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Register a new userOrdinary differential equations in Banach spaces and the spectral flow
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Authors
Daniele Garrisi
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We give a definition of the spectral flow for continuous paths in the space of bounded and essentially hyperbolic operators. We provide a homotopical characterization of the spectral flow in terms of a group homomorphism of the fundamental group of the projectors of the Calkin algebra with the infinite cyclic group Z. This characterization helps us to exhibit examples of infinite-dimensional Banach spaces where the spectral flow is not injective nor surjective. We prove that a path with spectral flow equal to an integer m exists if and only if there exists a projector P connected by an arc to a projector Q such that Range(Q) has co-dimension m in Range(P). We prove that if A is an asymptotically hyperbolic and essentially splitting path the differential operator F(u) = du/dt - Au is Fredholm. Moreover if A is also essentially hyperbolic the Fredholm index coincides with minus the spectral flow of A.
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We study some fundamental properties of semicocycles over semigroups of self-mappings of a domain in a Banach space. We prove that any semicocycle over a jointly continuous semigroup is itself jointly continuous. For semicocycles over semigroups which have generator, we establish a sufficient condition for differentiablity with respect to the time variable, and hence for the semicocycle to satisfy a linear evolution problem, giving rise to the notion of `generator of a semicocycle. Bounds on the growth of a semicocycle with respect to the time variable are given in terms of this generator. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator.
The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is presented for functions defined on $mathbb{R}^N$. In this article we present a more general and simplified construction of the Bochner integral on abstract measure spaces. An extension of the construction to functions with values in a locally convex space is also considered.
Assume that $mathcal{I}$ is an ideal on $mathbb{N}$, and $sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(mathcal{I}):=left{t in {0,1}^{mathbb{N}} colon sum_n t(n)x_n textrm{ is } mathcal{I}textrm{-convergent}right}$. In the category case, we assume that $mathcal{I}$ has the Baire property and $sum_n x_n$ is not unconditionally convergent, and we deduce that $A(mathcal{I})$ is meager. We also study the smallness of $A(mathcal{I})$ in the measure case when the Haar probability measure $lambda$ on ${0,1}^{mathbb{N}}$ is considered. If $mathcal{I}$ is analytic or coanalytic, and $sum_n x_n$ is $mathcal{I}$-divergent, then $lambda(A(mathcal{I}))=0$ which extends the theorem of Dindov{s}, v{S}alat and Toma. Generalizing one of their examples, we show that, for every ideal $mathcal{I}$ on $mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $lambda(A(Fin))=0$ and $lambda(A(mathcal{I}))=1$.
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.
We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $ell_1$ and that operators which do not fix copies of $ell_1$ on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.
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