For a Lagrange distribution of order zero we consider a quadratic integral which has logarithmic divergence at the singular locus of the distribution. The residue of the asymptotics is a Hermitian form evaluated in the space of positive distributions supported in the locus. An asymptotic analysis of the residue density is given in terms of the curvature form of the locus. We state a conservation law for the residue of the impulse-energy tensor of solutions of the wave equation which extends the...

An elliptic system in ${\mathbb{R}}^{n}$, which is invariant under the action of the group ${\mathbb{Z}}^{n}$ is considered. We construct a holomorphic family of finite-dimensional subrepresentations of the group in the space of solutions (Floquet solutions), such that any solution of the growth $O\left(\mathrm{exp}\right(a\left|x\right|\left)\right)$ at infinity can be rewritten in the form of an integral over the family.

In this paper we prove that the projective dimension of ${\mathcal{M}}_{n}={R}^{4}/\u27e8{A}_{n}\u27e9$ is $2n-1$, where $R$ is the ring of polynomials in $4n$ variables with complex coefficients, and $\u27e8{A}_{n}\u27e9$ is the module generated by the columns of a $4\times 4n$ matrix which arises as the Fourier transform of the matrix of differential operators associated with the regularity condition for a function of $n$ quaternionic variables. As a corollary we show that the sheaf $\mathcal{R}$ of regular functions has flabby dimension $2n-1$, and we prove a cohomology vanishing theorem for open...

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