A Solution of Nonlinear Boundary Value Problem of System With Rectangular Coefficients

This paper discusses a nonlinear boundary value problem of system with rectangular coefficients of the form with boundary conditions of the form A(t)x' + B(t)x = f(t,x) and which is is a real matrix with whose entries are continuous on the form B1x(to)=a and B2x(T)=b which is A(t) is a real m n matri with m > n matrix with m > n whose entries are continuous on J = [to,T] and f E C[J x Rn, Rn]. B1, B2 are nonsingular matrices such that and are constant vectors, especially about the proof of the uniqueness of its solution. To prove it, we use Moore-Penrose generalized inverse and method of variation of parameters to find its solution. Then we show the uniqueness of it by using fixed point theorem of contraction mapping. As the result, under a certain condition, the boundary value problem has a unique solution.

Partitions on Finite Projective Lines

The goal of this paper is to split the finite projective line into disjoint sublines by method of subgeometries where the order of line is not a prime number. The correspondence between the points on a line and the points on a conic has been described. The stabilizer group of some lines has been constructed using the fundamental theory of projective lines. All calculations are done using the GAP program. Also primitive polynomials over Galois filed are classified. Some examples with groups which are the fixed points of lines and study the properties of these groups are introduced. The nonsingular matrices which generate the points of conic and belong to groups of projectivities have been constructed.

ON THE MINIMAL COSET COVERINGS OF THE SET OF SINGULAR AND OF THE SET OF NONSINGULAR MATRICES

It is determined minimum number of cosets over linear subspaces in $ \mathbb{F}_q $ necessary to cover following two sets of $ A (n \mathclose{\times} n) $ matrices. For one of the set of matrices $ \det{A} = 0 $ and for the other set $ \det{A} \neq 0 $. It is proved that for singular matrices this number is equal to $ 1 \mathclose{+} q \mathclose{+} q^2 \mathclose{+} \ldots \mathclose{+} q^{n-1} $ and for the nonsingular matrices it is equal to $ (q^n \mathclose{-} 1)(q^n \mathclose{-} q)(q^n \mathclose{-} q^2) \cdots (q^n \mathclose{-} q^{n-1}) / q^{\large{\binom{n}{2}}} $.