On the density of integral sets with missing differences from sets related to arithmetic progressions

For a given set M of positive integers, a problem of Motzkin asks for determining the maximal density μ(M) among sets of nonnegative integers in which no two elements differ by an element of M. The problem is completely settled when |M|?2, and some partial results are known for several families of M for |M|?3, including the case where the elements of M are in arithmetic progression. We consider some cases when M either contains an arithmetic progression or is contained in an arithmetic progression.     

Mean value estimates of the error terms of Lehmer problem

Let p be an odd prime and a be an integer coprime with p. Denote by N(a, p) the number of pairs of integers b, c with bc ≡ a(mod p), 1 ≤ b, c < p and with b, c having different parity. The main purpose of this paper is to study the mean square value problem of (N(a, p) − 1/2 (p−1)) over interval (N, N + M] with M, N positive integers by using the analytic methods, and finally by obtaining a sharp asymptotic formula.     

Class number one quadratic fields and solvability of some Pellian equations

Consider these two types of positive square-free integers d≠ 1 for which the class number h of the quadratic field Q(√d) is odd: (1) d is prime∈ 1(mod 8), or d=2q where q is prime ≡ 3 (mod 4), or d=qr where q and r are primes such that q≡ 3 (mod 8) and r≡ 7 (mod 8); (2) d is prime ≡ 1 (mod 8), or d=qr where q and r are primes such that q≡r≡ 3 or 7 (mod 8). For d of type (2) (resp. (1)), let Π be the set of all primes (resp. odd primes) p∈N satisfying (d/p) = 1. Also, let δ :=0 (resp. δ :=1) if d≡ 2,3 (mod 4) (resp. d≡ 1 (mod 4)). Then the following are equivalent: (a) h=1; (b) For every p∈П at least one of the two Pellian equations Z ²-dY ² = ±4^δ p is solvable in integers. (c) For every p∈П the Pellian equation W ²-dV ² = 4^δ p ² has a solution (w,v) in integers such that gcd (w,v) divides 2^δ.     

Bounds on Exponential Sums and the Polynomial Waring Problem Mod p

Estimates are given for the exponential a prime and f a nonzero integer polynomial, of interestin cases where the Weil bound is worse than trivial. The resultsextend those of Konyagin for monomials to a general polynomial.Such bounds readily yield estimates for the corresponding polynomialWaring problem mod p, namely the smallest such that f(x₁)+...+f(x)N(mod p) is solvable in integers for any N.     

Betti and Tachibana numbers

The Tachibana numbers t _r (M), the Killing numbers k _r (M), and the planarity numbers p _r (M) are considered as the dimensions of the vector spaces of, respectively, all, coclosed, and closed conformal Killing r-forms with 1 ≤ r ≤ n ? 1 “globally” defined on a compact Riemannian n-manifold (M,g), n >- 2. Their relationship with the Betti numbers b _r (M) is investigated. In particular, it is proved that if b _r (M) = 0, then the corresponding Tachibana number has the form t _r (M) = k _r (M) + p _r (M) for t _r (M) > k _r (M) > 0. In the special case where b ₁(M) = 0 and t ₁(M) > k ₁(M) > 0, the manifold (M,g) is conformally diffeomorphic to the Euclidean sphere.     

The Elimination of All but Two from a Tournament

In a tournament with N participants, the smallest number Q(N)of decisive matches which may be necessary for the identificationof the runner-up has been observed by Schreier (1932) and Steinhaus(1950) to be the number M(N) defined in Section 1. Here we determinethe smallest number P(N) of such matches which may be necessaryfor the identification of the top pair, champion and runner-up,without necessarily distinguishing the champion. In particularwe show that P(N) = M(N)–1 if and only if N is of theform 2ⁿ+1, and that the optimal strategy has some unexpectedfeatures.     

Divisibility of Certain Partition Functions by Powers of Primes

Let be the prime factorization of a positive integer k and let b _k (n) denote the number of partitions of a non-negative integer n into parts none of which are multiples of k. If M is a positive integer, let S _k (N; M) be the number of positive integers N for which b _k(n ) 0(mod M). If we prove that, for every positive integer j In other words for every positive integer j, b _k(n) is a multiple of for almost every non-negative integer n. In the special case when k=p is prime, then in representation-theoretic terms this means that the number ofp -modular irreducible representations of almost every symmetric groupS _n is a multiple of p ^j. We also examine the behavior of b _k(n) (mod ) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n (mod t) satisfies b _k(n) 0 (mod ), we show that there are infinitely many non-negative integers n r (mod t) for which b _k(n) 0 (mod ) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 .     

On the oscillation of a second-order delay equation

The oscillatory nature of two equations (r(t) y′(t))′ + p₁(t)y(t) = f(t), (r(t) y′(t))′ + p₂(t) y(t ? τ(t))= 0, is compared when positive functions p₁ and p₂ are not “too close” or “too far apart.” Then the main theorem states that if h(t) is eventually negative and a twice continuously differentiable function which satisfies (r(t) h′(t))′ + p₁(t) h(t) ? 0, then this inequality is necessary and sufficient for every bounded solution of (r(t) y′(t))′ + p₂(t) y(t ? τ(t)) = 0 to be nonoscillatory.     

The Analogue of Buchi's Problem for Rational Functions

Büchi's problem asked whether there exists an integer Msuch that the surface defined by a system of equations of theform has no integer pointsother than those that satisfy ±x_n = ± x₀ + n (the± signs are independent). If answered positively, itwould imply that there is no algorithm which decides, givenan arbitrary system Q = (q1,...,q_r) of integral quadratic formsand an arbitrary r-tuple B = (b₁,...,b_r) of integers, whetherQ represents B (see T. Pheidas and X. Vidaux, Fund. Math. 185(2005) 171–194). Thus it would imply the following strengtheningof the negative answer to Hilbert's tenth problem: the positive-existentialtheory of the rational integers in the language of additionand a predicate for the property ‘x is a square’would be undecidable. Despite some progress, including a conditionalpositive answer (depending on conjectures of Lang), Büchi'sproblem remains open. In this paper we prove the following: (A) an analogue of Büchi's problem in rings of polynomialsof characteristic either 0 or p 17 and for fields of rationalfunctions of characteristic 0; and (B) an analogue of Büchi's problem in fields of rationalfunctions of characteristic p 19, but only for sequences thatsatisfy a certain additional hypothesis. As a consequence we prove the following result in logic. Let F be a field of characteristic either 0 or at least 17 andlet t be a variable. Let L_t be the first order language whichcontains symbols for 0 and 1, a symbol for addition, a symbolfor the property ‘x is a square’ and symbols formultiplication by each element of the image of [t] in F[t].Let R be a subring of F(t), containing the natural image of[t] in F(t). Assume that one of the following is true: (i) R F[t]; (ii) the characteristic of F is either 0 or p 19. Then multiplication is positive-existentially definable overthe ring R, in the language L_t. Hence the positive-existentialtheory of R in L_t is decidable if and only if the positive-existentialring-theory of R in the language of rings, augmented by a constant-symbolfor t, is decidable.     

A note on the Lehmer problem over short intervals

Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N ≤ p − 1. Denote by r(N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 ∤ a + ā, where ā is an integer with 1 ≤ ā ≤ p − 1, aā ≡ c (mod p). It is well known that r(N, c; p) = 1/2N + O(p ^1/2log² p). The main purpose of this paper is to give an asymptotic formula for Σ _c=1 ^p−1(r(N, c; p) − 1/2N)².     

Toroidal and Annular Dehn Fillings

Suppose that M is a hyperbolic 3-manifold which admits two Dehnfillings M(r₁) and M(r₂) such that M(r₁) contains an essentialannulus, and M(r₂) contains an essential torus. It is knownthat = (r₁, r₂) 5. We will show that if = 5 then M is theWhitehead sister link exterior, and if = 4 then M is the exteriorof either the Whitehead link or the 2-bridge link associatedto the rational number . There are infinitely many examples with = 3. 1991 Mathematics SubjectClassification: 11D25, 11G05, 14G05.     

Existence of unique solutions for perturbed autonomous differential equations

We consider the existence of unique absolutely continuous solutionsfor x' = p(t)f(x) + p(t)h(t), t 0, x(0) = 0, where p, f, andh are positive almost everywhere, but none of them needs becontinuous or monotone. Moreover, p and f can be unbounded aroundzero. Our uniqueness results are not based on assumptions onthe differences f(x) – f(y), as it is usual in most uniquenessresults, and they are new even when p, f, and h are continuous.     

On maximal t-linearly independent sets

Consider a finite t + r ? 1 dimensional projective space PG(t + r ? 1, s) over a Galois field GF(s) of order s = ?^h, where ? and h are positive integers and ? is the prime characteristic of the field. A collection of k points in PG (t + r ? 1, s) constitutes an L(t, k)-set if no t of them are linearly dependent. An L(t, k)-set is maximal if there exists no other L(t, k′)-set with k′ > k. The largest k for which an L(t, k)-set exists is denoted by M_t(t + r, s). K. A. Bush [3] established that M_t(t, s) = t + 1 for t ? s. The purpose of this paper is to generalize this result and study M_t(t + r, s) for t, r, and s in certain relationships.     

Boundary-value problems for shallow elastic membrane caps

Consider the general nonlinear boundary-value problem (p(t)y' (t))' = p(t)q(t) f (t, y(t), y' (t)), t 1, g(y(1), y' (1))= 0, where the function f may be singular at the point y(1)= 0 and p(1) 0. We obtain conditions which guarantee existenceof positive and bounded solutions of the above problem. As anapplication we prove existence and uniqueness of rotationallysymmetric solutions to a nonlinear boundary-value problem, representingthe elastic deformation of a spherical cap.     

Optimal Regularity and Fredholm Properties of Abstract Parabolic Operators in Lp Spaces on the Real Line

We study the operator Lu(t):= u'(t) – A(t) u(t) on L^p(R; X) for sectorial operators A(t), with t R, on a Banachspace X that are asymptotically hyperbolic, satisfy the Acquistapace–Terreniconditions, and have the property of maximal L^p-regularity.We establish optimal regularity on the time interval R showingthat L is closed on its minimal domain. We further give conditionsfor ensuring that L is a semi-Fredholm operator. The Fredholmproperty is shown to persist under A(t)-bounded perturbations,provided they are compact or have small A(t)-bounds. We applyour results to parabolic systems and to generalized Ornstein–Uhlenbeckoperators. 2000 Mathematics Subject Classification 35K20, 35K90,47A53.     

Scattering Matrix for Manifolds with Conical Ends

Let M be a manifold with conical ends. (For precise definitionssee the next section; we only mention here that the cross-sectionK can have a nonempty boundary.) We study the scattering forthe Laplace operator on M. The first question that we are interestedin is the structure of the absolute scattering matrix S(s).If M is a compact perturbation of Rⁿ, then it is well-knownthat S(s) is a smooth perturbation of the antipodal map on asphere, that is, S(s)f(·)=f(–·) (mod C) On the other hand, if M is a manifold with a scattering metric(see [8] for the exact definition), it has been proved in [9]that S(s) is a Fourier integral operator on K, of order 0, associatedto the canonical diffeomorphism given by the geodesic flow atdistance . In our case it is possible to prove that S(s) isin fact equal to the wave operator at a time t = plus C terms.See Theorem 3.1 for the precise formulation. This result isnot too difficult and is obtained using only the separationof variables and the asymptotics of the Bessel functions. Our second result is deeper and concerns the scattering phasep(s) (the logarithm of the determinant of the (relative) scatteringmatrix).     

The distribution of values of the partition function in residue classes

Making use of an identity of Euler's involving the partition function p(n), Kolberg (Math. Scand.7 (1959), 377–378) showed that p(n) assumes both even and odd values infinitely often. His method admits of refinement, and as a consequence we are led to the following more comprehensive result: Let q ? 2, 0 ? r < q and denote by E_r,q(N) the number of positive integers n? N such that p(n)  r (mod q). The there exist at least two distinct values of r such that, for all sufficiently large N,$\text{E}{}_{\mathrm{r,q}}\text{(N) >}\text{log log}\text{N}\text{q}\text{log}\text{2}$.     

Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay

In this note we propose a method for the integration of y'(t) = f(t, y(t), y(rt)), 0 t t_f y(0) = y₀, where 0 < r < 1, by a superconvengent s-stage continuousRK method of discrete global order p and continuous uniformorder q < p – 1 for the approximation of the delayedterm y(rt). We prove that, although the maximum attainable orderof the method on an arbitrary mesh is q' = min{p, q + 1}, byusing a quasi-geometric mesh, introduced by Bellen et al. (1997,Appl. Numer. Math. 24, 1997, 279–293), the optimal accuracyorder p is preserved.     

Bifurcations of Periodic Points of Holomorphic Maps From C2 into C2

Let F:Cⁿ Cⁿ be a holomorphic map, F^k be the kth iterate ofF, and p Cⁿ be a periodic point of F of period k. That is,F^k(p) = p, but for any positive integer j with j < k, F^j(p) p. If p is hyperbolic, namely if DF^k(p) has no eigenvalue ofmodulus 1, then it is well known that the dynamical behaviourof F is stable near the periodic orbit = {p, F(p),..., F^k–1(p)}.But if is not hyperbolic, the dynamical behaviour of F near may be very complicated and unstable. In this case, a veryinteresting bifurcational phenomenon may occur even though may be the only periodic orbit in some neighbourhood of : forgiven M N\{1}, there may exist a C^r-arc {F_t: t [0,1]} (wherer N or r = ) in the space H(Cⁿ) of holomorphic maps from Cⁿinto Cⁿ, such that F₀ = F and, for t (0,1], F_t has an Mk-periodicorbit _t with as t 0. Theperiod thus increases by a factor M under a C^r-small perturbation!If such an F_t does exist, then , as well as p, is said to beM-tupling bifurcational. This definition is independent of r. For the above F, there may exist a C^r-arc in H(Cⁿ), with t [0,1], such that and, for t (0,1], has two distinct k-periodic orbits _t,1 and _t,2 with d(_t,i, ) 0 as t 0 for i = 1,2. If such an does exist, then , as well as p, is said to be 1-tupling bifurcational. In recent decades, there have been many papers and remarkableresults which deal with period doubling bifurcations of periodicorbits of parametrized maps. L. Block and D. Hart pointed outthat period M-tupling bifurcations cannot occur for M >2 in the 1-dimensional case. There are examples showing thatfor any M N, period M-tupling bifurcations can occur in higher-dimensionalcases. An M-tupling bifurcational periodic orbit as defined here actsas a critical orbit which leads to period M-tupling bifurcationsin some parametrized maps. The main result of this paper isthe following. Theorem. Let k N and M N, and let F: C² C² be a holomorphicmap with k-periodic point p. Then p is M-tupling bifurcationalif and only if DF^k(p) has a non-zero periodic point of periodM. 1991 Mathematics Subject Classification: 32H50, 58F14.     

Amplitude Bounds on Periodic and Aperiodic Oscillations: Second-order Systems

The global behaviour of the control systems described by thepair of differential equations x{dot} = –f(x)±g(y)+p(t), y{dot} = –f(x)±g(y)+p(t) has been investigated. Here f(x), g(y), h(x) and k(y) are polynomialsof odd degree with leading coefficients positive and p(t) andq(t) are bounded functions of time. Sufficient conditions havebeen found under which the trajectories of the above systemmay eventually be confined in a subset of (x, y, t)-space, thusgiving bounds on the amplitude of periodic as well as aperiodicoscillations. Further bounds on the amplitude of oscillationshave been investigated by finding regions in (x,y,t)-space fromwhich all trajectories eventually leave and into which no trajectoriesenter. Thus sufficient conditions have been derived for theexistence of an annulus in which oscillatory behaviour may beconfined.