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1.
Consider the equation is small. For λ = 0, there is a homoclinic orbit Γ through zero. For λ ≠ 0 and small, there can be “strange” attractors near Γ. The purpose of this paper is to determine the curves in λ-space of bifurcation to “strange” attractors and to relate this to hyperbolic subharmonic bifurcations. 相似文献
2.
Stephen Bancroft 《Journal of Mathematical Analysis and Applications》1975,50(2):384-414
In this paper we discuss the problem of determining a T-periodic solution of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space Rk. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?0 > 0 such that the differential equation has a T-periodic solution for all ? satisfying 0 < ? < ?0. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ0 where λ0 is a fixed vector in Rk. This means that attention is confined in the perturbation procedure to examining the dependence of on λ as λ varies along a line segment terminating at the origin in the parameter-space Rk. The results established here generalize this previous work by allowing one to study the dependence of on λ as λ varies through a “conical-horn” whose vertex rests at the origin in Rk. In the process an implicit-function formula is developed which is of some interest in its own right. 相似文献
3.
Morris L Eaton 《Journal of multivariate analysis》1976,6(3):422-425
Let Σ be an n × n positive definite matrix with eigenvalues λ1 ≥ λ2 ≥ … ≥ λn > 0 and let M = {x, y | x?Rn, y?Rn, x ≠ 0, y ≠ 0, x′y = 0}. Then for x, y in M, we have that and the inequality is sharp. If is a partitioning of Σ, let θ1 be the largest canonical correlation coefficient. The above result yields . 相似文献
4.
P.J. Cook 《Journal of Number Theory》1977,9(1):142-152
It is shown that λ1, λ2,…, λ6, μ are not all of the same sign and at least one ratio is irrational then the values taken by for integer values of x1 ,…, x6, y are everywhere dense on the real line. A similar result holds for expressions of the form . 相似文献
5.
Hans G Weidner 《Journal of Number Theory》1976,8(1):99-108
Let g = (g1,…,gr) ≥ 0 and h = (h1,…,hr) ≥ 0, g?, h? ∈ , be two vectors of nonnegative integers and let λ ? , λ ≥ 0, λ ≡ 0 mod d, where d denotes g.c.d. (g1,…,gr). Define It is shown in this paper that Λ(λ) is periodic in λ with constant jump. If i? {1,…,r} is such that then holds true for all sufficiently large λ, λ ≡ 0 mod d. 相似文献
6.
Robert S Strichartz 《Journal of Functional Analysis》1982,49(1):91-127
The composition of two Calderón-Zygmund singular integral operators is given explicitly in terms of the kernels of the operators. For φ?L1(Rn) and ε = 0 or 1 and ∝ φ = 0 if ε = 0, let Ker(φ) be the unique function on Rn + 1 homogeneous of degree ?n ? 1 of parity ε that equals φ on the hypersurface x0 = 1. Let Sing(φ, ε) denote the singular integral operator , which exists under suitable growth conditions on ? and φ. Then Sing(φ, ε1) Sing(ψ, ε2)f = ?2π2(∝ φ)(∝ ψ)f + Sing(A, ε1, + ε2)f, where (with notation ). This result is used to show that the mapping ψ → A is a classical pseudo-differential operator of order zero if φ is smooth, with top-order symbol , where θ(ξ) is a cut-off function. These results are generalized to singular integrals with mixed homogeneity. 相似文献
7.
Let λ1 and λN be, respectively, the greatest and smallest eigenvalues of an N×N hermitian matrix H=(hij), and x=(x1,x2,…,xN) with (x,x)=1. Then, it is known that (1) λ1?(x,Hx)?λN and (2) if, in addition, H is positive definite, . Assuming that y=(y1,y2,…, yN) and |yi|?1, i=1,2,…,N, it is shown in this paper that these inequalities remain true if H and H?1 are, respectively, replaced by the Hadamard products and , where M(y) is a matrix defined by . Subsequently, these results are extended to improve the spectral bounds of . 相似文献
8.
Lawrence Turyn 《Journal of Differential Equations》1980,38(2):239-259
We consider the regular linear Sturm-Liouville problem (second-order linear ordinary differential equation with boundary conditions at two points x = 0 and x = 1, those conditions being separated and homogeneous) with several real parameters λ1,…,λN. Solutions to this problem correspond to eigenvaluesλ = (λ1,…,λN) forming sets N determined by the number of zeroes in (0, 1) of solutions. We describe properties of these sets including: boundedness, and when unbounded, asymptotic directions. Using these properties some results are given for the system of N Sturm-Liouville problems which share only the parameters λ. Sharp results are given for the system of two problems sharing two parameters. The eigensurfaces for a single problem are closely related to the cone , particularly in questions of boundedness. The cone K and related objects are discussed, and a result is given which relates cones with two oscillation conditions known as “Right-Definiteness” and “Left-Definiteness.” 相似文献
9.
Richard Askey Deborah Tepper Haimo 《Journal of Mathematical Analysis and Applications》1977,59(1):119-129
We study degeneration for ? → + 0 of the two-point boundary value problems , and convergence of the operators T?+ and T?? on 2(?1, 1) connected with them, T?±u := τ?±u for all for all . Here ? is a small positive parameter, λ a complex “spectral” parameter; a, b and c are real ∞-functions, a(x) ? γ > 0 for all x? [?1, 1] and h is a sufficiently smooth complex function. We prove that the limits of the eigenvalues of T?+ and of T?? are the negative and nonpositive integers respectively by comparison of the general case to the special case in which a 1 and b c 0 and in which we can compute the limits exactly. We show that (T?+ ? λ)?1 converges for ? → +0 strongly to (T0+ ? λ)?1 if . In an analogous way, we define the operator T?+, n (n ? in the Sobolev space H0?n(? 1, 1) as a restriction of τ?+ and prove strong convergence of (T+?,n ? λ)?1 for ? → +0 in this space of distributions if . With aid of the maximum principle we infer from this that, if h?1, the solution of τ?+u ? λu = h, u(±1) = A ± B converges for ? → +0 uniformly on [?1, ? ?] ∪ [?, 1] to the solution of xu′ ? λu = h, u(±1) = A ± B for each p > 0 and for each λ ? if ? ?.Finally we prove by duality that the solution of τ??u ? λu = h converges to a definite solution of the reduced equation uniformly on each compact subset of (?1, 0) ∪ (0, 1) if h is sufficiently smooth and if 1 ? ?. 相似文献
10.
The system is investigated, where x and y are scalar functions of time (t ? 0), and n space variables , and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution , which is bounded for r ? 0 and satisfies x(0) >x0, y(0) > y0, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique. 相似文献
11.
Ulrich Faigle 《Discrete Applied Mathematics》1984,9(2):209-211
Generalizing the multiple basis exchange property for matroids, the following theorem is proved: If x and y are vectors of a submodular system in and such that x = x1 + x2, then there are such that y = y1 + y2 and both x1 + y1 and x2 + y2 belong to the submodular system.An integral analogue holds for the integral submodular systems and a non-negative analogue for polymatroids. 相似文献
12.
R.J. Cook 《Journal of Number Theory》1979,11(1):49-68
It is shown that if λ1, …, λ5 are non-zero real numbers, not all of the same sign, and at least one of the ratios (1 ≤ j ≤ 3) is irrational then the values taken by for integer values of x1, …, x5 are everywhere dense on the real line. Similar results are proved for the polynomials and . 相似文献
13.
Jean-Louis Tassoul Gilles Aubin 《Journal of Mathematical Analysis and Applications》1974,45(1):116-126
This paper deals with finite-amplitude axisymmetric disturbances in a self-gravitating fluid column of finite radius R. It is shown that the cutoff wavelength λnl above which gravitational breakup occurs now depends on the relative amplitude of the initial perturbation. Actually, for small-but finite-amplitude disturbances, , where λl ( = 5.8898R) designates the cutoff wavelength predicted in the linear approximation. 相似文献
14.
Daniel J. Madden 《Journal of Number Theory》1978,10(3):303-323
If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions Kn over k(x) such that [K : k(x)] = pn using the concept of Witt vectors. This is accomplished in the following way; if [β1, β2,…, βn] is a Witt vector over k(x) = K0, then the Witt equation generates a tower of extensions through where . In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in Kn. This alternate generation has the form Ki = Ki?1(yi); yip ? yi = Bi, where, as a divisor in Ki?1, Bi has the form . In this form q is prime to Πpjλj and each λj is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field Kn over an algebraically closed field of constants. 相似文献
15.
James W Thomas David W Zachmann 《Journal of Mathematical Analysis and Applications》1977,58(3):647-652
We consider the differential equation ?(py′)′ + qy + λay + μby + f(x, y, y′) = 0, x? (α, γ) subject to the boundary conditions cos(α1) y(α) ? sin(α1) y′(α) = 0cos(β1) y(β) ? sin(β1) y′(β) = 0 β? (α, γ)cos(γ1) y(γ) ? sin(γ1) y′(γ) = 0. The functions p, g, a, b, and f are well-behaved functions of x; f is smooth and of “higher order” in y and y′; the scalars λ and μ are eigenparameters. With mild restrictions on a and b it is known that the linearized problem, f ≡ 0, has eigensolutions, (). In this paper we use an Implicit Function Theorem argument to establish the existence of a local branch of solutions, bifurcating from (), to the above nonlinear two-parameter eigenvalue problem. 相似文献
16.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
17.
Rudolf Wegmann 《Journal of Mathematical Analysis and Applications》1976,56(1):113-132
For an n × n Hermitean matrix A with eigenvalues λ1, …, λn the eigenvalue-distribution is defined by · number {λi: λi ? x} for all real x. Let An for n = 1, 2, … be an n × n matrix, whose entries aik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, , p) with the same distribution Fa. Suppose that all moments | a | k, k = 1, 2, … are finite, a=0 and | a | 2. Let with complex numbers θσ and finite products Pσ of factors A and (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G0(x) = G1x) + G2(x), where G1 is a step function and G2 is absolutely continuous, such that with probability converges to G0(x) as n → ∞ for all continuity points x of G0. The density g of G2 vanishes outside a finite interval. There are only finitely many jumps of G1. Both, G1 and G2, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples , , , r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form of Schur's inequality for the eigenvalues λi(n) of An holds. Consequently random matrices do not tend to be normal matrices for large n. 相似文献
18.
Paul Binding 《Journal of Mathematical Analysis and Applications》1984,102(1):233-243
The multiparameter eigenvalue problem Wm(λ) xm = xm, , m = 1,…, k, where /gl /gE k, xm is a nonzero element of the separable Hilbert space Hm, and Tm and Vmn are compact symmetric is studied. Various properties, including existence and uniqueness, of λ = λi ? k for which the imth greatest eigenvalue of Wm(λi) equals one are proved. “Right definiteness” is assumed, which means positivity of the determinant with (m, n)th entry (ym, Vmnym) for all nonzero ym?Hm, m = 1 … k. This gives a “Klein oscillation theorem” for systems of an o.d.e. satisfying a definiteness condition that is usefully weaker than in previous such results. An expansion theorem in terms of the corresponding eigenvectors xmi is also given, thereby connecting the abstract oscillation theory with a result of Atkinson. 相似文献
19.
Arthur Lubin 《Journal of Functional Analysis》1974,17(4):388-394
Let m and vt, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral = ⊕L2(vt) dm(t) and the operator on , where e(s, t) = exp ∫st ∫Tdvλ(θ) dm(λ). Let μt be the measure defined by for all continuous ?, and let ?t(z) = exp[?∫ (eiθ + z)(eiθ ? z)?1dμt(gq)]. Call {vt} regular iff for all for 1 a.e. 相似文献
20.
Hansjörg Kielhöfer 《Journal of Functional Analysis》1980,38(3):416-441
Let X and Y be Banach spaces, Y ?X, and let V be a neighborhood of zero in Y. We consider the equation G(λ, u) ≡ A(λ)u + F(λ, u) = 0, where G: [?d1, d1] × V → X, G(λ, 0) = 0, and A(λ) is the Fréchet derivative of G with respect to u at (λ, 0). Furthermore, we assume that G is analytic with respect to λ and u. Bifurcation at a simple eigenvalue means that zero is a simple eigenvalue of A (0). Let μ(λ) be the simple eigenvalue of the perturbed operator A(λ) for λ near zero. Let . Under the nondegeneracy condition m = 1 the existence of a unique curve of solutions intersecting the trivial solution (λ, 0) at (0, 0) is well known. Furthermore the “Principle of Exchange of Stability” was established in this case. We show that in the degenerate case (m > 1) up to m bifurcating curves of solutions can exist and that at least one nontrivial curve exists if m is odd. Our approach supplies all curves of solutions near (0, 0) together with their direction of bifurcation and their linearized stability. The decisive fact is that Am is also the leading term of the bifurcation equation. A consequence is a “Generalized Principle of Exchange of Stability”, which means that adjacent solutions for the same λ have opposite stability properties in a weakened sense. For practical use we give a criterion for asymptotic stability or instability which follows from the construction of the curves of solutions themselves. 相似文献