In continuation of the work in Leventides and Petroulakis (Adv Appl Clifford Algebras 27:1503–1515, 2016), et al. (J Optim Theory 169(1):1–16, which defines extremal varieties \(\mathbb {P}\left( {{ \bigwedge ^2}{\mathbb {R}^n}} \right) \), we define a more general concept real Grassmannian \({G_p}\left( {{\mathbb \) ^p}{\mathbb \). This is based on minimization sums squares quadratic Plücker r...

S u m m a r y . A functional integral for a particle obeying the Dirac equation is presented. In earlier work (reviewed here) we showed that 1) such a particle could be described as a massless particle randomly flipping direction and helicity at a complex rate i/m and 2) its between-flips propagation could be written as a sum over paths for a Grassmann variable valued stochastic process. We her...

The matrix pencil algebraic characterisation of the families of invariant subspaces of an implicit system S(F, G) : Fz = Gz F, G € K m x n , is further developed by using tools from Exterior Algebra and in particular the Grassmann Representative g(V) of the subspace V of the domain of (F,G). Two different approaches are considered: The first is based on the compound of the pencil C'd(sF — G), w...

An appropriate definition of the Hodge duality ⋆ operation on any arbitrary dimensional supermanifold has been a long-standing problem. We define a working rule for the Hodge duality ⋆ operation on the (2+2)-dimensional supermanifold parametrized by a couple of even (bosonic) spacetime variables x(μ = 0, 1) and a couple of odd (fermionic) variables θ and θ̄ of the Grassmann algebra. The Minkowsk...

An appropriate definition of the Hodge duality ⋆ operation on any arbitrary dimensional supermanifold has been a long-standing problem. We define a working rule for the Hodge duality ⋆ operation on the (2+2)-dimensional supermanifold parametrized by a couple of even (bosonic) spacetime variables x(μ = 0, 1) and a couple of odd (fermionic) variables θ and θ̄ of the Grassmann algebra. The Minkowsk...

We present a new approach for accurate and fast reconstruction of 3D models from hand-drawn perspective sketches and imposed geometric constraints. A distinctive feature of the approach is the decomposition of the reconstruction process into the stages of correction of the 2D sketch and elevation of the 3D model. All 3D constraints that describe the spatial structure of the model are strictly s...

We present a novel approach to Gaussian Berezin correlation functions. A formula well known in the literature expresses these quantities in terms of submatrices of the inverse matrix appearing in the Gaussian action. By using a recently proposed method to calculate Berezin integrals as an expectation of suitable functionals of Poisson processes, we obtain an alternative formula which allows one...

Exploring the spatial information contained in a single photo is an important research topic in computer vision and photogrammetry. Theoretically, more than two photos covering the same area are needed to restore the 3D scene of the imaged areas, while those photos were taken. This research focuses on collecting the 3D spatial information from a single photo by using a developed 3D perspective ...

In the framework of superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism, the derivation of the (anti-)BRST nilpotent symmetry transformations for the matter fields, present in any arbitrary interacting gauge theory, has been a long-standing problem. In our present investigation, the local, covariant, continuous and off-shell nilpotent (anti-)BRST symmetry transformations for the D...

We consider an effective Majorana neutrino mass operator with the FriedbergLee symmetry; i.e., it is invariant under the transformation να → να + z (for α = e, μ, τ) with z being a space-time independent constant element of the Grassmann algebra. We show that this new flavor symmetry can be broken in such a nontrivial way that the lightest neutrino remains massless but an experimentally-favored...