Volume 23, Issue 6

This article establishes the almost global existence of solutions for three-dimensional nonlinear wave equations with quadratic, divergence-form nonlinearities and time-independent inhomogeneous terms. The approach used here can be applied to the system of homogeneous, isotropic hyperelasticity with time-independent external force. The development for the scalar and vector cases will be presented in parallel. We first prove the existence and uniqueness of the stationary solutions. Then it suffices to prove the almost global existence of the original solutions minus the stationary solutions, which is carried out in line with Klainerman and Sideris [Comm. Pure Appl. Math. 49: 307–321, 1996], by using the classical invariance of the equations under translations, rotations and changes of scale.

We give an alternative computation of the twisted second moment of critical values of class group L -functions attached to an imaginary quadratic field. The method avoids long calculations and yields the polynomial growth in the s -parameter for the remaining term.

We show that complementary series representations of SO( n , 1), which are sufficiently close to a cohomological representation contain discretely , complementary series of SO( m , 1) also sufficiently close to cohomological representations, provided that the degree of the cohomological representation does not exceed m /2. We prove, as a consequence, that the cohomological representation of degree i of the group SO( n , 1) contains discretely, the cohomological representation of degree i of the subgroup SO( m , 1) if i ≤ m /2. As a global application, we show that if G /ℚ is a semisimple algebraic group such that G (ℝ) = SO( n , 1) up to compact factors, and if we assume that for all n , the tempered cohomological representations are not limits of complementary series in the automorphic dual of SO( n , 1), then for all n , non-tempered cohomological representations are isolated in the automorphic dual of G . This reduces conjectures of Bergeron to the case of tempered cohomological representations.

A positive definite even Hermitian lattice is called even universal if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields for all positive square-free integers m and we list optimal criterions on even universality of Hermitian lattices over which admits even universal binary Hermitian lattices.

We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension.

Weighted energy estimates including the Keel, Smith and Sogge estimate is obtained for solutions of exterior problem of the wave equation in three or higher dimensional Euclidean spaces. For the solutions of the Cauchy problem, which is corresponding to the free system in scattering theory, the estimates are given by using the ideas introduced by Morawetz and summarized by Mochizuki for the Dirichlet problem in the outside of star shaped obstacles. From the estimates for the free system, the corresponding estimates for exterior domains are given if it is assumed that the local energy decays uniformly with respect to initial data, which depends on the structures of propagation of singularities.

We prove that weak invariance is equivalent to strong invariance in the framework of quasi-regular local positivity preserving forms with lower bounds. As an application, we give an ergodic decomposition of Markov processes associated with quasi-regular local semi-Dirichlet forms with lower bounds. As consequences, criteria of transience and recurrence for Markov processes associated with quasi-regular semi-Dirichlet forms with lower bounds are presented.

We consider a weak solution to the non-linear, parabolic systems of the form u t – div A ( x, t, u, Du ) = 0, where the vector field A satisfies a Dini-type continuity condition with respect to the variables ( x, t, u ), and we prove a partial regularity result for such a solution. Moreover, we give an estimate of the size of the singular set of a solution in terms of a generalized parabolic Hausdorff measure associated to an appropriate modulus of continuity naturally associated to the coefficients of the system.

We produce a new proof of the reciprocity law for the twisted second moment of Dirichlet L -functions that was recently proved by Conrey. Our method is to analyze certain two-variable sums where the variables satisfy a linear congruence. We show that these sums satisfy an elegant reciprocity formula. In the case that the modulus is prime, these sums are closely related to the twisted second moment, and the reciprocity formula for these sums implies Conrey's reciprocity formula. We also extend the range of uniformity of Conrey's formula.