Is Including Under God in The Pledge of Allegiance Lawful?: An Impeccably Correct Ruling

On June 26, 2002, in Newdow v. U.S. Congress, a divided panel of the United States Court of Appeals for the Ninth Circuit held that the 1954 Congressional amendment adding the words “under God” to the Pledge of Allegiance violated the First Amendment’s proscription that, “Congress shall make not law respecting an establishment of religion.” Because the First Amendment’s Establishment Clause applies to the States via the due process clause of the Fourteenth Amendment, the Ninth Circuit likewise found unlawful a California school district’s policy encouraging public school students to utter the words “under God” as part of teacher-led daily recitals of the Pledge. Eight months later, the still divided Ninth Circuit panel issued an amended opinion reaffirming its ruling that the school district’s policy coerces students to perform a “religious act” in contravention of the Establishment Clause. However, holding that it had exceeded the legal analysis necessary to review the lawfulness of the policy, the Newdow Court vacated its determination that the words “under God” in the Pledge are per se unconstitutional. This article urges that the original Newdow decision rightly understood that adding the words “under God” to the Pledge violates the Constitution’s anti-establishment principles. Accordingly, government policy encouraging public school students to avow via the Pledge that ours is a nation dependent on or ruled by God, likewise contravenes the First Amendment

Brief Response to Attorney Albright\u27s Article

This article is a brief response to another article arguing that the words “under God” do not render the Pledge of Allegiance unconstitutional. Attorney D. Chris Allbright’s provocative plea that the phrase “under God” in the Pledge of Allegiance is insufficiently religious to offend contemporary Establishment Clause principles rests on three wobbly premises: (1) a limited perspective of some of the Framers, one which the Supreme Court rightly has eschewed; (2) Supreme Court dicta reflecting at best certain justices’ cursory suppositions about the religiosity of the words “under God;” and, (3) the wholly irrelevant, and possibly inaccurate argument that the words “under God” have had scant influence on schoolchildren

On the probability density function of baskets

The state price density of a basket, even under uncorrelated Black-Scholes dynamics, does not allow for a closed from density. (This may be rephrased as statement on the sum of lognormals and is especially annoying for such are used most frequently in Financial and Actuarial Mathematics.) In this note we discuss short time and small volatility expansions, respectively. The method works for general multi-factor models with correlations and leads to the analysis of a system of ordinary (Hamiltonian) differential equations. Surprisingly perhaps, even in two asset Black-Scholes situation (with its flat geometry), the expansion can degenerate at a critical (basket) strike level; a phenomena which seems to have gone unnoticed in the literature to date. Explicit computations relate this to a phase transition from a unique to more than one "most-likely" paths (along which the diffusion, if suitably conditioned, concentrates in the afore-mentioned regimes). This also provides a (quantifiable) understanding of how precisely a presently out-of-money basket option may still end up in-the-money.Comment: Appeared in: Large Deviations and Asymptotic Methods in Finance, Springer proceedings in Mathematics & Statistics, Editors: Friz, P.K., Gatheral, J., Gulisashvili, A., Jacquier, A., Teichmann, J., 2015, with minor typos remove

Multipliers for Continuous Frames in Hilbert Spaces

In this paper we examine the general theory of continuous frame multipliers in Hilbert space. These operators are a generalization of the widely used notion of (discrete) frame multipliers. Well-known examples include Anti-Wick operators, STFT multipliers or Calder\'on- Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as well as their discrete counterparts. Nonetheless, many results similar to the discrete case are proven for continuous frame multipliers as well, for instance compactness and Schatten class properties. Furthermore, the concepts of controlled and weighted frames are transferred to the continuous setting

Inhomogeneous minima of mixed signature lattices

We establish an explicit upper bound for the Euclidean minimum of a number field which depends, in a precise manner, only on its discriminant and the number of real and complex embeddings. Such bounds were shown to exist by Davenport and Swinnerton-Dyer. In the case of totally real fields, an optimal bound was conjectured by Minkowski and it is proved for fields of small degree. In this note we develop methods of McMullen in the case of mixed signature in order to get explicit bounds for the Euclidean minimum.Comment: To appear in the Journal of Number Theor

Living Lives of Faith (Holiness)

In lieu of an abstract, below is the essay\u27s first paragraph. Each year on November 1st, The Roman Catholic Church celebrates the Feast of All Saints and encourages us to reflect on men and women of all ages who are models of holiness and heroes in the faith

From rough path estimates to multilevel Monte Carlo

New classes of stochastic differential equations can now be studied using rough path theory (e.g. Lyons et al. [LCL07] or Friz--Hairer [FH14]). In this paper we investigate, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense. Our focus lies on numerical implementations, and more specifically on the saving possible via multilevel methods. Our analysis relies on a subtle combination of pathwise estimates, Gaussian concentration, and multilevel ideas. Numerical examples are given which both illustrate and confirm our findings.Comment: 34 page