that for every finitely generated subgroup Y of G containing X the D–Y submodule WY has a D–Y submodule M with and completely reducible. We also give some examples and state without proof some stronger conclusions valid for various special subgroup G. (Received 21 December 1998; in revised form 31 May 1999)     

Reducible subgroups of exceptional algebraic groups

Let G be a simple algebraic group over an algebraically closed field. A closed subgroup H of G is called G-completely reducible (G-cr) if, whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi factor of P. In this paper we complete the classification of connected G-cr subgroups when G has exceptional type, by determining the $L_0$-irreducible connected reductive subgroups for each simple classical factor $L_0$ of a Levi subgroup of G. As an illustration, we determine all reducible, G-cr semisimple subgroups when G has type $F_4$ and various properties thereof. This work complements results of Lawther, Liebeck, Seitz and Testerman, and is vital in classifying non-G-cr reductive subgroups, a project being undertaken by the authors elsewhere.     

Completely Reducible Infinite-Dimensional Skew Linear Groups

 Let V be a left vector space over a division ring D and the group of all D-automorphisms of V. A subgroup G of is completely reducible of V is completely reducible as D–G bimodule. Our aim in this brief note is to point out that in a sense the very useful notion of a local marker extends from V finite-dimensional to V infinite-dimensional. (A local marker of a subgroup G of is any finitely generated subgroup X of G such that row n-space has least composition length as D–X bimodule. A local marker of G controls to a considerable extent the local behaviour of G.) Our main result is the following. Let G be a completely reducible subgroup of and let W be any finite-dimensional D-subspace of V. Then G has a finitely generated subgroup X such that for every finitely generated subgroup Y of G containing X the D–Y submodule WY has a D–Y submodule M with and completely reducible. We also give some examples and state without proof some stronger conclusions valid for various special subgroup G.     

Modules with Reducible Complexity,II

We continue studying the class of modules having reducible complexity over a local ring. In particular, a method is provided for computing an upper bound of the complexity of such a module, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which all modules have finite complexity.