Global Behavior of Solutions of x n + 1 = max {x n , A}/x n x n – 1

We investigate the asymptotic behavior, the oscillatory character, and theperiodic nature of solutions of the difference equation

On Globally Periodic Solutions of the Difference Equation x n + 1 = f ( x n )/ x n − 1

We study the second-order difference equation x n +1 = f ( x n ) x n m 1 where f ] C 1 ([0, X ),[0, X )) and x n ] (0, X ) for all n ] Z . For the cases p h 5, we find necessary and sufficient conditions on f for all solutions to be periodic with period p . We answer some questions and conjectures of Kulenovi ' and Ladas.     

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 Behavior of Solutions of x n+1 = p+(x n−1/xn )

Our goal in this article is to complete the study of the behavior of solutions of the equation in the title when the parameter p is positive and the initial conditions are arbitrary positive numbers. Our main focus is the case 0 < p < 1. We will show that in this case, all solutions which do not monotonically converge to the equilibrium have a subsequence which converges to p and a subsequence which diverges to infinity. For the sake of completeness, we will also present the results (which were previously known) with alternative proofs for the case p = 1 and the case p > 1.     

On the law of the iterated logarithm for the discrepancy of 〈n k x〉

By a well known result of Philipp (1975), the discrepancy D _N (ω) of the sequence (n _k ω) _k≥1 mod 1 satisfies the law of the iterated logarithm under the Hadamard gap condition n _{k + 1}/n _k ≥ q > 1 (k = 1, 2, …). Recently Berkes, Philipp and Tichy (2006) showed that this result remains valid, under Diophantine conditions on (n _k ), for subexpenentially growing (n _k ), but in general the behavior of (n _k ω) becomes very complicated in the subexponential case. Using a different norming factor depending on the density properties of the sequence (n _k ), in this paper we prove a law of the iterated logarithm for the discrepancy D _N (ω) for subexponentially growing (n _k ) without number theoretic assumptions. C. Aistleitner, Research supported by FWF grant S9603-N13. I. Berkes, Research supported by FWF grant S9603-N13 and OTKA grants K 61052 and K 67961. Authors’ addresses: C. Aistleitner, Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria; I. Berkes, Institute of Statistics, Graz University of Technology, Steyrergasse 17/IV, 8010 Graz, Austria     

On the asymptotic behaviour of sums Σg(n){x/n} k

In this paper we estimate the difference between the sum given in the title (whereg(t) is an arbitrary real-valued non-decreasing function,k is a positive integer and summation is extended over all positive integersnx) and the corresponding integral, obtaining the boundO(g(x)x ^1/3logx). Furthermore, we show that these differences (for giveng and varyingk) are all approximately equal, with an error term ofO(g(x)x ^3/10). Finally it is remarked without proof that these estimates can be refined toO(g(x)x ) (with any >0,329...) resp.O(g(x)x ^109/382).     

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.     

A note on the number of solutions of the generalized Ramanujan-Nagell equation x 2 ? D = p n

Let D be a positive integer, and let p be an odd prime with p ? D. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M.A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for N(D, p), and also prove that if the equation U ² ? DV ² = ?1 has integer solutions (U, V), the least solution (u ₁, v ₁) of the equation u ² ? pv ² = 1 satisfies p ? v ₁, and D > C(p), where C(p) is an effectively computable constant only depending on p, then the equation x ² ? D = p ⁿ has at most two positive integer solutions (x, n). In particular, we have C(3) = 10⁷.     

O n the number of positive i nteger solutio ns ( x,n) of the ge neralized Rama nuja n–Nagell equatio n $$ x^2-2^r=p^ n$$ x2-2r=p n

Let p be a fixed odd prime, and let r be a fixed positive integer. Further let \(N(2^r,p)\) denote the number of positive integer solutions (x, n) of the generalized Ramanujan–Nagell equation \(x^2-2^r=p^n\). In this paper, we use the elementary method and properties of Pell’s equation to give a sharp upper bound estimate for \(N(2^r,p)\). That is, we prove that \(N(2^r,p)\le 1\).     

A Note on the Recursive Sequence x n + 1 = p k x n + p k − 1 x n − 1 +...+ p 1 x n − k + 1

We present some comments on the behavior of solutions of the difference equation where p _i 0, i = 1,..., k, k N, and x _–k ,..., x _–1 R.     

About the Cohomology of the Lie Superalgebra of Vector Fields on ℝ n|n

In this article, we compute the first space of cohomology of Vect (? ^n|n ), the Lie superalgebra of vector fields on the supermanifold ? ^n|n with coefficients in 𝒻 (? ^n|n ), the space of smooth functions on ? ^n|n . We give a super analog of the cohomologies of vector fields that where studied for instance by Fuchs [2 Fuchs , D. B. ( 1986 ). Cohomology of Infinite Dimensional Lie Algebras . New York : Consultants Bureau . [Google Scholar]]. This work allows us to classify the deformations of the action of Vect(? ^n|n ) on 𝒻 (? ^n|n ).     

On the (n − 2)-transversals of n convex subsets of the plane

In this paper we consider families of distinct ovals in the plane, with the property that certain subfamilies have stabbing lines (transversals). Our main result says that if any k member of the family can be stabbed by a line avoiding all the other ovals and k is large enough, then the family consists of at most k+1 ovals. For any n4 we show a family of n ovals, whose n–2 element subfamilies have, but the n–1 element subfamilies do not have, transversals.     

An Elementary Approach to Quasi-Isometries of Tree x ℝ n

We prove by elementary means a regularity theorem for quasi-isometries of T x ⁿ (where T denotes an infinite tree), and of many other metric spaces with similar combinatorial properties, e.g. Cayley graphs of Baumslag–Solitar groups. For quasi-isometries of T x ⁿ, it states that the image of {x} x ⁿ (xT) is uniformly close to {y} x ⁿ for some yT, and there is a well-defined surjection . Even stronger, the image of a quasi-isometric embedding of ⁿ⁺¹ in T x ⁿ is close to (a geodesic in T)T)x ⁿ.     

On the neutrix composition of the distributions x −s ln m |x| and x r

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {F _n (f)}, where F _n (x) = F(x) * δ _n (x) and {δ _n (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x ^?s ln ^m |x| and x ^r is proved to exist and be equal to r ^m x ^?rs ln ^m |x| for r, s, m = 2, 3….     

On the diophantine equation x 2 − Dy 2 = n

We propose a method to determine the solvability of the diophantine equation x2-Dy2=n for the following two cases:(1) D = pq,where p,q ≡ 1 mod 4 are distinct primes with(q/p)=1 and(p/q)4(q/p)4=-1.(2) D=2p1p2 ··· pm,where pi ≡ 1 mod 8,1≤i≤m are distinct primes and D=r2+s2 with r,s ≡±3 mod 8.