Iterated Differential Forms III: Integral Calculus

Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms are computed. Various applications and the integral calculus over the algebra $\Lambda_{\infty}$ will be discussed in subsequent notes.Comment: 7 pages, submitted to Math. Dok

Iterated Differential Forms VI: Differential Equations

We describe the first term of the $\Lambda_{k-1}\mathcal{C}$--spectral sequence (see math.DG/0610917) of the diffiety (E,C), E being the infinite prolongation of an l-normal system of partial differential equations, and C the Cartan distribution on it.Comment: 8 pages, to appear in Dokl. Mat

Iterated Differential Forms V: C-Spectral Sequence on Infinite Jet Spaces

In the preceding note math.DG/0610917 the $\Lambda_{k-1}\mathcal{C}$--spectral sequence, whose first term is composed of \emph{secondary iterated differential forms}, was constructed for a generic diffiety. In this note the zero and first terms of this spectral sequence are explicitly computed for infinite jet spaces. In particular, this gives an explicit description of secondary covariant tensors on these spaces and some basic operations with them. On the basis of these results a description of the $\Lambda_{k-1}\mathcal{C}$--spectral sequence for infinitely prolonged PDE's will be given in the subsequent note.Comment: 9 pages, to appear in Math. Dok

On the Strong Homotopy Lie-Rinehart Algebra of a Foliation

It is well known that a foliation F of a smooth manifold M gives rise to a rich cohomological theory, its characteristic (i.e., leafwise) cohomology. Characteristic cohomologies of F may be interpreted, to some extent, as functions on the space P of integral manifolds (of any dimension) of the characteristic distribution C of F. Similarly, characteristic cohomologies with local coefficients in the normal bundle TM/C of F may be interpreted as vector fields on P. In particular, they possess a (graded) Lie bracket and act on characteristic cohomology H. In this paper, I discuss how both the Lie bracket and the action on H come from a strong homotopy structure at the level of cochains. Finally, I show that such a strong homotopy structure is canonical up to isomorphisms.Comment: 41 pages, v2: almost completely rewritten, title changed; v3: presentation partly changed after numerous suggestions by Jim Stasheff, mathematical content unchanged; v4: minor revisions, references added. v5: (hopefully) final versio

Government Speech Doctrine—Legislator-Led Prayer\u27s Saving Grace

(Excerpt) This Note argues that Lund was decided incorrectly in part because the Fourth Circuit failed to analyze the type of speech at issue before assessing the constitutionality of the prayer practice. This Note is composed of four parts. Part I surveys the Supreme Court’s legislative prayer jurisprudence—Marsh and Town of Greece. Part II outlines Lund and Bormuth, and the Fourth and Sixth Circuits’ dissimilar applications of the Supreme Court’s precedent. Part III argues that courts must first classify legislative prayers as either government or private speech before assessing whether a prayer practice violates the Establishment Clause. It further argues that legislator-led prayer is a form of government speech. Lastly, Part IV, the most extensive of this Note, argues that because legislator-led prayer is government speech, courts must focus on the intent underlying legislator-led prayer practices, and only practices motivated by impermissible purposes should be deemed unconstitutional. It then proposes a framework to determine whether a legislative prayer practice classified as government speech is motivated by impermissible intent and analyzes under this framework the legislator-led prayer practices in Lund and Bormuth

Iterated Differential Forms I: Tensors

We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed context and, in particular, enriches it with new natural operations. Applications will be considered in subsequent notes.Comment: 9 pages, extended version of the published note Dokl. Math. 73, n. 2 (2006) 16

Does postoperative radiation therapy represent a contraindication to expander-implant based immediate breast reconstruction? An update 2012-2014

Post-mastectomy radiotherapy (PMRT) is well known in the plastic surgery community for having a negative impact on expander-implant based immediate breast reconstruction (IBBR), although recently some technical improvements allow better results. Very recent papers would suggest that there is no difference in postoperative complications in patients receiving post-mastectomy radiotherapy using modern techniques. However, study results are often biased by small groups of patients and by heterogeneity of radiotherapy timing, different surgical techniques and measured outcomes

On Higher Derivatives as Constraints in Field Theory: a Geometric Perspective

We formalize geometrically the idea that the (de Donder) Hamiltonian formulation of a higher derivative Lagrangian field theory can be constructed understanding the latter as a first derivative theory subjected to constraints.Comment: 7 page

Iterated Differential Forms II: Riemannian Geometry Revisited

A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.Comment: 12 pages, extended version of the published note Dokl. Math. 73, n. 2 (2006) 18

Decaying Leptophilic Dark Matter at IceCube

We present a novel interpretation of IceCube high energy neutrino events (with energy larger than 60 TeV) in terms of an extraterrestrial flux due to two different contributions: a flux originated by known astrophysical sources and dominating IceCube observations up to few hundreds TeV, and a new flux component where the most energetic neutrinos come from the leptophilic three-body decays of dark matter particles with a mass of few PeV. Differently from other approaches, we provide two examples of elementary particle models that do not require extremely tiny coupling constants. We find the compatibility of the theoretical predictions with the IceCube results when the astrophysical flux has a cutoff of the order of 100 TeV (broken power law). In this case the most energetic part of the spectrum (PeV neutrinos) is due to an extra component such as the decay of a very massive dark matter component. Due to the low statistics at our disposal we have considered for simplicity the equivalence between deposited and neutrino energy, however such approximation does not affect dramatically the qualitative results. Of course, a purely astrophysical origin of the neutrino flux (no cutoff in energy below the PeV scale - unbroken power law) is still allowed. If future data will confirm the presence of a sharp cutoff above few PeV this would be in favor of a dark matter interpretation.Comment: 19 pages, 3 figures. Version published in JCAP. The analysis was performed in terms of the number of neutrino events instead of the neutrino flux, using a multi-Poisson likelihood approac