An estimation for the hyperbolic region of elliptic Lagrangian solutions in the planar three-body problem

It is well known that the linear stability of elliptic Lagrangian solutions depends on the mass parameter β = 27(m ₁ m ₂ + m ₂ m ₃ + m ₃ m ₁)/(m ₁ + m ₂ + m ₃)² ∈ [0, 9] and the eccentricity e ∈ [0, 1). Based on new techniques for evaluating the hyperbolicity and the recently developed trace formula for Hamiltonian systems [9], we identify regions for (β, e) such that elliptic Lagrangian solutions are hyperbolic. Consequently, we have proven that the elliptic relative equilibrium of square central configurations is hyperbolic with any eccentricity.     

Linear Stability of the Elliptic Lagrangian Triangle Solutions in the Three-Body Problem

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.     

HLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATIONS

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].     

A Supplement to Hölder’s Inequality. The Resonance Case. I

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 Δγ_k ∈ \({L^{{p_k}}}\)(?¹) under which the resonance set of each function γ_k + Δγ_k coincides with that of γ_k 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.     

Notes on Homoclinic Solutions of the Steady Swift-Hohenberg Equation

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 ....     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(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 = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

Schrödinger operators with singular potentials

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.     

Global structure of positive solutions for superlinear 2mth-boundary value problems

We consider boundary value problems for nonlinear 2mth-order eigenvalue problem $$ \begin{gathered} ( - 1)^m u^{(2m)} (t) = \lambda a(t)f(u(t)),0 < t < 1, \hfill \\ u^{(2i)} (0) = u^{(2i)} (1) = 0,i = 0,1,2,...,m - 1. \hfill \\ \end{gathered} $$ . where a ∈ C([0, 1], [0, ∞)) and a(t ₀) > 0 for some t ₀ ∈ [0, 1], f ∈ C([0, ∞), [0, ∞)) and f(s) > 0 for s > 0, and f ₀ = ∞, where $ \mathop {\lim }\limits_{s \to 0^ + } f(s)/s $ . We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.     

On the difference between an integer and its m-th power mod n

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.     

Existence of Infinitely Many Non-Trivial Bifurcation Points

We consider the non-linear two point boundary value problem where λ > 0,f ∈ C², f′ ≥ 0, f(0) < 0 and lim_{u → ∞} 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.     

On some properties of schlicht functions

One considers the classes S _β ^* (α),S _β (λ),, and S of functionsf (z)=z+ ..., which are respectivelyα-starlike of orderβ, γ-spirallike of orderβ, and regular schlicht in ¦z ¦ < 1. It is proved that forα? 0, 0 < β < 1 fromf (z) ∈S _β ^* (α) followsf (z) ∈S _β ^* (0); this generalizes appropriate results of [1–5]. A converse result is also obtained. For certainα andβ the exact value of the radius ofα-starlikeness of orderβ for the class S is given. An equation is found, whose unique root gives the radiusγ-spirallikeness of orderβ for the class S.     

Regularized traces of higher-order singular differential operators

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.     

A polarographic study of methionine complexes of cadmium in mixed solvents

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.     

Rate of stability of solutions of matrix polynomial and quadratic equations

In this paper the rate of stability of solutions of matrix polynomial equations of the typeA ₀+A ₁ X+A ₂ X ²+...+A _m X ^m =0 is studied. Particular attention is given to the case where the matrix polynomialL(λ):=A ₀+A ₁ λ+A ₂ λ ²+...+A _m λ ^m is weakly hyperbolic, i.e., for every non-zero vectorx the scalar polynomial 〈L(λ)x, x〉 has only real roots. Also the rate of stability of solutions of matrix quadratic equations of the typeXBX+XA-DX-C=0 is studied. Here the special case that is of interest to continuous-time optimal control theory, that is, the case whereB=B ^* is positive semidefinite andC=C ^*,A=?D ^*, is discussed in detail. The analogous theory for the discrete-time optimal control leads to the equation $$X = A^* XA + Q - (B^* XA)^* (R + B^* XB)^{ - 1} B^* XA,$$ and the rate of stability of solutions of this equation is also studied. Most of the problems are discussed in both real and complex settings.     

On Substructure Densities of Hypergraphs

A real number α ∈ [0, 1) is a jump for an integer r ≥ 2 if there exists c > 0 such that for any ∈ > 0 and any integer m ≥ r, there exists an integer n ₀ such that any r-uniform graph with n > n ₀ vertices and density ≥ α + ∈ contains a subgraph with m vertices and density ≥ α + c. It follows from a fundamental theorem of Erdös and Stone that every α ∈ [0, 1) is a jump for r = 2. Erdös also showed that every number in [0, r!/r ^r ) is a jump for r ≥ 3 and asked whether every number in [0, 1) is a jump for r ≥ 3 as well. Frankl and Rödl gave a negative answer by showing a sequence of non-jumps for every r ≥ 3. Recently, more non-jumps were found for some r ≥ 3. But there are still a lot of unknowns on determining which numbers are jumps for r ≥ 3. The set of all previous known non-jumps for r = 3 has only an accumulation point at 1. In this paper, we give a sequence of non-jumps having an accumulation point other than 1 for every r ≥ 3. It generalizes the main result in the paper ‘A note on the jumping constant conjecture of Erdös’ by Frankl, Peng, Rödl and Talbot published in the Journal of Combinatorial Theory Ser. B. 97 (2007), 204–216.     

On the equivalence of the Euler-Pommier operators in spaces of analytic functions

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.     

Quantization causes waves: Smooth finitely computable functions are affine

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.     

A note on the conditional moments of a multivariate normal distribution confined to a convex set

Let Y be an N(μ, Σ) random variable on R^m, 1 ≤ m ≤ ∞, where Σ is positive definite. Let C be a nonempty convex set in R^m with closure $\text{C}$. Let (·,-·) be the Eculidean inner product on R^m, and let μ_c be the conditional expected value of Y given Y ∈ C. For v ∈ R^m and s ≥ 0, let β_s(v) be the expected value of |(v, Y) ? (v, μ)|^s and let γ_s(v) be the conditional expected value of |(v, Y) ? (v, μ_c)|^s given Y ∈ C. For s ≥ 1, γ_s(v) < β_s(v) if and only if $\text{C}\text{+ Σ v ≠}\text{C}$, and γ_s(v) < β_s(v) for all v ≠ 0 if and only if $\text{C}\text{+ v ≠}\text{C}$ for any v ∈ R^m such that v ≠ 0.     

On The Postulation of The Multiples of The Nodes for Generic Nodal Curves on Generic Rational Surfaces

Fix integers x > 0, m₁ ≥ … ≥ ^m _x > 0 and P₁,…,P_x ∈ P² such that no 3 of them are collinear. Let C ? P² a “ general ” degree d plane curve with an ordinary point with multiplicity m _i at each P _i and y further singularities which are ordinary nodes. Fix any A ? Sing(C){P₁,…, P_x} and any integer m > 0. Here we study the postulation of the fat points m A ? ∪_Q∈AmQ.     

Periodic solutions of a nonlinear propagation equation via a fixed point argument

In this work the initial value problem for the equation $$u_t + \beta u_x + yf(u)_x - \delta u_{xxt} = g,\forall x \in R, \forall t \in [0,T],$$ with periodic boundary conditions is interpreted in the sense of periodic distributions and studied via fixed point arguments. Weak solutions exist iff∈C ⁰ (R) andg∈L ^∞(L ²(0,1)). Moreover, regularity inf, g and the initial data implies regularity of solutions.