Volume 2008, Issue 616

The classic example of parametric resonance for ordinary differential equations is a vibrating spring with time varying spring constant This example is studied using Floquet theory. A classic example is κ ( t ) = 1 + ε cos ωt , where one finds regions of instability in the ε , ω plane for which there are solutions which grow exponentially in time. The instability regions are open and with closures touching ε = 0 at critical frequencies (see [ V. Arnold , Ordinary Differential Equations, MIT Press, 1973.], [ W. Magnus and S. Winkler , Hill's equation, Intersc. Tracts Pure Appl. Math. 20, Interscience Publishers John Wiley and Sons, New York-London-Sydney 1966.]). For any t , the constant coefficient problem with κ frozen at κ ( t ) is conservative and has no such growth.

In this paper we derive new simple estimates for ordinary differential equations with Sobolev coefficients. These estimates not only allow to recover some old and recent results in a simple direct way, but they also have some new interesting corollaries.

In a paper recently published in Crelle's Journal , Deschamps and Grekos [ B. Deschamps et G. Grekos , Estimation du nombre d'exceptions à ce qu'un ensemble de base privé d'un point reste un ensemble de base, J. reine angew. Math. 539 (2001), 45–53.] study asymptotically (when h tends to infinity) the quantity E ( h ), introduced by Erdős and Graham, and defined as the maximal number of elements which are necessary to the basicity of an additive basis of order h . They show that the maximal order of this function is ( h /log h ) ½ . The aim of this article is to show that the E function does not have oscillations but, to the contrary, does possess a regular asymptotic behaviour, that we determine explicitly. More precisely, we prove that E ( h ) ~ 2( h /log h ) ½ .

In 1956, Alder conjectured that the number of partitions of n into parts differing by at least d is greater than or equal to that of partitions of n into parts ≡ ±1 (mod d + 3). The Euler identity, the first Rogers-Ramanujan identity, and a theorem of Schur show that the conjecture is true for d = 1, 2, 3, respectively. In 1971, Andrews proved that the conjecture holds for d = 2 r – 1, r ≧ 4. In this paper, we prove the conjecture for all d ≧ 32 and d = 7.

We derive identities for general flows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the first author for mean curvature flow and other nonlinear diffusions. Our results apply in particular to Ricci flow, where they yield a local monotone quantity directly analogous to Perelman's reduced volume V and a local identity related to Perelman's average energy F .

We describe Serre functors for (generalisations of) the category associated with a semisimple complex Lie algebra. In our approach, projective-injective modules, that is modules which are both, projective and injective, play an important role. They control the Serre functor in the case of a quasi-hereditary algebra having a double centraliser with respect to a projective-injective module whose endomorphism ring is a symmetric algebra. As an application of the double centraliser property together with our description of Serre functors, we prove three conjectures of Khovanov about the projective-injective modules in the parabolic category .

In this paper we are concerned with the monodromy of Picard-Fuchs differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy relative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of Sp(4, ℤ) of finite index and whose levels are related to the geometric invariants of the Calabi-Yau threefolds.

We show that the viscous Burgers equation u t + uu x = u xx considered for complex valued functions u develops finite-time singularities from compactly supported smooth data. By means of the Cole-Hopf transformation, the singularities of u are related to zeros of complex-valued solutions v of the heat equation v t = v xx . We prove that such zeros are isolated if they are not present in the initial data.

Following the work of Harris and Kudla, we prove a general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain L -function. As a consequence, we deduce a theorem relating the existence of GL 2 ( K ) -invariant linear forms on irreducible, admissible representations of GL 2 () for a commutative semi-simple cubic algebra over a non-archimedean local field k in terms of local epsilon factors which was proved only in some cases by the first author in his earlier work in [ D. Prasad , Invariant forms for representations of GL 2 over a local field, Amer. J. Math. 114 (1992), no. 6, 1317–1363.]. This has been achieved by globalising a locally distinguished supercuspidal representation to a globally distinguished representation, a result of independent interest which is proved by an application of the relative trace formula.