共查询到20条相似文献,搜索用时 31 毫秒
1.
OU YuWei 《中国科学 数学(英文版)》2014,57(7):1539-1544
The linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three body problem depends on the mass parameterβ=27(m1m2+m2m3+m3m1)/(m1+m2+m3)2∈[0,9]and the eccentricity e∈[0,1).In this paper we use Maslov-type index to study the stability of these solutions and prove that the elliptic Lagrangian solutions is hyperbolic forβ8 with any eccentricity. 相似文献
2.
Gareth E. Roberts 《Journal of Differential Equations》2002,182(1):191-218
This paper concerns the linear stability of the well-known periodic orbits of Lagrange in the three-body problem. Given any three masses, there exists a family of periodic solutions for which each body is at the vertex of an equilateral triangle and travels along an elliptic Kepler orbit. Reductions are performed to derive equations which determine the linear stability of the periodic solutions. These equations depend on two parameters - the eccentricity e of the orbit and the mass parameter β=27(m1m2+m1m3+m2m3)/(m1+m2+m3)2. A combination of numerical and analytic methods is used to find the regions of stability in the βe-plane. In particular, using perturbation techniques it is rigorously proven that there are mass values where the truly elliptic orbits are linearly stable even though the circular orbits are not. 相似文献
3.
We show the existence of dissipative H¨older continuous solutions of the Boussinesq equations. More precise, for any β∈(0,1/5), a time interval [0, T ] and any given smooth energy profile e : [0, T ] →(0, ∞), there exist a weak solution(v, θ) of the 3 d Boussinesq equations such that(v, θ) ∈ Cβ(T~3× [0, T ]) with e(t) =′his T~3|v(x, t)|~2 dx for all t ∈ [0, T ]. Textend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30]. 相似文献
4.
This paper considers the steady Swift-Hohenberg equation u'''+β2u''+u^3-u=0.Using the dynamic approach, the authors prove that it has a homoclinic solution for each β∈ (4√8-ε0,4 √8), where ε0 is a small positive constant. This slightly complements Santra and Wei's result [Santra, S. and Wei, J., Homoclinic solutions for fourth order traveling wave equations, SIAM J. Math. Anal., 41, 2009, 2038-2056], which stated that it admits a homoclinic solution for each β∈C (0,β0) where β0 = 0.9342 .... 相似文献
5.
Mohamed Ben Alaya Thierry Huillet 《Annals of the Institute of Statistical Mathematics》2005,57(4):817-831
With ?(p),p≥0 the Laplace-Stieltjes transform of some infinitely divisible probability distribution, we consider the solutions to the functional equation ?(p-e ?pβΠ i=1 m ?γi (c i p) for somem≥1,c i>0, γ i >0,i=1., …,m, β ε ®. We supply its complete solutions in terms of semistable distributions (the ones obtained whenm=1). We then show how to obtain these solutions as limit laws (r → ∞) of normalized Poisson sums of iid samples when the Poisson intensity λ(r) grows geometrically withr. 相似文献
6.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1998,326(3):307-310
Let be the Clifford algebra of with a quadratic form of negative signature, D = ei ∂/∂xi, Δ the ordinary Laplacian. The holomorphic cliffordian functions are solutions of D Δmƒ = 0. We study the polynomial and singular solutions, representation integral formulas, and the foundation of the Cliffordian elliptic function theory. 相似文献
7.
B. F. Ivanov 《Vestnik St. Petersburg University: Mathematics》2018,51(1):49-56
Suppose that m ≥ 2, numbers p1, …, p m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ1 ∈ \({L^{{p_1}}}\)(?1), …, γ m ∈ \({L^{{p_m}}}\)(?1) are given. It is proved that if the set of “resonance points” of each of these functions is nonempty and the so-called “resonance condition” holds, then there are arbitrarily small (in norm) perturbations Δγk ∈ \({L^{{p_k}}}\)(?1) under which the resonance set of each function γk + Δγk coincides with that of γk for 1 ≤ k ≤ m, but \({\left\| {\int\limits_0^t {\prod\limits_{k = 0}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty \). The notion of a resonance point and the resonance condition for functions in the spaces L p (?1), p ∈ (1, +∞], were introduced by the author in his previous papers. 相似文献
8.
David Kalaj 《Potential Analysis》2014,41(3):931-943
Let H 3 be the hyperbolic space identified with the unit ball B 3 = {x ∈ R 3 : |x| < 1} with the Poincaré metric d h and let ??(x 0, p, q) : = {x : p < d h (x, x 0) < q} ? H 3 be a hyperbolic annulus with the inner and outer radii 0 < p < q < ∞. We prove that if there exists a hyperbolic harmonic diffeomorphism between annuli ??(x 0, a, b) and ?? (y 0, α, β) in the hyperbolic space H 3, then β/α>1+ψ(a,b), where ψ is a positive function. In addition, for given two annuli in the hyperbolic space H 3, satisfying certain conditions, we construct radial harmonic mappings between them. 相似文献
9.
Tosis Kato 《Israel Journal of Mathematics》1972,13(1-2):135-148
Recently B. Simon proved a remarkable theorem to the effect that the Schrödinger operatorT=?Δ+q(x) is essentially selfadjoint onC 0 ∞ (R m if 0≦q ∈L 2(R m). Here we extend the theorem to a more general case,T=?Σ j =1/m (?/?x j ?ib j(x))2 +q 1(x) +q 2(x), whereb j, q1,q 2 are real-valued,b j ∈C(R m),q 1 ∈L loc 2 (R m),q 1(x)≧?q*(|x|) withq*(r) monotone nondecreasing inr ando(r 2) asr → ∞, andq 2 satisfies a mild Stummel-type condition. The point is that the assumption on the local behavior ofq 1 is the weakest possible. The proof, unlike Simon’s original one, is of local nature and depends on a distributional inequality and elliptic estimates. 相似文献
10.
We consider the non-linear two point boundary value problem where λ > 0,f ∈ C2, f′ ≥ 0, f(0) < 0 and limu → ∞ f(u) > 0. By considering the non-negative as well as all sign changing solutions, we establish the existence of infinitely many non-trivial bifurcation points. Further, when f is superlinear, we prove that there exists a constant λ* > 0, such that for each λ ∈ (0, λ*) there are exactly two solutions with m interior zeros for every m = 1,2, …We apply our results to the case when f(u) = u 3 - k; k > 0, and also discuss the evolution of the bifurcation diagram as k → 0. 相似文献
11.
ZheFeng Xu 《中国科学 数学(英文版)》2013,56(8):1597-1606
For any real constants λ 1, λ 2 ∈ (0, 1], let $n \geqslant \max \{ [\tfrac{1} {{\lambda _1 }}],[\tfrac{1} {{\lambda _2 }}]\} $ , m ? 2 be integers. Suppose integers a ∈ [1, λ 1 n] and b ∈ [1, λ 2 n] satisfy the congruence b ≡ a m (mod n). The main purpose of this paper is to study the mean value of (a ? b)2k for any fixed positive integer k and obtain some sharp asymptotic formulae. 相似文献
12.
The nonlinear Klein-Gordon equation ?μ?μΦ + M2Φ + λ1Φ1?m + λ2Φ1?2m = 0 has the exact formal solution Φ = [u2m ?λ1um/(m ? 2)M2+λ12/(m?2)2M4?λ2/4(m ? 1)M2]1/mu?1, m ≠ 0, 1, 2, where u and v?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [aum + b(uv)m/2 + cvm]k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0. 相似文献
13.
M. I. Nagnybida 《Ukrainian Mathematical Journal》1996,48(7):1084-1098
In the space A (θ) of all one-valued functions f(z) analytic in an arbitrary region G ? ? (0 ∈ G) with the topology of compact convergence, we establish necessary and sufficient conditions for the equivalence of the operators L 1=α n z n Δ n + ... + α1 zΔ+α0 E and L 2= z n a n (z)Δ n + ... + za 1(z)Δ+a 0(z)E, where δ: (Δ?)(z)=(f(z)-?(0))/z is the Pommier operator in A(G), n ∈ ?, α n ∈ ?, a k (z) ∈ A(G), 0≤k≤n, and the following condition is satisfied: Σ j=s n?1 α j+1 ∈ 0, s=0,1,...,n?1. We also prove that the operators z s+1Δ+β(z)E, β(z) ∈ A R , s ∈ ?, and z s+1 are equivalent in the spaces A R, 0?R?-∞, if and only if β(z) = 0. 相似文献
14.
Dr. Klaus Deimling 《manuscripta mathematica》1971,4(3):201-212
By a well known theorem of H. Kneser [4] the set U of all solutions of the initial value problem $$u' = f(x,u)forx\varepsilon [0,a],u(0) = u_o $$ has the following property: If f is continuous and bounded then U(x0)={u(x0): u∈U} is a continuum (i.e. a compact and connected subset) for every x0∈[0,a]. In the present paper we claim to extend this theorem to a system of Volterra integral equations in several variables of the form x∈B∞Rm, ν=1,...,n that had been investigated in [8]. In fact we shall prove that U is a continuum of the Banach space Cn(B) of all ‘vector functions’ u(x)=(u1(x),...,un(x)), continuous on B. It is an immediate consequence from this that U(x0) is a continuum of Rn. These results will be established by the help of a suitable modification of a method used by M. Müller [5] to prove Kneser's theorem. Especially, we obtain new theorems for some initial value problems for hyperbolic equations. 相似文献
15.
Methionine complexes of cadmium in 25 and 50 per cent aqueous mixtures of ethyl and methyl alcohol and dioxan have been studied. The half-wave potentials measured in both the alcohols were the same and the reduction was reversible. Three complex species withβ 1=1·0×104,β 2=1·1×107 andβ 3=1·2×109 were found in 25 per cent alcohol while four complexes withβ 1=3·0×104,β 2=4·3×107,β 3=4·0×109 andβ 4=1·6×1011 were observed in 50 per cent solutions. In the case of dioxan, the reduction was quasi-reversible (k s=1·0×10?3 cm sec?1) in 25 per cent and irreversible (k s=2·0×10?4 cm sec?1) in 50 per cent solutions. The stability constants, evaluated using the formal potentials, wereβ 1=7·0×103,β 2-3·9×105;β 2=3·9×108 andβ 4=3·4×1010 in 25 per cent dioxan andβ 1=1·5×104,β 2=3·4×107.β 3=7·5×109 andβ 4=9·0×1011 in 50 per cent solutions. 相似文献
16.
Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are F p = {0, 1,..., p - 1} produces a 1-Lipschitz map f A from the space Z p of p-adic integers to Z p . The map fA can naturally be plotted in a unit real square I2 ? R2: To an m-letter non-empty word v = γ m-1γ m-2... γ0 there corresponds a number 0.v ∈ R with base-p expansion 0.γ m-1γ m-2... γ0; so to every m-letter input word w = α m-1α m-2 ··· α0 of A and to the respective m-letter output word a(w) = β m-1β m-2 ··· β0 of A there corresponds a point (0.w; 0.a(w)) ∈ R2. Denote P(A) a closure of the point set (0.w; 0.a(w)) where w ranges over all non-empty words.We prove that once some points of P(A) constitute a C 2-smooth curve in R2, the curve is a segment of a straight line with a rational slope. Moreover, when identifying P(A) with a subset of a 2-dimensional torus T2 ∈ R3, the smooth curves from P(A) constitute a collection of torus windings which can be ascribed to complex-valued functions ψ(x, t) = e i(Ax-2πBt) (x, t ∈ R), i.e., to matter waves. As automata are causal discrete systems, the main result may serve a mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter. 相似文献
17.
The linear equation Δ2u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?2 has the particularly interesting nonlinear generalization Δg2u = 1, where Δg = e?2u Δ is the Laplace‐Beltrami operator for the metric g = e2ug0, with g0 the standard Euclidean metric on ?2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+Kg(x) exp(2u(x)) = 0 and Δ Kg(x) + exp(2u(x)) = 0, with x ∈ ?2, describing a conformally flat surface with a Gauss curvature function Kg that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ Kg exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K+ ∈ L1(B1(y)), uniformly w.r.t. y ∈ ?2. We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc. 相似文献
18.
Wolfgang Walter 《Results in Mathematics》1994,26(3-4):399-402
The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλi x (i = 1, 2, …), where the λi∈ ? are the fixed points of the map z ? sinh z, and (ii) C∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?(j)(? 1) = ?(j)(0) = ?(j)(1) = 0 for all j = 0, 1, 2 … 相似文献
19.
We consider singular differential operators of order 2m, m ∈ ?, with discrete spectrum in L 2[0, + ∞). For self-adjoint extensions given by the boundary conditions y(0) = y″(0) = ? = y (2m?2)(0) = 0 or y′(0) = y?(0) = ? = y (2m?1)(0) = 0, we obtain regularized traces. We present the explicit form of the spectral function, which can be used for calculating regularized traces. 相似文献
20.
Mordechay B. Levin 《Israel Journal of Mathematics》2010,178(1):61-106
In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ? ? s , we construct Kronecker-like and van der Corput-like ergodic transformations T 1,Γ and T 2,Γ of [0, 1) s . We prove that for admissible lattices Γ, (T ν,Γ n (x))n≥0 is a low discrepancy sequence for all x ∈ [0, 1) s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ? [0, 1) s , for almost all lattices Γ ∈ L s = SL(s,?)/SL(s, ?) (in the sense of the invariant measure on L s ), the following asymptotic formula holds with arbitrary small ? > 0, for all x ∈ [0, 1) s , and ν ∈ {1, 2}.
相似文献
$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$