Assessing Conceptual Knowledge of Differential Equations

The differential equations and linear algebra math classes at Valparaiso University participate in an online tutoring survey consisting of conceptual questions from the field. The test was originally constructed by a team of math professors from San Diego State University. The questions are available in an online format and most questions provide scaffolding, or a tutoring set of questions when a primary question is answered incorrectly. This project analyzes the effectiveness of the scaffolding on a subset of questions with specific focus on areas of separable variables and Euler’s method. Special attention has been given to questions with multiple knowledge components, which may complicate the effectiveness of the scaffolding. In several questions, we have found that the scaffolding is not impacting student understanding of the subject area. We are also discovering surprising anomalies in students’ conceptions that the test creators did not predict. Thus, in some cases, it may be necessary to adapt the scaffolding or question wording to maximize test efficiency and overall student comprehension

Properly stratified algebras and tilting

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras, whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties, for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.Comment: A revised version of the preprint 2003:31, Department of Mathematics, Uppsala Universit

Autocrat of the Armchair

Structural analysis is a standard tool to identify submodels that can be used to design model based diagnostic tests. Structural approaches typically operate on models described by a set of equations. This work extends such methods to be able to handle models with constraints, e.g. inequality constraints on state variables. The objective is to improve isolability properties of a diagnosis system by extending the class of redundancy relations. An algorithm is developed that identifies which are the constraints and equations that can be used together to derive a new test that can not be found using previous approachesCADIC

Designing frequency-dependent relaxation rates and Lamb shift for a giant artificial atom

In traditional quantum optics, where the interaction between atoms and light at optical frequencies is studied, the atoms can be approximated as point-like when compared to the wavelength of light. So far, this relation has also been true for artificial atoms made out of superconducting circuits or quantum dots, interacting with microwave radiation. However, recent and ongoing experiments using surface acoustic waves show that a single artificial atom can be coupled to a bosonic field at several points wavelengths apart. Here, we theoretically study this type of system. We find that the multiple coupling points give rise to a frequency dependence in the coupling strength between the atom and its environment, and also in the Lamb shift of the atom. The frequency dependence is given by the discrete Fourier transform of the coupling point coordinates and can therefore be designed. We discuss a number of possible applications for this phenomenon, including tunable coupling, single-atom lasing, and other effects that can be achieved by designing the relative coupling strengths of different transitions in a multi-level atom.Comment: 14 pages, 8 figure

Undoing measurement-induced dephasing in circuit QED

We analyze the backaction of homodyne detection and photodetection on superconducting qubits in circuit quantum electrodynamics. Although both measurement schemes give rise to backaction in the form of stochastic phase rotations, which leads to dephasing, we show that this can be perfectly undone provided that the measurement signal is fully accounted for. This result improves upon that of Phys. Rev. A, 82, 012329 (2010), showing that the method suggested can be made to realize a perfect two-qubit parity measurement. We propose a benchmarking experiment on a single qubit to demonstrate the method using homodyne detection. By analyzing the limited measurement efficiency of the detector and bandwidth of the amplifier, we show that the parameter values necessary to see the effect are within the limits of existing technology

Simple preparation of Bell and GHZ states using ultrastrong-coupling circuit QED

The ability to entangle quantum systems is crucial for many applications in quantum technology, including quantum communication and quantum computing. Here, we propose a new, simple, and versatile setup for deterministically creating Bell and Greenberger-Horne-Zeilinger (GHZ) states between photons of different frequencies in a two-step protocol. The setup consists of a quantum bit (qubit) coupled ultrastrongly to three photonic resonator modes. The only operations needed in our protocol are to put the qubit in a superposition state, and then tune its frequency in and out of resonance with sums of the resonator-mode frequencies. By choosing which frequency we tune the qubit to, we select which entangled state we create. We show that our protocol can be implemented with high fidelity using feasible experimental parameters in state-of-the-art circuit quantum electrodynamics. One possible application of our setup is as a node distributing entanglement in a quantum network.Comment: 15 pages, 7 figure

Modeling an integrated market for sawlogs, pulpwood and forest bioenergy

Traditionally, most applications in the initial stage of forest supply chain deal with sawlogs to sawmills, pulpwood to pulp or paper mills and forest residues to heating plants. However, in the past decades, soaring prices of fossil fuel, global awareness about CO2 emission and increasing attention to domestic resource security have boosted the development of alternative renewable energy, among which forest bioenergy is the most promising and feasible choice for medium- and large-scale heating and electricity generation. Different subsidies and incentive policies for green energy further promote the utilization of forest bioenergy. As a result, there is a trend that pulpwood may be forwarded to heating plants as complementary forest bioenergy. Though pulpwood is more expensive than forest residues, it is more efficient to transport and has higher energy content. The competition between traditional forest industries and wood-energy facilities, expected to grow in the future, is very sensitive for the forest companies as they are involved in all activities. In this paper, we develop a model that all raw materials in the forest, i.e. sawlogs, pulpwood and forest residues, and byproducts from sawmills, i.e. wood chips and bark, exist in an integrated market where pulpwood can be sent to heating plants as bioenergy. It represents a multi-period multi-commodity network planning problem with multiple sources of supply, i.e. pre-selected harvest areas, and multiple kinds of destination, i.e. sawmills, pulp mills and heating plants. The decisions incorporate purchasing the raw materials in harvest areas, reassigning byproducts from sawmills, transporting those assortments to different points for chipping, storing, wood-processing or wood-fired, and replenishing fossil fuel when necessary. Moreover, different from the classic wood procurement problem, we take the unit purchasing costs of raw materials as variables, on which the corresponding supplies of different assortments linearly depend. With this price mechanism, the popularity of harvest areas can be distinguished. The objective of the problem is to minimize the total cost for the integrated market including the purchasing cost of raw materials. Therefore, the model is a quadratic programming (QP) problem with a quadratic objective function and linear constraints. A large case study in southern Sweden under different scenario assumptions is implemented to simulate the integrated market and to study how price restriction, market regulation, demand fluctuation, policy implementation and exogenous change in price for fossil fuel will influence the entire wood flows. Pair-wise comparisons show that in the integrated market, competition for raw materials between forest bioenergy facilities and traditional forest industries pushes up the purchasing costs of pulpwood. The results also demonstrate that resources can be effectively utilized with the price mechanism in supply market. The overall energy value of forest bioenergy delivered to heating plants is 23% more than the amount in the situation when volume and unit purchasing cost of raw materials are fixed.Forest supply chain; integrated market; bioenergy; wood procurement; wood distribution; quadratic programming