This work is motivated by the relation between KP and BKP integrable hierarchies, whose τ-functions may be viewed as flows of sections dual determinantal Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate Cartan map, which, for a vector space V dimension N, embeds Grassmannian GrV0(V+V*) maximal isotropic subspaces + V*, with respect natur...

Let F∈Z[x1,…,xn] be a homogeneous form of degree d≥2, and let VF∗ denote the singular locus affine variety V(F)={z∈Cn:F(z)=0}. In this paper, we prove existence integer solutions with prime coordinates to equation F(x1,…,xn)=0 provided that F satisfies suitable local conditions n−dimVF∗≥283452d3(2d−1)24d. Our result improves on what was known previously due Cook Magyar, which required n−dimVF∗ ...

Discrete kinetic equations describing binary processes of agglomeration and fragmentation are considered using formal equivalence between the geodesic some affinely connected space A associated with equation called affine connection. The geometric properties treated locally in coordinate chart (x;U). peculiarity is that coordinates (x) selected local chart, Christoffel symbols defining connecti...

We derive the exact equations of motion (in Newtonian, F = ma, form) for test masses in Schwarzschild and Gullstrand-Painlevé coordinates. These equations of motion are simpler than the usual geodesic equations obtained from Christoffel tensors in that the affine parameter is eliminated. The various terms can be compared against tests of gravity. In force form, gravity can be interpreted as res...

A. M. Youssef and S. E. Tavares Department Of Electrical and Computer Engineering Queen’s University Kingston, Ontario, Canada, K7L 3N6 Phone : (613) 545–2945 Fax : (613) 545–6615 email : [email protected] Indexing terms: Cryptography, Private-key cryptosystems, Substitution boxes, Combinatorics. Abstract Nonlinearity is a crucial requirement for the substitution boxes in secure block ciphe...

In this work we define, for the first time, affine and projective plane over real Okubo algebra, showing a concrete geometrical interpretation of its Spin group. algebra is flexible, composition which also not unital division algebra. Even though, has been known more than 40 years, believe that time was used geometry. After all axioms geometry are verified, define as completion directly through...

locally extended affine lie algebras were introduced by morita and yoshii in [j. algebra 301(1) (2006), 59-81] as a natural generalization of extended affine lie algebras. after that, various generalizations of these lie algebras have been investigated by others. it is known that a locally extended affine lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight...

"A picture is worth one thousand words". This proverb comes from Confucius a Chinese philosopher about 2500 years ago. Now, the essence of these words is universally understood. A picture can be magical in its ability to quickly communicate a complex story or a set of ideas that can be recalled by the viewer later in time. Visual information plays an important role in our society, it will play ...

The problem is to determine the homography H up to similarity which maps a point or line from a plane in the world to the image plane x́ = Hx. This is to be done using two different methods. For the first method, the projective error is first removed followed by the affine error using the provided coordinates of a rectangle and an additional orthogonal line pair. For the second method, five orth...