Introduction: Religious Accomodation in the Age of Civil Rights

The papers in this symposium grow out of a conference on Religious Accommodation in the Age of Civil Rights that was held at Harvard Law School in April 2014. Co-sponsored by Harvard, the Williams Institute at UCLA School of Law, the American Civil Liberties Union (ACLU), and the University of Southern California Center for Law, History and Culture, the conference was convened at a moment of intense agitation as the nation\u27s attention was trained on the case of Burwell v. Hobby Lobby Stores, Inc., then before the Supreme Court. Hobby Lobby posed the question of whether closely held for-profit corporations had the right to a religious exemption from provisions of the Affordable Care Act (ACA) requiring coverage of contraceptive services in their employees\u27 health insurance plans. More broadly, Hobby Lobby raised important questions about the meaning and contours of the Religious Freedom Restoration Act (RFRA), including whether businesses are persons covered by RFRA, how RFRA\u27s strict scrutiny standard should be implemented, and what counts as a substantial burden on religion (and who decides). At the time our conference took place, the Supreme Court had heard oral argument in the case, but had not yet issued its decision

Introduction: Religious Accomodation in the Age of Civil Rights

The papers in this symposium grow out of a conference on Religious Accommodation in the Age of Civil Rights that was held at Harvard Law School in April 2014. Co-sponsored by Harvard, the Williams Institute at UCLA School of Law, the American Civil Liberties Union (ACLU), and the University of Southern California Center for Law, History and Culture, the conference was convened at a moment of intense agitation as the nation\u27s attention was trained on the case of Burwell v. Hobby Lobby Stores, Inc., then before the Supreme Court. Hobby Lobby posed the question of whether closely held for-profit corporations had the right to a religious exemption from provisions of the Affordable Care Act (ACA) requiring coverage of contraceptive services in their employees\u27 health insurance plans. More broadly, Hobby Lobby raised important questions about the meaning and contours of the Religious Freedom Restoration Act (RFRA), including whether businesses are persons covered by RFRA, how RFRA\u27s strict scrutiny standard should be implemented, and what counts as a substantial burden on religion (and who decides). At the time our conference took place, the Supreme Court had heard oral argument in the case, but had not yet issued its decision

Energy bounds for codes and designs in Hamming spaces

We obtain universal bounds on the energy of codes and for designs in Hamming spaces. Our bounds hold for a large class of potential functions, allow unified treatment, and can be viewed as a generalization of the Levenshtein bounds for maximal codes.Comment: 25 page

Integration of Product, Package, Process, and Environment: A Food System Optimization

The food systems slated for future NASA missions must meet crew nutritional needs, be acceptable for consumption, and use resources efficiently. Although the current food system of prepackaged, moderately stabilized food items works well for International Space Station (ISS) missions, many of the current space menu items do not maintain acceptability and/or nutritive value beyond 2 years. Longer space missions require that the food system can sustain the crew for 3 to 5 years without replenishment. The task "Integration of Product, Package, Process, and Environment: A Food System Optimization" has the objective of optimizing food-product shelf life for the space-food system through product recipe adjustments, new packaging and processing technologies, and modified storage conditions. Two emergent food processing technologies were examined to identify a pathway to stable, wet-pack foods without the detrimental color and texture effects. Both microwave-assisted thermal sterilization (MATS) and pressure-assisted thermal stabilization (PATS) were evaluated against traditional retort processing to determine if lower heat inputs during processing would produce a product with higher micronutrient quality and longer shelf life. While MATS products did have brighter color and better texture initially, the advantages were not sustained. The non-metallized packaging film used in the process likely provided inadequate oxygen barrier. No difference in vitamin stability was evident between MATS and retort processed foods. Similarly, fruit products produced using PATS showed improved color and texture through 3 years of storage compared to retort fruit, but the vitamin stability was not improved. The final processing study involved freeze drying. Five processing factors were tested in factorial design to assess potential impact of each to the quality of freeze-dried food, including the integrity of the microstructure. The initial freezing rate and primary freeze drying temperature and pressure were linked to final product quality in freeze-dried corn, indicating processing modifications that could lead to improved product shelf life. Storage temperatures and packaging systems were also assessed for the impact to food quality. Reduced temperature storage had inconclusive impact to the progression of rancidity in butter cookies. Frozen storage was detrimental to fruit and vegetable textural attributes but refrigerated storage helped to sustain color and organoleptic ratings for plant-based foods. With regard to packaging systems, the metallized film overwrap significantly decreased the progression of the rancidity of butter cookies as compared to the highest barrier non-metallized film. The inclusion of oxygen scavengers resulted in noticeable moisture gains in butter cookies over time, independent of packaging film systems. Neither emergent processing technology nor the freeze dry optimization resulted in compelling quality differences from current space food provisions such that a five-year shelf life is likely with these processing changes alone. Using a combination of refrigeration and PATS processing is expected to result in organoleptically-acceptable fruit quality for most fruits through five years. The vitamin degradation will be aided somewhat by the cold temperatures but, given the labile nature of vitamin C, a more stable fortification method, such as encapsulation, should also be investigated to ensure vitamin delivery throughout the product life. Similarly, significant improvement to the packaging film used in the MATS processing, optimization of formulation for dielectric properties, vitamin fortification, and reduced temperature storage should be investigated as a hurdle approach to reach a five year shelf life in wet-pack entrees and soups. Baked goods and other environmentally-sensitive spaceflight foods will require an almost impenetrable barrier to protect the foods from oxygen and moisture ingress but scavengers and reduced storage temperature did not improve baked good shelf life and are not recommended at this time for these foods

Next levels universal bounds for spherical codes: the Levenshtein framework lifted

We introduce a framework based on the Delsarte-Yudin linear programming approach for improving some universal lower bounds for the minimum energy of spherical codes of prescribed dimension and cardinality, and universal upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum separation. Our results can be considered as next level universal bounds as they have the same general nature and imply, as the first level bounds do, necessary and sufficient conditions for their local and global optimality. We explain in detail our approach for deriving second level bounds. While there are numerous cases for which our method applies, we will emphasize the model examples of $24$ points ($24$-cell) and $120$ points ($600$-cell) on $\mathbb{S}^3$. In particular, we provide a new proof that the $600$-cell is universally optimal, and furthermore, we completely characterize the optimal linear programing polynomials of degree at most $17$ by finding two new polynomials, which together with the Cohn-Kumar's polynomial form the vertices of the convex hull that consists of all optimal polynomials. Our framework also provides a conceptual explanation of why polynomials of degree $17$ are needed to handle the $600$-cell via linear programming.Comment: 30 pages, 4 figures, 5 tables, submitte

On polarization of spherical codes and designs

In this article we investigate the $N$-point min-max and the max-min polarization problems on the sphere for a large class of potentials in $\mathbb{R}^n$. We derive universal lower and upper bounds on the polarization of spherical designs of fixed dimension, strength, and cardinality. The bounds are universal in the sense that they are a convex combination of potential function evaluations with nodes and weights independent of the class of potentials. As a consequence of our lower bounds, we obtain the Fazekas-Levenshtein bounds on the covering radius of spherical designs. Utilizing the existence of spherical designs, our polarization bounds are extended to general configurations. As examples we completely solve the min-max polarization problem for $120$ points on $\mathbb{S}^3$ and show that the $600$-cell is universally optimal for that problem. We also provide alternative methods for solving the max-min polarization problem when the number of points $N$ does not exceed the dimension $n$ and when $N=n+1$. We further show that the cross-polytope has the best max-min polarization constant among all spherical $2$-designs of $N=2n$ points for $n=2,3,4$; for $n\geq 5$, this statement is conditional on a well-known conjecture that the cross-polytope has the best covering radius. This max-min optimality is also established for all so-called centered codes