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1.
Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots
Yasuko Chikuse 《Journal of multivariate analysis》1976,6(2):237-249
An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n?2, for the 0F0 function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root. 相似文献
2.
Takafumi Isogai 《Annals of the Institute of Statistical Mathematics》1977,29(1):235-246
S
e
andS
n
are independent central and noncentral Wishart matrices having Wishart distributionsW
p
(n
e
, Σ) andW
p
(n
h
, Σ; Ω) respectively. Asymptotic expansions are given for the distributions of latent roots ofS
h
S
e
−1
and of certain test statistics in MANOVA under the assumption thatn=n
e
+n
h
becomes large with a fixed ration
e
∶n
h
=e∶h(e+h=1,e>0,h>0) andΩ=O(n). 相似文献
3.
It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1 in the form of blades
that square to –1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research
has been done [1] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the
Clifford (geometric) algebra Cℓ
3 of
\mathbb R3{{\mathbb R^3}} . All these roots of –1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations. 相似文献
4.
Bui Minh Phong 《Acta Mathematica Hungarica》2006,113(1-2):63-71
Summary We prove that if a complex valued completely multiplicative function F and a positive integer ℓ≦5 satisfy the condition F(N) = Uℓ, where Uℓ is the set of all ℓ-th roots of unity, then {F(n+1) F(n) ∣ nε N} = Uℓ. 相似文献
5.
This paper presents a fifth-order iterative method as a new modification of Newton’s method for finding multiple roots of
nonlinear equations with unknown multiplicity m. Its convergence order is analyzed and proved. Moreover, several numerical examples demonstrate that the proposed iterative
method is superior to the existing methods. 相似文献
6.
Esa Antero Seuranen 《Designs, Codes and Cryptography》2007,45(1):91-94
A code , where Z
2 = {0,1}, is said to be a binary μ-fold R-covering code, if for any word there are at least μ distinct codewords which differ from v in at most R coordinates. The size of the smallest binary μ-fold R-covering code of length n is denoted by K(n, R, μ). In this paper we use integer programming and exhaustive search to improve 57 lower bounds on K(n, R, μ) for 6 ≤ n ≤ 16, 1 ≤ R ≤ 4 and 2 ≤ μ ≤ 4.
相似文献
7.
Summary The joint density function of the latent roots ofS
1
S
2
−1
under violations is obtained whereS
1 has a complex non-central Wishart distributionW
c
(p,n
1,Σ
1,Ω) andS
2, an independent complex central Wishart,W
c
(p,n
2,Σ
2, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions
of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's
trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for
tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and
Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical. 相似文献
8.
Shinji Yamashita 《Israel Journal of Mathematics》1985,50(3):201-206
Iff is a nonconstant holomorphic function with finite Dirichlet integralD(f) on a Riemann surfaceR, then |f|2 has the least harmonic majorantf
2 onR. We show Σf
2(a ≦π
−1
D(f)), wherea runs over all the roots off = 0 onR. The equality holds if and only iff is of type ℬℓ1 fromR onto a disk of center 0. A consideration is proposed for the non-Euclidean case. 相似文献
9.
Iterative root problem can be regarded as a weak version of the problem of embedding a homeomorphism into a flow. There are
many results on iterative roots of monotone functions. However, this problem gets more diffcult in non-monotone cases. Therefore,
it is interesting to find iterative roots of linear fractional functions (abbreviated as LFFs), a class of non-monotone functions
on ℝ. In this paper, iterative roots of LFFs are studied on ℂ. An equivalence between the iterative functional equation for
non-constant LFFs and the matrix equation is given. By means of a method of finding matrix roots, general formulae of all
meromorphic iterative roots of LFFs are obtained and the precise number of roots is also determined in various cases. As applications,
we present all meromorphic iterative roots for functions z and 1/z.
This work was supported by the Youth Fund of Sichuan Provincial Education Department of China (Grant No. 07ZB042) 相似文献
10.
Ishai Oren 《Israel Journal of Mathematics》1982,42(4):353-360
LetT be the unit circle, α irrational andF: T → R a step function. A necessary and sufficient condition for the skew of the α-rotation byf (considered as taking values mod 1) to be minimal is given. Also, the boundedness of Σ
i=1
n
f(x+iα asn → α is resolved. 相似文献
11.
Larry Guth 《Inventiones Mathematicae》2008,172(3):477-489
If P and P
′ are symplectic polydisks of radii R
1≤...≤R
n
and R
1
′≤...≤R
n
′, respectively, then we prove that P symplectically embeds in P
′ provided that C(n)R
1≤R
1
′ and C(n)R
1...R
n
≤R
1
′...R
n
′. Up to a constant factor, these conditions are optimal. 相似文献
12.
In the study of the Sparre Andersen risk model with phase‐type (n) inter‐claim times (PH (n) risk model), the distinct roots of the Lundberg fundamental equation in the right half of the complex plane and the linear independence of the eigenvectors related to the Lundberg matrix Lδ(s) play important roles. In this paper, we study the case where the Lundberg fundamental equation has multiple roots or the corresponding eigenvectors are linearly dependent in the PH (n) risk model. We show that the multiple roots of the Lundberg fundamental equation det[Lδ(s)] = 0 can be approximated by the distinct roots of the generalized Lundberg equation introduced in this paper and that the linearly dependent eigenvectors can be approximated by the corresponding linearly independent ones as well. Using this result we derive the expressions for the Gerber–Shiu penalty function. Two special cases of the generalized Erlang(n) risk model and a Coxian(3) risk model are discussed in detail, which illustrate the applicability of main results. Finally, we consider the PH(2) risk model and conclude that the roots of the Lundberg fundamental equation in the right half of the complex plane are distinct and that the corresponding eigenvectors are linearly independent. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
13.
Armin Rainer 《Israel Journal of Mathematics》2011,184(1):157-182
We prove that the roots of a definable C
∞ curve of monic hyperbolic polynomials admit a definable C
∞ parameterization, where ‘definable’ refers to any fixed o-minimal structure on (ℝ,+, ·). Moreover, we provide sufficient
conditions, in terms of the differentiability of the coefficients and the order of contact of the roots, for the existence
of C
p
(for p ∈ ℕ) arrangements of the roots in both the definable and the non-definable case. These conditions are sharp in the definable
and, under an additional assumption, also in the non-definable case. In particular, we obtain a simple proof of Bronshtein’s
theorem in the definable setting. We prove that the roots of definable C
∞ curves of complex polynomials can be desingularized by means of local power substitutions t ↦ ±t
N
. For a definable continuous curve of complex polynomials we show that any continuous choice of roots is actually locally
absolutely continuous. 相似文献
14.
Ferruccio Colombini Nicola Orrù Ludovico Pernazza 《Israel Journal of Mathematics》2012,191(2):923-944
We prove that a hyperbolic monic polynomial whose coefficients are functions of class C r of a parameter t admits roots of class C 1 in t, if r is the maximal multiplicity of the roots as t varies. Moreover, if the coefficients are functions of t of class C 2r , then the roots may be chosen two times differentiable at every point in t. This improves, among others, previous results of Bronšteĭn, Mandai, Wakabayashi and Kriegl, Losik and Michor. 相似文献
15.
A method is put forward to establish the lower bounds for somen-color classical Ramsey numbers
. With this method six new explicit lower boundsR
4(4) ≥458,R
3(5) ≥ 242,R
3(6)≥1070,R
3(7) ≥ 1214,R
3(8) ≥2834 andR
3(9) ≥ 5282 are obtained using a computer.
Project supported by Guangxi Natural Science Foundation 相似文献
16.
O. K. Ivanova 《Mathematical Notes》1999,65(6):694-700
We prove relations in the braid group that generalize the relations of the formaba=bab. These relations are obtained by means of the generalization to the higher Bruhat orders of the well-known relation between
the decompositions of the element of maximal length in the Weyl groupA
n−1
and some specific linear orders on the system of positive roots ofA
n−1
.
Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 840–848, December, 1999. 相似文献
17.
We optimise a distribution of two isotropic materials α
I and β
I (α < β) occupying the given body in R
d
. The optimality is described by an integral functional (cost) depending on temperatures u
1, . . . , u
m
of the body obtained for different source terms f
1, . . . ,f
m
with homogeneous Dirichlet boundary conditions. The relaxation of this optimal design problem with multiple state equations
is needed, introducing the notion of composite materials as fine mixtures of different phases, mathematically described by
the homogenisation theory. The necessary conditions of optimality are derived via the Gateaux derivative of the cost functional.
Unfortunately, there could exist points in which necessary conditions of optimality do not give any information on the optimal
design. In the case m < d we show that there exists an optimal design which is a rank-m sequential laminate with matrix material α
I almost everywhere on Ω. Contrary to the optimality criteria method, which is commonly used for the numerical solution of optimal design problems
(although it does not rely on a firm theory of convergence), this result enables us to effectively use classical gradient
methods for minimising the cost functional.
相似文献
18.
IBN rings and orderings on grothendieck groups 总被引:2,自引:0,他引:2
Tong Wenting 《数学学报(英文版)》1994,10(3):225-230
LetR be a ring with an identity element.R∈IBN means thatR
m⋟Rn impliesm=n, R∈IBN
1 means thatR
m⋟Rn⊕K impliesm≥n, andR∈IBN
2 means thatR
m⋟Rm⊕K impliesK=0. In this paper we give some characteristic properties ofIBN
1 andIBN
2, with orderings on the Grothendieck groups. In addition, we obtain the following results: (1) IfR∈IBN
1 and all finitely generated projective leftR-modules are stably free, then the Grothendieck groupK
0(R) is a totally ordered abelian group. (2) If the pre-ordering of the Grothendieck groupK
0(R) of a ringR is a partial ordering, thenR∈IBN
1 orK
0(R)=0.
Supported by National Nature Science Foundation of China. 相似文献
19.
LetR be a commutative noetherian ring and ƒ1, …, ƒr ∃ R. In this article we give (cf. the Theorem in §2) a criterion for ƒ1, …, ƒr to be regular sequence for a finitely generated module overR which strengthens and generalises a result in [2]. As an immediate consequence we deduce that if V(g
1, …,g
r
) ⊆ V(ƒ1, …, ƒr) in SpecR and if ƒ1, …, ƒr is a regular sequence inR, theng
1, …,g
r is also a regular sequence inR. 相似文献
20.
Takashi Noiri 《Rendiconti del Circolo Matematico di Palermo》2006,55(1):29-42
Cammaroto and Noiri [14] introduced a separation axiom calledm-R
0 in anm-space (X, m). In this paper, we introduce the notion ofm-R
1 spaces and offer many characterizations ofm-R
0 (resp.m-R
1) spaces which enable us to obtain unified characterizations of separation axiomsR
0, semi-R
0, pre-R
0,α-R
0,δ-semiR
0, (δ, p)-R
0 (resp.R
1, semi-R
1, pre-R
1,α-R
1,δ-semiR
1, (δ, p)-R
1). 相似文献