Optimal discrimination of single-qubit mixed states

We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities.Comment: 8 pages, 1 figur

The extensional realizability model of continuous functionals and three weakly non-constructive classical theorems

We investigate wether three statements in analysis, that can be proved classically, are realizable in the realizability model of extensional continuous functionals induced by Kleene's second model $K_2$. We prove that a formulation of the Riemann Permutation Theorem as well as the statement that all partially Cauchy sequences are Cauchy cannot be realized in this model, while the statement that the product of two anti-Specker spaces is anti-Specker can be realized

The Combinatorial World (of Auctions) According to GARP

Revealed preference techniques are used to test whether a data set is compatible with rational behaviour. They are also incorporated as constraints in mechanism design to encourage truthful behaviour in applications such as combinatorial auctions. In the auction setting, we present an efficient combinatorial algorithm to find a virtual valuation function with the optimal (additive) rationality guarantee. Moreover, we show that there exists such a valuation function that both is individually rational and is minimum (that is, it is component-wise dominated by any other individually rational, virtual valuation function that approximately fits the data). Similarly, given upper bound constraints on the valuation function, we show how to fit the maximum virtual valuation function with the optimal additive rationality guarantee. In practice, revealed preference bidding constraints are very demanding. We explain how approximate rationality can be used to create relaxed revealed preference constraints in an auction. We then show how combinatorial methods can be used to implement these relaxed constraints. Worst/best-case welfare guarantees that result from the use of such mechanisms can be quantified via the minimum/maximum virtual valuation function

A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

The main result of this paper is a bijective proof showing that the generating function for partitions with bounded differences between largest and smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied through analytic methods by Andrews, Beck, and Robbins. Our approach is geometric: We model partitions with bounded differences as lattice points in an infinite union of polyhedral cones. Surprisingly, this infinite union tiles a single simplicial cone. This construction then leads to a bijection that can be interpreted on a purely combinatorial level.Comment: 12 pages, 5 figure

Constructive aspects of Riemann's permutation theorem for series

The notions of permutable and weak-permutable convergence of a series $\sum_{n=1}^{\infty}a_{n}$ of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann's two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara's principle \BDN implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for series holds but \BDN does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies \BDN. We show that this is the case when the property is weak-permutable convergence

Influence of Materials and Design Parameters on Zinc Oxide Surface Acoustic Devices

This thesis presents research into Zinc Oxide (ZnO) based resonators to include Width Extensional Mode (WEM), Length Extensional Mode (LEM), and Surface Acoustic Wave (SAW) devices. The design and operation of ZnO based SAW devices are investigated further to characterize design parameters and operating modes. Their design, fabrication, and results are discussed in detail. SAW device testing in conjunction with X-Ray Diffractometry (XRD) and Atomic Force Microscopy (AFM) are utilized to characterize ZnO and its deposition parameters on a variety of different substrates and interlayers, with different deposition temperatures and annealing parameters. These substrates include silicon, silicon oxide-on-silicon, and sapphire wafers with interlayers including titanium, tungsten, and silicon oxide. Fabrication methods are discussed to explain all processing steps associated with SAW and released contour mode resonators. The SAW devices in this research test different design parameters to establish better reflector design spacing for higher frequency Sezawa wave modes. The characterization and design of ZnO based SAW devices establishes the potential for prototyping high frequency SAW designs using standard lithography techniques. These devices are desired for space-based operations for use in GPS filters, signal processing, and sensing in satellites and space vehicles