Infinite-dimensional linear algebra and solvability of partial differential equations

We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective.

Hierarchical and Multi-resolution Neuron Network Computing for Complex Inverse Problems-An Approach Based on Brain-inspired Computing and Learning Method

Background: Complex inverse problems such as Radar Imaging and CT/EIT imaging are well investigated in mathematical algorithms with various regularization methods. However it is difficult to obtain stable inverse solutions with fast convergence and high accuracy at the same time due to the ill-posed property and non-linear property. Objective: In this paper, we propose a hierarchical and multi-resolution scalable method from both algorithm perspective and hardware perspective to achieve fast and accurate solu-tions for inverse problems by taking radar and EIT imaging as examples. Method: We present an extension of discussion on neuromorphic computing as brain-inspired computing method and the learning/training algorithm to design a series of problem specific AI “brains” (with different memristive values) to solve a general complex ill-posed inverse problems that are traditionally solved by mathematical regular operators. We design a hierarchical and multi-resolution scalable method and an algorithm framework to train AI deep learning neuron network and map into the memristive circuit so that the memristive val-ues are optimally obtained. We propose FPGA as an emulation implementation for neuro-morphic circuit as well. Result: We compared the methodology between our approach and traditional regulariza-tion method. In particular we use Electrical Impedance Tomography (EIT) and Radar imaging as typical examples to compare how to design an AI deep learning neuron network architec-tures to solve inverse problems. Conclusion: With EIT imaging as a typical example, we show that any moderate complex inverse problem, as long as it can be described as combinational problem, AI deep learning neuron network is a practical alternative approach to try to solve the inverse problems with any given expected resolution accuracy, as long as the neuron network width is large enough and computational power is strong enough for all combination samples training purpose.

A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems

Central in entanglement theory is the characterization of local transformations among pure multipartite states. As a first step towards such a characterization, one needs to identify those states which can be transformed into each other via local operations with a non-vanishing probability. The classes obtained in this way are called SLOCC classes. They can be categorized into three disjoint types: the null-cone, the polystable states and strictly semistable states. Whereas the former two are well characterized, not much is known about strictly semistable states. We derive a criterion for the existence of the latter. In particular, we show that there exists a strictly semistable state if and only if there exist two polystable states whose orbits have different dimensions. We illustrate the usefulness of this criterion by applying it to tripartite states where one of the systems is a qubit. Moreover, we scrutinize all SLOCC classes of these systems and derive a complete characterization of the corresponding orbit types. We present representatives of strictly semistable classes and show to which polystable state they converge via local regular operators.

The Kasparov product on submersions of open manifolds

We study the Kasparov product on (possibly non-compact and incomplete) Riemannian manifolds. Specifically, we show on a submersion of Riemannian manifolds that the tensor sum of a regular vertically elliptic operator on the total space and an elliptic operator on the base space represents the Kasparov product of the corresponding classes in [Formula: see text]-theory. This construction works in general for symmetric operators (i.e. without assuming self-adjointness), and extends known results for submersions with compact fibers. The assumption of regularity for the vertically elliptic operator is not always satisfied, but depends on the topology and geometry of the submersion, and we give explicit examples of non-regular operators. We apply our main result to obtain a factorization in unbounded [Formula: see text]-theory of the fundamental class of a Riemannian submersion, as a Kasparov product of the shriek map of the submersion and the fundamental class of the base manifold.