Using the Bockstein spectral sequence developed previously by the authors, we compute the ring ER(n)∗(BO(q)) explicitly. We then use this calculation to show that the ring spectrum MO[2n+1] is ER(n)-orientable (but not ER(n + 1)-orientable), where MO[2n+1] is defined as the Thom spectrum for the self-map of BO given by multiplication by 2n+1.

Understanding how cultural values influence undergraduate students’ science research experiences and career interest is important in efforts to broaden participation and to diversify the biomedical research workforce. The results from our prospective longitudinal study demonstrated that underrepresented minority student (URM) research assistants who see the altruistic value of conducting biom…

Let d > 1 be an integer. In 1986, Shen defined a class of weight modules Fα b (V) over the Witt algebra Wd for α ∈ Cd, b ∈ C, and an irreducible module V over the general linear Lie algebra gld on which the identity matrix acts as multiplication by b. In 1996, Eswara Rao determined necessary and sufficient conditions for these modules to be irreducible when V is finite-dim…

Let (Mn, g) be a compact n-dimensional (n >= 2) manifold with nonnegative Ricci curvature, and if n >= 3, then we assume that (Mn, g) × R has nonnegative isotropic curvature. The lower bound of the Ricci flow’s existence time on (Mn, g) is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows’ maximal existence times, which was first prove…

In this paper, we investigate the exactness of the Grassmannian Bernstein-Gel’fand-Gel’fand complexes introduced in a previous work of the author, and obtain some inequalities between some Hodge numbers of some irregular varieties. In particular, we obtain sharp lower bounds for the Hodge numbers of smooth subvarieties of Abelian varieties, as well as some improvements of results of Lazarsf…

Let A be an m × m complex matrix with zero trace. Then there are m × m matrices B and C such that A = [B,C] and |B||C|2 <= (logm + O(1))1/2|A|2 where |D| is the norm of D as an operator on m2 and |D|2 is the Hilbert–Schmidt norm of D. Moreover, the matrix B can be taken to be normal. Conversely, there is a zero trace m × m matrix A such that whenever A = [B,C], |B||C|2 >= | logm − …

Genomic approaches are revolutionizing the biomedical sciences, providing new ways in which to approach the development of therapeutic practices and the understanding of the causal bases of disease. So, too, do genomics approaches have the potential to transform the way in which ecological studies can be conducted by providing powerful tools in which genotype and phenotype data of diverse “wi…

The ethical position underpinning decision making is an important concern for conservation biologists when setting priorities for interventions. The recent debate on how best to protect nature has centered on contrasting intrinsic and aesthetic values against utilitarian and economic values, driven by an inevitable global rise in conservation conflicts. These discussions have primarily been tar…

Society’s growing demand for clean and abundant energy has repercussions for biodiversity and human well-being. Directives for renewable energy, energy security, and technological advancements such as horizontal drilling in conjunction with hydraulic fracturing have spurred a rapid increase in alternative and unconventional energy production over the last decade. Given the projected increases…

We discuss definability of henselian valuation rings in the Macintyre language LMac, the language of rings expanded by nth power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly ∃-∅-definable in LMac, and henselian valuation rings with value group Z are uniformly ∃∀-∅-definable in the ring langu…