Related papers: On linear equations arising in Combinatorics (Part…
In one of his papers, the author introduces the class of Farkas-related vectors for which a version of Farkas' lemma over integers is derived. In this paper, two similar classes are introduced and studied.
The main point of this paper is to present a class of equations over integers that one can check if they have a solution by checking a set of inequalities. The prototype of such equations is the equations appearing in the well-known…
Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over…
A new version of Farkas lemma of alternative linear systems is proposed. One and the same matrix $A$ and vector $b$ have always been used in alternative linear systems. The paper shows a different way of alternative systems involving…
Farkas' Lemma is a foundational result in linear programming, with implications in duality, optimality conditions, and stochastic and bilevel programming. Its generalizations are known as theorems of the alternative. There exist theorems of…
In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear…
The following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second…
We study a class of overdetermined algebraic systems of equations. We prove that the number of distinct solutions equals to the maximal possible if and only if certain matrices are commuting and semisimple. This gives a characterization of…
Given $A\in \Z^{m\times n}$ and $b\in\Z^m$, we consider the issue of existence of a nonnegative integral solution $x\in \N^n$ to the system of linear equations $Ax=b$. We provide a discrete and explicit analogue of the celebrated Farkas…
A linear polyomial non-negative on the non-negativity domain of finitely many linear polynomials can be expressed as their non-negative linear combination. Recently, under several additional assumptions, Helton, Klep, and McCullough…
Program termination is a hot research topic in program analysis. The last few years have witnessed the development of termination analyzers for programming languages such as C and Java with remarkable precision and performance. These…
The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence…
Although it is easy to prove the sufficient conditions for optimality of a linear program, the necessary conditions pose a pedagogical challenge. A widespread practice in deriving the necessary conditions is to invoke Farkas' lemma, but…
A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach.…
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…
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…
The principle of finding an integrating factor for a none exact differential equations is extended to a class of third order differential equations. If the third order equation is not exact, under certain conditions, an integrating factor…
The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…
The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation…