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1.
A nonnegative, infinitely differentiable function defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and
0
1
( t) dt=1. In this article, the following problem is considered. Determine
k
=inf
0
1
| (k)( t)| dt, k=1, 2, ..., where (k) denotes the kth derivative of and the infimum is taken over the set of all mollifier functions , which is a convex set. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. The problem is reducible to three equivalent problems, a nonlinear programming problem, a problem on the functions of bounded variation, and an approximation problem involving Tchebycheff polynomials. One of the results of this article shows that
k
= k!2 2k–1, k=1, 2, .... The numerical values of the optimal solutions of the three problems are obtained as a function of k. Some inequalities of independent interest are also derived.This research was supported in part by the National Science Foundation, Grant No. GK-32712. 相似文献
2.
Let E be a linear space, let K
E and f:K . We formulate in terms of the lower Dini directional derivative problem GMVI ( f
,K
), which can be considered as a generalization of MVI ( f
,K
), the Minty variational inequality of differential type. We investigate, in the case of K star-shaped (SS), the existence of a solution x
* of GMVI ( f
K
) and the property of f to increase-along-rays starting at x
*, fIAR ( K, x
*). We prove that the GMVI ( f
,K
) with radially l.s.c. function f has a solution x
* ker K if and only if fIAR ( K, x
*). Further, we prove that the solution set of the GMVI ( f
,K
) is a convex and radially closed subset of ker K. We show also that, if the GMVI ( f
,K
) has a solution x
*K, then x
* is a global minimizer of the problem min f( x), xK. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function f, these sets coincide. 相似文献
3.
The following problem is studied: Given a compact set S in R
n
and a Minkowski functional p(x), find the largest positive number r for which there exists x S such that the set of all y R
n
satisfying p(y–x) r is contained in S. It is shown that when S is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this design centering problem can be reformulated as the minimization of some d.c. function (difference of two convex functions) over R
n
. In the case where, moreover, p(x) = (x
T
Ax) 1/2, with A being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set. 相似文献
4.
Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the function xf( x) –1f( x) on the domain where it is defined is deduced from some continuity properties of the normal cone N to the level sets of the quasiconvex function f. We also prove that, under a pseudoconvexity-type condition, the normal cone N( 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 to x and contained in a dense subset D. In particular, when f is pseudoconvex, D can be taken equal to the set of points where f 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.
There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix n-monotone functions and matrix n-convex functions, and we focus the following three assertions at each label n among them: - (i) f(0)0 and f is n-convex in [0,α),
- (ii) For each matrix a with its spectrum in [0,α) and a contraction c in the matrix algebra Mn,
| (iii) The function is n-monotone in (0,α).We show that for any nN two conditions (ii) and (iii) are equivalent. The assertion that f is n-convex with f(0)0 implies that g(t) is (n-1)-monotone holds. The implication from (iii) to (i) does not hold even for n=1. We also show in a limited case that the condition (i) implies (ii). 相似文献
7.
The main result is that for sets , the following are equivalent:
(1) |
The shuffle sum σ(S) is computable.
|
(2) |
The set S is a limit infimum set, i.e., there is a total computable function g(x, t) such that enumerates S.
|
(3) |
The set S is a limitwise monotonic set relative to 0′, i.e., there is a total 0′-computable function satisfying such that enumerates S.
|
Other results discuss the relationship between these sets and the sets.
The author’s research was partially supported by a VIGRE grant fellowship. The author thanks Denis Hirschfeldt and Steffen
Lempp for an insightful conversation about L
IMI
NF sets; Christopher Alfeld and Robert Owen for numerous comments and suggestions; and his thesis advisor Steffen Lempp for
his guidance.
相似文献
8.
Spread sets of projective planes of order
q
3 are represented as sets of
q
3 points in
A AG(3,
q
3). A line through the origin in
A can be interpreted as a space
A
0 AG(3,
q), and the spread set induces a cubic surface
L in
A
0. If the projective plane is a semifield plane of dimension 3 over its kernel, then
L has the property that it misses a plane of
A
0. Determining all such surfaces
L leads to a complete classification of the semifield planes of order
q
3, whose spread sets are division algebras of dimension 3.An alternative proof of a result due to Menichetti, that finite division algebras of dimension 3 are associative or are twisted fields, follows with the classification.
相似文献
10.
The strict lower semicontinuity property (slsc property) of the level sets of a real-valued function
f defined on a subset
CR
n
was introduced by Zang, Choo, and Avriel (Ref. 1). They showed a class of functions for which the slsc property is equivalent to invexity, i.e., the statement that every stationary point of
f over
C is a global minimum. In this paper, we study the relationship between the slsc property of the level sets and invexity for another class of functions. Namely, we consider the class formed by all locally Lipschitz real-valued functions defined on an open set containing
C. For these functions, invexity implies the slsc property of the level sets, but not conversely.The authors would like to thank Dr. B. D. Craven and the referees for helpful comments and suggestions.
相似文献
11.
We characterize the Julia sets of certain exponential functions. We show that the Julia sets
J(
Fλn) of
Fλn(
z) = λ
nezn where λ
n > 0 is the whole plane
, provided that lim
k → ∞ Fkλn(0) = ∞. In particular, this is true when λ
n are real numbers such that
. On the other hand, if
, then
J(
Fλn) is nowhere dense in
and is the complement of the basin of attraction of the unique real attractive fixed point of
Fλn. We then prove similar results for the functions[formula] where λ
i
− {0}, 1 ≤
i ≤
n + 1,
aj > 1, 1 ≤
j ≤
n, and
m,
n ≥ 1.
相似文献
12.
A
d.c. set is a set which is the difference of two convex sets. We show that any set can be viewed as the image of a d.c. set under an appropriate linear mapping. Using this universality we can convert any problem of finding an element of a given compact set in
n
into one of finding an element of a d.c. set. On the basis of this approach a method is developed for solving a system of nonlinear equations—inequations. Unlike Newton-type methods, our method does not require either convexity, differentiability assumptions or an initial approximate solution.The revision of this paper was produced during the author's stay supported by a Sophia lecturing-research grant at Sophia University (Tokyo, Japan).
相似文献
13.
This paper provides new exponent and rank conditions for the existence of abelian relative (
p
a,
p
b,
p
a,
p
a–b)-difference sets. It is also shown that no splitting relative (2
2c,2
d,2
2c,2
2c–d)-difference set exists if d > c and the forbidden subgroup is abelian. Furthermore, abelian relative (16, 4, 16, 4)-difference sets are studied in detail; in particular, it is shown that a relative (16, 4, 16, 4)-difference set in an abelian group
G Z
8 × Z
4 × Z
2 exists if and only if exp(
G) 4 or
G = Z
8 × (Z
2)
3 with
N Z
2 × Z
2.
相似文献
14.
Given a continuous map
F:
R
n
R
n
and a lower semicontinuous positively homogeneous convex function
h:
R
n
R, the nonlinear complementarity problem considered here is to find
xR
+
n
and
yh(
x), the subdifferential of
h at
x, such that
F(
x)+
y0 and
x
T
(
F(
x)+
y)=0. Some existence theorems for the above problem are given under certain conditions on the map
F. An application to quasidifferentiable convex programming is also shown.The authors are grateful to Professor O. L. Mangasarian and the referee for their substantive suggestions.
相似文献
15.
If {
n
} is an orthonormal system and {
a
n} is a sequence of random variables such that
n
(
a
n
)
2=1 a.s. then
f(
t)=|
n
a
n
n
(
t)|
2 produces a randomly selcted density function. We study the properties of
f under the assumptions that |
a
n| is decreasing to zero at a geometric rate and {
n
} is one of the following four function systems: trigonometric Jacobi, Hermite, or Laguerre. It is shown that, with probability one,
f is an analytic function,
f has at most a finite number of zeros in any finite interval, and the tail of
f goes to zero rapidly.
相似文献
16.
Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing
h, as translates of the “master” function
(
x;
α,
h)
≡exp(-[
α2/
h2]
x2) where
α is a user-choosable constant. Unfortunately, computing the coefficients of
(
x-
jh;
α,
h) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis
Cj(
x;
α,
h) is defined by the set of linear combinations of the Gaussians such that
Cj(
kh)=1 when
k=
j and
Cj(
kh)=0 for all integers
. We show that the cardinal functions for the uniform grid are
Cj(
x;
h,
α)=
C(
x/
h-
j;
α) where
C(
X;
α)≈(
α2/
π)sin(
πX)/sinh(
α2X). The
relative error is only about 4exp(-2
π2/
α2) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed
α as
h→0, but only to an “error saturation” proportional to exp(-
π2/
α2). Because the error in our approximation to the master cardinal function
C(
X;
α) is the
square of the error saturation, there is no penalty for using our new approximations to obtain
matrix-free interpolating RBF approximations to an arbitrary function
f(
x). The master cardinal function on a uniform grid in
d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions
. We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.
相似文献
17.
Given a convex function
f:
p
×
q
(–, +], the marginal function is defined on
p
by (
x)=inf{
f(
x, y)|
y
q
}. Our purpose in this paper is to express the approximate first-order and second-order directional derivatives of at
x
0 in terms of those of
f at (
x
0,
y
0), where
y
0 is any element for which (
x
0)=
f(
x
0,
y
0).The author is indebted to one referee for pointing out an inaccuracy in an earlier version of Theorem 4.1.
相似文献
18.
In this work we study the necessary and sufficient conditions for a generalized trigonometric series in order for it to be the series of a Stepanoff almost-periodic function
fS
q
(
R),1
q<. We consider analogous conditions for functions belonging to
D(,
R). Finally, we characterize the multipliers of invariance of the (
B
1(),
B
1()) type.
相似文献
19.
We estimate some sums of the shape
S(
X
1,...,
X
m
):=
1 d1 X1...1 dm Xm
f(
d
1,...,
d
m
)when
m N and
f is a nonnegative arithmetical function. We relate them to the behaviour of the associated Dirichlet series
F(
s
1,...,
s
m
) =
d1 = 1 ...
dm = 1
f(
d
1,...,
d
m
)/
d
1
s1 ...
d
m
sm.The main aim of this work is to develop analytic tools to count the rational points of bounded height on toric varieties.
相似文献
20.
By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((
q
3k
–1)/(
q–1),
q–1,
q
3k–1,
q
3k–2) relative difference sets, where
q=3
e
. These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute
p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when
q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.
相似文献