Hereditarily indecomposable subcontinua of inverse limits of Souslin arcs are metric

We prove that if Si is a Souslin arc (a Hausdorff arc that is the compactification of a Souslin line) for each i and , then every hereditarily indecomposable subcontinuum of X is metric. Since every non-degenerate hereditarily indecomposable continuum that is an inverse limit on metric arcs is a pseudo-arc, it follows that such an X would be a pseudo-arc or a point.     

A Note on Range-Inclusive Maps in Rings with Idempotents

Let A be a ring, let M be an A-bimodule, and let C be the center of M. A map F: A → M is said to be range-inclusive if [F(x), A] ? [x, M] for every x ? A. Recently, Bre?ar proved that if A is a unital ring and M a unital A-bimodule such that A contains wide idempotents, then every range-inclusive additive map F: A → M is of the form F(x) = λx + μ(x) for some λ ?C and μ: A → C. Our main purpose is to remove the assumption of unitality in the above result.     

Locally strongly convex hypersurfaces with constant affine mean curvature

Let be a locally strongly convex hypersurface, given by a strictly convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂An. We consider the Riemannian metric G# on M, defined by . In this paper we prove that if M is a locally strongly convex surface with constant affine mean curvature and if M is complete with respect to the metric G#, then M must be an elliptic paraboloid.     

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.     

Indecomposables as Elements in Affine Composition Algebras

LetAbe a path algebra of tame type over a finite field, letMbe an indecomposableA-module, and let (A) be the composition algebra ofA. The main result in this paper is that [M] ∈ (A) if and only ifMis a stone, i.e., Ext¹_A(M, M) = 0.     

Minimum cycle cover and Chinese postman problems on mixed graphs with bounded tree-width

Let M=(V,E,A) be a mixed graph with vertex set V, edge set E and arc set A. A cycle cover of M is a family C={C₁,…,C_k} of cycles of M such that each edge/arc of M belongs to at least one cycle in C. The weight of C is . The minimum cycle cover problem is the following: given a strongly connected mixed graph M without bridges, find a cycle cover of M with weight as small as possible. The Chinese postman problem is: given a strongly connected mixed graph M, find a minimum length closed walk using all edges and arcs of M. These problems are NP-hard. We show that they can be solved in polynomial time if M has bounded tree-width.     

Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ～ i = 1,2,…,M is used where x_i are arbitrary‐spaced. This approximation is a linear system for the derivatives u^(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A ^?1 where A = [A_ik] = (x_i ? x₀)^k is found. A finite‐difference approximation of derivatives u^(k) of a given function u(x) at point x₀ is derived in terms of the values u(x_i). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Recent hypertopologies and continuity of the value function and of the constrained level sets

Given a continuous function f: X→R, sufficient conditions are offered for the continuity of the value function v(A):=inf{f{x): x ε A} and of the level set multifunction Lev(A, α) := {x ε A: f(x)?α}, with respect to recently defined topologies on the closed sets of a metric space.     

Multidimensional version of M. A. Krasnoseľskii’s generalized contraction principle

Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → R ⁿ , where K is the cone of nonnegative vectors in R ⁿ . A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)^-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x _k = Fx _k-1, k = 1, 2,..., starting from an arbitrary point x ₀ in M, and the following error estimates hold: ρ (x*, x _k ) ⩽ Q ^k (I - Q)^-1ρ(x ₁, _x ⁰) ⩽ (I - Q)^-1 Q ^k ρ(x ₁, x ₀), k = 1, 2,.... Generally the mappings (I - Q)^-1 and Q ^k do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle.     

The Hilbert function of a maximal Cohen-Macaulay module

We study Hilbert functions of maximal CM modules over CM local rings. When A is a hypersurface ring with dimension d>0, we show that the Hilbert function of M with respect to is non-decreasing. If A=Q/(f) for some regular local ring Q, we determine a lower bound for e₀(M) and e₁(M) and analyze the case when equality holds. When A is Gorenstein a relation between the second Hilbert coefficient of M, A and S^A(M)= (Syz^A₁(M*))* is found when G(M) is CM and depthG(A)≥d−1. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyze when equality holds. We also give good bounds on Hilbert coefficients of M when M is maximal CM and G(M) is CM.     

Best Proximity Pair Results for Relatively Nonexpansive Mappings in Geodesic Spaces

Given A and B two nonempty subsets in a metric space, a mapping T: A ∪ B → A ∪ B is relatively nonexpansive if d(Tx, Ty) ≤ d(x, y) for every x ∈ A, y ∈ B. A best proximity point for such a mapping is a point x ∈ A ∪ B such that d(x, Tx) = dist(A, B). In this work, we extend the results given in Eldred et al. (2005) [A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171, 283–293] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. Namely, we give existence results of best proximity points for cyclic and noncyclic relatively nonexpansive mappings in the context of Busemann convex reflexive metric spaces. Moreover, particular results are proved in the setting of CAT(0) and uniformly convex geodesic spaces. Finally, we show that proximal normal structure is a sufficient but not necessary condition for the existence in A × B of a pair of best proximity points.     

On Absorbing Cycles in Min–Max Digraphs

We consider smooth finite dimensional optimization problems with a compact, connected feasible set M and objective function f. The basic problem, on which we focus, is: how to get from one local minimum to all the other ones. To this aim we introduce a bipartite digraph as follows. Its nodes are formed by the set of local minima and maxima of f|_M, respectively. Given a smooth Riemannian (i.e. variable) metric, there is an arc from a local minimum x to a local maximum y if the ascent (semi-)flow induced by the projected gradients of f connects points from a neighborhood of x with points from a neighborhood of y. The existence of an arc from y to x is defined with the aid of the descent (semi-)flow. Strong connectedness of ensures that, starting from one local minimum, we may reach any other one using ascent and descent trajectories in an alternating way. In case that no inequality constraints are present or active, it is well known that for a generic Riemannian metric the resulting min-max digraph is indeed strongly connected. However, if inequality constraints are active, then there might appear obstructions. In fact, we show that may contain absorbing two-cycles. If one enters such a cycle, one cannot leave it anymore via ascent and descent trajectories. Moreover, the cycles being constructed are stable with respect to small perturbations (in the C¹-topology) of the Riemannian metric.     

Hypersurfaces with Flat Centroaffine Metric and Equations of Associativity

It is demonstrated that hypersurfaces M ⁿ A ⁿ⁺¹ with a flat centroaffine metric are governed by a system of nonlinear PDEs known as the equations of associativity of 2-dimensional topological field theory. In the case of surfaces M ²A ³ this system reduces to a single third-order PDE, f _{x x x} f _{y y y} –f _{x x y} f _{x y y} =1 where x and y are the asymptotic coordinates on M ².     

Choosing a sheltered middle path

We say that point x∈R² is sheltered by a continuum S⊂R² if x does not belong to the unbounded component of R²\S. Suppose that points a and b are the endpoints of each of three arcs A₀, A₁ and A₂ contained in R². We prove that there is an arc B⊂A₀∪A₁∪A₂ with its endpoints a and b such that each point of B is sheltered by the union of each two of the arcs A₀, A₁ and A₂.     

An analogue of Hoffman's circulation conditions for max-balanced flows

LetD=(V, A) be a directed graph. A real-valued vectorx defined on the arc setA is amax-balanced flow forD if for every cutW the maximum weight over arcs leavingW equals the maximum weight over arcs enteringW. For vectorslu defined onA, we describe an analogue of Hoffman's circulation conditions for the existence of a max-balanced flowx satisfyinglxu. We describe an algorithm for computing such a vector, but show that minimizing a linear function over the set of max-balanced flows satisfyinglxu is NP-hard. We show that the set of all max-balanced flows satisfyinglxu has a greatest element under the usual coordinate partial order, and we describe an algorithm for computing this element. This allows us to solve several related approximation problems. We also investigate the set of minimal elements under the coordinate partial order. We describe an algorithm for finding a minimal element and show that counting the number of minimal elements is #P-hard. Many of our algorithms exploit the relationship between max-balanced flows and bottleneck paths.Research supported in part by NSF grant DMS 89-05645.Research supported in part by NSF grant ECS 87-18971.     

On Maps Preserving Zero Jordan Products

Let R be a ring, A = M _n (R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism. The third author is Corresponding author.     

An Exceptional Set in the Ergodic Theory of Expanding Maps on Manifolds

Under a general hypothesis an expanding map T of a Riemannian manifold M is known to preserve a measure equivalent to the Liouville measure on that manifold. As a consequence of this and Birkhoff’s pointwise ergodic theorem, the orbits of almost all points on the manifold are asymptotically distributed with regard to this Liouville measure. Let T be Lipschitz of class τ for some τ in (0,1], let Ω(x) denote the forward orbit closure of x and for a positive real number δ and let E(x₀, δ) denote the set of points x in M such that the distance from x₀ to Ω is at least δ. Let dim A denote the Hausdorff dimension of the set A. In this paper we prove a result which implies that there is a constant C(T) > 0 such that dimE(x₀,d) 3 dimM - \fracC(T)|logd| \dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\vert\!\log \delta \vert} if τ = 1 and dimE(x₀,d) 3 dimM - \fracC(T)log|logd|\dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\log \vert \log \delta \vert} if τ < 1. This gives a quantitative converse to the above asymptotic distribution phenomenon. The result we prove is of sufficient generality that a similar result for expanding hyperbolic rational maps of degree not less than two follows as a special case.     

Spanning k‐arc‐strong subdigraphs with few arcs in k‐arc‐strong tournaments

Given a k‐arc‐strong tournament T, we estimate the minimum number of arcs possible in a k‐arc‐strong spanning subdigraph of T. We give a construction which shows that for each k ≥ 2, there are tournaments T on n vertices such that every k‐arc‐strong spanning subdigraph of T contains at least arcs. In fact, the tournaments in our construction have the property that every spanning subdigraph with minimum in‐ and out‐degree at least k has arcs. This is best possible since it can be shown that every k‐arc‐strong tournament contains a spanning subdigraph with minimum in‐ and out‐degree at least k and no more than arcs. As our main result we prove that every k‐arc‐strong tournament contains a spanning k‐arc‐strong subdigraph with no more than arcs. We conjecture that for every k‐arc‐strong tournament T, the minimum number of arcs in a k‐arc‐strong spanning subdigraph of T is equal to the minimum number of arcs in a spanning subdigraph of T with the property that every vertex has in‐ and out‐degree at least k. We also discuss the implications of our results on related problems and conjectures. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 265–284, 2004     

Weighted inequalities for iterated maximal functions in Orlicz spaces

Let M be the classical Hardy‐Littlewood maximal operator. The object of our investigation in this paper is the iterated maximal function M^kf(x) = M(M^k?1f) (x) (k ≥ 2). Let Φ be a φ‐function which is not necessarily convex and Ψ be a Young function. Suppose that w is an A′_∞ weight and that k is a positive integer. If there exist positive constants C₁ and C₂ such that ((I)) then there exist positive constants C₃ and C₄ such that ((II)) where the functions a(t) and b(t) are the right derivatives of Φ(t) and Ψ(t), respectively. Conversely, if w is an A₁ weight, then (II) implies (I). Another necessary and sufficient condition will be given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)