Circuit preserving edge maps II

In Section 1 of this article we prove the following. Let f: G → G′ be a circuit surjection, i.e., a mapping of the edge set of G onto the edge set of G′ which maps circuits of G onto circuits of G′, where G, G′ are graphs without loops or multiple edges and G′ has no isolated vertices. We show that if G is assumed finite and 3-connected, then f is induced by a vertex isomorphism. If G is assumed 3-connected but not necessarily finite and G′ is assumed to not be a circuit, then f is induced by a vertex isomorphism. Examples of circuit surjections f: G → G′ where G′ is a circuit and G is an infinite graph of arbitrarily large connectivity are given. In general if we assume G two-connected and G′ not a circuit then any circuit surjection f: G → G′ may be written as the composite of three maps, f(G) = q(h(k(G))), where k is a 1-1 onto edge map which preserves circuits in both directions (the “2-isomorphism” of Whitney (Amer. J. Math. 55 (1933), 245–254) when G is finite), h is an onto edge map obtained by replacing “suspended chains” of k(G) with single edges, and G is a circuit injection (a 1-1 circuit surjection). Let f: G → M be a 1-1 onto mapping of the edges of G onto the cells of M which takes circuits of G onto circuits of M where G is a graph with no isolated vertices, M a matroid. If there exists a circuit C of M which is not the image of a circuit in G, we call f nontrivial, otherwise trivial. In Section 2 we show the following. Let G be a graph of even order. Then the statement “no trivial map f: G → M exists, where M is a binary matroid,” is equivalent to “G is Hamiltonian.” If G is a graph of odd order, then the statement “no nontrivial map f: G → M exists, where M is a binary matroid” is equivalent to “G is almost Hamiltonian,” where we define a graph G of order n to be almost Hamiltonian if every subset of vertices of order n − 1 is contained in some circuit of G.     

Linear maps preserving invariants

Let be a complex reductive group. Let denote for all . We show that, ``in general', . In case is the adjoint group of a simple Lie algebra , we show that is an order 2 extension of . We also calculate for all representations of .

     

Non-linear maps preserving solvability

Let M_n be the algebra of all n×n complex matrices and let L be the general linear Lie algebra gl(n,C) or the special linear Lie algebra sl(n,C). A bijective (not necessarily linear) map preserves solvability in both directions if both and ⁻¹ map every solvable Lie subalgebra of L into some solvable Lie subalgebra. If n3 then every such map is either a composition of a bijective lattice preserving map with a similarity transformation and a map [a_ij][f(a_ij)] induced by a field automorphism , or a map of this type composed with the transposition. We also describe the general form of such maps in the case when n=2. Using Lie's theorem we will reduce the proof of this statement to the problem of characterizing bijective maps on M_n preserving triangularizability of matrix pairs in both directions. As a byproduct we will characterize bijective maps on M_n that preserve inclusion for lattices of invariant subspaces in both directions.     

Discriminant preserving linear maps

Consider the n-square matrices over an infiniie field Kas an n²-dimcnsional vector space M( nK). We determine all linear maps Ton M(nK) such that discriminant TX- discriminant Xfor all Xin M(nK)     

Linear maps preserving commutativity

Commutativity-preserving maps on the real space of all real symmetric or complex self-adjoint matrices are characterized. Related results are given for adjoint-preserving maps defined on all n × n matrices. These results are extended to infinite dimensions in the case of invertible maps.     

Linear maps preserving group majorization

A necessary and sufficient condition for a linear map to preserve group majorizations is given. The condition is applied to prove some preservation results.     

Enumeration of order preserving maps

Three results are obtained concerning the number of order preserving maps of an n-element partially ordered set to itself. We show that any such ordered set has at least 2 ^2n/3 order preserving maps (and 2 ² in the case of length one). Precise asymptotic estimates for the numbers of self-maps of crowns and fences are also obtained. In addition, lower bounds for many other infinite families are found and several precise problems are formulated.Supported by ONR Contract N00014-85-K-0769.Supported by NSF Grant DMS-9011850.Supported by NSERC Grants 69-3378 and 69-0259.     

保持二次算子的可加映射

设X是具有无限重复度的无限维或维数不小于3的有限维复Banach空间,B(X)是X上全体有界线性算子组成的Banach代数.首先证明了单位算子不能表示成3个平方幂零算子之和,利用算子分块矩阵技巧获得了平方幂零算子的本质特征.以此特征为基础,刻画了B(X)上双边保持二次算子可加满射的结构.     

Linear maps preserving reduced norms

If a linear map between central simple algebras preserves reduced norms, it is an isomorphism or antiisomorphism followed by multiplication by an element of reduced norm 1.     

Graphs for cone preserving maps

Let K be a closed, pointed, full cone in a finite dimensional real vector space. We associate with a linear map A for which AK?K four directed graphs. For two of the graphs the vertex set is the collection of all faces of K, and for two the vertices are all the extreme rays of K. We relate the irreducibility and primitivity of A to the strong connectedness of some of these graphs.     

Zero product preserving maps on

The main result of the paper characterizes continuous bilinear maps from C¹[0,1]×C¹[0,1] into a Banach space X with the property that (f,g)=0 whenever fg=0. This is applied to the study of zero product preserving operators on C¹[0,1], and operators on C¹[0,1] satisfying a version of the condition of the locality of an operator.     

Area preserving maps and volume preserving maps between a class of polyhedrons and a sphere

We construct a bijective continuous area preserving map from a class of elongated dipyramids to the sphere, together with its inverse. Then we investigate for which such solid polyhedrons the area preserving map can be used for constructing a bijective continuous volume preserving map to the 3D-ball. These maps can be further used in constructing uniform and refinable grids on the sphere and on the ball, starting from uniform and refinable grids on the elongated dipyramids. In particular, we show that HEALPix grids can be obtained from these maps. We also study the optimality of the logarithmic energy of the configurations of points obtained from these grids.     

Maps completely preserving idempotents and maps completely preserving square-zero operators

Let X, Y be real or complex Banach spaces with dimension greater than 2 and let A, B be standard operator algebras on X and Y, respectively. In this paper, we show that every map completely preserving idempotence from A onto B is either an isomorphism or (in the complex case) a conjugate isomorphism; every map completely preserving square-zero from A onto B is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism.     

Similarity preserving linear maps on matrices

We determine all similarity preserving linear maps on the space of n x n complex matrices and all unitary equivalence preserving linear maps on the space of n x n Hermitian matrices. (Sub)majorization preserving linear maps on Hermitian matrices are also determined.     

Surjective linear maps preserving local spectra

Let B(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize linear surjective and continuous maps on B(X) preserving different local spectral quantities at a nonzero fixed vector.