Distributedly Solving Boolean Equations over Networks

Typically, simultaneous Boolean equations are expressed by a set of relations Fi = fi, i = 1,2 ?, where the Fi and fi are switching functions of N binary variables. In many logical design problems, a special type of Boolean equation is often encountered; namely, F(A0, A1 ?, An?1, X0, X1 ?, Xm?1) = f(a1, a2 ?, ak, X0, X1 ?, Xm?1), in which the arguments Ai's and Xi's are binary variables and the a1's are implicit functions of the Ai's only. This paper presents a somewhat new approach to solving such special types of Boolean functional equations. The solution function (see reference 1) can be formulated and mapped on a Veitch chart, especially tailored to the present problem, and all possible sets of solutions of the implicit functions ai's which satisfy the original functional equation are obtained simultaneously. Other possible applications and extensions of this method are discussed also.

It is studied a regularization of systems of linear Volterra integral equations of the third kind with continuous kernel degenerated at points of the given segment on the diagonal. The determinant of the matrix outside the integral disappears at the starting point or at the end of the segment. Regularizing Lavrent’ev type operator is constructed that preserves the property of Volterra equation. It is proved the uniform convergence of the regularized solution to the exact solution, and defined the conditions of uniqueness of the solution in the Holder space. Also it is considered the case of systems of third kind Volterra linear integral equations with the approximated right hand side of equation. It is established conditions under which a regularized solution of the equation with approximately given right-hand side can serve as an approximate solution to the original equation.

This paper presents an efficient algorithm to solve the radial distribution power flow problem in complex mode. The relationship between the complex branch powers and complex bus powers is derived as a non singular square matrix known as element incidence matrix. The power flow equations are rewritten in terms of a new variable as linear recursive equations. The linear equations are solved to determine the bus voltages and branch currents in terms of new variable as complex numbers. The advantage of this algorithm is that it does not need any initial value and easier to develop the code since all the equations are expressed in matrix format. It is tested on the distribution systems available in the literature. This proposed method could be applied to distribution systems having voltage-controlled buses also. The results prove the efficiency of the proposed method. Keywords: radial distribution power flow, element incidence matrix, transmission loss, linear recursive equations.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

The Forward Problem (FB) of Boolean equations consists of finding solutions of a system of Boolean equations, or equivalently, a single Boolean equation of the form f(X) = 0 where f(X): Bn → B and B is an arbitrary Boolean algebra. By contrast, the Inverse Problem (IB) of Boolean equations aims to reconstruct the equation f (X) = 0 given the set of solutions and hence to verify the correctness of this set. This study derives methods that handle this inverse problem for the main types of solutions of Boolean equations. These include: (a) Subsumptive general solutions, in which each of the variables is expressed as an interval by deriving successive conjunctive or disjunctive eliminants of the original function, (b) Parametric general solutions, in which each of the variables is expressed via arbitrary parameters which are freely chosen elements of the underlying Boolean algebra and (c) Particular solutions, each of which is an assignment from the underlying Boolean algebra to every pertinent variable that makes the Boolean equation an identity. The reconstructed function f(X) in every case is set in a canonical form, such as the complete-sum form, to facilitate proving its equivalence to the original function. The methods presented herein are demonstrated with carefully-chosen illustrative examples over big Boolean algebras of various sizes. Among the methods utilized in handling the inverse problem for Boolean equations, the ones utilizing the variable-entered Karnaugh map offered pictorial insight and exhibited an efficient divide-and-conquer strategy.

In previous chapters, we have learned to solve a wide range of differential equations. Primarily, our focus has been on linear differential equations: first-order linear equations, higher-order linear equations with constant coefficients, and systems of linear equations with constant coefficients. Indeed, we have learned through a variety of techniques that under the proviso that a differential equation or system is linear, we can almost always find a solution.

Systems of Fredholm integral equations are essential in engineering fields for modeling complex systems involving dynamic behavior, material properties, and interactions between different forces or fluids. In this research, the Taylor expansion approach is utilized to solve the systems of linear Fredholm integral equations (SLFIEs) of the second kind in a novel manner. By combining the m th-order Taylor polynomial of unknown functions at an arbitrary point and employing the repeated integration method; the given system of linear Fredholm integral equations is transformed into a new system of linear equations of unknown functions and their derivatives. Eventually, this new system is solved to obtain demanded m th-order approximate solutions. An error analysis is given as well as several numerical examples to demonstrate the efficiency, accuracy, and ability of the proposed method. Furthermore, this method leads always to the exact solution if the exact solution is a polynomial of degree less than or equal to m .

We present two new parallel algorithms for exact (error-free) solution of a system of linear equations on a distributed-memory multiprocessor. The exact solution is obtained using the congruence technique which consists of two steps: First, the system of linear equations is converted to systems of linear congruence equations with respect to several prime moduli, and each of these systems is solved on a separate processor. Then, these solutions are combined using the mixed-radix conversion algorithm to obtain the exact solution. The first step is completely (embarrassingly) parallel with no communication requirements among the processors. We improve our previous work and describe two efficient parallel algorithms for the second step. We present the results of our experiments on an Intel iPSC/860 with 8 processors. A linear system of dimension 128 with integer entries as large as 10577 is solved in about 195 seconds on 8 processors with an efficiency of 99.76%.

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations.

A method for solving systems of linear interval equations applied to the Leontief input–output model of economics

The integral-differential equations describe many phenomena in different field of mechanical and nuclear engineering, chemistry, astronomy, biology and potential theory. The resolution of boundary problems for these equations are the subject of several papers in which the authors have approached in most cases numerical methods: Monte Carlo, truncated series of Chebyshev polynomials, the fictitious domain method, S N method and the finite element method (1-4). An exact solution was found only in the particular cases. Generally, these are obtained with the help of the methods of mathematical analysis, abstract functional analysis and the spectral methods (5). In recent years, the basic ideas of the homotopy, which is a concept of the topology and differential geometry, were used to obtain the approximate solutions for a wide class of differential, integral and integral-differential equations. We mention here the homotopy perturbation method (HPM) proposed by Ji-Huan He in 1998 and the homotopy analysis method (HAM) proposed by Liao in 1992. The perturbation methods approximate the solution of given problem by a series of small parameters. Unfortunately, the majority of non-linear problems have no small parameters and an unsuitable choice of these parameters can lead to bad effects. The new homotopy perturbation technique (HPM) embeds a parameter p that ranges from zero to one (6,7). When the embedding parameter is zero, we get a linear equation and if it is equal to one, we get the original transport equation. This embedding parameter that belongs to the interval (0, 1) can be considered as a small parameter. The homotopy analysis method (HAM) is one of the most effective methods to obtain the exact and approximate series solutions for the integral-differential equations (8). Being a great freedom in choosing of the initial guess functions and the auxiliary functions, are proposed for these such forms that correspond to the conditions imposed at the ends of the spatial interval and thus the algorithm becomes fast convergent. In the papers (9-11) is shown that the homotopy perturbation method, variational iteration method and Adomian decomposition method are only special cases of the homotopy analysis method solutions. Today, a new homotopy method is added to the above. With this we can solve the multipoint boundary problem for a stationary transport equation in the non-homogeneous media. Comparative studies between the solutions obtained with the help of mentioned algorithms for a numerical example with the appropriate exact solutions illustrate the accuracy and computational efficiency of these new methods.

The paper is devoted to establishing an approximate formula for the number of solutions of one boolean linear equation. The formula is deduced by using a technique of analytical number theory.

This paper proposes a distributed dynamics that solves the least-squares problem associated with a network system of linear algebraic equations. We consider static directed multi-agent networks. Each agent in the network has access to a private subset of the linear equations. Furthermore, we assume that agents cannot acquire any information about their "out-degrees" at any time. Under the strong connectivity condition on the underlying communication network, we show that the local estimated solution of each agent converges exponentially to the exact least-squares solution of the associated network system of linear algebraic equations.