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Let F be either the real number field R or the complex number field C and RPn the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F-vector bundle over RPn to be stably extendible to RPm for every m?n. In this paper, we simplify the theorems and apply them to the tangent bundle of RPn, its complexification, the normal bundle associated to an immersion of RPn in Rn+r(r>0), and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5]. 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
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A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence x over a finite alphabet is ultimately periodic if and only if, for some n, the number of different factors of length n appearing in x is less than n+1. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let d≥2. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of Zd definable by a first order formula in the Presburger arithmetic 〈Z;<,+〉. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse–Hedlund theorem to an arbitrary dimension d and characterize sets of Zd definable in 〈Z;<,+〉 in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often. 相似文献
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For any symmetric function f:Rn?Rn, one can define a corresponding function on the space of n×n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Frechet differentiability, continuous differentiability. 相似文献
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Hadwiger’s Theorem states that En-invariant convex-continuous valuations of definable sets in Rn are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable R-valued functions on Rn. This generalizes intrinsic volumes to (dual pairs of) non-linear valuations on functions and provides a dual pair of Hadwiger classification theorems. 相似文献
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Let Ω⊂Rn be an open, connected subset of Rn, and let F:Ω−Ω→C, where Ω−Ω={x−y:x,y∈Ω}, be a continuous positive definite function. We give necessary and sufficient conditions for F to have an extension to a continuous positive definite function defined on the entire Euclidean space Rn. The conditions are formulated in terms of existence of a unitary representations of Rn whose generators extend a certain system of unbounded Hermitian operators defined on a Hilbert space associated to F. Different positive definite extensions correspond to different unitary representations. 相似文献
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In this paper, we study degenerate CR embeddings f of a strictly pseudoconvex hypersurface M⊂Cn+1 into a sphere S in a higher dimensional complex space CN+1. The degeneracy of the mapping f will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d of the second fundamental form and all of its covariant derivatives is <n (here, n is the CR dimension of M), then f(M) is contained in a complex plane of dimension n+d+1. The converse of this statement is also true, as is easy to see. When the total rank d exceeds n, it is no longer true, in general, that f(M) is contained in a complex plane of dimension n+d+1, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n, then partial rigidity may still persist, but there is a “defect” k that arises from the ranks exceeding n such that f(M) is only contained in a complex plane of dimension n+d+k+1. Moreover, this defect occurs in general, as is illustrated by examples. 相似文献
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In the present article we provide a sufficient condition for a closed set F∈Rd to have the following property which we call c -removability: Whenever a continuous function f:Rd→R is locally convex on the complement of F , it is convex on the whole Rd. We also prove that no generalized rectangle of positive Lebesgue measure in R2 is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) [5]: Assume the closed set F⊂Rd is such that any locally convex function defined on Rd?F has a unique convex extension on Rd. Is F necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R2. 相似文献
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We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z/5Z-quotient of the Fermat quintic surface in P3. This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z11⊕Z/5Z. In particular, the result answers Kuznetsov’s Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type E8. We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence. 相似文献
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