共查询到20条相似文献,搜索用时 31 毫秒
1.
Jia-Feng Tang 《Journal of Mathematical Analysis and Applications》2007,334(1):517-527
In this paper, we study the differential equations of the following form w2+R(z)2(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w2+P2(z)2(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w2+(z−z0)P2(z)2(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w2+A(z−z0)2(w′)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab2=1, a2=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B2=C and q′(z)=±P(z). 相似文献
2.
Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ2Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ2S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V3 into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ3T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation
F(t)+t3G(1/t)=0, 相似文献
3.
Matania Ben-Artzi 《Israel Journal of Mathematics》1981,40(3-4):259-274
LetH=?Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely,V(r)=V S(r)+V L(r), whereV S(r) is short-range and the exploding partV L(r) satisfies the following assumptions: (a) Λ=lim sup r→∞ V L(r)<∞ (but Λ=?∞ is possible). Denote Λ+= max(Λ,0). (b)V L(r)∈C 2k (r 0, ∞) and, with someδ>0 such that 2kδ>1: (d/dr) j V L(r) · (Λ+?V L(r))?1=O(r jδ) asr → ∞,j=1, ..., 2k. (c) ∫ r0 ∞ dr|V L(r|1/2 dr|V L(r)|1/2=∞. (d) (d/dr)V L(r)≦0. Under these assumptions a limiting absorption principle forR(z)=(H?z)?1 is established. More specifically, ifK ?C +={zImz≧0} is compact andK ∩ (?∞, Λ]=Ø thenR (z) can be extended as a continuous map ofK intoB (Y, Y*) (with the uniform operator topology), whereY ?L 2(R n) is a weighted-L 2 space. To ensure uniqueness of solutions of (H?z)u=f, z ∈K, a suitable radiation condition is introduced. 相似文献
4.
Matania Ben Artzi 《Journal of Differential Equations》1984,52(3):327-341
Let be a Schrödinger operator in Rn. Here is an “exploding” radially symmetric potential which is at least C2 monotone nonincreasing and O(r2) as r → ∞. V is a general potential which is short range with respect to VE. In particular, VE 0 leads to the “classical” short-range case (V being an Agmon potential). Let Λ = limr → ∞VE(r) and R(z) = (H ? z)?1, 0 < Im z, Λ < Re z < ∞. It is shown that R(z) can be extended continuously to Im z = 0, except possibly for a discrete subset ?(Λ, ∞), in a suitable operator topology . And L ? L2(Rn) is a weighted L2-space; H is then absolutely continuous over (Λ, ∞), except possibly for a discrete set of eigenvalues. The corresponding eigenfunctions are shown to be rapidly decreasing. 相似文献
5.
G. V. Rozenblyum 《Mathematical Notes》1977,21(3):222-227
The Schrödinger operator Hu = -Δu + V(x)u, where V(x) → 0 as ¦x¦ → ∞, is considered in L2(Rm) for m?3. The asymptotic formula $$N(\lambda ,V) \sim \Upsilon _m \int {(\lambda - V(x))_ + ^{{m \mathord{\left/ {\vphantom {m {2_{dx} }}} \right. \kern-\nulldelimiterspace} {2_{dx} }}} ,} \lambda \to ---0,$$ is established for the number N(λ, V) of the characteristic values of the operator H which are less than λ. It is assumed about the potential V that V = Vo + V1; Vo < 0, ¦Vo =o (¦Vo¦3/2) as ¦x¦ → ∞; σ (t/2, Vo) ?cσ (t. Vo) and V1∈Lm/2,loc, σ(t, V1) =o (σ (t, Vo)), where σ (t,f)= mes {x:¦f (x) ¦ > t). 相似文献
6.
Minoru Murata 《Journal of Functional Analysis》1982,49(1):10-56
Let ?iA = ?i(p(D) + V) be a dissipative operator in L2(Rn), where p(D) is an elliptic differential operator of order m with real constant coefficients and V is a compact operator from the weighted Sobolev space Hm′,s (Rn) to H0, p + s (Rn), s?R, for some m′ < m ? 1 and p > 1. Let R(z) be the resolvent of A. Then an asymptotic expansion of R(z) as z approaches a critical value of the polynomial p(ξ) is given; the coefficient operators in the expansion are computed explicitly. By using the resolvent expansion and the results of M. Murata [9], an asymptotic expansion of e?itA as t → ∞ is given. 相似文献
7.
Vesselin Petkov 《Journal of Functional Analysis》2006,235(2):357-376
We obtain global Strichartz estimates for the solutions u of the wave equation for time-periodic potentials V(t,x) with compact support with respect to x. Our analysis is based on the analytic properties of the cut-off resolvent Rχ(z)=χ(U−1(T)−zI)ψ1, where U(T)=U(T,0) is the monodromy operator and T>0 the period of V(t,x). We show that if Rχ(z) has no poles z∈C, |z|?1, then for n?3, odd, we have a exponential decal of local energy. For n?2, even, we obtain also an uniform decay of local energy assuming that Rχ(z) has no poles z∈C, |z|?1, and Rχ(z) remains bounded for z in a small neighborhood of 0. 相似文献
8.
John Wermer 《Arkiv f?r Matematik》2008,46(1):183-196
Let X be a rationally convex compact subset of the unit sphere S in ?2, of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the $\bar{\delta}_{b}Let X be a rationally convex compact subset of the unit sphere S in ℂ2, of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)?
Our work makes use of the kernel function for the operator on S, introduced by Henkin in [5] and builds on results obtained in Anderson–Izzo–Wermer [3].
We define a real-valued function ε
X
on the open unit ball intB, with ε
X
(z,w) tending to 0 as (z,w) tends to X. We give a growth condition on ε
X
(z,w) as (z,w) approaches X, and show that this condition is sufficient for R(X)=C(X) (Theorem 1.1).
In Section 4, we consider a class of sets X which are limits of a family of Levi-flat hypersurfaces in intB.
For each compact set Y in ℂ2, we denote the rationally convex hull of Y by . A general reference is Rudin [8] or Aleksandrov [1]. 相似文献
9.
Entire functions that share a polynomial with their derivatives 总被引:1,自引:1,他引:0
Jian-Ping Wang 《Journal of Mathematical Analysis and Applications》2006,320(2):703-717
Let f be a nonconstant entire function, k and q be positive integers satisfying k>q, and let Q be a polynomial of degree q. This paper studies the uniqueness problem on entire functions that share a polynomial with their derivatives and proves that if the polynomial Q is shared by f and f′ CM, and if f(k)(z)−Q(z)=0 whenever f(z)−Q(z)=0, then f≡f′. We give two examples to show that the hypothesis k>q is necessary. 相似文献
10.
We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u0 on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u0∈L∞(Ω), f∈L∞(Q), a∈L∞((0,T)×∂Ω). In the L1-setting, a corresponding result is proved for the more general notion of renormalised entropy solution. 相似文献
11.
Leonard Gross 《Journal of Functional Analysis》2002,190(1):38-92
A Hermitian metric, g, on a complex manifold, M, together with a smooth probability measure, μ, on M determine minimal and maximal Dirichlet forms, QD and Qmax, given by Q(f)=∫M g(grad f(z), grad f(z)) dμ(z). QD is the form closure of Q on C∞c(M) and Qmax is the form closure of Q on C1b(M). The corresponding operators, AD and Amax, generate semigroups having standard hypercontractivity properties in the scale of Lp spaces, p>1, when the corresponding form, Q, satisfies a logarithmic Sobolev inequality. It was shown by the author (1999, Acta Math.182, 159-206) that the semigroup e−tAD has even stronger hypercontractivity properties when restricted to certain holomorphic subspaces of Lp. These results are extended here to Amax. When (M, g) is not complete it is necessary that the elliptic differential operator Amax degenerate on the boundary of M. A second proof of these strong hypercontractive inequalities for both AD and Amax is given, which depends on an extension of the submean value property of subharmonic functions. The Riemann surface for z1/n and the weighted Bergman spaces in the unit disc are given as examples. 相似文献
12.
David E. Dobbs 《Rendiconti del Circolo Matematico di Palermo》2005,54(3):396-408
SoientR ?T des anneaux intègres. D’après Dobbs-Mullins, on pose Λ(T/R) ? sup{λ(k Q (T)/k Q∩R (R)) |Q ∈ Spec(T)} où, pour des corpsK?L,λ(L/K) est la longueur maximale d’une chaîne de corps contenus entreK etL. On introduitσ(R):=sup{Λ(T/R)|T est un suranneau deR\. On détermineσ(R) siR′, la clôture intégrale deR, est un anneau de Prüfer et également siR est un anneau de pseudo-valuation. On considère le cas oùσ(R)=1, en particulier siR′ est une extension minimale deR. Plusieurs calculs sont facilités par un résultat sur les carrés cartésiens, et il y a des exemples divers. 相似文献
13.
François Nicoleau 《Journal of Differential Equations》2004,205(2):354-364
We study an inverse scattering problem for a pair of Hamiltonians (H,H0) on L2(Rn), where H0=-Δ and H=H0+V, V being a short- or long-range potential. By an elementary constructive method, we show that the scattering operator S, which is localized near a fixed energy λ>0, determines the asymptotics of the potential V at infinity, in dimension n?3. This is done by studying the action of the scattering operator on suitable wave packets. 相似文献
14.
The Nevanlinna characteristic of a nonconstant elliptic function φ (z) satisfiesT(r, φ)=Kr 2 (1+o(1)) asr→∞ whereK is a nonzero constant. In this paper, we completely answer the following question: For which polynomialsQ(z, u 0,...,u n ) inu 0,...,u n , having coefficientsa(z) satisfyingT(r, a)=o(r 2) asr→∞, will the meromorphic functionh Q (z)=Q(z, ?(z),...,?(n)(z)) either be identically zero or satisfyN(r, 1/h Q )=o(r 2) asr→∞? In fact, we answer this question for rational functionsQ(z, u 0,...,u n ) inu 0,...,u n , and also obtain analogous results for the Weierstrass functions ζ(z) and σ(z). 相似文献
15.
Ognjen Milatovic 《Differential Geometry and its Applications》2004,21(3):361-377
We consider a family of Schrödinger-type differential expressions L(κ)=D2+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L2(E) and L2(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula. 相似文献
16.
C.E. Langenhop 《Linear algebra and its applications》1977,16(3):267-284
An explicit representation is obtained for P(z)?1 when P(z) is a complex n×n matrix polynomial in z whose coefficient of the highest power of z is the identity matrix. The representation is a sum of terms involving negative powers of z?λ for each λ such that P(λ) is singular. The coefficients of these terms are generated by sequences uk, vk of 1×n and n×1 vectors, respectively, which satisfy u1≠0, v1≠0, ∑k?1h=0(1?h!)uk?hP(h)(λ)=0, ∑k?1h=0(1?h!)P(h)(λ)vk?h=0, and certain orthogonality relations. In more general cases, including that when P(z) is analytic at λ but not necessarily a polynomial, the terms in the representation involving negative powers of z?λ provide the principal part of the Laurent expansion for P(z)?1 in a punctured neighborhood of z=λ. 相似文献
17.
On a certain type of nonlinear differential equations admitting transcendental meromorphic solutions
We study the differential equations w 2+R(z)(w (k))2 = Q(z), where R(z),Q(z) are nonzero rational functions. We prove
- if the differential equation w 2+R(z)(w′)2 = Q(z), where R(z), Q(z) are nonzero rational functions, admits a transcendental meromorphic solution f, then Q ≡ C (constant), the multiplicities of the zeros of R(z) are no greater than 2 and f(z) = √C cos α(z), where α(z) is a primitive of $\tfrac{1} {{\sqrt {R(z)} }}$ such that √C cos α(z) is a transcendental meromorphic function.
- if the differential equation w 2 + R(z)(w (k))2 = Q(z), where k ? 2 is an integer and R,Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C (constant), R(z) ≡ A (constant) and f(z) = √C cos (az + b), where $a^{2k} = \tfrac{1} {A}$ .
18.
A distributed control problem for the hyperbolic operator with an infinite number of variables is considered. The controls are allowed to be in the space L2(0, T; L2(R∞)) = L2(Q). The necessary and sufficient condition for the control to be optimal is obtained and the set of inequalities that characterize this condition is also obtained. 相似文献
19.
Yan Meng 《Journal of Mathematical Analysis and Applications》2007,335(1):314-331
Under the assumption that μ is a non-doubling measure on Rd, the author proves that for the multilinear Calderón-Zygmund operator, its boundedness from the product of Hardy space H1(μ)×H1(μ) into L1/2(μ) implies its boundedness from the product of Lebesgue spaces Lp1(μ)×Lp2(μ) into Lp(μ) with 1<p1,p2<∞ and p satisfying 1/p=1/p1+1/p2. 相似文献
20.
Françoise Tisseur Seamus D. Garvey Christopher Munro 《Linear algebra and its applications》2011,435(3):464-479
Given a pair of distinct eigenvalues (λ1,λ2) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form having the same eigenvalue s as Q(λ), with Qd(λ) an (n-1)×(n-1) quadratic matrix polynomial and q(λ) a scalar quadratic polynomial with roots λ1 and λ2. This block diagonalization cannot be achieved by a similarity transformation applied directly to Q(λ) unless the eigenvectors corresponding to λ1 and λ2 are parallel. We identify conditions under which we can construct a family of 2n×2n elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of Q(λ), (c) preserve the block structure of a large class of block symmetric linearizations of Q(λ), thereby defining new quadratic matrix polynomials Q1(λ) that have the same eigenvalue s as Q(λ), (d) yield quadratics Q1(λ) with the property that their eigenvectors associated with λ1 and λ2 are parallel and hence can subsequently be deflated by a similarity applied directly to Q1(λ). This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action. 相似文献