Related papers: On linear equations arising in Combinatorics (Part…
Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…
This preprint is a text for students and teachers on inequalities. Some standard topics are covered on application of calculus to inequality proving. Many examples are considered, stated, solved or partially solved. Some problems are…
In this article we determine several theorems and methods for solving linear congruences and systems of linear congruences, and we find the number of distinct solutions. Many examples of solving congruences are given.
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
By introducing a kind of special functions namely exponent-like function, cosine-like function and sine-like function, we obtain explicitly the basic structures of solutions of initial value problem at the original point for this kind of…
We study the conjugate gradient method for solving s system of linear equations with coefficients which are measurable functions and establish the rate of convergence of this method.
An algebraic criterion that is sufficient to establish the existence of certain a priori estimates for the solution of first-order homogeneous linear characteristic problems is derived. Estimates of such kind ensure the stability of the…
In this work we present some classes of models whose the corresponding two coupled first-order nonlinear equations can be put into a linear form, and consequently be solved completely. In these cases the so-called trial orbit method is…
To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…
Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and…
Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better understand and efficiently solve those…
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…
A full Lie point symmetry analysis of rational difference equations is performed. Non-trivial symmetries are derived and exact solutions using these symmetries are obtained.
In this book, there are five chapters: Systems of Linear Equations, Vector Spaces, Homogeneous Systems, Characteristic Equation of Matrix, and Matrix Dot Product. It has also exercises at the end of each chapter above to let students…
We consider a class of linear ODEs of second order with variable coefficients and construct its Lie algebra of Lie group of equivalence transformations. Further we find invariants and differential invariants of this Lie algebra and by using…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
This paper deals with the comparison of two common types of equivalence groups of differential equations, and this gives rise to a number of results presented in the form of theorems. It is shown in particular that one type can be…
A modified Gauss's algorithm for solving a system of linear equations in an integral ring is proposed, as well as an appropriate algorithm for calculating the elements of the adjoint matrix.
Based on previous work we consturct an equation (Lagrange equation) and relate it with a system of generalized integrals and differential equations in such a way to provide useful evaluations and connections between them.
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…