Let T be a subgroup of PSL (2, ℚ) generated by a pair of rational parabolic matrices P1, P2, and let ℐ be the Jøgensen number. We prove that T has a non-trivial element of finite order if and only if ℐ = 4/n2 or ℐ = 9/n2 for some non-zero integer n.

Consider a family of elliptic curves (A, A0, d0 fixed integers). We prove that, under certain conditions on A0 and d0, the rational torsion subgroup of E(B) is either cyclic of order ≤ 3 or non-cyclic of order 4. Also, assuming standard conjectures, we establish estimates for the order of the Tate-Shafarevich groups as B varies.

A.N. Kolmogorov showed that, if f, f′, …, f (n) are bounded continuous functions on ℝ, then when 0 < k < n. This result was extended by E.M. Stein to Lebesgue Lp-spaces and by H.H. Bang to Orlicz spaces. In this paper, the inequality is extended to more general function spaces.

In this paper we prove that if a, b, c, r are fixed positive integers satisfying a2 + b2 = cr, gcd(a, b) = 1, a ≡ 3(mod 8), 2 | b, r > 1, 2 ∤ r, and c is a (x,y,z) = (2, 2,r) satisfying x > 1, y > 1 and z > 1.

In this paper we discuss in some detail the difference equations arising in the discretization of some second-order differential equations. We also show how such difference problems can be solved exactly.

Dans [1, Section 3], on a montré comment les conjectures les plus définitives sur les formes linéaires de logarithmes entraînent simplement la conjecture abc (dans sa forme faible). De façcon plus pragmatique on peut également chercher les conjectures les plus accessibles sur les formes linéaires de logarithmes qui entraînent encore la conjecture abc. De ce point de vue l'énoncé suivant semble être l'amélioration la plus minime de l'inégalité de Liouville qu'il conviendrait d'établir. On y note | · |p la valeur absolue p-adique sur Q, normalisée de la façon habituelle (c'est-à-dire, |P|p = 1/P), et h (·) la hauteur logarithmique sur Q (c'est-à-dire, le logarithme du maximum des valeurs absolues des numérateur et dénominateur dans une écriture en fraction réduite).

We prove an algebraic formula for the Euler characteristic of the Milnor fibres of functions with critical locus a smooth curve on a space which is a weighted homogeneous complete intersection with isolated singularity.

In this paper, by using the technique of product nets, we are able to prove a weak convergence theorem for an almost-orbit of right reversible semigroups of nonexpansine mappings in a general Banach space X with Opial's condition. This includes many well known results as special cases. Let C be a weakly compact subset of a Banach space X with Opial's condition. Let G be a right reversible semitopological semigroup,  = {T (t): t ∈ G} a nonexpansive semigroup on C, and u (·) an almost-orbit of . Then {u (t): t ∈ G} is weakly convergent (to a common fixed point of ) if and only if it is weakly asymptotically regular (that is, {u (ht) − u (t)} converges to 0 weakly for every h ∈ G).