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A combination of elementary methods and Dirichlet’s theorem on the infinitude of primes in arithmetic progressions is used to prove that a certain interesting set is dense in the unit square. The construction of the set involves how an element is related to its multiplicative inverse in the group (Z/pZ)∗.

We study a robust and efficient eigensolver for computing the positive dense spectrum of the two-dimensional transmission eigenvalue problem (TEP) which is derived from the Maxwell’s equation with complex media in pseudo-chiral model and the transverse magnetic mode. The discretized governing equations by the Nédélec edge element result in a large-scale quadratic eigenvalue problem (QEP). We es...

We propose an incomplete multifrontal LU-factorization (IMF) preconditioner that extends supernodal multifrontal methods to incomplete factorizations. It can be used as a preconditioner in a Krylov-subspace method to solve large-scale sparse linear systems with an element structure; e.g., those arising from a finite element discretization of a partial differential equation. The fact that the el...

Non-overlapping domain decomposition techniques are used both to solve the finite element equations and to couple them with a boundary element method. A suitable approach dealing with finite element nodes common to more than two subdomains, the so-called cross-points, endows the method with the following advantages. It yields a robust and efficient procedure to solve the equations resulting fro...

In this thesis, I explore an approach called “active libraries”. These are libraries that take part in their own optimisation, enabling both high-performance code and the presentation of intuitive abstractions. I investigate the use of active libraries in two domains. Firstly, dense and sparse linear algebra, particularly, the solution of linear systems of equations. Secondly, the specification...

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind-MacNeille completion of P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of clos...

This paper presents the first volumetric images obtained using a 2D CMUT array with throughwafer via interconnects. An 8×16-element portion of a 32×64-element array flip-chip bonded onto a glass fanout chip was used in the experiments. This study experimentally demonstrates that 2D CMUT arrays can be fabricated with high yield using silicon micromachining processes, individual electrical connec...