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The purpose of this paper is to introduce and investigate some basic properties of mixed homogeneous Herz-Hardy spaces $H\dot{K}_{\vec{p}}^{\alpha, q}(\mathbb{R}^n)$ and mixed non-homogeneous Herz-Hardy spaces $HK_{\vec{p}}^{\alpha,…
We introduce a notion of equivariant coarse cohomology of the complement of a subspace in a metric space. We use this cohomology to define a notion of coarse cohomology of the configuration space of a metric space and develop tools to…
We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a positive selfadjoint operator $H$, that is called the Hamiltonian of the…
We present an algebraic method for constructing a highly effective coarse grid correction to accelerate domain decomposition. The coarse problem is constructed from the original matrix and a small set of input vectors that span a low-degree…
We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial…
We present a framework for computing the solution to Hamiltonian eigenproblems in a subspace defined by bit-strings sampled from a quantum computer. Hamiltonians are represented using an extended alphabet that includes projection and ladder…
In several geophysical applications, such as full waveform inversion and data modelling, we are facing the solution of inhomogeneous Helmholtz equation. The difficulties of solving the Helmholtz equa- tion are two fold. Firstly, in the case…
Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable…
We explore complex Riemannian geometry and Hermitian metrics on complex algebraic varieties and analytic spaces, respectively. In particular, we introduce Hermitian metrics on holomorphic Lie algebroids and examine the associated…
We introduce Hopf algebroid covariance on Woronowicz's differential calculus. Using it, we develop quite a general framework of noncommutative complex geometry that subsumes the one in [2]. We present transverse complex and K\"ahler…
This paper is concerned with two extremal problems from matrix analysis. One is about approximating the top eigenspaces of a Hermitian matrix and the other one about approximating the orthonormal polar factor of a general matrix. Tight…
In this paper, we propose a domain decomposition method for multiscale second order elliptic partial differential equations with highly varying coefficients. The method is based on a discontinuous Galerkin formulation. We present both a…
The theory uses methods and language of linear algebra to study nonlinear spaces. These techniques can be used particularly to describe analytic geometry of non-linear elliptic, hyperbolic, De Sitter and Anti de Sitter spaces. The main…
For a family of elliptic operators with periodically oscillating coefficients, $-\text{div}( A(\cdot/\varepsilon) \nabla) $ with tiny $\varepsilon>0$, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions…
In this paper we design, analyse and test domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations, namely for $\mathbf{H}(\mathbf{curl})$…
In this work we introduce and analyze a new multiscale method for strongly nonlinear monotone equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local…
Continuous mappings between compact Hausdorff spaces can be studied using homomorphisms between algebraic structures (lattices, Boolean algebras) associated with the spaces. This gives us more tools with which to tackle problems about these…
We present an auxiliary space theory that provides a unified framework for analyzing various iterative methods for solving linear systems that may be semidefinite. By interpreting a given iterative method for the original system as an…
Motivated by applications to probability and mathematical finance, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable…
The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree…