It is well known that real points of the Study quadric (sliced along a 3-dimensional generator space) correspond to displacements of the Euclidean 3-space. But we still lack of a kinematic meaning for the points of the ambient 7-dimensional projective space P7. This paper gives one possible interpretation in terms of displacements of the Euclidean 4-space. From this point of view we also discus...

Understanding and modeling human preferences is one of the key problems in applications ranging from marketing to automated recommendation. In this paper, we focus on learning and analyzing the preferences of consumers regarding food products. In particular, we explore machine learning methods that embed consumers and products in an Euclidean space such that their relationship to each other mod...

Observer that for any three points, if the three distances between them are given, then the three angles are fixed. Given n−1 vectors, the vectors together with the origin form a set of n points. In fact, given any n points in Euclidean space (in n−1 dimensions), the Johnson-Lindenstrauss Lemma states that the n points can be placed in O( logn 2 ) dimensions such that distances are preserved wi...

We give a simple proof of a characterization of euclidean space due to Aronszajn and derive a well-known characterization due to Jordan & von Neumann as a corollary. A norm || || on a vector space V is euclidean if there is an inner product 〈 , 〉 on V such that ||v|| = √ 〈v,v〉. Characterizations of euclidean normed spaces abound. Amir [1] surveys some 350 characterizations, starting with a well...

We give an exposition of the result that there is no closed exact Lagrangian submanifold L of (C, ω0) where ω0 is the standard symplectic structure. We show that the assertion is equivalent to the statement that the perturbed Cauchy-Riemann equation ∂̄J0u = g for maps u from the unit disc D to C which map the boundary circle ∂D to L has no solution for some function g0. To do this, we follow [1]...

In this paper we studied the translation surfaces according to a new frame called q-frame in three dimensional Euclidean space. The curvatures of the translation surface are obtained in terms of q-frame curvatures. Finally some special cases are investigated for these surfaces.