Julian Edward

On the resonances of the Laplacian on waveguides

Abstract The resonances for the Dirichlet and Neumann Laplacian are studied on compactly perturbed waveguides. In the absence of resonances, an upper bound is proven for the localised resolvent. This is then used to prove that the existence of a quasimode whose asymptotics is bounded away from the thresholds implies the existence of resonances converging to the real axis. The following upper bound to the number of resonances is also proven: # k j ∈ Res (Δ), dist (k j , physical plane ) |k j | /2, |k j | 3+ϵ .

An inequality for Steklov eigenvalues for planar domains

Study of the zeta function associated to the Neumann operator on planar domains yields an inequality for Steklov eigenvalues for planar domains.

Determinant of the Neumann operator on smooth Jordan curves

Using the method of heat kernel expansion, the determinant of the Neumann operator on an arbitrary smooth Jordan curve is shown to be equal to the circumference.

The Properties of Simple Vs. Absolute Majority Rule: Cases Where Absences and Abstentions Are Important:

Little attention has been paid to the differences between absolute majority rule and simple majority rule, which differ in their treatment of absences and ‘votes to abstain’. This article fills that gap by undertaking a probabilistic analysis of the two voting rules assuming two alternatives and a quorum requirement for simple majority rule. The rules are compared in both a modified sincere setting and a strategic setting using five criteria: (1) the Pareto criterion, (2) the BT criterion (Buchanan and Tullock, 1962), (3) the Expected Social Gain criterion, (4) the Responsiveness criterion, and (5) a modified version of Rae’s criterion. In the sincere setting, we find that simple majority rule (with and without a quorum) outperforms absolute majority rule under most conditions for four out of the five criteria. In the strategic setting, we find that the voting rules perform much more similarly.

The Pareto Efficiency and Expected Costs of k-Majority Rules

Several authors have analyzed the optimal k-majority rule based on a variety of criteria. Buchanan and Tullock argued that, in constitutional settings, the criterion should be that all changes meet the Pareto criterion; otherwise the status quo should be preferred (we call this the BT criterion). They then asserted that unanimity rule would be the preferred voting rule in this setting. In parliamentary settings, they claimed that a near majority rule would be preferred because it minimizes the sum of decision costs and external costs. This article investigates both claims in an N-voter, two-alternative setting. We show the conditions under which unanimity rule is less likely to select BT preferred alternatives than other k-majority rules and prove that the difference in performance can be negligible when Nis large and certain weak conditions are met. Furthermore, if we define external costs as the expected number of losers from a BT-inferior vote, then external costs become negligible for a range of supermajority rules. This implies that unanimity rule and a range of supermajority rules should be equally preferred when decision costs are added. Finally, we show that the external cost function can actually increase for certain populations. Many of the broader conclusions should also hold for multiple alternatives.

The Calculus of Consent and Constitutional Design

Introduction.- Original Theories and Current Studies.- Clarifying Concepts.- Constitutional Decision Making.- Legislative Decision Making.- Electoral Decision Making.- Conclusion.

SPECTRAL THEORY FOR THE NEUMANN LAPLACIAN ON PLANAR DOMAINS WITH HORN-LIKE ENDS

The spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of su ch domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at 0 or . The proof uses Mourre theory.

Odd or Even: Assembly Size and Majority Rule*

This paper provides a probabilistic comparison of small assemblies with an even number of members versus an adjacent odd number of members. We show that assemblies with an odd number of members are more likely to pass proposals under absolute majority rule than assemblies with an adjacent even number of members. We also show they are more likely to attain quorum and to pass proposals under simple majority rule with a quorum. The first claim is supported by data on the U.S. appellate courts, 1940–84. Finally, we discover similar effects in the probability of making truthful judgements (Condorcet 1785) and the ability to satisfy common welfare criteria (Rae 1969). The results have important implications for institutional design.

Legislative Decision Making

How many individuals must agree before a collective decision is imposed on a community? Buchanan and Tullock (1962) raised that question roughly fifty years ago and answered that it depends on how a community weighs decision costs and external costs. At the constitutional stage decision costs are less consequential. Hence, voting rules that produce Pareto superior and Pareto optimal outcomes (or just Pareto optimal outcomes) should be promoted. The only voting rule that could guarantee such results, and minimize external costs, is unanimity rule. At the legislative stage, the optimal k-majority rule may depend on both external costs and decision costs. With decision costs considered, the sum of decision costs and external costs might be minimized closer to majority rule.

Electoral Decision Making

Everyone remembers the 2000 U.S. presidential election between George W. Bush and Al Gore. Bush won more Electoral College votes than Gore, and with it the presidency. Nevertheless, Gore won more popular votes than Bush. Many argued that Gore should have been elected the president because he won the plurality of the popular vote. Making such an argument implies that one voting rule (plurality rule) is more desirable than another voting rule (the Electoral College) and opens up a discussion about the desirable properties of voting rules and which voting rule is best. Although one voting rule may be particularly adept at fulfilling one criterion, other voting rules may be particularly adept at fulfilling other criteria. Since no voting rule satisfies a small subset of reasonable criteria (Arrow, 1951), the natural question is which rules are most likely to satisfy common norms?