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1.
M. E. Mikhailov 《Mathematical Notes》1998,63(2):197-204
Here we define decomposable pseudometrics. A pseudometric is decomposable if it can be represented as the sum of two pseudometrics
that are obtained in a way other than the multiplication all distances by a positive factor. We consider spaces consisting
ofn points. We prove that there exist a finite number of indecomposable pseudometrics (that is, a basis) such that any pseudometric
is a linear combination of basic pseudometrics with nonnegative coefficients. Forn ≤ 7, the basic pseudometrics are listed. A decomposability test is derived for finite pseudometric spaces. We also establish
some other conditions of decomposability and indecomposability.
Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 225–234, February, 1998. 相似文献
2.
Thomas Lorenz 《Set-Valued Analysis》2008,16(1):1-50
The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order
geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this
paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional
analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied
with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler
approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological
vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ
N
evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity
assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions
for the uniqueness of these solutions.
相似文献
3.
The class of stable Banach spaces, inspired by the stability theory in mathematical logic, was introduced by Krivine and Maurey
and provided the proper context for the abstract formulation of Aldous’ result of subspaces ofL
1. In this paper we study the wider class of weakly stable Banach spaces, where the exchangeability of the iterated limits
occurs only for sequences belonging to weakly compact subsets, introduced independently by Garling (in an earlier unpublished
version of his expository paper on stable Banach spaces brought recently to our attention) and by the authors. Taking into
account Rosenthal’s application of the study of pointwise compact sets of Baire-1 functions (Rosenthal compact spaces) in
the study of Banach spaces (for whichl
1 does not embed isomorphically) and of the study of Rosenthal compact sets by Rosenthal and Bourgain-Fremlin-Talagrand, we
prove the following analogue of the Krivine-Maurey theorem for weakly stable spaces:If X is infinite dimensional and weakly stable then either l
p for some p≧1or co embeds isomorphically in X (§1). Garling (in the above reference) proved this result under the additional assumption thatX* is separable. We also construct an example of a Banach spaceX which is weakly stable, without an equivalent stable norm, and such thatl
2 embeds isomorphically in every infinite dimensional subspace ofX (§3). 相似文献
4.
We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator Tμ of a group algebra L1(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras. 相似文献
5.
Takemi Yanagimoto Masaaki Sibuya 《Annals of the Institute of Statistical Mathematics》1976,28(1):329-342
Summary One-sample test problem for ‘stochastically more (or less) spread’ is defined and a family of tests with isotonic power is
given. The problem is closely related to that for ‘longer (or shorter) tail’ in the reliability theory and the correspondence
between them is shown.
To characterize the tests three spread preorders inR
n
and corre-sponding tail preorders inR
+
n
are introduced. Functions which are ‘monotone’ in these orders, and subsets which are ‘centrifugal’ or ‘centripetal’ with
respect to these orders are studied. These notions generalize the Schur convexity.
The Institute of Statistical Mathematics 相似文献
6.
V. I. Polovinkin 《Siberian Advances in Mathematics》2012,22(1):41-49
We establish the necessary and sufficient conditions for the boundedness of the cubature formulas error functionals in spaces
of type L
p
m
corresponding to the considered sets of integrable functions defined on bounded subsets of cylindrical and conical surfaces. 相似文献
7.
The tensor structure of spaces L
p
(R
n
) of summable functions of several variables is described. A scale of Hardy-type spaces of analytic functionals defined in the unit ball of the space L
p
(R
1) of summable functions of one variable is introduced. One-parameter groups of isometries of such spaces of analytic functionals are investigated. 相似文献
8.
Giovanni Porcello 《Mathematische Nachrichten》2014,287(14-15):1733-1744
In this paper we introduce the notion of decomposability in the space of Henstock‐Kurzweil‐Pettis integrable (for short HKP‐integrable) functions. We show representations theorems for decomposable sets of HKP‐integrable or Henstock integrable functions, in terms of the family of selections of suitable multifunctions. 相似文献
9.
Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems 总被引:1,自引:0,他引:1
In this paper, we introduce a new concept of ϵ-efficiency for vector optimization problems. This extends and unifies various notions of approximate solutions in the literature.
Some properties for this new class of approximate solutions are established, and several existence results, as well as nonlinear
scalarizations, are obtained by means of the Ekeland’s variational principle. Moreover, under the assumption of generalized
subconvex functions, we derive the linear scalarization and the Lagrange multiplier rule for approximate solutions based on
the scalarization in Asplund spaces. 相似文献
10.
Multidimensional ultrametric pseudodifferential equations 总被引:1,自引:1,他引:0
We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined
as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and the
space D′0(X) of generalized functions on a multidimensional ultrametric space. We also consider some family of pseudodifferential operators
on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of
ultrametric characteristics are introduced. We prove an existence theorem and describe all solutions for the Cauchy problem
(an analog of the Kovalevskaya theorem). 相似文献
11.
Summary In [6] we introduced and investigated the notions of fI -sets and fI -continuous functions in ideal topological spaces. In this paper, we investigate their further important properties. 相似文献
12.
We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence (f
n
) of functions from a measure space to a Banach space. In one result the equi-integrability of f
n
’s is involved and we assume f
n
→ f almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence
of (f
n
) to f is assumed. 相似文献
13.
Branko Dragovich Zoran Rakić 《P-Adic Numbers, Ultrametric Analysis, and Applications》2010,2(4):322-340
Feynman’s path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes K(x″, t″; x′, t′) for two-dimensional systems with quadratic Lagrangians are evaluated analytically and obtained expressions are generalized
to any finite-dimensional spaces. These general formulas are presented in the form which is invariant under interchange of
the number fields ℝ ↔ ℚ
p
and ℚ ↔ ℚ
p
, p ≠ p′. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum
phenomena. 相似文献
14.
We prove a generalization of the Krasnosel’ski theorem on star-shaped sets. Usingd-segments inn-dimensional Minkowski spaces instead of usual segments, the notions “d-visibility” and “d-star-shapedness” are introduced. Our main aim is to give necessary and sufficient conditions ford-star-shapedness in finite-dimensional normed spaces. 相似文献
15.
We establish a decomposability criterion for linear sheaves on ℙ
n
. Applying it to instanton bundles, we show, in particular, that every rank 2n instanton bundle of charge 1 on ℙ
n
is decomposable. Moreover, we provide an example of an indecomposable instanton bundle of rank 2n − 1 and charge 1, thus showing that our criterion is sharp. 相似文献
16.
Given a compact metric space X and a strictly positive Borel measure ν on X, Mercer’s classical theorem states that the spectral decomposition of a positive self-adjoint integral operator T
k
:L
2(ν)→L
2(ν) of a continuous k yields a series representation of k in terms of the eigenvalues and -functions of T
k
. An immediate consequence of this representation is that k is a (reproducing) kernel and that its reproducing kernel Hilbert space can also be described by these eigenvalues and -functions.
It is well known that Mercer’s theorem has found important applications in various branches of mathematics, including probability
theory and statistics. In particular, for some applications in the latter areas, however, it would be highly convenient to
have a form of Mercer’s theorem for more general spaces X and kernels k. Unfortunately, all extensions of Mercer’s theorem in this direction either stick too closely to the original topological
structure of X and k, or replace the absolute and uniform convergence by weaker notions of convergence that are not strong enough for many statistical
applications. In this work, we fill this gap by establishing several Mercer type series representations for k that, on the one hand, make only very mild assumptions on X and k, and, on the other hand, provide convergence results that are strong enough for interesting applications in, e.g., statistical
learning theory. To illustrate the latter, we first use these series representations to describe ranges of fractional powers
of T
k
in terms of interpolation spaces and investigate under which conditions these interpolation spaces are contained in L
∞(ν). For these two results, we then discuss applications related to the analysis of so-called least squares support vector machines,
which are a state-of-the-art learning algorithm. Besides these results, we further use the obtained Mercer representations
to show that every self-adjoint nuclear operator L
2(ν)→L
2(ν) is an integral operator whose representing function k is the difference of two (reproducing) kernels. 相似文献
17.
A. G. Smirnov 《Theoretical and Mathematical Physics》2007,151(2):591-603
We study the localization properties of generalized functions defined on a broad class of spaces of entire analytic test functions.
This class, which includes all Gelfand-Shilov spaces S
α
β
(R
k
) with β < 1, provides a convenient language for describing quantum fields with a highly singular infrared behavior. We show
that the carrier cone notion, which replaces the support notion, can be correctly defined for the considered analytic functionals.
In particular, we prove that each functional has a uniquely determined minimal carrier cone.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 179–194, May, 2007. 相似文献
18.
Summary A new class of sets called πgs-closed sets is introduced and its properties are studied. Moreover the notions of πgs-T1/2 spaces and πgs-continuity are introduced. 相似文献
19.
In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range
is a dual space and satisfies the Radon–Nikodym property, Clarke’s generalized Jacobian will be extended to this setting.
Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness,
the β-compactness, the β-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using
key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the
generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set.
Research of the first author is supported by the Hungarian Scientific Research Fund (OKTA) under grant K62316.
Research of the second author is supported by the National Science Foundation under grant DMS-0306260. 相似文献
20.
Fumihiro Sato 《Proceedings Mathematical Sciences》1994,104(1):99-135
The theory of zeta functions associated with prehomogeneous vector spaces (p.v. for short) provides us a unified approach
to functional equations of a large class of zeta functions. However the general theory does not include zeta functions related
to automorphic forms such as the HeckeL-functions and the standardL-functions of automorphic forms on GL(n), even though they can naturally be considered to be associated with p.v.’s. Our aim is to generalize the theory to zeta
functions whose coefficients involve periods of automorphic forms, which include the zeta functions mentioned above.
In this paper, we generalize the theory to p.v.’s with symmetric structure ofK
ε-type and prove the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character.
In another paper, we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary
representations of compact groups.
Dedicated to the memory of Professor K G Ramanathan 相似文献