共查询到20条相似文献,搜索用时 24 毫秒
1.
Albert G. Buckley 《Mathematical Programming》1986,36(3):256-275
This work concerns the derivation of formulae for updating quasi-Newton matrices used in algorithms for computing approximate
minima of smooth unconstrained functions. The paper concentrates strictly on the techniques used to derive update formulae.
It demonstrates a technique in which problems of finding matrices in ℝ
n ×n
of minimum Frobenius norm are converted to equivalent problems, using vector representations in ℝ
n2
and ℝ
n(n+1)/2 of these matrices, and then solvingl
2-minimization problems. These problems are more directly dealt with, and indeed, the paper demonstrates how this technique
may be used to handle weighted sparse updates. 相似文献
2.
LeiE(ℝn) be the space of all functions on ℝn which can continue to the entire holomorphic functions on ℂn. We define Riesz transformation Rj of distributions as a linear transformation of the quotient spaceD′(ℝn)/E(ℝn) to itself, j=1,2,..., n. These generalized Riesz transformations share the same properties with the classical ones, such
as
. As applications we generalize further a theorem of F. & M. Riesz generalized by Stein and Weiss, and then define a generalized
Hardy space, of which some properties are studied. 相似文献
3.
T. V. Budnyts’ka 《Ukrainian Mathematical Journal》2009,61(1):164-170
We consider affine mappings from ℝ
n
into ℝ
n
, n ≥ 1. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding
linear mapping. We give a classification, up to topological conjugacy, for affine mappings from R into R and also for affine
mappings from ℝ
n
into ℝ
n
, n > 1, having at least one fixed point and the nonperiodic linear part.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 134–139, January, 2009. 相似文献
4.
James V. Burke 《Mathematical Programming》1987,38(3):287-302
In convex composite NDO one studies the problem of minimizing functions of the formF:=h ○f whereh:ℝ
m
→ ℝ is a finite valued convex function andf:ℝ
n
→ ℝ
m
is continuously differentiable. This problem model has a wide range of application in mathematical programming since many
important problem classes can be cast within its framework, e.g. convex inclusions, minimax problems, and penalty methods
for constrained optimization. In the present work we extend the second order theory developed by A.D. Ioffe in [11, 12, 13]
for the case in whichh is sublinear, to arbitrary finite valued convex functionsh. Moreover, a discussion of the second order regularity conditions is given that illuminates their essentially geometric nature. 相似文献
5.
We propose a class of non-interior point algorithms for solving the complementarity problems(CP): Find a nonnegative pair
(x,y)∈ℝ
2n
satisfying y=f(x) and x
i
y
i
=0 for every i∈{1,2,...,n}, where f is a continuous mapping from ℝ
n
to ℝ
n
. The algorithms are based on the Chen-Harker-Kanzow-Smale smoothing functions for the CP, and have the following features;
(a) it traces a trajectory in ℝ
3n
which consists of solutions of a family of systems of equations with a parameter, (b) it can be started from an arbitrary
(not necessarily positive) point in ℝ
2n
in contrast to most of interior-point methods, and (c) its global convergence is ensured for a class of problems including
(not strongly) monotone complementarity problems having a feasible interior point. To construct the algorithms, we give a
homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition, and propose a class
of algorithms involving suitable neighborhoods of the trajectory. We also give a sufficient condition on the neighborhoods
for global convergence and two examples satisfying it.
Received April 9, 1997 / Revised version received September 2, 1998? Published online May 28, 1999 相似文献
6.
Under very minimal regularity assumptions, it can be shown that 2n−1 functions are needed to generate an orthonormal wavelet basis for L2(ℝn). In a recent paper by Dai et al. it is shown, by abstract means, that there exist subsets K of ℝn such that the single function ψ, defined by
, is an orthonormal wavelet for L2(ℝn). Here we provide methods for construucting explicit examples of these sets. Moreover, we demonstrate that these wavelets
do not behave like their one-dimensional couterparts. 相似文献
7.
Liguang Liu 《Frontiers of Mathematics in China》2007,2(4):599-611
Let ℐ(ℝn) be the Schwartz class on ℝn and ℐ∞(ℝn) be the collection of functions ϕ ∊ ℐ(ℝn) with additional property that
for all multiindices γ. Let (ℐ(ℝn))′ and (ℐ∞(ℝn))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝn))′ and (ℐ∞(ℝn))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley
function of f belongs to L
p(ℝn) for some p ∊ (0,1], the condition f ∊ (ℐ∞(ℝn))′ is equivalent to that f ∊ (ℐ(ℝn))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝn) and ℐ∞(ℝn), also including their corresponding Hardy spaces.
相似文献
8.
Sérgio Alvarez Lev Birbrair João Carlos Ferreira Costa Alexandre Fernandes 《Central European Journal of Mathematics》2010,8(2):338-345
We study the topological K-equivalence of function-germs (ℝ
n
, 0) → (ℝ, 0). We present some special classes of piece-wise linear functions and prove that they are normal forms for equivalence
classes with respect to topological K-equivalence for definable functions-germs. For the case n = 2 we present polynomial models for analytic function-germs. 相似文献
9.
E. Miglierina E. Molho M. Rocca 《Journal of Optimization Theory and Applications》2008,138(3):479-496
In this work, we study the critical points of vector functions from ℝ
n
to ℝ
m
with n≥m, following the definition introduced by Smale in the context of vector optimization. The local monotonicity properties of
a vector function around a critical point which are invariant with respect to local coordinate changes are considered. We
propose a classification of critical points through the introduction of a generalized Morse index for a critical point, consisting
of a triplet of nonnegative integers. The proposed index is based on the sign of an appropriate invariant vector-valued second-order
differential. 相似文献
10.
Francis J. Narcowich Xinping Sun Joseph D. Ward 《Advances in Computational Mathematics》2007,27(1):107-124
Error estimates for scattered data interpolation by “shifts” of a conditionally positive definite function (CPD) for target
functions in its native space, which is its associated reproducing kernel Hilbert space (RKHS), have been known for a long
time. Regardless of the underlying manifold, for example ℝn or S
n, these error estimates are determined by the rate of decay of the Fourier transform (or Fourier series) of the CPD. This
paper deals with the restriction of radial basis functions (RBFs), which are radial CPD functions on ℝn+1, to the unit sphere S
n. In the paper, we first strengthen a result derived by two of us concerning an explicit representation of the Fourier–Legendre
coefficients of the restriction in terms of the Fourier transform of the RBF. In addition, for RBFs that are related to completely
monotonic functions, we derive a new integral representation for these coefficients in terms of the measure generating the
completely monotonic function. These representations are then utilized to show that if an RBF has a native space equivalent
to a Sobolev space H
s(ℝn+1), then the restriction to S
n has a native space equivalent to H
s−1/2(S
n). In addition, they are used to recover the asymptotic behavior of such coefficients for a wide variety of RBFs. Some of
these were known earlier.
Joseph D. Ward: Francis J. Narcowich: Research supported by grant DMS-0204449 from the National Science Foundation. 相似文献
11.
I. Ginchev A. Guerraggio M. Rocca 《Journal of Optimization Theory and Applications》2009,143(1):87-105
The present paper studies the following constrained vector optimization problem: min
C
f(x), g(x)∈−K, h(x)=0, where f:ℝ
n
→ℝ
m
, g:ℝ
n
→ℝ
p
and h:ℝ
n
→ℝ
q
are locally Lipschitz functions and C⊂ℝ
m
, K⊂ℝ
p
are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient
conditions are obtained for a point x
0 to be a w-minimizer (weakly efficient point) or an i-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the
implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven.
Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo. 相似文献
12.
V. A. Zalgaller 《Journal of Mathematical Sciences》2000,100(3):2209-2227
Let Dn be a convex compact set in ℝn. If a function
admits a representation of the form f=g−h, where g and are convex and h is bounded from above, then there exists a representation
of the same form which is “minimal” in some sense. A recurrent procedure converging to this minimal representation is described.
For piecewise-linear functions f (in the cases n=1,2), finite algorithms giving minimal representations are found. A number
of examples clarifying some unexpected effects are given. Some problems are formulated. Bibliography: 5 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 36–65.
Translated by S. Yu. Pilyugin. 相似文献
13.
A refinable function φ(x):ℝn→ℝ or, more generally, a refinable function vector Φ(x)=[φ1(x),...,φr(x)]T is an L1 solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding
integer matrix. A refinable function vector is called orthogonal if {φj(x−α):α∈ℤn, 1≤j≤r form an orthogonal set of functions in L2(ℝn). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and
multiwavelet bases of L2(ℝn). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported
refinable functions and refinable function vectors. 相似文献
14.
Swanhild Bernstein 《Advances in Applied Clifford Algebras》1998,8(1):31-46
Using the properties of the monogenic extension of the Fourier transform, we state a Paley-Wiener-type theorem for monogenic
functions. Based on an multiplier algebra related to boundary values of monogenic functions we consider integral equations
of Wiener-Hopf-typeK±u
±=f on ℝ
n
, whereK ∈S′ andu
± are boundary values of monogenic functions in ℝ+
n+1 and ℝ_
n+1 respectivly. 相似文献
15.
The authors establish the boundedness of Marcinkiewicz integrals from the Hardy space H
1 (ℝ
n
× ℝ
m
) to the Lebesgue space L
1(ℝ
n
× ℝ
m
) and their commutators with Lipschitz functions from the Hardy space H
1 (ℝ
n
× ℝ
m
) to the Lebesgue space L
q
(ℝ
n
× ℝ
m
) for some q > 1. 相似文献
16.
Area,coarea, and approximation in <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript>
David Swanson 《Arkiv f?r Matematik》2007,45(2):381-399
Let Ω⊂ℝ
n
be an arbitrary open set. We characterize the space W
1,1
loc(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation
and trace theorems for functions in the space W
1,1(ℝ
n
). 相似文献
17.
We show that in the worst case, Ω(n
d
) sidedness queries are required to determine whether a set ofn points in ℝ
d
is affinely degenerate, i.e., whether it containsd+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on
the explicit construction of a point set containing Ω(n
d
) “collapsible” simplices, any one of which can be made degenerate without changing the orientation of any other simplex.
As an immediate corollary, we have an Ω(n
d
) lower bound on the number of sidedness queries required to determine the order type of a set ofn points in ℝ
d
. Using similar techniques, we also show that Ω(n
d+1) in-sphere queries are required to decide the existence of spherical degeneracies in a set ofn points in ℝ
d
.
An earlier version of this paper was presented at the 34th Annual IEEE Symposium on Foundations of Computer Science [8]. This
research has been supported by NSF Presidential Young Investigator Grant CCR-9058440. 相似文献
18.
Alexander Koldobsky 《Israel Journal of Mathematics》2011,185(1):277-292
We say that a random vector X = (X
1, …, X
n
) in ℝ
n
is an n-dimensional version of a random variable Y if, for any a ∈ ℝ
n
, the random variables Σa
i
X
i
and γ(a)Y are identically distributed, where γ: ℝ
n
→ [0,∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that
embeds in L
0. This result is almost optimal, as the norm of any finite-dimensional subspace of L
p
with p ∈ (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P. Lèvy. An equivalent formulation is that if a function of the form f(‖ · ‖
K
) is positive definite on ℝ
n
, where K is an origin symmetric star body in ℝ
n
and f: ℝ → ℝ is an even continuous function, then either the space (ℝ
n
, ‖·‖
K
) embeds in L
0 or f is a constant function. Combined with known facts about embedding in L
0, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions. 相似文献
19.
Bilal Atfeh Laurent Baratchart Juliette Leblond Jonathan R. Partington 《Journal of Fourier Analysis and Applications》2010,16(2):177-203
In this work, we develop a theory of approximating general vector fields on subsets of the sphere in ℝ
n
by harmonic gradients from the Hardy space H
p
of the ball, 1<p<∞. This theory is constructive for p=2, enabling us to solve approximate recovery problems for harmonic functions from incomplete boundary values. An application
is given to Dirichlet–Neumann inverse problems for n=3, which are of practical importance in medical engineering. The method is illustrated by two numerical examples. 相似文献
20.
We develop a necessary and sufficient condition for the Bedrosian identity in terms of the boundary values of functions in
the Hardy spaces. This condition allows us to construct a family of functions such that each of which has non-negative instantaneous
frequency and is the product of two functions satisfying the Bedrosian identity. We then provide an efficient way to construct
orthogonal bases of L
2(ℝ) directly from this family. Moreover, the linear span of the constructed basis is norm dense in L
p
(ℝ), 1 < p < ∞. Finally, a concrete example of the constructed basis is presented. 相似文献