Infinite Roots Research Articles

Decomposition-based multiobjective optimization for nonlinear equation systems with many and infinitely many roots

Although the development of Pareto-dominance-based multiobjective optimization algorithms has enabled the solution of nonlinear equation systems, few studies have been conducted on the use of multiobjective optimization techniques for the solution of nonlinear equation systems. Accordingly, in this study, we use decomposition-based multiobjective optimization to solve nonlinear equation system, an attempt is made at deploying the decomposition-based multiobjective optimization to solve nonlinear equation systems, including those with infinite roots. In our novel approach, a given system is transformed into a bi-objective optimization problem using reference points. An improved decomposition-based multiobjective optimization is then applied to solve a transformed bi-objective optimization problem. To ensure this optimization suits the characteristics of the problem, we develop an adaptive multiobjective differential evolution and local search approach. The roots of the original nonlinear equation system can then be identified, together with the Pareto-optimal solutions of the transformed problem. We conducted extensive experiments to compare the performance of our novel approach with that of 16 state-of-the-art algorithms in solving 30 nonlinear equation systems derived from industrial applications. The results clearly show that our novel approach achieves better performance with a shorter execution time than that of the selected single-objective-based evolutionary approaches or Pareto-dominance-based multiobjective optimization algorithms.

Nested Square Roots of 2 Revisited

We show how various recently established results on finite and infinite nested square roots of 2 can be derived elegantly via a topological conjugacy linking the tent map and the logistic map. We also prove an additional new and striking consequence of this relationship.

Finite and infinite nested square roots convergent to unity

We investigate nested square root formulas convergent to \textbf{1} by two methods. In the first method, we derive a general infinitely nested square root formula. In the second method we derive a general finitely nested square root formula.

Calculation of the Fine-Structure Constant

The fine-structure constant α [1] is a constant in physics that plays a fundamental role in the electromagnetic interaction. It is a dimensionless constant, defined as: (1) being q the elementary charge, e0 the vacuum permittivity, h the Planck constant and c the speed of light in vacuum. The value shown in (1) is according CODATA 2014 [2]. In this paper, it will be explained that the fine-structure constant is one of the roots of the following equation: (2) being e the mathematical constant e (the base of the natural logarithm). One of the solutions of this equation is: (3) This means that it is equal to the CODATA value in nine decimal digits (or the seven most significant ones if you prefer). And therefore, the difference between both values is: (4) This coincidence is higher in orders of magnitude than the commonly accepted necessary to validate a theory towards experimentation. As the cosine function is periodical, the Equation (2) has infinite roots and could seem the coincidence is just by chance. But as it will be shown in the paper, the separation among the different solutions is sufficiently high to disregard this possibility. It will also be shown that another elegant way to show Equation (2) is the following (being i the imaginary unit): (5) having of course the same root (3). The possible meaning of this other representation (5) will be explained.

Finding Only Finite Roots to Large Kinematic Synthesis Systems

In this work, a new method is introduced for solving large polynomial systems for the kinematic synthesis of linkages. The method is designed for solving systems with degrees beyond 100,000, which often are found to possess quantities of finite roots that are orders of magnitude smaller. Current root-finding methods for large polynomial systems discover both finite and infinite roots, although only finite roots have meaning for engineering purposes. Our method demonstrates how all infinite roots can be precluded in order to obtain substantial computational savings. Infinite roots are avoided by generating random linkage dimensions to construct startpoints and start systems for homotopy continuation paths. The method is benchmarked with a four-bar path synthesis problem.

Nonlinear and additive white noise perturbations of linear delay differential equations at the verge of instability: An averaging approach

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

Numerical solution of linear differential-algebraic systems of partial differential equations

A linear differential-algebraic system of partial differential equations of an arbitrary index is examined. The index of a differential-algebraic system is determined by the maximum degree (over the corresponding domain) of the elementary divisors corresponding to the zero and infinite roots of the characteristic polynomial of the matrix pencil constructed from the coefficients of this system. An iterative algorithm for the numerical solution of a differential-algebraic system is proposed. It is based on a special splitting of the associated matrix pencil and an application of the method of successive approximations to the split system. The stability of this method is shown, and its efficiency is demonstrated through test examples.

On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions

AbstractWe recently introduced a weighted Banach algebra of functions that are holomorphic on the unit disc D, continuous up to the boundary, and of the class C(n) at all points where the function G does not vanish. Here, G refers to a function of the disc algebra without zeros on D. Then we proved that all closed ideals in with at most countable hull are standard. In this paper, on the assumption that G is an outer function in C(n) having infinite roots in and countable zero set h0(G), we show that all the closed ideals I with hull containing h0(G) are standard.

The hydrodynamics of arrays of truncated elliptical cylinders

The hydrodynamic scattering problem by arrays of elliptical truncated cylinders subjected to regular waves is investigated. Linear potential theory is employed that is based on the assumption of inviscid/incompressible fluid and irrotational flow. Effort is made to provide an analytical solution of the associated hydrodynamic boundary value problem. The solution is based on the semi-analytical formulation of the diffraction potentials and the application of the so-called Mathieu functions addition theorem. The latter is decomposed into even and odd Mathieu functions whereas it considers arbitrary angular orientations of the semi-major axes of the bodies. It was also extended to account for negative Mathieu parameters which correspond to the infinite real roots of the transcendental equation. The details of the interaction phenomena are investigated with the aid of the transfer functions of the exciting forces on two-body configurations. Indicative results for the free surface elevation around the bodies are also provided.

Low-Order Stabilization of LTI Systems With Time Delay

This paper considers the problem of stabilizing a single-input-single-output (SISO) linear time-invariant (LTI) plant with known time delay using a low-order controller, such as a Proportional (P), a Proportional-Integral (PI), or a proportional-integral-derivative (PID) controller. For the SISO LTI system with time delay, the closed-loop characteristic function is a quasipolynomial that possesses the following features: all its infinite roots are located on the left of certain vertical line of the complex plane, and the number of its unstable roots is finite. Necessary and sufficient conditions for the stability of LTI systems with time delay are first presented by employing an extended Hermite-Biehler Theorem applicable to quasi-polynomials. Based on the conditions, analytical algorithms are then proposed to compute the stabilizing sets of P, PI and PID controllers. The resulting characterizations of the stabilizing sets for P, PI and PID controllers are analogous to the Youla parameterization of all stabilizing controllers for plants without time delay. Numerical examples are provided to illustrate the proposed algorithm.

Infinite roots of the quadratic equation and tangents to conic sections at infinity

Infinite roots of the quadratic equation and tangents to conic sections at infinity

Most maps of the pseudo-arc are homeomorphisms

We prove the following results. (1) If M ( P ) M(P) is the space of maps of the pseudo-arc into itself with the sup metric, then the subset H ^ ( P ) \hat H(P) of maps of the pseudo-arc into itself which are homeomorphisms onto their images is a dense G δ {G_\delta } in M ( P ) M(P) . (2) Every homeomorphism of the pseudo-arc onto itself is a product of ∈ \in -homeomorphisms. (3) There exists a nonidentity homeomorphism of the pseudo-arc with an infinite sequence of p p th roots. (4) Every map between chainable continua can be lifted to a homeomorphism of pseudo-arcs.

Notes on the Solution of certain Equations

The equations considered are equations of the third and fourth degree in a single variable, and systems consisting of two equations in two variables. This is a comparatively small class; but if there be added systems consisting of a single equation of the third or fourth degree and a number of others all of the first degree, it will include all equations which admit, in general, of an algebraic solution.The object of these notes is to point out the advantage of making greater use, than is generally the custom, of the discriminant in the solution of equations, and to emphasise the importance of looking for geometrical illustrations of analytical methods whenever this is possible. The usefulness of such illustrations is well known in considering, for example, limiting cases in the solution of equations, such as the cases of infinite roots and of equal roots, of the meaning of which it is almost impossible to form an adequate conception, without the use of such illustrations.