Author index for volume 39

The distribution γ(c, n) of de Bruijn sequences of order n and linear complexity c is investigated. Some new results are proved on the distribution of de Bruijn sequences of low complexity, i.e., their complexity is between 2^n?1 + n and 2^n?1 + 2^n?2. It is proved that for n ? 5 and 2^n?1 + n?c<2^n?1 + 2^n?2, γ(c, n) ≡ 0 (mod 4). It is shown that for n ? 11, γ(2^n?1 + n, n) > 0. It is also proved that γ(2^n?1 + 2^n?2, n) ? 4γ(2^n?2 ? 1, n ? 2) and we give a recursive method to generate de Bruijn sequences of complexity 2^n?1 + 2^n?2.     

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.     

Characterization results on small blocking sets of the polar spaces Q +(2n + 1, 2) and Q +(2n + 1, 3)

In De Beule and Storme, Des Codes Cryptogr 39(3):323–333, De Beule and Storme characterized the smallest blocking sets of the hyperbolic quadrics Q ⁺(2n + 1, 3), n ≥ 4; they proved that these blocking sets are truncated cones over the unique ovoid of Q ⁺(7, 3). We continue this research by classifying all the minimal blocking sets of the hyperbolic quadrics Q ⁺(2n + 1, 3), n ≥ 3, of size at most 3 ⁿ + 3 ^n–2. This means that the three smallest minimal blocking sets of Q ⁺(2n + 1, 3), n ≥ 3, are now classified. We present similar results for q = 2 by classifying the minimal blocking sets of Q ⁺(2n + 1, 2), n ≥ 3, of size at most 2 ⁿ + 2 ^n-2. This means that the two smallest minimal blocking sets of Q ⁺(2n + 1, 2), n ≥ 3, are classified.     

On simultaneous representations of primes by binary quadratic forms

Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M² + 7N², and knowing M and N makes it possible to “predict” whether p = A² + 14B² is solvable or p = 7C² + 2D² is solvable. More generally, let q and r be distinct primes, and let an integral solution of H²p = M² + qN² be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K²q = A² + qrB² is solvable and the possible values of K′ for which K′²p = qC² + rD² is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant ?4qr from representing p.     

On a Żołądek theorem

We prove the existence of cubic systems of the form $$ \begin{gathered} \dot x = y[1 - 2r(5 + 3r^2 )x + \gamma \lambda ^2 x^2 ] + a_0 x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 , \hfill \\ \dot y = - x(1 - 8rx)(1 - 3r\gamma x) - 2x[2(1 - 3r^2 ) - r\gamma (7 - 15r^2 )x]y \hfill \\ - [r(11 + r^2 ) + \gamma (1 - 22r^2 - 3r^4 )x]y^2 \hfill \\ - 2r\gamma \delta y^3 + a_0 y + a_7 x^2 + a_8 xy + a_9 y^2 + a_{10} x^3 + a_{11} x^2 y, \hfill \\ \end{gathered} $$ where α = 3r ² + 17, γ = r ² + 3, δ = 1 ? r ², and λ = 3r ² + 1, that have at least eleven limit cycles in a neighborhood of the point O(0, 0).     

An ?-dependent formulation of the Kadomtsev-Petviashvili hierarchy

We briefly review a recursive construction of ?-dependent solutions of the Kadomtsev-Petviashvili hierarchy. We give recurrence relations for the coefficients X_n of an ?-expansion of the operator X = X ₀ + ?X ₁ + ? ² X ₂ + ... for which the dressing operator W is expressed in the exponential form W = e^X/?. The wave function ?? associated with W turns out to have the WKB (Wentzel-Kramers-Brillouin) form ?? = e^S/kh, and the coefficients S_n of the ?-expansion S = S ₀ + ?S ₁ + ? ² S ₂ + ... are also determined by a set of recurrence relations. We use this WKB form to show that the associated tau function has an ?-expansion of the form log ?? = ?^?2 F ₀ + ?^?1 F ₁ + F ₂ + ....     

Computing the number of ways of representing primes by a norm form

Formulae for the number of different integral solutions ofa ²+b²+c²+d²+ac+bd=p are given wherep is a prime and the solution satisfies certain natural congruence conditions. Similar formulae are given for the case of the quadratic forma ²+b²+2c²+2d²+ac+bd.     

Optimal Preemptive Online Algorithms for Scheduling with Known Largest Size on Two Uniform Machines

In this paper, we consider the seml-online preemptive scheduling problem with known largest job sizes on two uniform machines. Our goal is to maximize the continuous period of time （starting from time zero） when both machines are busy, which is equivalent to maximizing the minimum machine completion time if idle time is not introduced. We design optimal deterministic semi-online algorithms for every machine speed ratio s ∈ [1, ∞）, and show that idle time is required to achieve the optimality during the assignment procedure of the algorithm for any s 〉 （s^2 ＋ 3s ＋ 1）/（s^2 ＋ 2s ＋ 1）. The competitive ratio of the algorithms is （s^2 ＋ 3s ＋ 1）/（s^2 ＋ 2s ＋ 1）, which matches the randomized lower bound for every s ≥ 1. Hence randomization does not help for the discussed preemptive scheduling problem.     

Solution of a problem of Skolem

T. Skolem shows that there are at most six integer solutions to the Diophantine equation x⁵ + 2y⁵ + 4z⁵ ? 10xy³z + 10x²yz² = 1. The author shows here that there are precisely three integer solutions.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

Generalized balancing numbers

The positive integer x is a (k, l) -balancing number for y(x ≤ y — 2) if 1^k + 2^k + … + (x — 1)^k = (x + 1)^l + … + (y — 1)^l for fixed positive integers k and l. In this paper, we prove some effective and ineffective finiteness statements for the balancing numbers, using certain Baker-type Diophantine results and Bilu—Tichy theorem, respectively.     

The exponential Diophantine equation x2 + (3n 2 + 1) y = (4n 2 + 1) z

Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x ² + (3n ² + 1) ^y = (4n ² + 1) ^z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n ³ + 3n, 1, 3).     

Pure powers in recurrence sequences and some related diophantine equations

We prove that there are only finitely many terms of a non-degenerate linear recurrence sequence which are qth powers of an integer subject to certain simple conditions on the roots of the associated characteristic polynomial of the recurrence sequence. Further we show by similar arguments that the Diophantine equation ax^2t + bx^ty + cy² + dx^t + ey + f = 0 has only finitely many solutions in integers x, y, and t subject to the appropriate restrictions, and we also treat some related simultaneous Diophantine equations.     

New proof of a theorem of Szekeres

A Hadamard matrix H of order 16t² is constructed for all t for which there is a Hadamard matrix of order 4t, in such a way that each row of H contains exactly 8t² + 2t ones. As a consequence a new method of constructing the symmetric block designs with parameters (16t², 8t² + 2t, 4t² + 2t) for all t for which there is a Hadamard matrix of order 4t is given.     

Geometric, topological and differentiable rigidity of submanifolds in space forms

Let M be an n-dimensional submanifold in the simply connected space form F ^n+p (c) with c + H ² > 0, where H is the mean curvature of M. We verify that if M ⁿ (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric _M ≥ (n ? 2)(c + H ²), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or ${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$ C P 2 4 3 ( c + H 2 ) in S 7 1 c + H 2 . In particular, if Ric _M > (n ? 2)(c + H ²), then M is a totally umbilic sphere. We then prove that if M ⁿ (n ≥ 4) is a compact submanifold in F ^n+p (c) with c ≥ 0, and if Ric _M > (n ? 2)(c + H ²), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.     

Approximate solutions of a nonlinear differential equation

Approximate solutions of a nonlinear differential equation ẍ + α^mx² = β^mx^{m + 2} (m⩾1) are approximate solution is exact for a particular initial value.     

Chromatic numbers of the strong product of odd cycles

The problem of determining the chromatic numbers of the strong product of cycles is considered. A construction is given proving χ (G) = 2^p +1 for a product of p odd cycles of lengths at least 2^p +1. Several consequences are discussed. In particular, it is proved that the strong product of p factors has chromatic number at most 2^p +1 provided that each factor admits a homomorphism to sufficiently long odd cycle C_mi, m_i ≥ 2^p +1.     

There are no de bruijn sequences of span n with complexity 2n − 1 + n + 1

If s = (s₀, s₁,…, s_2ⁿ?1) is a binary de Bruijn sequence of span n, then it has been shown that the least length of a linear recursion that generates s, called the complexity of s and denoted by c(s), is bounded for n ? 3 by 2^{n ? 1} + n ? c(s) ? 2ⁿ ?1. A numerical study of the allowable values of c(s) for 3 ? n ? 6 found that all values in this range occurred except for 2^n?1 + n + 1. It is proven in this note that there are no de Bruijn sequences of complexity 2^n?1 + n + 1 for all n ? 3.     

Extending symmetric designs

If a symmetric 2-design with parameters (v, k, λ) is extendable, then one of the following holds: v = 4λ + 3, k = 2λ + 1; or v = (λ + 2)(λ² + 4λ + 2), k = λ² + 3λ + 1; or v = 111, k = 11, λ = 1; or v = 495, k = 39, λ = 3. In particular, there are at most three sets of extendable symmetric design parameters with any given value of λ. As a consequence, the only twice-extendable symmetric design is the 21-point projective plane.     

Zeros of the functions f(n) = Σi=0(−1)i(in−2i)

The main result of this paper is the following: the only zeros of the title function are at n = 3 and n = 12. This is achieved by means of the recursion function for f(n), viz. F(x) = x³ ? x ? 1 which has only one real root w. This turns out to be the fundamental unit of Q(w). From the norm equation of the units, N(w) = x³ + y³ + z³ ? 3xyz + 2x²z + xz² ? xy² ? yz² = 1, and the negative powers of w which are of binary form, the result follows. The paper concludes with two remarkable combinatorial identities.