Almost invariant half-spaces for operators on Hilbert space. II: operator matrices

This paper is a sequel to [6]. In that paper we transferred the discussions in [1] and [13] concerning almost invariant half-spaces for operators on complex Banach spaces to the context of operators on Hilbert space, and we gave easier proofs of the main results in [1] and [13]. In the present paper we discuss consequences of the above-mentioned results for the matricial structure of operators on Hilbert space

Unbounded quasinormal operators revisited

Various characterizations of unbounded closed densely defined operators commuting with the spectral measures of their moduli are established.In particular, Kaufman's definition of an unbounded quasinormal operator is shown to coincide with that given by the third-named author and Szafraniec. Examples demonstrating the sharpness of results are constructed.Comment: 13 page

Finding branch-decompositions of matroids, hypergraphs, and more

Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $\mathcal F$, we wish to find a "branch-decomposition" of these subspaces of width at most $k$, that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated with leaves in one component of $T-e$ and the sum of subspaces associated with leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of $\mathcal F$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over $\mathcal F$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of $\mathcal F$-represented matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time $O(n^3)$ where $n$ is the number of elements of the input $\mathcal F$-represented matroid. But their method is highly indirect. Their algorithm uses the non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic second-order formulas on $\mathcal F$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.Comment: 73 pages, 10 figure

Computational Analysis and Design Optimization of Convective PCR Devices

Polymerase Chain Reaction (PCR) is a relatively novel technique to amplify a few copies of DNA to a detectable level. PCR has already become common in biomedical research, criminal forensics, molecular archaeology, and so on. Many have attempted to develop PCR devices in numerous types for the purpose of the lab-on-chip (LOC) or point-of-care (POC). To use PCR devices for POC lab testing, the price must be lower, and the performance should be comparable to the lab devices. For current practices with the existing methods, the price is pushed up higher partially due to too much dependence on numerous developmental experiments. Our proposition herein is that the computational methods can make it possible to design the device at lower cost and less time, and even improved performance. In the present dissertation, a convective PCR, that is the required flow circulation is driven by the buoyancy forces, is researched towards the use in POC testing. Computational Fluid Dynamics (CFD) is employed to solve the nonlinear equations for the conjugate momentum and heat transfer model and the species transport model. The first application of the models considers four reactors in contact with two separate heaters, but with different heights. Computational analyses are carried out to study the nature of buoyancy-driven flow for DNA amplification and the effect of the capillary heights on the performance. The reactor performance is quantified by the doubling time of DNA and the results are experimentally verified. The second application includes a novel design wherein a reactor is heated up by a single heater. A process is established for low-developmental cost and high-performance design. The best is searched for and found by evaluating the performance for all possible candidates. The third application focuses on the analysis of the performance of single-heater reactors affected by positions of a capillary tube: (1) horizontal, and (2) vertical. In the last application, numerous double-heater reactor designs are considered to find the one that assure the optimal performance. Artificial Neural Network (ANN) is employed to approximate the CFD results for optimization. In summary, through the four segments of our studies, the results show significant possibilities of increasing the performance and reducing the developmental cost and time. It is also demonstrated that the proposed methodology is advantageous for the development of cPCR reactors for the purpose of POC applications