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No abstract uploaded!

The linear congruence equations are the ancient and significant research contents. Most discussions of the linear congruence equations focus on the special cases, for example, there is a linear congruent theorem for solving the congruent linear equation in one unknown and the Chinese remainder theorem for solving the simultaneous congruent linear equations in one unknown, which are involved to find a special solution using the properties of the integer number, and some papers discuss the equation in unknowns. But all the results are not convenient and efficient to solve the equations and not adapt to solving the general system of congruent linear equations in n unknowns. There is no uniform, convenient and efficient technique and theory for the general system of congruent linear equations, like the theory of linear equations over real numbers. The inspiration arose from the elimination of variables when solving the linear equations over real numbers, Generalized the elementary transformations of matrix over real numbers to the integer numbers modulo m, the paper discussed the properties of the modular matrix under the elementary transformations and a similar equivalent transforming theorem for matrix modulom theorem was obtained that any matrix modulo m can be transformed into a canonical diagonal form by means of a finite number of elementary row and column operations. furthermore, by means of the equivalent transforming theorem, the solution criterion theorem and structure theorem were proposed for the general congruent linear equations based on the modular matrix transformations, which extended the theories of the congruent linear equations, and finally, the detailed steps of the uniform method were given for solving the general system of congruent linear equations based on the elementary transformations of matrix modulo m, it can be easily written out the solutions of the system immediately as determining solutions of the system conveniently by elementary matrix transformations. Analysis and discussions indicate that the results are most valuable in science and the proposed technique for solving the system is convenient, efficient and adaptable.

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In Ramsey theory (a branch of combinatorics) one often proves that a function exists by giving enormous bounds on it. The function's actual values may be much smaller; however, this is not known. In this paper we explore variants and special cases of these functions where we obtain much smaller bounds. The following is known: no matter how the lattice points of a coordinate plane are colored red and blue, there exists a square whose corners are the same color (a monochromatic square). In fact, using more than just two colors will still guarantee a monochromatic square (one whose vertices are the same color). So for all c (the number of colors), there is a number G(c) where all colorings with c colors of the lattice points of a G(c)*G(c) grid will contain a monochromatic square. Unfortunately, the necessary number of points is unknown, but bounds are known. These bounds are enormous,however. So we look at a variant of this problem in order to work with more manageable bounds. Looking for rectangles or right triangles rather than squares makes these bounds polynomial. We explore which grids contain a specic number of rectangles or right triangles. It is known that if the naturals are colored with two colors, there will be two of the same color that are a square apart. Once again, this is true for any number of colors. It is also true for any polynomial whose constant term is 0 and for multiple polynomials. This is known as the Polynomial Van der Waerden Theorem, or PolyVDW. The bounds on the number of naturals that have to be colored grow very quickly. However, for quadratic polynomials and 2 or 3 colors, there are some cases where the theorem is false, and others where it is true with polynomial bounds. Since PolyVDW does not hold in its entirety for all such polynomials, we try to classify for which of these polynomials and for which numbers of colors the theorem holds and try to nd bounds for them. We explore monochromatic rectangles and right triangles rather than squares in order to get reasonable bounds on the grid sizes necessary to guarantee the existence of these monochromatic shapes. We also explore the necessary grid sizes for multiple rectangles and right triangles. In fact, we nd many exact numbers. In addition, we look at variants of PolyVDW and obtain much better upper bounds for these numbers than are known for the standard PolyVDW numbers.