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1.
LetR be a ring and σ an automorphism ofR. We prove the following results: (i)J(R
σ[x])={Σiri
x
i:r0∈I∩J(R]),
r
i∈I for alliε 1} whereI↪ {r∈R:rx ∈J(R
Σ[x])|s= (ii)J(R
σ<x>)=(J(R
σ<x>)∩R)σ<x>. As an application of the second result we prove that ifG is a solvable group such thatG andR, + have disjoint torsions thenJ(R)=0 impliesJ(R(G))=0. 相似文献
2.
Juan Luis Vázquez 《Israel Journal of Mathematics》1982,43(3):255-272
The semilinear perturbation of Poisson’s equation (E): −Δu+β(u)∋f, where β is a maximal monotone graph inR, has been investigated by Ph. Bénilan, H. Brézis and M. Crandall forf∈L
1(R
N
),N≧1, under the assumptions 0∈β(0) ifN≧3 and 0∈β(0) ∩ Int β(R) ifN=1,2. We discuss in this paper the solvability and well-posedness of (E) in terms of any maximal monotone graph β. In particular,
if β takes only positive values andN≧3 we prove that no solution exists; ifN=2 we give necessary and sufficient conditions on β andf for (E) to be solvable in a natural sense. 相似文献
3.
Gerd Grubb 《Israel Journal of Mathematics》1971,10(1):32-95
The paper treats coerciveness inequalities (of the form Re(Au, u)≧c ‖u‖
s
2
−λ ‖u‖
0
2
,c>0,λ ∈ R) and semiboundedness inequalities (of the form Re (Au, u)≧−λ ‖u‖2) for the general boundary problems associated with an elliptic 2m-order differential operatorA in a compactn-dimensional manifold with boundary. In particular, we study the normal pseudo-differential boundary conditions, for which
we determine necessary and sufficient conditions for coerciveness withs=m, and for semiboundedness with ‖u‖ = ‖u‖m, in explicit form. 相似文献
4.
An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed
convex proper cone inR
n and −Γ′ be the antipodes of the dual cone of Γ. Let
be a partial differential operator with constant coefficients inR
n, whereQ(ζ)≠0 onR
n−iΓ′ andP
i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R
n−iΓ′;P
j(ζ)=0, gradP
j(ζ)≠0} contains some real point on which gradP
j≠0 and gradP
j∉Γ∪(−Γ). LetC be an open cone inR
n−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in
{ξ∈R
n;P(ξ)=0}. Ifu∈ℒ′∩L
loc
2
(R
n−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition
implies that the support ofu is contained in Γ. 相似文献
5.
We prove the existence of a completely integrable Pfaff system ∂x/∂t
i
= A
i
(t)x, x ∈ R
n
, t = (t
1, t
2, t
3) ∈ R
+3, i = 1, 2, 3, with infinitely differentiable bounded coefficient matrices and with lower characteristic set being the union
of countably many segments in the space R
3. 相似文献
6.
Vincenzo De Filippis 《Proceedings Mathematical Sciences》2010,120(3):285-297
Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, H and G non-zero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that (H(u)u − uG(u))
n
= 0, for all u ∈ L, then one of the following holds: (1) there exists c ∈ U such that H(x) = xc, G(x) = cx; (2) R satisfies the standard identity s
4 and char (R) = 2; (3) R satisfies s
4 and there exist a, b, c ∈ U, such that H(x) = ax+xc, G(x) = cx+xb and (a − b)
n
= 0. 相似文献
7.
Tetsutaro Shibata 《Journal d'Analyse Mathématique》2001,83(1):109-120
We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u
λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu
λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu
λ as λ→∞.
This research has been supported by the Japan Society for the Promotion of Science. 相似文献
8.
Emilien Tarquini 《Monatshefte für Mathematik》2007,243(1):333-339
In this paper we consider the Gross-Pitaevskii equation iu
t
= Δu + u(1 − |u|2), where u is a complex-valued function defined on
\Bbb RN×\Bbb R{\Bbb R}^N\times{\Bbb R}
, N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x
1 − ct, x
2, …, x
N
), where
c ? \Bbb Rc\in{\Bbb R}
is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence
result for non-constant travelling waves of fixed speed having small energy. 相似文献
9.
H. Brézis 《Israel Journal of Mathematics》1971,9(4):513-534
Let φ be a convex l.s.c. function fromH (Hilbert) into ] - ∞, ∞ ] andD(φ)={u ∈H; φ(u)<+∞}. It is proved that for everyu
0 ∈D(φ) the equation − (du/dt)(t ∈ ∂φ(u(t)),u(0)=u
0 has a solution satisfying ÷(du(t)/dt)÷ ≦(c
1/t)+c
2. The behavior ofu(t) in the neighborhood oft=0 andt=+∞ as well as the inhomogeneous equation (du(t)/dt)+∂φ(u(t)) ∈f(t) are then studied. Solutions of some nonlinear boundary value problems are given as applications.
相似文献
10.
A. O. Botyuk 《Ukrainian Mathematical Journal》1997,49(7):1120-1124
We study the boundary-value perlodic problem u
tt
−u
xx
=F(x, t), u(0, t)=u(π, t)=0, u(x, t+T)=u(x, t), (x, t) ∈ R
2. By using the Vejvoda-Shtedry operator, we determine a solution of this problem.
Ternopol Pedagogical Institute, Temopol. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 998–1001,
July, 1997. 相似文献
11.
MiaoLI QuanZHENG 《数学学报(英文版)》2004,20(5):821-828
Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups. 相似文献
12.
Let n ≥ 1 be an integer and π a permutation of I = {1, ⋯ ,n}. For any ring R, we provide a systematic construction of rings A which contain R as a subring and enjoy the following properties: (a) 1 = ∑
i ∈ I
e
i
with the e
i
orthogonal idempotents; (b) e
i
x = xe
i
for all i ∈ I and x ∈ R; (c) e
i
A
e
j
≠ 0 for all i, j ∈ I; (d) e
i
A
A
≇ e
j
A
A
unless i = j; (e) every e
i
Ae
i
is a local ring whenever R is; (f) e
i
A
A
≅ Hom
R
(Ae
π(i),R
R
) and
A
Ae
π(i) ≅
A
Hom
R
(e
i
A,
R
R) for all i ∈ I; and (g) there exists a ring automorphism η ∈ Aut(A) such that η(e
i
) = e
π(i) for all i ∈ I. Furthermore, for any nonempty π-stable subset J of I, the mapping cone of the multiplication map is a tilting complex.
Dedicated to Takeshi Sumioka on the occasion of his 60th birthday. 相似文献
13.
LetA be a (nonlinear) operator in an ordered linear spaceX with resolvantJ
λ=(I+λA)-1 well-defined onX and non-decreasing for any smallλ>0, andν ∈X. We define sub-potential ofν with respect toA, as anyu ∈X satisfyingu≧J
λ(u+λv) for smallλ>0, and show that this coincides with the notion of sub-solution of the equationAu∋ν in some abstract cases where such notion is defined in a natural way. At last, we give some general properties of sub-potentials,
in particular an extension of the Kato inequality whenX is a lattice, and, for good set of constraintsU, existence of a largest solution for the control problem:u ∈U andu is a sub-potential ofν with respect toA.
相似文献
14.
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
15.
We investigate an initial value problem which is closely related to the Williams-Bjerknes tumour model for a cancer which
spreads through an epithelial basal layer modeled onI ⊂ Z
2. The solution of this problem is a familyp = (p
i(t)), where eachp
i(t)could be considered as an approximation to the probability that the cell situated ati is cancerous at timet. We prove that this problem has a unique solution, it is defined on [0, +∞[, and, for some relevant situations, limt→∞
P
i(t) = 1 for alli ∈ I. Moreover, we study the expected number of cancerous cells at timet. 相似文献
16.
We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu
t
= Δu + |Δu|
p
,t>0,x ∈ ℝ
N
, wherep≥1 andu(0,.)=u
0≥0,u
0≢0,u
0∈L
1. DenotingI
∞=lim
t→∞‖u(t)‖1≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:
We also consider a similar question for the equationu
t=Δu+u
p
. 相似文献
– | • ifp≤(N+2)/(N+1), thenI ∞=∞ for allu 0; |
– | • if (N+2)/(N+1)<p<2, then bothI ∞=∞ andI ∞<∞ occur; |
– | • ifp≥2, thenI ∞<∞ for allu 0. |
17.
C. Boldrighini R. A. Minlos A. Pellegrinotti 《Probability Theory and Related Fields》1997,109(2):245-273
Summary We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ
t
(x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X
t
+1= y|X
t
= x,ξ
t
=η) =P
0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ
t
(x):(t,x) ∈ℤν+1} are i.i.d., that both P
0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P
0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X
t
, and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X
t
and a corresponding correction of order to the C.L.T.. Proofs are based on some new L
p
estimates for a class of functionals of the field.
Received: 4 January 1996/In revised form: 26 May 1997 相似文献
18.
We consider the asymptotic stability problems by Lyapunov functionals V for a class of functional differential equations with impulses of the form x′(t)=f(t,x
t
), x∈R
n
, t≧t
0, t≠t
k
; △x=I
k
(t,x(t
−)), t=t
k
, k∈Z
+
. Some new asymptotic stability results are presented by using an idea originated by Burton and Makay [6] and developed by
Zhang [1]. We generalize some known results about impulsive functional differential equations in the literature in which we
only require the derivative of V to be negative definite on a sequence of intervals I
n
=[s
n
,ξ
n
] which may or may not be contained in the sequence of impulsive time intervals [t
n
,t
n+1). 相似文献
19.
Let {ξ(t), t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E
ξ(t) ≡ 0, D
ξ(t) ≡ 1, and continuous trajectories defined on the m-dimensional interval
T ì \mathbbRm T \subset {\mathbb{R}^m} . The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{−v(t) < ξ(t) < u(t), t ∈ T}, when, for all t ∈ T, u(t), v(t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions
of empirical processes and fields, Sov. Math., Dokl., 45(1):226–228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e−Q
+ Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R(t) = cov(ξ′(t), ξ′(t)). 相似文献
20.
Pierre Collet Servet Martínez Jaime San Martín 《Probability Theory and Related Fields》2000,116(3):303-316
We study the asymptotic behaviour of the transition density of a Brownian motion in ?, killed at ∂?, where ?
c
is a compact non polar set. Our main result concern dimension d = 2, where we show that the transition density p
?
t
(x, y) behaves, for large t, as
u(x)u(y)(t(log t)2)−1 for x, y∈?, where u is the unique positive harmonic function vanishing on (∂?)
r
, such that u(x) ∼ log ∣x∣.
Received: 29 January 1999 / Revised version: 11 May 1999 相似文献