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).     

How often is 84(g−1) achieved?

For any finitely generated group Γ, the asymptotics of the set of orders of finite quotient groups of Γ are determined by the minimum dimension of a complex linear group containing an infinite quotient of Γ. We give a proof and an application to the asymptotic behavior of the set of integersg for which the Hurwitz bound is sharp. Partially supported by NSF Grant DMS-97-27553.     

Uniqueness theorems for entire functions having growth [2, π/2)

It is shown that if f is any entire function in the class [2,π/2), which along with finitely many of its successive derivatives, vanishes at the integer lattice points, suitably scaled, then f is identically zero. It is then shown that if f is any entire function in a proper subclass of [2,π/2), which along with finitely many of its successive derivatives, is bounded at the integer lattice points, suitably scaled, then f is constant. A heuristic argument in support of the conjecture that this latter result holds for the full class [2, π/2) is given.     

The recursive sequence xn+1 = g(xn,xn−1)/(A + xn)

In this note, we investigate the periodic character of solutions of the nonlinear, second-order difference equation

where the parameter A and the initial conditions x₀ and x₁ are positive real numbers. We give sufficient conditions under which every positive solution of this equation converges to a period two solution.     

(S − 1,S) Perishable systems with stochastic leadtimes

This article discusses a one-to-one ordering perishable system, in which reorders are processed in the order of their arrival and the processing times are arbitrarily distributed, and as such, the leadtimes are not independent. The Markov renewal techniques are employed to obtain the various operating characteristics for the case of Poisson demand and exponential lifetimes. The problem of minimizing the steady state expected cost rate is also discussed, and in the special case of exponential processing times, the optimal stock level is derived explicitly.     

Padé tables of entire functions of very slow and smooth growth

Letf(z)=σ _j?o ^∞ a _j z ^j be entire with $$|a_{j - 1} a_{j + 1} /a_j^2 | \leqslant \rho _0^2 ,j = 1,2,3, \ldots ,$$ whereρ ₀=0.4559... is the positive root of the equation $$2\sum\limits_{j = 1}^\infty {\rho ^{j^2 } = 1.}$$ . It is shown that the Padé table off is normal, and asL→∞, [L/M _L ](z) converges uniformly in compact subsets ofC tof, for any sequence of nonnegative integers {M _L } _L=1 ^∞. In particular, the diagonal sequence {[L/L]} converges uniformly in compact subsets ofC tof. Furthermore, the constantρ ₀ is shown to be best possible in a strong sense.     

Compositions of analytic functions of the form Fn(z) = Fn−1(fn(z)), fn(z) → f(z)

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {F_n(z)} is studied, where F₁(z) = f₁(z) and F_n(z) = F_n−1(f_n(z)), with f_n → f. Many infinite arithmetic expansions exhibit this form, and the fixed point, α, of f may be used as a modifying factor (z = α) to influence the convergence behaviour of these expansions. Thus one employs, rather than seeks the fixed point of the function f.     

Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space

In a previous work (Adimurthi and Yang, 2010 [2]), Adimurthi–Yang proved a singular Trudinger–Moser inequality in the entire Euclidean space ${\mathbb{R}}^N$$(N\ge 2)$. Precisely, if $0\le \beta <1$ and $0<\gamma \le 1?\beta$, then there holds for any $\tau >0$,

$\underset{u\in W^{1,N}({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}(|\mathrm{?}u{|}^N+\tau |u{|}^N)dx\le 1}{\sup }?\underset{{\mathbb{R}}^N}{\int }\frac{1}{|x{|}^{N\beta }}(e^{{\alpha }_N\gamma |u{|}^{\frac{N}{N?1}}}?\sum _{k=0}^{N?2}\frac{{{\alpha }_N}^k{\gamma }^k|u{|}^{\frac{kN}{N?1}}}{k!})dx<\infty ,$

where ${\alpha }_N=N{\omega }_{N?1}^{1/(N?1)}$ and ${\omega }_{N?1}$ is the area of the unit sphere in ${\mathbb{R}}^N$. The above inequality is sharp in the sense that if $\gamma >1?\beta$, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case $\beta =0$ has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11]. We shall investigate the singular case $0<\beta <1$ and prove that for all $\tau >0$, $0<\beta <1$ and $0<\gamma \le 1?\beta$, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.