The integral points on elliptic curves y 2 = x 3 + (36n 2 − 9)x − 2(36n 2 − 5)

Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6n ² ? 1 and 12n ² + 1 are odd primes, then the general elliptic curve y ² = x ³+(36n ²?9)x?2(36n ²?5) has only the integral point (x, y) = (2, 0). By this result we can get that the above elliptic curve has only the trivial integral point for n = 3, 13, 17 etc. Thus it can be seen that the elliptic curve y ² = x ³ + 27x ? 62 really is an unusual elliptic curve which has large integral points.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

Regularity and explicit representation of (0,1,...,m−2,m) interpolation on the zeros of (1−x 2)P n −2/(α,β) (x)

Necessary and sufficient conditions for the regularity andq-regularity of (0,1,...,m–2,m) interpolation on the zeros of (1–x ²)P _n ^–2/(,) (x) (,>–1) in a manageable form are established, whereP _n ^–2/(,) (x) stands for the (n–2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,...,m–2,m) interpolation has an infinity of solutions then the general form of the solutions isf ₀(x)+C f(x) with an arbitrary constantC.This work is supported by the National Natural Science Foundation of China.     

One-Sided Boundary Control for the Process of Oscillations Described by the Equation k(x)[k(x)u x (x,t)] x − u tt (x,t) = 0

This article concerns the boundary control on the side x = 0 when the other side x = l is fixed. The corresponding process of oscillations is studied for the time interval T > 0 and is described by the generalized solution of the equation

Paramétrisation des points algébriques de degré donné sur la courbe d'équation affine y3=x(x−1)(x−2)(x−3)

We give a parameterization of the algebraic points of given degree over $\mathbb{Q}$ on the curve$y^3=x(x?1)(x?2)(x?3)$ This result extends a previous result of E.F. Schaefer who described in Schaefer (1998) [1] the set of algebraic points of degree ?3 over $\mathbb{Q}$.     

The topological entropy of the transformationx ↦ax (1−x)

A well-known conjecture about the transformationT _a :x ax (1–x) on [0, 1], where 2a4, says that the mapa h _top (T _a ) is monotone. In this paper we show that this is connected with a property of the polynomialsP _k (t) (4t8) given byP ₀ (t)=0 andP _k+1 (t)=(t–P _k t)²)/2, namely that they have in some sense a minimal number of zeros. Furthermore we show for a countable subset of [2, 4], whose limit points form a sequence converging to 4, to be in {a[2,4]:h _top (T _c ), ifc<a andd>a}.With 2 Figures     

Group Laws [x,y −1] ≡ u(x,y) and Varietal Properties

Let F = ?x, y? be a free group. It is known that the commutator [x, y ^?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [x, y ^?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y ^?1] ≡ u, where u ∈ [F′, ?x?].     

On the recursive sequencex n +1=α-(x n /x n −1)

We study the global asymptotic stability, global attractivity, boundedness character, and periodic nature of all positive solutions and all negative solutions of the difference equation $$x_{n + 1} = \alpha - \frac{{x_n }}{{x_{n - 1} }}, n = 0,1,...,$$ where α∈R is a real number, and the initial conditionsx?1,x ₀ are arbitrary real numbers.     

Q_k(x)与F_k(x)的渐近公式

设ｎ，ｋ为自然数，Ｇ（ｎ）阶群中的同构类数，Ｆｋ（ｘ）与Ｑｋ（ｘ）分别表示不超过ｘ的自然数中使Ｇ（ｎ）＝ｋ的自然数、无平方因子自然数的个数．本文的目的是用Ｂｒｕｎ筛法证明Ｑｋ（ｘ）的条件渐近公式并对Ｆｋ（ｘ）的渐近性质做出了推测．     

Boundedness properties of the difference equation x n+1=(α+βx n +δx n−2)/(x n−1)

Consider the third-order difference equation x _n+1 = (α+βx _n +δx _n ? 2)/(x _n ? 1) with α ∈ [0,∞) and β,δ ∈ (0,∞). It is shown that this difference equation has unbounded solutions if and only if δ>β.     

On the non-commutative neutrix product (x + r lnx +) ox − −s

The non-commutative neutrix product of the distributionsx ₊ ^r lnx ₊ andx _– ^–s is evaluated forr=0, 1, 2, ands=1, 2, . Further neutrix products are then deduced.     

A note on the diophantine equation (a n − 1)(b n − 1) = x 2

Let a, b be fixed positive integers such that a ≠ b, min(a, b) > 1, ν(a?1) and ν(b ? 1) have opposite parity, where ν(a ? 1) and ν(b ? 1) denote the highest powers of 2 dividing a ? 1 and b ? 1 respectively. In this paper, all positive integer solutions (x, n) of the equation (a ⁿ ? 1)(b ⁿ ? 1) = x ² are determined.     

求证f(x)>g(x)的四种情形

<正>求证f(x)>g(x)是高考中经常出现的问题,大家直接想到的是通过作差构造新函数h(x)=f(x)-g(x)找到f(x)>g(x)的充要条件,但有时由于h(x)的单调性很难确定,往往陷入困境.所以有时我们要找到f(x)>g(x)的充分条件,即通过最值之间的比较.本文意在通过四道例题将求证f(x)>g(x)的所有情况细分开来归纳成四种情形.     

The integrable nonlinear degenerate diffusion equationu t=[f(u)u x −1 x and its relatives

The nonlinear diffusion equationu _t=[f(u)g(u _x )] arises in recent models of turbulent transport and of stress dissipation in rock blasting. A Lie point symmetry analysis produces many similarity reductions of exponential and power-law forms, and reveals that for all choices off the equation is always integrable wheng(u _x )=1/u _x . We identify the functionsf(u) which guarantee equivalence to the linear heat equation. For all other choices off, the linear canonical form leads to a self-adjoint differential equation by separation of variablesx andt. We construct a number of explicit solutions with simple boundary conditions, which illustrate behavior in the vicinity of the degenerate region withu _x =. If zero flux and constant concentration are maintained on free boundaries, then steep concentration gradients may evolve from smooth initial conditions. For other boundary conditions, unlike the examples of strong degeneracy, smoothing will occur at initial step discontinuities.     

方程y″(x)+p(x)y(x)=y(x)解的有界性与振动性

设P(x)、f(x)∈C~1[0,+∞),在[0,+∞)上,P(x)>0,P′(x)≤0且(?)P(x)=ρ>0,intejral form 0 to +∞。|f′(t)|dt<+∞。我们给出了方程y″+P(x)y=f(x)解的有界性与振动性结果。