Finite Number Of Computations Research Articles

Dynamical cancellation of polynomials

AbstractExtending the work of Bell, Matsuzawa and Satriano, we consider a finite set of polynomials over a number field and give a necessary and sufficient condition for the existence of an and a finite set such that for any , we have the cancellation result: If and are maps in such that , then in fact . Moreover, the conditions we give for this cancellation result to hold can be checked by a finite number of computations from the given set of polynomials.

Joint User Association, Power Allocation, and Throughput Maximization in 5G H-CRAN Networks

In order to get worthwhile gain in coverage and capacity required by fifth generation networks, dense heterogeneous networks are being considered as promising solution. Heterogeneous networks consists of small cells overlaid on macrocells. Introduction of small cells within the coverage of macrocell give birth to problems such as user association, power allocation, and admission control. To address these problems and to maximize overall network throughput heterogeneous networks are integrated with cloud radio access network (CRAN). We formulated a joint user association, admission control, and power allocation problem in heterogeneous CRAN with an objective to maximize overall throughput of network. By examining the structure of the problem, we categorize it under mixed integers nonlinear problems that are generally NP-hard. This paper presents an outer approximation approach (OAA) based on linear programming that simplifies the NP-hard problem, which requires an exhaustive search. OAA ensures $\varepsilon$ -optimal solution with $\varepsilon =10^{-3}$ . OAA converges in a finite number of computations. Effectiveness of proposed algorithm is verified by simulation results.

Study of the Kepler’s conjecture: the problem of the closest packing

In the first and second parts the problem of the Kepler’s conjecture is reduced from a problem with an infinite number of parameters to a problem with a finite number of parameters. In the third part is it is shown that the latter problem can be solved by a numerical verification using only a finite number of computations. However that finite number remains large, even if modern computers can do it. The method of analysis is classical: to each sphere of an arbitrary packing is associated a corresponding large enough “associated domain”, disjoint from all other similar domains, in the hope of obtaining an interesting upper limit of the space occupation coefficient. Many methods of association have been tried in the past and the new method presented in this study is complex, but very well adapted to the problem of interest.

Observability and Forward-Backward Observability of Discrete-Time Nonlinear Systems

In this paper we study the observability properties of nonlinear discrete-time systems. Two types of contributions are given. First we present observability criteria in terms of appropriate codistributions. For particular, but significant, classes of systems we provide criteria that require only a finite number of computations. Then we consider invertible systems (which includes discrete-time models obtained by sampling continuous-time systems) and prove that the weaker notion of forward–backward observability is equivalent to the stronger notion of (forward) observability.

Damping ratio of linear uncertain continuous time systems

Polytopes of (characteristic) polynomials arise when systems experience real parameter variations. A sufficient condition (requiring only a finite number of computations) is presented for determining whether all the roots of a polytope of polynomials have the prescribed damping ratios of continuous time systems. It also gives a measure of robustness for a given polytope of polynomials.

Necessary and sufficient conditions for interval polynomials to have only real distinct roots

This paper presents new necessary and sufficient conditions for a family of interval polynomials to have only real distinct roots. These new conditions require only a finite number of computations.

Damping margin of continuous-time systems with polytope uncertainties

Polytopes of (characteristic) polynomials arise when systems experience real parameter variations. This paper presents a sufficient condition (requiring only a finite number of computations) for determining if all the roots of a polytope of polynomials have their real parts less than a ≤ 0. It also gives a robustness measure for a given polytope of polynomials.

Disturbance Decoupling, $(f,g)$-Invariant and Controllability Subspaces of a Class of Homogeneous Polynomial Systems

This paper discusses control systems of the type \[\dot x = f(x) + \sum_{j = 1}^m {b_i u_i + \sum_{j = 1}^k {d_j \omega _j ,\quad x \in \mathbb{R}^n } } \] where $f(x)$ is a homogeneous polynomial vector field, $b_i $ and $d_i $ are constant vectors, and the output function is linear. It is shown that the well-known theory of disturbance decoupling of linear systems extends to this class in a very natural way. The resulting theory is far simpler than the general nonlinear theory. More important, all computations needed can be done using very simple algorithms, which require only a finite number of computations and use only methods from linear algebra.

Damping ratio of linear uncertain discrete-time systems

Poly topes of (characteristic) polynomials arise when systems experience realparameter variations. The paper presents a sufficient condition, requiring only a finite number of computations, for determining if all the roots of a polytope of polynomials have prescribed damping ratios of discrete-time systems. It also gives a measure of robustness for a given polytope of polynomials.

A necessary and sufficient condition for Schur invariance and generalized stability of polytopes of polynomials

Polytopes of (characteristic) polynomials arise when systems experience real parameter variations. A necessary and sufficient condition (requiring only a finite number of computations) is presented for determining whether all the roots of a polytope of polynomials are contained in a desired region. This can be any region that is the image of the open left-half plane under a real linear fractional transformation. Since the open unit cycle is such a region, this result establishes a finite algorithm for strict Schur invariance. Finite algorithms for generalized stability regions such as maximum bandwidth and maximum setting time are also established.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

New methods for evaluating the validity of the results of mathematical computations

From a mathematical standpoint, numerical methods can be divided in two classes: (i) Direct methods, which give an exact result after some finite number of computations, and (ii) approximate methods, which only give an approximate result after any finite number of computations. When these numerical methods are carried out on a computer, they are projected into discrete space and performed with a limited precision arithmetic. Consequently they yield approximate results. A result given by a direct method contains only a computing error while a result given by an approximate method contains both a computing error and a method error. It is absolutely necessary to evaluate these errors. In this paper, we present new methods for estimating these errors and for evaluating the exact number of significant decimal digits appearing in computed results. Also presented here is a general methodology for improving the precision of computed results.