Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the General Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, due to Klivans and Swartz, is generalized and some applications are presented.Comment: Survey, 23 page

Biotechnology, environmental forcing, and unintended trophic cascades

A long ongoing discussion between scientists and policy decision-makers seems to have entered recently into a new phase. The consequences of release of transgenic crops into the environment are being discussed not only by scientists but also by farmers, environmental groups and politicians, while an increasing amount of data is becoming available at all biological scales, including the field level. However, data still rely on experiments designed to capture direct consumer¿resource interactions. Here we argue that we should attempt to concentrate on the ecosystem functioning of soil biota under genetically-modified (GM) plants, because functional and mechanistic analysis of the multitrophic effects of GM plants on soil biota is still lacking. It is our opinion that we should avoid addressing taxa and soil communities separately, but link them at their functional level. We shall explain why, using examples from ecosystem services, allometric scaling, and soil food webs. The energy flow of any food web under stress incorporates several factors and pooled information on ecosystem services and on the different responses of soil invertebrates to induced perturbations in other trophic levels. Therefore, we will systematically focus on the complementarities of these approache

Gordon's inequality and condition numbers in conic optimization

The probabilistic analysis of condition numbers has traditionally been approached from different angles; one is based on Smale's program in complexity theory and features integral geometry, while the other is motivated by geometric functional analysis and makes use of the theory of Gaussian processes. In this note we explore connections between the two approaches in the context of the biconic homogeneous feasiblity problem and the condition numbers motivated by conic optimization theory. Key tools in the analysis are Slepian's and Gordon's comparision inequalities for Gaussian processes, interpreted as monotonicity properties of moment functionals, and their interplay with ideas from conic integral geometry

Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps

We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem to multiply a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp. 305-306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix. In addition, we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications.Comment: 19 page

The coexistence of commodity money and fiat money

In reaction to the monetary turmoil created by the financial crisis of September 2008, both legislative and constitutional reforms have been proposed in different Countries to introduce Commodity Money longside existing National Fiat Currency. A thorough evaluation of the Economic consequences of these new proposals is warranted. This paper surveys some of the existing knowledge in Monetary and Financial Economics for the purpose of answering the significant Economic questions raised by these new political initiatives.Currency competition, commodity money, fiat money, gold, safe haven, search models

Effective Condition Number Bounds for Convex Regularization

We derive bounds relating Renegar's condition number to quantities that govern the statistical performance of convex regularization in settings that include the $\ell_1$-analysis setting. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection, or restriction, of the analysis operator to a lower dimensional space, and can still be effective if these operators are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian's inequality and the kinematic formula from integral geometry.Comment: 17 pages, 4 figures . arXiv admin note: text overlap with arXiv:1408.301

Lurking and learning: Making learning visible in a Virtual Design Studio

This paper explores certain types of student behaviour in design courses presented through an online distance learning environment and using a virtual design studio. It demonstrates that types of behaviour often considered to be passive, and therefore negative or less valuable than obviously active behaviours, can be significant evidence of student learning. Specifically, viewing other students’ work is demonstrated to be a stronger (or equal) correlation of student success compared to any other behaviour measured in the virtual design studios studied. It is hypothesised that this activity is part of a larger set of social learning behaviours that contribute to a general social press or ‘ecology’ of studio learning. This finding has important implications for the design and implementation of virtual studios (technically and in learning design) and these are reported specifically for the interest and use of learning designers