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1.
We obtain a generalization of the complete Perron effect whereby the characteristic exponents of all solutions change their
sign from negative for the linear approximation system to positive for a nonlinear system with perturbations of higher-order
smallness [Differ. Uravn., 2010, vol. 46, no. 10, pp. 1388–1402]. Namely, for arbitrary parameters λ
1 ≤ λ
2 < 0 and m > 1 and for arbitrary intervals [b
i
, d
i
) ⊂ [λ
i
,+∞), i = 1, 2, with boundaries d
1 ≤ b
2, we prove the existence of (i) a two-dimensional linear differential system with bounded coefficient matrix A(t) infinitely differentiable on the half-line t ≥ 1 and with characteristic exponents λ
1(A) = λ
1 ≤ λ
2(A) = λ
2 < 0; (ii) a perturbation f(t, y) of smallness order m > 1 infinitely differentiable with respect to time t > 1 and continuously differentiable with respect to y
1 and y
2, y = (y
1, y
2) ∈ R
2 such that all nontrivial solutions y(t, c), c ∈ R
2, of the nonlinear system .y = A(t)y + f(t, y), y ∈ R
2, t ≥ 1, are infinitely extendible to the right and have characteristic exponents λ[y] ∈ [b
1, d
1) for c
2 = 0 and λ[y] ∈ [b
2, d
2) for c
2 ≠ 0. 相似文献
2.
Let y = y(x) be a function defined by a continued fraction. A lower bound for |Λ| = |β
1
y
1 + β
2
y
2 + α| is given, where y
1 = y(x
1), y
2 = y(x
2), x
1 and x
2 are positive integers, α, β
1 and β
2 are algebraic irrational numbers. 相似文献
3.
We realize the Perron effect of change of values of characteristic exponents: for arbitrary parameters λ
1 <- λ
2 < 0, β
2 ≥ β
1 ≥ λ
2, and m > 1, we prove the existence of a linear differential system $
\dot x
$
\dot x
= A(t)x, x ∈ R
2, t ≥ t
0, with bounded infinitely differentiable coefficients and with characteristic exponents λ
1(A) = λ
1 <- λ
2(A) = λ
2 and of an m-perturbation f: [t
0,+∞) × R
2 → R
2 infinitely differentiable in time, continuously differentiable with respect to the phase variables y
1 and y
2, (y
1, y
2) = y ∈ R
2 (infinitely differentiable with respect to the variables y
1 ≠ 0 and y
2 ≠ 0 and with respect to all of these variables in the case of a positive integer m > 1), satisfying the condition ‖f(t, y)‖ ≤ const × ‖y‖
m
, y ∈ R
2, t ≥ t
0, and such that all nontrivial solutions y(t, c) of the perturbed system
$
\dot y = A(t)y + f(t,y), y \in R^2
$
\dot y = A(t)y + f(t,y), y \in R^2
相似文献
4.
LetF(u, v) be a symmetric real function defined forα<u, v<β and assume thatG(u, v, w)=F(u, v)+F(u, w)−F(v, w) is decreasing inv andw foru≦min (u, v). For any set (y)=(y
1, …,y
n
),α<y
i
<β, given except in arrangement Σ
i
=1/n
F(y
i
,y
i+1) wherey
n+1=y
1) is maximal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order. Examples are given and an additional result is proved on the productΠ
i
=1/n
[(y2i−1y2i)
m
+α
1(y
2i−1
y
2i
)
m−1+ … +a
m
] wherea
k
≧0 and where the set (y)=(y
1, ..,y
n
),y
i
≧0 is given except in arrangement. The problems considered here arose in connection with a theorem by A. Lehman [1] and a
lemma of Duffin and Schaeffer [2].
This paper is part of the author’s Master of Science dissertation at the Technion-Israel Institute of Technology.
The author wishes to thank Professor B. Schwarz and Professor E. Jabotinsky for their help in the preparation of this paper. 相似文献
5.
Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming 总被引:1,自引:0,他引:1
We investigate the characteristics that have to be possessed by a functional mapping f:ℝ↦ℝ so that it is suitable to be employed in a variable transformation of the type x→f(y) in the convexification of posynomials. We study first the bilinear product of univariate functions f
1(y
1), f
2(y
2) and, based on convexity analysis, we derive sufficient conditions for these two functions so that ℱ2(y
1,y
2)=f
1(y
1)f
2(y
2) is convex for all (y
1,y
2) in some box domain. We then prove that these conditions suffice for the general case of products of univariate functions;
that is, they are sufficient conditions for every f
i
(y
i
), i=1,2,…,n, so as ℱ
n
(y
1,y
2,…,y
n
)=∏
i=1
n
f
i
(y
i
) to be convex. In order to address the transformation of variables that are exponentiated to some power κ≠1, we investigate under which further conditions would the function (f)
κ
be also suitable. The results provide rigorous reasoning on why transformations that have already appeared in the literature,
like the exponential or reciprocal, work properly in convexifying posynomial programs. Furthermore, a useful contribution
is in devising other transformation schemes that have the potential to work better with a particular formulation. Finally,
the results can be used to infer the convexity of multivariate functions that can be expressed as products of univariate factors,
through conditions on these factors on an individual basis.
The authors gratefully acknowledge support from the National Science Foundation. 相似文献
6.
Let R be a local ring and let (x
1, …, x
r) be part of a system of parameters of a finitely generated R-module M, where r < dimR
M. We will show that if (y
1, …, y
r) is part of a reducing system of parameters of M with (y
1, …, y
r) M = (x
1, …, x
r) M then (x
1, …, x
r) is already reducing. Moreover, there is such a part of a reducing system of parameters of M iff for all primes P ε Supp M ∩ V
R(x
1, …, x
r) with dimR
R/P = dimR
M − r the localization M
P of M at P is an r-dimensional Cohen-Macaulay module over R
P.
Furthermore, we will show that M is a Cohen-Macaulay module iff y
d is a non zero divisor on M/(y
1, …, y
d−1) M, where (y
1, …, y
d) is a reducing system of parameters of M (d:= dimR
M). 相似文献
7.
J. -C. Aval 《Theoretical and Mathematical Physics》2009,161(3):1582-1589
We consider the partition function Z(N; x
1
, …, xN, y
1
, …, yN) of the square ice model with domain wall boundary conditions. We give a simple proof that Z is symmetric with respect to
all its variables when the global parameter a of the model is set to the special value a = eiπ/3
. Our proof does not use any determinant interpretation of Z and can be adapted to other situations (e.g., to some symmetric
ice models). 相似文献
8.
Yong Ge Tian 《数学学报(英文版)》2009,25(11):1907-1920
This paper studies relationships between the best linear unbiased estimators (BLUEs) of an estimable parametric functions Kβunder the Gauss-Markov model {y, Xβ, σ^2]E} and its misspecified model {y, X0β,σ^2∑0}. In addition, relationships between BLUEs under a restricted Gauss Markov model and its misspecified model are also investigated. 相似文献
9.
O. V. Lopotko 《Ukrainian Mathematical Journal》2011,63(6):981-992
We obtain an integral representation of even functions of two variables for which the kernel [k
1(x + y) + k
2(x − y)], x, y ∈ R
2, is positive definite. 相似文献
10.
We consider the problem of nonparametric identification for a multi-dimensional functional autoregression y
t
= f(y
t
−1, …,y
t−d
) + e
t
on the basis of N observations of y
t
. In the case when the unknown nonlinear function f belongs to the Barron class, we propose an estimation algorithm which provides approximations of f with expected L
2 accuracy O(N
1/4ln1/4
N). We also show that this approximation rate cannot be significantly improved.
The proposed algorithms are “computationally efficient”– the total number of elementary computations necessary to complete
the estimate grows polynomially with N.
Received: 23 September 1997 / Revised version: 28 January 1999 相似文献
11.
Nearrings here are right nearrings. LetN be a nearring and fix an element α εN. Form another nearring Nα by taking addition to be the same as the addition inN but define the productx ⊗y of two elementsx, y ε Nα byx ⊗y =xay. The nearring Nα is referred to as a laminated nearring ofN andN is referred to as the base nearring. The element α is called the laminating element or the laminator. An elementx of a nearingN is a left zero ifxy =x for ally εN. A homomorphismϕ from a nearringN
1 into a nearringN
2 is a left zero covering homomorphism if for each left zeroy εN
2,ϕ(x) =y for somex εN
1. The left zero covering homomorphisms from one laminated nearring into another are investigated where the base nearring is
the nearring of all continuous selfmaps of the Euclidean group ℝ2 under pointwise addition and composition and the laminators are complex polynomials. Finally, it is shown that one can determine
whether or not two such laminated nearrings are isomorphic simply by inspecting the coefficients of the two laminating polynomials. 相似文献
12.
Yu. K. Sabitova 《Russian Mathematics (Iz VUZ)》2009,53(12):41-49
We consider the equation y
m
u
xx
− u
yy
− b
2
y
m
u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u
y
(x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u
x
(0, y) = 0 or u
x
(0, y) = u
x
(1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems 相似文献
13.
E. Preissmann 《Aequationes Mathematicae》1987,32(1):195-212
We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ2=aθ4+bθ2+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x
r
−x
s
)) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex
r
are equidistant.
Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux 相似文献 14.
We consider an Abel equation (*)y’=p(x)y
2 +q(x)y
3 withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane)
is thaty
0=y(0)≡y(1) for any solutiony(x) of (*).
Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y
2 +εq(x)y
3
p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1)..
We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm
k
(1), wherem
k
(x)=∫
0
x
pk
(t)q(t)(dt),P(x)=∫
0
x
p(t)dt. We investigate the structure of zeroes ofm
k
(x) and generalize a “canonical representation” ofm
k
(x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric
center problem.
The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the
Minerva Foundation. 相似文献
15.
We consider an Abel equation (*)y’=p(x)y
2 +q(x)y
3 withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane)
is thaty
0=y(0)≡y(1) for any solutiony(x) of (*).
We introduce a parametric version of this condition: an equation (**)y’=p(x)y
2 +εq(x)y
3
p, q as above, ℂ, is said to have a parametric center, if for any ε and for any solutiony(ε,x) of (**),y(ε,0)≡y(ε,1).
We show that the parametric center condition implies vanishing of all the momentsm
k
(1), wherem
k
(x)=∫
0
x
pk
(t)q(t)(dt),P(x)=∫
0
x
p(t)dt. We investigate the structure of zeroes ofm
k
(x) and on this base prove in some special cases a composition conjecture, stated in [10], for a parametric center problem.
The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the
Minerva Foundation. 相似文献
16.
An increasing sequence of realsx=〈x
i
:i<ω〉 is simple if all “gaps”x
i
+1−x
i
are different. Two simple sequencesx andy are distance similar ifx
i
+1−x
i
<x
j
+1−x
j
if and only ify
i
+1−y
i
<y
j
+1−y
j
for alli andj. Given any bounded simple sequencex and any coloring of the pairs of rational numbers by a finite number of colors, we prove that there is a sequencey distance similar tox all of whose pairs are of the same color. We also consider many related problems and generalizations.
Partially supported by OTKA-4269.
Partially supported by NSF grant STC-91-19999.
Partially supported by OTKA-T-020914, NSF grant CCR-9424398 and PSC-CUNY Research Award 663472. 相似文献
17.
O. V. Lopotko 《Ukrainian Mathematical Journal》2010,62(2):320-324
We obtain an integral representation of even positive-definite functions of one variable for which the kernel [k
1(x + y) + k
2 (x − y)] is positive definite. 相似文献
18.
Strongly Closed Subgraphs in a Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1 总被引:1,自引:1,他引:0
Akira Hiraki 《Graphs and Combinatorics》2008,24(6):537-550
Let Γ be a distance-regular graph of diameter d ≥ 3 with c
2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:
19.
20.
P. K. SAHO 《数学学报(英文版)》2005,21(5):1159-1166
In this paper, we determine the general solution of the functional equation f1 (2x + y) + f2(2x - y) = f3(x + y) + f4(x - y) + f5(x) without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 : R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f(2x + y) + f(2x - y) = g(x + y) + g(x - y) + h(x) and f(2x + y) - f(2x - y) = g(x + y) - g(x - y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi. 相似文献
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