共查询到20条相似文献,搜索用时 0 毫秒
1.
2.
3.
4.
5.
6.
7.
8.
In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to Rn. A natural question arises, whether these approaches are equivalent or not. In this paper we study this problem and prove that, while the answer to the previous question is negative in the general case, the approach by continuous functions is not restrictive with respect to the other, provided that some natural stability and completeness assumptions are made. In particular, we show that every compact and stable 1-dimensional filtration of a compact metric space is induced by a continuous function. Moreover, we extend the previous result to the case of multi-dimensional filtrations, requiring that our filtration is also complete. Three examples show that we cannot drop the assumptions about stability and completeness. Consequences of our results on the definition of a distance between filtrations are finally discussed. 相似文献
9.
Let be a Tychonoff space, let be the space of all continuous real-valued functions defined on and let be the hyperspace of all nonempty closed subsets of . We prove the following result. Let be a locally connected, countably paracompact, normal -space without isolated points, and let . Then is in the closure of in with the locally finite topology if and only if is the graph of a cusco map. Some results concerning an approximation in the Vietoris topology are also given.
10.
11.
Gaetano Fichera 《Annali di Matematica Pura ed Applicata》1970,84(1):375-386
Summary Viene data la condizione necessaria e sufficiente perchè le funzioni razionali di una variabile, aventi poli di ordine prefissato
in assegnati punti del piano complesso, costituiscano un sistema completo iu Co (0, 1).
A Bruno Finzi nel suo70
mo
compleanno.
This research has been sponsored in part by the Aerospace Research Laboratoires through the European Office of Aerospace Research,
OAR, United States Air Force, under Grant EOOAR-69-0066.
Entrata in Redazione il 22 aprile 1970. 相似文献
12.
V. L. Velikin 《Mathematical Notes》1970,8(1):492-495
An investigation of the approximation on [0, 1] of functionsf (x) by spline functions s(f,; x) of degree 2r-1 and of deficiency r (r>1) depending on the vector function =
1 (x),...,
r-1(x) and interpolatingf (x) at fixed points. For the optimal choice of the vector
0, exact estimates are obtained of the norms f(x)-s (f,
0; x)C[0,1] and f (x)-s (f,
0; x)L[0, 1] on the function classes H
Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 41–46, July, 1970.In conclusion we would like to thank N. P. Korneichuk for suggesting this problem and for his valuable advice. 相似文献
13.
Ben Andrews 《Calculus of Variations and Partial Differential Equations》2010,39(3-4):649-657
A convex surface contracting by a strictly monotone, homogeneous degree one function of its principal curvatures remains smooth until it contracts to a point in finite time, and is asymptotically spherical in shape. No assumptions are made on the concavity of the speed as a function of principal curvatures. We also discuss motion by functions homogeneous of degree greater than 1 in the principal curvatures. 相似文献
14.
V. A. Kaminskii 《Mathematical Notes》1972,11(4):226-231
We discuss the problem of the best approximation in the mean of functions, which are continuous on a segment, by the polynomials in a differentiable Chebyshev system that do not exceed a given differentiable function. We prove that the extremal polynomial is unique and develop its characteristic properties.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 365–374, April, 1972.The author thanks A. L. Garkavi for posing the problem. 相似文献
15.
J. M. Aldaz 《Rendiconti del Circolo Matematico di Palermo》1996,45(2):289-302
We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a
separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization
one can easily obtain results that have appeared recently in the literature, in a more general form. We also examine what
happens when the range is a subset of an arbitrary Banach space, and show that this case often reduces to the separable case. 相似文献
16.
Mathematical Notes - 相似文献
17.
Jesú s M. F. Castillo Yolanda Moreno 《Proceedings of the American Mathematical Society》2008,136(7):2417-2423
We present two complementary results on the splitting of exact sequences having the form . The first one characterizes the Banach spaces such that for every compact space . The second is a nonlinear generalization of Zippin's criterion for the extension of -valued operators.
18.
19.
Let X be a Tychonoff space, C(X) be the space of all continuous real-valued functions defined on X and CL(X×R) be the hyperspace of all nonempty closed subsets of X×R. We prove the following result. Let X be a countably paracompact normal space. The following are equivalent: (a) dimX=0; (b) the closure of C(X) in CL(X×R) with the Vietoris topology consists of all F∈CL(X×R) such that F(x)≠∅ for every x∈X and F maps isolated points into singletons; (c) each usco map which maps isolated points into singletons can be approximated by continuous functions in CL(X×R) with the locally finite topology. From the mentioned result we can also obtain the answer to Problem 5.5 in [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] and to Question 5.5 in [R.A. McCoy, Comparison of hyperspace and function space topologies, Quad. Mat. 3 (1998) 243-258] in the realm of normal, countably paracompact, strongly zero-dimensional spaces. Generalizations of some results from [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] are also given. 相似文献