et M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that a(M)≤k, where a(M) is the algebraic dimension a(M) (i.e. the transcendence degree of the field of meromorphic functions) and k is the dimension of the space of holomorphic differentials. We prove a similar result about meromorphic maps to Kahler manifolds.
We consider for $\eps\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force together with the averaged equation. Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\eps(t,\tau)$ acting on the natural weak energy space $\H$ are shown to possess uniform attractors $\A^\eps$. Within the further assumption, the family $\A^\eps$ turns out to be bounded in $\H$, uniformly with respect to $\eps\in[0,1]$. The convergence of the attractors $\A^\eps$ to the attractor $\A^0$ of the averaged equation as $\eps\to 0$ is also established.
We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any non-linear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters as we show by giving a counterexample. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associated parameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we also prove categorically. We illustrate our main results by a series of examples, using, in particular, a relation between the Gauss–Manin connection and parameterized differential Galois groups.
We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets--Ingham 1/4 theorem for Paley--Wiener spaces. Contrarily to the situation in Paley--Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density stricly different from the critical one then either it can be completed or reduced to a complete interpolating sequence. In particular, this allows to give necessary and sufficient conditions for interpolation or sampling in terms of densities.
The double ramification hierarchy is a new integrable hierarchy of hamiltonian PDEs introduced recently by the first author. It is associated to an arbitrary given cohomological field theory. In this paper we study the double ramification hierarchy associated to the cohomological field theory formed by Witten's r-spin classes. Using the formula for the product of the top Chern class of the Hodge bundle with Witten's class, found by the second author, we present an effective method for a computation of the double ramification hierarchy. We do explicit computations for r=3,4,5 and prove that the double ramification hierarchy is Miura equivalent to the corresponding Dubrovin--Zhang hierarchy. As an application, this result together with a recent work of the first author with Paolo Rossi gives a quantization of the r-th Gelfand--Dickey hierarchy for r=3,4,5.