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2.
We prove that if for a continuous map f on a compact metric space X, the chain recurrent set, R(f) has more than one chain component, then f does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f on a compact metric space X has the asymptotic average shadowing property and if A is an attractor for f, then A is the single attractor for f and we have A=R(f). We also study diffeomorphisms with asymptotic average shadowing property and prove that if M is a compact manifold which is not finite with dimM=2, then the C1 interior of the set of all C1 diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω-stable diffeomorphisms. 相似文献
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The ball hull mapping β associates with each closed bounded convex set K in a Banach space its ball hull β(K), defined as the intersection of all closed balls containing K. We are concerned in this paper with continuity and Lipschitz continuity (with respect to the Hausdorff metric) of the ball hull mapping. It is proved that β is a Lipschitz map in finite dimensional polyhedral spaces. Both properties, finite dimension and polyhedral norm, are necessary for this result. Characterizing the ball hull mapping by means ofH-convexity we show, with the help of a remarkable example from combinatorial geometry, that there exist norms with noncontinuous β map, even in finite dimensional spaces. Using this surprising result, we then show that there are infinite dimensional polyhedral spaces (in the usual sense of Klee) for which the map β is not continuous. A property known as ball stability implies that β has Lipschitz constant one. We prove that every Banach space of dimension greater than two can be renormed so that there is an intersection of closed balls for which none of its parallel bodies is an intersection of closed balls, thus lacking ball stability. 相似文献
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Let v1,…,vm be a finite set of unit vectors in Rn. Suppose that an infinite sequence of Steiner symmetrizations are applied to a compact convex set K in Rn, where each of the symmetrizations is taken with respect to a direction from among the vi. Then the resulting sequence of Steiner symmetrals always converges, and the limiting body is symmetric under reflection in any of the directions vi that appear infinitely often in the sequence. In particular, an infinite periodic sequence of Steiner symmetrizations always converges, and the set functional determined by this infinite process is always idempotent. 相似文献
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A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G∈X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word wq is indeed concise in the class of residually finite groups. Further, we show that in the case where w=γk the word wq is boundedly concise in the class of residually finite groups. It remains unknown whether the word wq is actually concise in the class of all groups. 相似文献
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We give an elementary proof for Lewis Bowen’s theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups K and every free product Γ of infinite amenable groups, the factor Γ?KΓ/K of the Bernoulli action Γ?KΓ by the diagonal K-action is isomorphic with a Bernoulli action of Γ. 相似文献
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The Morse–Sard theorem states that the set of critical values of a Ck smooth function defined on a Euclidean space Rd has Lebesgue measure zero, provided k≥d. This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of Ck functions: it is shown that these functions are Lipschitz continuous and the set of their Clarke critical values is null. 相似文献
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By means of a certain module V and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R of a finite-dimensional Hopf algebra H is finite. The module V is the counit representation induced from R to H, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V is either semisimple with R∗ pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R, then the depth of R in H is finite. One assigns a nonnegative integer depth to V, or any other H-module, by comparing the truncated tensor algebras of V in a finite tensor category and so obtains upper and lower bounds for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutation module of a corefree subgroup has depth less than the number of values of its character. 相似文献
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The Severi variety parameterizes plane curves of degree d with δ nodes. Its degree is called the Severi degree. For large enough d, the Severi degrees coincide with the Gromov–Witten invariants of CP2. Fomin and Mikhalkin (2010) [10] proved the 1995 conjecture that for fixed δ, Severi degrees are eventually polynomial in d. 相似文献
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In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X in a bounded κ-fat open set; if u is a positive harmonic function with respect to X in a bounded κ-fat open set D and h is a positive harmonic function in D vanishing on Dc, then the non-tangential limit of u/h exists almost everywhere with respect to the Martin-representing measure of h. 相似文献
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Let S be a positively graded polynomial ring over a field of characteristic 0, and I⊂S a proper graded ideal. In this note it is shown that S/I is Golod if ∂(I)2⊂I. Here ∂(I) denotes the ideal generated by all the partial derivatives of elements of I. We apply this result to find large classes of Golod ideals, including powers, symbolic powers, and saturations of ideals. 相似文献
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For a non-degenerate convex subset Y of the n -dimensional Euclidean space Rn, let F(Y) be the family of all fuzzy sets of Rn which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y . We show that the space F(Y) with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?2 if Y is compact; and the space F(Rn) is also homeomorphic to ?2. 相似文献
16.
Let C be a nonempty subset of a topological vector space E. We state and prove new various fixed point theorems of Fan–Browder type for set-valued maps F:C→2E such that C⊂F(C) (called expansive), without assuming that the sets C and F(C) are convex or compact or equal, and E is Hausdorff. Let K be a convex subset of E and let C be a nonempty subset of K. Our proofs use a technique based on the investigations of the images of maps and restated for maps f:C×K→R∪{−∞,+∞} of G.X.-Z. Yuan’s results concerning the existence of equilibrium points and minimax inequalities for maps f:K×K→R∪{−∞,+∞}. Examples are provided. 相似文献
17.
It is shown that if a sequence of open n-sets Dk increases to an open n-set D then reflected stable processes in Dk converge weakly to the reflected stable process in D for every starting point x in D. The same result holds for censored α-stable processes for every x in D if D and Dk satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains. 相似文献
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In this paper, we consider a continuous map f:X→X, where X is a compact metric space, and prove that for any positive integer N, f is Schweizer–Smital chaotic if and only if fN is too. 相似文献
20.
A tournament of order n is usually considered as an orientation of the complete graph Kn. In this note, we consider a more general definition of a tournament that we call aC-tournament, where C is the adjacency matrix of a multigraph G, and a C-tournament is an orientation of G. The score vector of a C-tournament is the vector of outdegrees of its vertices. In 1965 Hakimi obtained necessary and sufficient conditions for the existence of a C-tournament with a prescribed score vector R and gave an algorithm to construct such a C-tournament which required, however, some backtracking. We give a simpler and more transparent proof of Hakimi’s theorem, and then provide a direct construction of such a C-tournament which works even for weighted graphs. 相似文献