In this paper I am going to present the way to define fermionic field in terms of three orthogonal vector fields of norm 1 together with two real valued scalar fields. This paper is based on a toy model where there are no Grassmann variables.

We prove that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors.

Let n be a positive integer and B be a non-degenerate symmetric bilinear form over Fq , where q is an odd prime power and Fq is the finite field with q elements. We determine the largest possible size of a subset S of Fq such that |{B(x,y) |x,y ∈ S and x 6= y}| = 1. We also pose some conjectures concerning nearly orthogonal subsets of Fq where a nearly orthogonal subset T of Fq is a set of vect...

A series of problems in different fields such as physics and chemistry are modeled by differential equations. Differential equations are divided into partial differential equations and ordinary differential equations which can be linear or nonlinear. One approach to solve those kinds of equations is using orthogonal functions into spectral methods. In this paper, we firstly describe Laguerre, H...

Linear codes with large minimal distances are important error correcting codes in information theory. Orthogonal codes have more applications in the other fields of mathematics. In this paper, we study the binary and ternary orthogonal codes generated by the weight matrices on finite-dimensional modules of simple Lie algebras. The Weyl groups of the Lie algebras act on these codes isometrically...

In orthogonal range reporting we are to preprocess N points in d dimensions such that the points inside an axis-aligned query box can be found efficiently. This is a fundamental problem in various fields, including spatial databases and computational geometry. In this paper we construct the first data structure for orthogonal range reporting in the I/O-model in higher dimensions and also signif...

We propose a duality for N = 2 d = 3 Chern-Simons gauge theories with orthogonal gauge groups and matter in the vector representation. This duality generalizes levelrank duality for pure Chern-Simons gauge theories with orthogonal gauge groups and is reminiscent of Seiberg duality in four dimensions. We perform extensive checks by comparing partition functions of theories related by dualities. ...