Relation between two-phase quantum walks and the topological invariant

We study a position-dependent discrete-time quantum walk (QW) in one dimension, whose time-evolution operator is built up from two coin operators which are distinguished by phase factors from $x\geq0$ and $x\leq-1$. We call the QW the ${\it complete\;two}$-${\it phase\;QW}$ to discern from the two-phase QW with one defect[13,14]. Because of its localization properties, the two-phase QWs can be considered as an ideal mathematical model of topological insulators which are novel quantum states of matter characterized by topological invariants. Employing the complete two-phase QW, we present the stationary measure, and two kinds of limit theorems concerning ${\it localization}$ and the ${\it ballistic\;spreading}$, which are the characteristic behaviors in the long-time limit of discrete-time QWs in one dimension. As a consequence, we obtain the mathematical expression of the whole picture of the asymptotic behavior of the walker in the long-time limit. We also clarify relevant symmetries, which are essential for topological insulators, of the complete two-phase QW, and then derive the topological invariant. Having established both mathematical rigorous results and the topological invariant of the complete two-phase QW, we provide solid arguments to understand localization of QWs in term of topological invariant. Furthermore, by applying a concept of ${\it\;topological\;protections}$, we clarify that localization of the two-phase QW with one defect, studied in the previous work[13], can be related to localization of the complete two-phase QW under symmetry preserving perturbations.Comment: 50 pages, 13 figure

The stationary measure for diagonal quantum walk with one defect

This study is motivated by the previous work [14]. We treat 3 types of the one-dimensional quantum walks (QWs), whose time evolutions are described by diagonal unitary matrix, and diagonal unitary matrices with one defect. In this paper, we call the QW defined by diagonal unitary matrices, "the diagonal QW", and we consider the stationary distributions of generally 2-state diagonal QW with one defect, 3-state space-homogeneous diagonal QW, and 3-state diagonal QW with one defect. One of the purposes of our study is to characterize the QWs by the stationary measure, which may lead to answer the basic and natural question, "What the stationary measure is for one-dimensional QWs ?". In order to analyze the stationary distribution, we focus on the corresponding eigenvalue problems and the definition of the stationary measure.Comment: 10 page

Measurement of Agriculture-Related Air Pollutant Emissions Using Point and Remote Sensors

Measuring air pollution emissions from agricultural activities is usually difficult because of their large area and variability. Traditional air quality sensors, called point samplers, measure conditions in one location, which may not adequately measure a plume. Remote sensors, instruments that measure pollution along a line rather than at a single point, are better able to measure conditions around large areas. This dissertation reports on four agricultural air emissions studies that used both point and remote sensors for comparison. The methods used to calculate the emissions are based on previous work and are further developed in these studies. In particular, an atmospheric dispersion model was developed and tested that can account for a particle behaving different than the surrounding gas due to gravity and inertia and depositing out of the flow. Particulate matter (PM) emissions values are reported for two agricultural tillage conservation management practices (CMPs)and the corresponding traditional tillage methods in order to determine how well the CMP reduces emissions. In addition, gas-phase ammonia (NH3) emissions for a dairy operation and PM emissions from a feedlot operation are reported. These studies can help us better measure emissions from agricultural operations and understand how much air pollution is being emitted