Is the Helmholtz equation really sign-indefinite?

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions, or the exterior of a star-shaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise just by multiplying the Helmholtz equation by particular test functions and integrating by parts.

Production of definite and indefinite articles in typically developing English-speaking children and children with SLI

The present paper examines the production of definite and indefinite articles in English-speaking typically developing (TD) children and children with Specific Language Impairment (SLI). Twenty four English-speaking children with SLI (mean age: 7;5), twenty nine TD age-matched (TD-AM) children (mean age: 7;5) and eleven younger (mean age: 5;5) TD vocabulary-matched (TD-VM) children participated in a production task involving short stories without picture props based on Schafer and de Villiers (2000). Article production was examined in two different semantic contexts for the definite article, namely in the anaphoric and the bridging context. In the anaphoric condition, definiteness is established via linguistic means, whereas in the bridging condition via shared world knowledge. Indefinite article production was examined in the referential specific, non-referential predicational, and non-referential instrumental contexts. The referential specific context involves [+speaker, −hearer] knowledge and the non-referential predicational and instrumental [−speaker, −hearer] knowledge. Results showed that in the definite article contexts, all three groups performed better on the bridging compared with the anaphoric condition; in the indefinite article contexts, they had better performance on the non-referential predicational vs. the referential specific and the non-referential instrumental conditions. In terms of errors, the TD-VM children and the children with SLI produced significantly more substitutions than the TD-AM children in the definite article contexts. In the indefinite article contexts, the three groups did not differ in terms of accuracy or error patterns. The present results point towards problems in the discourse integration of entities that are part of the speaker's and hearer's knowledge in children with SLI and TD-VM controls, especially in definite articles. These problems are accentuated in the children with SLI due to their grammatical impairment and suggest that children with SLI exhibit a delayed acquisition profile.

Accumulation of complex eigenvalues of an indefinite Sturm--Liouville operator with a shifted Coulomb potential

For a particular family of long-range potentials V, we prove that the eigenvalues of the indefinite Sturm–Liouville operator A = sign(x)(−Δ+V(x)) accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators.