Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity

We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in toric rings as a certain volume in real spaceComment: 19 page

The Supreme Court and Recent School Desegregation Cases: The Role of Social Science in a Period of Judicial Retrenchment

Through an analysis of the Supreme Court\u27s post-1973 rulings, argues that the Court has not repudiated any doctrine that it had adopted Before 1973 but has exhibited a reluctance to extend previously announced legal principles to claims for new remedies

Non-conformal entanglement entropy

We explore the behaviour of renormalized entanglement entropy in a variety of holographic models: non-conformal branes; the Witten model for QCD; UV conformal RG flows driven by explicit and spontaneous symmetry breaking and Schr\"odinger geometries. Focussing on slab entangling regions, we find that the renormalized entanglement entropy captures features of the previously defined entropic c-function but also captures deep IR behaviour that is not seen by the c-function. In particular, in theories with symmetry breaking, the renormalized entanglement entropy saturates for large entangling regions to values that are controlled by the symmetry breaking parameters.Comment: 33 pages, 4 figure

Inhomogeneity simplified

We study models of translational symmetry breaking in which inhomogeneous matter field profiles can be engineered in such a way that black brane metrics remain isotropic and homogeneous. We explore novel Lagrangians involving square root terms and show how these are related to massive gravity models and to tensionless limits of branes. Analytic expressions for the DC conductivity and for the low frequency scaling of the optical conductivity in phenomenological models are derived, and the optical conductivity is studied in detail numerically. The square root Lagrangians are associated with linear growth in the DC resistivity with temperature and also lead to minima in the optical conductivity at finite frequency, suggesting that our models may capture many features of heavy fermion systems.Comment: 58 pages; v2 references adde

The holographic F theorem

The F theorem states that, for a unitary three dimensional quantum field theory, the F quantity defined in terms of the partition function on a three sphere is positive, stationary at fixed point and decreases monotonically along a renormalization group flow. We construct holographic renormalization group flows corresponding to relevant deformations of three-dimensional conformal field theories on spheres, working to quadratic order in the source. For these renormalization group flows, the F quantity at the IR fixed point is always less than F at the UV fixed point, but F increases along the RG flow for deformations by operators of dimension $3/2 < \Delta < 5/2$. Therefore the strongest version of the F theorem is in general violated

Real deal or no deal? A comparative analysis of raw milk cheese regulation in Australia and France

Australia’s regulatory framework has resulted in the standardisation of cheese production based on pasteurisation. Up until early 2015, regulations effectively prohibited raw milk cheese-making in Australia and thus stifled artisanal on-farm production. Although the introduction of Food Standards Australia New Zealand Standard 4.2.4 has allowed the production of certain hard, low-moisture raw milk cheeses, the new standard is rigid and does not encourage new entrants into the emerging raw milk cheese consumer market. This article compares the Australian system with the French raw milk cheese regulation and production system, and argues that its approach in encouraging and supporting small farmhouse artisanal traditional raw milk cheese is beneficial to both producer and consumer, and has not resulted in any significant health risks. The Australian approach amounts to a missed opportunity to encourage the emergence of a value-added industry with local and export potential, and is at odds with important movements in food policy, such as recognition of the value of localism and terroir