Additive relative invariants and the components of a linear free divisor

A 'prehomogeneous vector space' is a rational representation $\rho:G\to\mathrm{GL}(V)$ of a connected complex linear algebraic group $G$ that has a Zariski open orbit $\Omega\subset V$. Mikio Sato showed that the hypersurface components of $D:=V\setminus \Omega$ are related to the rational characters $H\to\mathrm{GL}(\mathbb{C})$ of $H$, an algebraic abelian quotient of $G$. Mimicking this work, we investigate the 'additive functions' of $H$, the homomorphisms $\Phi:H\to (\mathbb{C},+)$. Each such $\Phi$ is related to an 'additive relative invariant', a rational function $h$ on $V$ such that $h\circ \rho(g)-h=\Phi(g)$ on $\Omega$ for all $g\in G$. Such an $h$ is homogeneous of degree $0$, and helps describe the behavior of certain subsets of $D$ under the $G$--action. For those prehomogeneous vector spaces with $D$ a type of hypersurface called a linear free divisor, we prove there are no nontrivial additive functions of $H$, and hence $H$ is an algebraic torus. From this we gain insight into the structure of such representations and prove that the number of irreducible components of $D$ equals the dimension of the abelianization of $G$. For some special cases ($G$ abelian, reductive, or solvable, or $D$ irreducible) we simplify proofs of existing results. We also examine the homotopy groups of $V\setminus D$.Comment: 27 pages. From v1, strengthen results in section 3, improve prose, and update contact informatio

Lifting free divisors

Let $\varphi:X\to S$ be a morphism between smooth complex analytic spaces, and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T^1_{X/S}$ of $\varphi$ is a Cohen-Macaulay $\mathcal{O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi$, then $f\circ \varphi=0$ defines a free divisor on $X$; this is generalized in several directions. Among applications we recover a result of Mond-van Straten, generalize a construction of Buchweitz-Conca, and show that a map $\varphi:\mathbb{C}^{n+1}\to \mathbb{C}^n$ with critical set of codimension $2$ has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi$ is the algebraic quotient $X\to S=X// G$ with $X// G$ smooth, we describe sufficient conditions for $T^1_{X/S}$ to be Cohen-Macaulay of codimension $2$. In one such case, a free divisor on $\mathbb{C}^{n+1}$ lifts under the operation of "castling" to a free divisor on $\mathbb{C}^{n(n+1)}$, partially generalizing work of Granger-Mond-Schulze on linear free divisors. We give several other examples of such representations.Comment: 30 pages. Many minor changes from v1 in response to a thorough review process. To appear in Proc. London Math. Soc. This version differs from the final published versio

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities II: Vanishing Topology

In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an extension of that introduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Instead of linear subspaces, we use free divisors arising from the geometric configuration and which correspond to subgroups of the solvable groups. Here the vanishing topology involves a singular version of the Milnor fiber; however, it still has the good connectivity properties and is homotopy equivalent to a bouquet of spheres, whose number is called the singular Milnor number. We give formulas for this singular Milnor number in terms of singular Milnor numbers of various free divisors on smooth subspaces, which can be computed as lengths of determinantal modules. In addition to being applied to symmetric, general and skew-symmetric matrix singularities, the results are also applied to Cohen--Macaulay singularities defined as 2 x 3 matrix singularities. We compute the Milnor number of isolated Cohen--Macaulay surface singularities of this type in $\mathbb{C}^4$ and the difference of Betti numbers of Milnor fibers for isolated Cohen--Macaulay 3--fold singularities of this type in $\mathbb{C}^5$.Comment: 53 pages. To appear in Geometry & Topology. Changes in response to helpful referee: replace the erroneous Corollary 6.2 with a new version, specify that we consider 2x3 Cohen-Macaulay singularities, calculate more entries of Table 5, improve wording, format for publicatio

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

We introduce a method for obtaining new classes of free divisors from representations $V$ of connected linear algebraic groups $G$ where $\dim(G)=\dim(V)$, with $V$ having an open orbit. We give sufficient conditions that the complement of this open orbit, the "exceptional orbit variety", is a free divisor (or a slightly weaker free* divisor) for "block representations" of both solvable groups and extensions of reductive groups by them. These are representations for which the matrix defined from a basis of associated "representation vector fields" on $V$ has block triangular form, with blocks satisfying certain nonsingularity conditions. For towers of Lie groups and representations this yields a tower of free divisors, successively obtained by adjoining varieties of singular matrices. This applies to solvable groups which give classical Cholesky-type factorization, and a modified form of it, on spaces of $m \times m$ symmetric, skew-symmetric or general matrices. For skew-symmetric matrices, it further extends to representations of nonlinear infinite dimensional solvable Lie algebras.Comment: 50 pages. Many changes from v1 in response to a thorough review, mostly concentrated in sections 2, 3, and 4. To appear in Annales de l'Institut Fourie

The origin and prevention of pandemics.

Despite the fact that most emerging diseases stem from the transmission of pathogenic agents from animals to humans, the factors that mediate this process are still ill defined. What is known, however, is that the interface between humans and animals is of paramount importance in the process. This review will discuss the importance of the human-animal interface to the disease emergence process. We also provide an overview of factors that are believed to contribute to the origin and global spread of emerging infectious diseases and offer suggestions that may serve as future prevention strategies, such as social mobilization, public health education, behavioral change, and communication strategies. Because there exists no comprehensive global surveillance system to monitor zoonotic disease emergence, the intervention measures discussed herein may prove effective temporary alternatives

You Saw THAT?: Social Networking Sites, Self-Presentation, and Impression Formation in the Hiring Process

Facebook, Twitter, LinkedIn – individuals have more opportunities than ever before to present themselves in public using social networking sites (SNSs). However, individuals tend to live segmented lives and often develop different self-presentations depending on the audience. The combination of opportunities to present publicly and presenting different self-presentations can have unforeseen impacts for job candidates. From employers’ perspectives, access to this public information represents a new source of information about job candidates. This paper studies how self-presentations of candidates in SNSs affect impressions formed of candidates by individuals faced with a hiring decision. Utilizing the self-presentation and impression management literature, a model is developed and tested utilizing data from an online survey-based experiment. Findings of the study suggest information from self-presentations is seen as valuable, yet can create ambiguity for decision makers. Implications for theory and hiring organizations assessing the influence of SNSs on hiring are discussed

Effective area calibration of the nuclear spectroscopic telescope array (NuSTAR)

The Nuclear Spectroscopic Telescope ARray (NuSTAR) has been in orbit for 6 years, and with the calibration data accumulated over that period we have taken a new look at the effective area calibration. The NuSTAR 10-m focal length is achieved using an extendible mast, which flexes due to solar illumination. This results in individual observations sampling a range of off-axis angles rather than a particular off-axis angle. In our new approach, we have split over 50 individual Crab observations into segments at particular off-axis angles. We combine segments from different observations at the same off-axis angle to generate a new set of synthetic spectra, which we use to calibrate the vignetting function of the optics against the canonical Crab spectrum

Growing Local Food Systems: Information Technology Use and Impacts in Geographically-Embedded Markets

Over recent decades, reliance on global food systems involving highly distributed supply chains has increased. However, as awareness of environmental, social, and health consequences of these arrangements has developed, so has interest in local food systems (LFSs) in which consumers are served by nearby producers and intermediaries. Yet, in spite of the purported benefits of LFSs, there are challenges which limit their impact. There is an opportunity for IS scholars to contribute by examining how technology is and could be used in geographically-embedded markets like LFSs. We draw on prior studies of IT use and impacts in markets to generate exploratory propositions regarding ways that IT might be used to in LFSs. The results have the potential to build a bridge between IS research and the study and development of LFSs and, thus, create opportunities for IS scholars to contribute directly to the economic health and quality of life of communities