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1.
In a general normed vector space, we study the minimal time function determined by a differential inclusion where the set-valued mapping involved has constant values of a bounded closed convex set U and by a closed target set S. We show that proximal and Fréchet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel sets of the function and level or suplevel sets of the support function of U. The known results in the literature require the set U to have the origin as an interior point or U be compact. (In particular, if the set U is the unit closed ball, the results obtained reduce to the subdifferential of the distance function defined by S.) 相似文献
2.
In general Banach space setting, we study the minimum time function determined by a closed convex set K and a closed set S (this function is simply the usual Minkowski function of K if S is the singleton consisting of the origin). In particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function. 相似文献
3.
《偏微分方程通讯》2013,38(9-10):2031-2053
ABSTRACT A Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicial choices of controls. The sublevel sets of the unique level set solution of the geometric equation is shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data. 相似文献
4.
5.
In this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems, where the preference relation is defined via an improvement set E. To be precise, we extend the notion of E-optimal point due to Chicco et al. in [4] to a general (non-necessarily Pareto) quasi ordered linear space and we study its properties. In particular, we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions. Moreover, we obtain necessary and sufficient conditions for E-optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general (nonconvex) case. By applying the obtained results to certain improvement sets we generalize well-known results of the literature referred to efficient, weak efficient and approximate efficient solutions of vector optimization problems. 相似文献
6.
G. Katz 《Commentarii Mathematici Helvetici》2002,77(1):39-77
For a closed 1-form with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which is harmonic. For a codimension 1 foliation , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of are minimal hypersurfaces. The conditions of Calabi and Sullivan are strikingly similar. If a closed form has no singularities, then both criteria are satisfied and, for an appropriate choice of metric, is harmonic and the associated foliation is comprised of minimal leaves. However, when has singularities, the foliation is not necessarily minimal.? We show that the Calabi condition enables one to find a metric in which is harmonic and the leaves of the foliation are minimal outside a neighborhood U of the -singular set. In fact, we prove the best possible result of this type: we construct families of metrics in which, as U shrinks to the singular set, the taut geometry of the foliation outside U remains stable. Furthermore, all compact leaves missing U are volume minimizing cycles in their homology classes. Their volumes are controlled explicitly.
Received: January 24, 2000 相似文献
7.
8.
Jan Chleboun 《Applications of Mathematics》2003,48(6):487-496
In practice, input data entering a state problem are almost always uncertain to some extent. Thus it is natural to consider a set U
ad of admissible input data instead of a fixed and unique input. The worst scenario method takes into account all states generated by U
ad and maximizes a functional criterion reflecting a particular feature of the state solution, as local stress, displacement, or temperature, for instance. An increase in the criterion value indicates a deterioration in the featured quantity. The method takes all the elements of U
ad as equally important though this can be unrealistic and can lead to too pessimistic conclusions. Often, however, additional information expressed through a membership function of U
ad is available, i.e., U
ad becomes a fuzzy set. In the article, infinite-dimensional U
ad are considered, two ways of introducing fuzziness into U
ad are suggested, and the worst scenario method operating on fuzzy admissible sets is proposed to obtain a fuzzy set of outputs. 相似文献
9.
Given an open set U in R
n
(n3) and a dense open subset V of U, it is shown that there is a finely harmonic function u on U such that V is the largest open subset of U on which u is harmonic. This result, which establishes the sharpness of a theorem of Fuglede, is obtained following a consideration of fine cluster sets of arbitrary functions. 相似文献
10.
A set of six axioms for sets of relations is introduced. All well-known sets of specific orderings, such as linear and weak orderings, satisfy these axioms. These axioms impose criteria of closedness with respect to several operations, such as concatenation, substitution and restriction. For operational reasons and in order to link our results with the literature, it is shown that specific generalizations of the transitivity condition give rise to sets of relations which satisfy these axioms. Next we study minimal extensions of a given set of relations which satisfy the axioms. By this study we come to the fundamentals of orderings: They appear to be special arrangements of several types of disorder. Finally we notice that in this framework many new sets of relations have to be regarded as a set of orderings and that it is not evident how to minimize the number of these new sets of orderings.Symbol Table
U
universe (infinite countable)
-
D
set of possible domains (finite and non-empty subsets of U)
-
R
set of all considered relations
- A
empty relation on A
- Id A
identity relation on A
- All A
all relation on A
-
c
complement operator (see Definition 2.1)
-
v
converse operator (see Definition 2.1)
-
s
symmetric part (see Definition 2.1)
-
asymmetric part (see Definition 2.1)
-
n
non-diagonal part (see Definition 2.1)
-
r
reflexive closure (see Definition 2.1)
We gratefully acknowledge the support by the Co-operation Centre of Tilburg and Eindhoven Universities. 相似文献
11.
Marshall A. Whittlesey 《Mathematische Annalen》2000,317(4):677-701
We say that a subset of is hypoconvex if its complement is the union of complex hyperplanes. Let be the closed unit disk in , . We prove two conjectures of Helton and Marshall. Let be a smooth function on whose sublevel sets have compact hypoconvex fibers over . Then, with some restrictions on , if Y is the set where is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector valued functions with boundary in Y. Furthermore, we show that the infimum is attained by a unique bounded analytic f which in fact is also smooth on . We also prove that if varies smoothly with respect to a parameter, so does the unique f just found.
Received: 18 December 1998 / Published online: 28 June 2000 相似文献
12.
Gonçalo Gutierres 《Mathematical Logic Quarterly》2010,56(3):331-336
It is well known that, in a topological space, the open sets can be characterized using ?lter convergence. In ZF (Zermelo‐Fraenkel set theory without the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultra?lter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultra?lter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski closure operator and u is the Ultra?lter Closure with uX (A):= {x ∈ X: (? U ultrafilter in X)[U converges to x and A ∈ U ]}. However, it is possible to built a topological space X for which uX ≠ kX, but the open sets are characterized by the ultra?lter convergence. To do so, it is proved that if every set has a free ultra?lter, then the Axiom of Countable Choice holds for families of non‐empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0‐spaces or {?} (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
13.
Classical reducibilities have complete sets U that any recursively enumerable set can be reduced to U. This paper investigates existence of complete sets for reducibilities with limited oracle access. Three characteristics of classical complete sets are selected and a natural hierarchy of the bounds on oracle access is built. As the bounds become stricter, complete sets lose certain characteristics and eventually vanish. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
Stephen J. Gardiner 《Potential Analysis》2011,34(1):81-88
This paper answers an old question of Fuglede by characterising those finely open sets U with the following property: any finely harmonic function on U must coincide with a harmonic function on some non-empty finely open subset. 相似文献
15.
A. Gómez-Tato E. Macías-Virgós M. J. Pereira-Sáez 《Annals of Global Analysis and Geometry》2011,39(3):325-335
The aim of this article is to use the so-called Cayley transform in order to compute the LS category of Lie groups and homogeneous
spaces by giving explicit categorical open coverings. When applied to U(n), U(2n)/Sp(n) and U(n)/O(n) this method is simpler than those formerly known. We also show that the Cayley transform is related to height functions
in Lie groups, allowing to give a local linear model of the set of critical points. As an application we give an explicit
covering of Sp(2) by categorical open sets. The obstacles to generalize these results to Sp(n) are discussed. 相似文献
16.
In this paper we show that any p-perverse sheaf on an arbitrary stratified topological space (p is a perversity function) is functorially determined by a system of usual sheaves on the open sets U
r
(r≥0) and certain gluing data, where U
r
is the union of strata of perversity ≤r.
Both authors are partially supported by BFM2001-3207 and MTM2004-07203-C02-01 and FEDER. 相似文献
17.
Peter Giesl 《Journal of Approximation Theory》2008,153(2):184-211
The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration xn+1=g(xn) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523–546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally. 相似文献
18.
Marcin Bilski 《Constructive Approximation》2012,35(3):273-291
Let X be an analytic subset of U×C
n
of pure dimension k such that the projection of X onto U is a proper mapping, where U⊂C
k
is a Runge domain. We show that X can be approximated by algebraic sets. Next we present a constructive method for local approximation of analytic sets by
algebraic ones. 相似文献
19.
Julien Demouth Olivier Devillers Marc Glisse Xavier Goaoc 《Discrete and Computational Geometry》2009,42(3):379-398
Let ℱ∪{U} be a collection of convex sets in ℝ
d
such that ℱ covers U. We prove that if the elements of ℱ and U have comparable size, then a small subset of ℱ suffices to cover most of the volume of U; we analyze how small this subset can be depending on the geometry of the elements of ℱ and show that smooth convex sets
and axis parallel squares behave differently. We obtain similar results for surface-to-surface visibility amongst balls in
three dimensions for a notion of volume related to form factor. For each of these situations, we give an algorithm that takes
ℱ and U as input and computes in time O(|ℱ|*|ℋ
ε
|) either a point in U not covered by ℱ or a subset ℋ
ε
covering U up to a measure ε, with |ℋ
ε
| meeting our combinatorial bounds.
The authors acknowledge support from the French–Korean Science and Technology amicable relationship program (STAR) 11844QJ. 相似文献
20.
Pascal Dingoyan 《Mathematische Zeitschrift》1999,232(2):217-240
In this article, we study univalent open subsets , , assuming to be pseudoconcave in Andreotti's sense. We prove an Hartogs's Kugelsatz theorem for such open sets: Let U an open subset in V such that is a pseudoconcave domain in the sense of Andreotti. Then U contains a maximal compact hypersurface H. Moreover, any meromorphic section s, of a vector bundle F over V, defined on (a neighborhood of) extends on , and, if s is holomorphic then s extends meromorphically to U, with a polar set in H.
Received November 30, 1997; in final form July 23, 1998 相似文献
Received November 30, 1997; in final form July 23, 1998 相似文献