This paper deals with the subject of infinitesimal variations Euclidean submanifolds arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar theory isometric immersions in space, we prove that system three equations certain pair tensors are integrability conditions differential equation determines variations. In addition, giv...

Our focus here is on the infinitesimal model. In this model, one or several quantitative traits are described as the sum of a genetic and a non-genetic component, the first being distributed within families as a normal random variable centred at the average of the parental genetic components, and with a variance independent of the parental traits. We first review the long history of the infinit...

Geometry of the wave function is a central pillar modern solid state physics. In this work, we unveil wave-function geometry two-dimensional semimetals with band crossing points (BCPs). We show that Berry phase BCPs are governed by quantum metric describing infinitesimal distance between states. For generic linear BCPs, corresponding determined either an angular integral metric, or equivalently...

Based on the recent work [4] we put forward a new type of transformation for Lorentzian manifolds characterized by mapping every causal future-directed vector onto a causal future-directed vector. The set of all such transformations, which we call causal symmetries, has the structure of a submonoid which contains as its maximal subgroup the set of conformal transformations. We find the necessar...

In our last work, we formulate a Fourier transformation on the infinitedimensional space of functionals. Here we first calculate the Fourier transformation of infinite-dimensional Gaussian distribution exp ( −πξ ∞ −∞ α 2(t)dt ) for ξ ∈ C with Re(ξ) > 0, α ∈ L2(R), using our formulated Feynman path integral. Secondly we develop the Poisson summation formula for the space of functionals, and defi...

between the K-theory of locally free modules on the non-commutative infinitesimal site and hypercohomology K-theory of free modules. These K-theory results are obtained as a very particular case of a general construction. The input of this construction consists of a category C equipped with a suitable notion of infinitesimal deformation, (e.g. C=rings, deformation=surjection with nilpotent kern...

in this paper, heir-equations method is applied to investigate nonclassical symmetries and new solutions of the black-scholes equation. nonlinear self-adjointness is proved and infinite number of conservation laws are computed by a new conservation laws theorem.