共查询到20条相似文献,搜索用时 31 毫秒
1.
R. Lucchetti P. Shunmugaraj Y. Sonntag 《Numerical Functional Analysis & Optimization》2013,34(1-2):103-115
Given a continuous function f: X→R, sufficient conditions are offered for the continuity of the value function v(A):=inf{f{x): x ε A} and of the level set multifunction Lev(A, α) := {x ε A: f(x)?α}, with respect to recently defined topologies on the closed sets of a metric space. 相似文献
2.
Moshe Stupel David Fraivert 《International Journal of Mathematical Education in Science & Technology》2013,44(6):892-904
Given a composite function of the form h(x) = f(g(x)), difficulties are often encountered in calculating the value of the nth derivative at some point x = x0 when one attempts to determine whether its nth derivative becomes zero at this point, or attempts to find the sign of the nth derivative by differentiating it n times and substituting x0.This present paper offers an alternative method that allows the investigation of the nth derivative of function h(x) based on the investigation of functions f?(x) and g(x) only.Several examples are given, which implement the conclusions on the properties of the relation. 相似文献
3.
We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume
that for any submodular function f: ?→R on a distributive lattice ?⊆2
V
with ?,V∈? and f(?)=0 and for any vector x∈R
V
where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|2) times, the descent method finds a sequence of subsets Z
1,Z
2,···,Z
k
of V such that f(Z
1)>f(Z
2)>···>f(Z
k
)=min{f(Y) | Y∈?}, where k is O(|V|2). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient
membership algorithms.
Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001 相似文献
4.
A comparative study of the functional equationsf(x+y)f(x–y)=f
2(x)–f
2(y),f(y){f(x+y)+f(x–y)}=f(x)f(2y) andf(x+y)+f(x–y)=2f(x){1–2f
2(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x–y)=2f(x){1–2f
2(y/2)} with D'Alembert's equation,f(x+y)+f(x–y)=2f(x)f(y) and the sine-cosine equationg(x–y)=g(x)g(y) +f(x)f(y) has also been investigated. 相似文献
5.
Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionxf(x)–1f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)f(x)} can be expressed as the convex hull of the limits of type {N(x
n)}, where {x
n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper. 相似文献
6.
Bernard Epstein 《Israel Journal of Mathematics》1966,4(3):145-152
An analysis is presented of the equationf(x+a)−f(x)=e
−x
{f(x)−f(x−b)}. Herea andb denote arbitrary positive constants, and a solution is sought which satisfies the following conditions:f(−∞)=0,f(+∞)=1, 0≦f(x)≦1. Existence and uniqueness of solution are established, and then an analytical form of the solution is obtained by use
of bilateral Laplace transform.
Research supported by the National Science Foundation, Grant GP-2558. 相似文献
7.
In this paper, we discuss the following inequality constrained optimization problem (P) minf(x) subject tog(x)0,g(x)=(g
1(x), ...,g
r
(x))
, wheref(x),g
j
(x)(j=1, ...,r) are locally Lipschitz functions. TheL
1 exact penalty function of the problem (P) is (PC) minf(x)+cp(x) subject tox R
n
, wherep(x)=max {0,g
1(x), ...,g
r
(x)},c>0. We will discuss the relationships between (P) and (PC). In particular, we will prove that under some (mild) conditions a local minimum of (PC) is also a local minimum of (P). 相似文献
8.
A function f : GF(2
r
) → GF(2
r
) is called crooked if the sets {f(x) + f(x + a)|x ∈ GF(2
r
)} is an affine hyperplane for any nonzero a ∈ GF(2
r
). We prove that a crooked binomial function f(x) = x
d
+ ux
e
defined on GF(2
r
) satisfies that both exponents d, e have 2-weights at most 2.
相似文献
9.
We consider the delay differential equation [(x)\dot](t) = - mx(t) + f(x(t - t))\dot x(t) = - \mu x(t) + f(x(t - \tau )), where μ, τ are positive parameters and f is a strictly monotone, nonlinear C
1-function satisfying f(0) = 0 and some convexity properties. It is well known that for prescribed oscillation frequencies (characterized by the
values of a discrete Lyapunov functional) there exists τ* > 0 such that for every τ > τ* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function
of τ > τ*. First we show that it is a monotone nondecreasing Lipschitz continuous function of τ with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu
[14] about uniqueness and existence of periodic solutions of a system of delay differential equations. 相似文献
10.
孙太祥 《高校应用数学学报(英文版)》2002,17(3):313-318
Let f be a tree map,P(f) the set of periodic points of f and CR(f) the set of chain recurrent points of f. In this paper,the notion of division for invariant closed subsets of a tree map is introduced. It is proved that: (1) fhas zero topological entropy if and only if for any x∈CR(f)-P(f) and each natural number s the orbit of x under f^5 has a division; (2) If f has zero topological entropy,then for any xECR(f)--P(f) the w-limit set of x is an infinite minimal set. 相似文献
11.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
12.
R. I. Hovsepyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2009,44(4):205-211
It is proved that any measurable, finite function f(x) has a smooth primitive F(x), i.e. there is a function F(x) such that F′(x) = f(x) almost everywhere, and particularly ω(δ; F) = o(δ ln δ). This is an improvement of N. N. Luzin’s theorem which states just the continuity of the primitive F(x). 相似文献
13.
This paper is concerned with sequential Monte Carlo methods for optimizing a system under constraints. We wish to minimize f(x), where qi(x) ? 0 (i = 1, …, m) must hold. We can calculate the qi(x), but f(x) can only be observed in the presence of noise. A general approach, based on an adaptation of a version of stochastic approximation to the penalty function method, is discussed and a convergence theorem is proved. 相似文献
14.
Zbigniew Grande 《Central European Journal of Mathematics》2011,9(4):772-777
A sequence (f
n
)
n
of functions f
n
: X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f
n+1(x) ≤ f
n
(x) (f
n+1(x) ≥ f
n
(x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions. 相似文献
15.
Another logarithmic functional equation 总被引:1,自引:0,他引:1
K. J. Heuvers 《Aequationes Mathematicae》1999,58(3):260-264
Summary. Let f : ]0,¥[? \Bbb R f :\,]0,\infty[\to \Bbb R be a real valued function on the set of positive reals. The functional equations¶¶f(x + y) - f(x) - f(y) = f(x-1 + y-1) f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1}) ¶and¶f(xy) = f(x) + f(y) f(xy) = f(x) + f(y) ¶are equivalent to each other. 相似文献
16.
In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:
f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)