Complementary bases of a matroid

Let e₁, e′₁, e₂, e′₂, …, e_n, e′_n be the elements of matroid M. Suppose that {e₁, e₂, …;, e_n} is a base of M and that every circuit of M contains at least m + 1 elements. We prove that there exist at least 2^m bases, called complementary bases, of M with the property that only one of each complementary pair e_j, e′_j is contained in any base.We also prove an analogous result for the case where E is partitioned into E₁, E₂, …, E_n and the initial base contains |E_j| ? 1 elements from partition E_j.     

Strongly e-movable convergence and spaces of ANR's

Let X be a metric space and let ANR(X) denote the hyperspace of all compact ANR's in X. This paper introduces a notion of a strongly e-movable convergence for sequences in ANR(X) and proves several characterizations of strongly e-movable convergence. For a (complete) separable metric space X we show that ANR(X) with the topology induced by strongly e-movable convergence can be metrized as a (complete) separable metric space. Moreover, if X is a finite-dimensional compactum, then strongly e-movable convergence induces on ANR(X) the same topology as that induced by Borsuk's homotopy metric.For a separable Q-manifold M, ANR(M) is locally arcwise connected and A, B ? ANR(M) can be joined by an arc in ANR(M) iff there is a simple homotopy equivalence ?: A → B homotopic to the inclusion of A into M.     

Almost-Graphic Matroids

A nongraphic matroid M is said to be almost-graphic if, for all elements e, either M\e or M/e is graphic. We determine completely the class of almost-graphic matroids, thereby answering a question posed by Oxley in his book “Matroid Theory.” A nonregular matroid is said to be almost-regular if, for all elements e, either M\e or M/e is regular. An element e for which both M\e and M/e are regular is called a regular element. We also determine the almost-regular matroids with at least one regular element.     

On minor-minimally-connected matroids

By a well-known result of Tutte, if e is an element of a connected matroid M, then either the deletion or the contraction of e from M is connected. If, for every element of M, exactly one of these minors is connected, then we call M minor-minimally-connected. This paper characterizes such matroids and shows that they must contain a number of two-element circuits or cocircuits. In addition, a new bound is proved on the number of 2-cocircuits in a minimally connected matroid.     

Dimension and embedding theorems for geometric lattices

Let G be an n-dimensional geometric lattice. Suppose that 1 ? e, f ? n ? 1, e + f ? n, but e and f are not both n ? 1. Then, in general, there are E, F? G with dim E = e, dim F = f, E ? F = 1, and dim E ∧ F = e + f ? n ? 1; any exception can be embedded in an n-dimensional modular geometric lattice M in such a way that joins and dimensions agree in G and M, as do intersections of modular pairs, while each point and line of M is the intersection (in M) of the elements of G containing it.     

Nonpositive curvature: A geometrical approach to Hilbert-Schmidt operators

We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into Σ. We give an intrinsic algebraic characterization of convex closed submanifolds M. We study the group of isometries of such submanifolds: we prove that GM, the Banach-Lie group generated by M, acts isometrically and transitively on M. Moreover, GM admits a polar decomposition relative to M, namely GM?M×K as Hilbert manifolds (here K is the isotropy of p=1 for the action ), and also GM/K?M so M is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds M. These decompositions are obtained via a nonlinear but analytic orthogonal projection , a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism NM?Σ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed ea, we obtain ea=exevex with ex∈M and v orthogonal to M at p=1. As a corollary we obtain decompositions for the full group of invertible elements G?M×exp(T₁M^⊥)×K.     

Block transitivity and degree matrices: (Extended abstract)

We say that a square matrix M is a degree matrix of a given graph G if there is a so called equitable partition of its vertices into r blocks such that whenever two vertices belong to the same block, they have the same number of neighbors inside any block.We ask now whether for a given degree matrix M, there exists a graph G such that M is a degree matrix of G, and in addition, for any two edges e, f spanning between the same pair of blocks there exists an automorphism of G that sends e to f. In this work, we fully characterize the matrices for which such a graph exists and show a way to construct one.     

A note on nongraphic matroids

Tutte found an excluded minor characterization of graphic matroids with five excluded minors. A variation on Tutte's result is presented here. Let {e, f, g} be a circuit of a 3-connected nongraphic matroid M. Then M has a minor using e, f, g isomorphic to either the 4-point line, the Fano matroid, or the bond matroid of K_3,3.     

Marriage in denumerable societies

A society is an ordered triple (M, W, K) of sets such that M, W are disjoint and K ? M × W. An espousal of (M, W, K) is a subset of K of the form {(a, e(a)) : a ∈ M} where e(a₁) ≠ e(a₂) whenever a₁ ≠ a₂. If M is countable, we associate with (M, W, K) and each ordinal α a function m_α from the set of subsets of W into the union of the set of integers and {? ∞, ∞}. Three different definitions of m_α (all fairly elaborate) are presented and their equivalence under suitable conditions is proved. Assuming M to be countable, we prove that (i) (M, W, K) has an espousal if and only if $\text{m}{}_{\mathrm{\Omega }}\text{(X) ? 0}$ for every subset X of W, where Ω is the first uncountable ordinal, and (ii) if X ? W and α ? β and m_α(X) < ∞ and m_α(Z) ? 0 for every subset Z of X then m_α(Z) = m_β(Z) for every subset Z of X. The result (i) is a theorem of Damerell and Milner, but the proof here presented differs somewhat in formulation and structure from theirs.     

Strong Splitter Theorem

The Splitter Theorem states that, if N is a 3-connected proper minor of a 3-connected matroid M such that, if N is a wheel or whirl then M has no larger wheel or whirl, respectively, then there is a sequence M ₀, . . . , M _n of 3-connected matroids with ${M_0 \cong N}$ , M _n = M and for ${i \in \{1, \ldots , n}\}$ , M _i is a single-element extension or coextension of M _i?1. Observe that there is no condition on how many extensions may occur before a coextension must occur. We give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, M starting with N and at each step doing a 3-connected single-element extension or coextension, such that at most two consecutive single-element extensions occur in the sequence (unless the rank of thematroids involved is r(M)). Moreover, if two consecutive single-element extensions by elements {e, f} are followed by a coextension by element g, then {e, f , g} form a triad in the resulting matroid.     

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group, π _e (G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(π _e (G)) = k < ∞, otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M ₁₂, M ₂₂, J ₂, He, Suz, M ^c L and O′N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M ₁₂, M ₂₂, He, Suz or O′N, then h(π _e (Aut(M))) ∈¸ {1,∞}.     

Two-Dimensional Graphs Moving by Mean Curvature Flow

A surface Σ is a graph in ?⁴ if there is a unit constant 2-form ω on ?⁴ such that <e ₁∧e ₂, ω≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if $ \vartheta _{0} \geqslant \frac{1} {{{\sqrt 2 }}} A surface Σ is a graph in ℝ⁴ if there is a unit constant 2-form ω on ℝ⁴ such that <e ₁∧e ₂, ω>≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if v ₀≥ on the initial surface, then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution. A surface Σ is a graph in M ₁×M ₂ where M ₁ and M ₂ are Riemann surfaces, if <e ₁∧e ₂, ω₁>≥v ₀>0 where ω₁ is a K?hler form on M ₁. We prove that, if M is a K?hler-Einstein surface with scalar curvature R, v ₀≥ on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R≥0 it converges to a totally geodesic surface which is holomorphic. Received July 25, 2001, Accepted October 11, 2001     

On the Hahn--Mazurkiewicz theorem

The following generalization of the Hahn-Mazurkiewicz theorem is proved: Let (E,e) be a locally compact locally connected metric space. Let M be a continuum in this space and let d,e&isin; M. Then there is a continuous mapping f: [0,1]→E such that f(0) = d, f(1)= e and M&sub;f([0,1]). Also some corollaries of this theorem are proved.     

On reformulated Zagreb indices

The first and second reformulated Zagreb indices are defined respectively in terms of edge-degrees as EM₁(G)=∑_e∈Edeg(e)² and EM₂(G)=∑_e∼fdeg(e)deg(f), where deg(e) denotes the degree of the edge e, and e∼f means that the edges e and f are adjacent. We give upper and lower bounds for the first reformulated Zagreb index, and lower bounds for the second reformulated Zagreb index. Then we determine the extremal n-vertex unicyclic graphs with minimum and maximum first and second Zagreb indices, respectively. Furthermore, we introduce another generalization of Zagreb indices.     

On minor-minimally 3-connected binary matroids

A matroid M is called minor-minimally 3-connected if M is 3-connected and, for each e∈E(M), either M?e or M/e is not 3-connected. In this paper, we prove a chain theorem for the class of minor-minimally 3-connected binary matroids. As a consequence, we obtain a chain theorem for the class of minor-minimally 3-connected graphs.     

Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary

Let e _λ (x) be an eigenfunction with respect to the Laplace-Beltrami operator Δ _M on a compact Riemannian manifold M without boundary: Δ _M e _λ = λ ² e _λ . We show the following gradient estimate of e _λ : for every λ ≥ 1, there holds ${\lambda\|e_\lambda\|_\infty/C\leq \|\nabla e_\lambda\|_\infty\leq C\lambda\|e_\lambda\|_\infty}$ , where C is a positive constant depending only on M.     

Monge-Ampère equations and moduli spaces of manifolds of circular type

A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point xo, so that: (a) it is a smooth solution on M?{xo} to the Monge-Ampère equation n(ddcu)=0; (b) xo is a singular point for u of logarithmic type and eu extends smoothly on the blow up of M at xo; (c) ddc(eu)>0 at any point of M?{xo}. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of Cn.A set of modular parameters for bounded manifolds of circular type is considered. In particular, for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form. It is also shown that the class of normalizing maps for an n-dimensional manifold M is a new holomorphic invariant with the following property: it is parameterized by the points of a finite dimensional real manifold of dimension n² when M is a (non-convex) circular domain while it is of dimension n²+2n when M is a strictly linearly convex domain. New characterizations of the circular domains and of the unit ball are also obtained.     

Finding a small 3-connected minor maintaining a fixed minor and a fixed element

LetN andM be 3-connected matroids, whereN is a minor ofM on at least 4 elements, and lete be an element ofM and not ofN. Then, there exists a 3-connected minor \(\bar M\) ofM that usese, hasN as a minor, and has at most 4 elements more thanN. This result generalizes a theorem of Truemper and can be used to prove Seymour’s 2-roundedness theorem, as well as a result of Oxley on triples in nonbinary matroids.     

A splitting theorem on smooth metric measure spaces

We consider a smooth metric measure space (M, g, e ^?f dv). Let ?? _f be its weighted Laplacian. Assuming that ??₁(?? _f ) is positive and the m-dimensional Bakry-émery curvature is bounded below in terms of ??₁(?? _f ), we prove a splitting theorem for (M, g, e ^?f dv). This theorem generalizes previous results by Lam and Li-Wang (Trans Am Math Soc 362:5043?C5062, 2010; J Diff Geom 58:501?C534, 2001; see also J Diff Geom 62:143?C162, 2002).