Double products and hypersymplectic structures on R4nR^{4n}

In this paper we give a procedure to construct hypersymplectic structures on $R^{4n}$ beginning with affine-symplectic data on $R^{2n}$. These structures are shown to be invariant by a 3-step nilpotent double Lie group and the resulting metrics are complete and not necessarily flat. Explicit examples of this construction are exhibited

Odd dimensional counterparts of abelian complex and hypercomplex structures

We introduce the notion of abelian almost contact structures on an odd dimensional real Lie algebra $\mathfrak g$. This a sufficient condition for the structure to be normal. We investigate correspondences with even dimensional real Lie algebras endowed with an abelian complex structure, and with K\"ahler Lie algebras when $\mathfrak g$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce and study abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$. These are given by triples of abelian almost contact structures, satisfying certain compatibility conditions, which are equivalent to the existence of a sphere of abelian almost contact structures. We obtain the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group $G$ endowed with a left invariant abelian almost 3-contact structure and a compatible left invariant Riemannian metric. We determine conditions for $G$ to admit a special metric connection with totally skew-symmetric torsion, called canonical, which plays the role of the Bismut connection for HKT structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is K\"ahler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.Comment: 14 page

Explosives identification by infrared spectrometry

In order to identify various explosives and their precursors, technicians worldwide rely on chemical analysis instruments for rapid specific identification results to help ensure a safe remediation. This is one of the central tasks for homeland security and public safety personnel, especially since the recent proliferation of improvised explosive devices (IEDs). These instruments that are being used in the field, are extremely important for first responders. For this paper and the experiments made, a FTIR spectrometer (Fourier-transform infrared spectroscopy) was used. This is a technique used to obtain an infrared spectrum of absorption or emission of a solid, liquid or gas. A FTIR spectrometer simultaneously collects high-resolution spectral data over a wide spectral range. They are essentially in identifying unknown chemicals on a wide range of colors. Given the fact that this spectrometer does not generate energy during the sampling process, makes it ideal for verifying substances such as: Semtex, smokeless powders, dynamite, TNT and hundreds of other colored materials. Since contact is required between the sample and the instrument, we took extreme caution measures while analyzing these pressure sensitive substances. In this paper, determinations were made for the identification of functional groups from a series of explosives for civil use, in order to establish the necessary steps in developing an ideal method of identification