Search

Our purpose is to shed some new light on problems arising from a study of the classical Single Ladder Problem (SLP). The basic idea is to convert the SLP to a corresponding Single Staircase Problem. The main result (Theorem 1) shows that this idea works fine and new results can be obtained by just calculating rational solutions of an algebraic equation. Some examples of such concrete calculations are given and examples of new results are also given. In particular, we derive a number of integer SLPs with congruent ladders, where a set of rectangular boxes with integer sides constitutes a staircase along a common ladder. Finally, the case with a regular staircase along a given ladder is investigated and illustrated with concrete examples.

This paper is devoted to the analytical treatment of trinomial equations of the form \(y^n+y=x,\) where \(y\) is the unknown and \(x\in\mathbb{C}\) is a free parameter. It is well-known that, for degree \(n\geq 5,\) algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [1,2] for example) requires the use of power series, following the seminal work of Lagrange [3]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance.

The aim is to put new light on the single ladder problem (SLP). Some new methods for finding complete integer solutions to the corresponding quartic equation z 4 &minus, 2 L z 3 + ( L 2 &minus, a 2 &minus, b 2 ) z 2 + 2 L a 2 z &minus, L 2 a 2 = 0 are developed. For the case L &ge, L min , these methods imply a complete parametric representation for integer solutions of SLP in the first quadrant. Some corresponding (less complete) results for the case L &gt, L min are also pointed out.

In this paper, we present a new root-finding algorithm to compute a non-zero real root of the transcendental equations using exponential series. Indeed, the new proposed algorithm is based on the exponential series and in which Secant method is special case. The proposed algorithm produces better approximate root than bisection method, regula-falsi method, Newton-Raphson method and secant method. The implementation of the proposed algorithm in Matlab and Maple also presented. Certain numerical examples are presented to validate the efficiency of the proposed algorithm. This algorithm will help to implement in the commercial package for finding a real root of a given transcendental equation. The author is thankful to the editor and reviewers for providing valuable inputs to improve the present format of manuscript.

With the help of the method of Lagrange multipliers and KKT theory, we investigate the structure and existence of optimal solutions of the continuous-time model of consumption with satiation. We show that the differential equations have no solutions in the C1 class but that solutions exist in a wider space of functions, namely, the space of functions of bounded variation with non-negative Borel measures as controls. We prove our theorems with no additional assumptions about the structure of the control Borel measures. We prove the conjecture made in the earlier literature, that there are only three types of solutions: I-shaped solutions, with a gulp of consumption at the end of the interval and no consumption at the beginning or in the interior; U-shaped solutions, with consumption in the entire interior of the interval and gulps at the beginning and the end; and intermediate (J-shaped) solutions, with an initial interval of abstinence followed by a terminal interval of distributed consumption at rates and a gulp at the end. We also establish the criteria that permit determination of the solution type using the problem's parameters. When the solution structure is known, we reduce the problem of the existence of a solution to algebraic equations and discuss the solvability of these equations. We construct explicit solutions for logarithmic utility and CRRA utility.

In this paper we deal with the resolution of third and fourth degree equations by means of radicals. We present some historical details about this fundamental problem. We also discuss practical methods for the resolution of third and fourth degree equations from the algebraic point of view. Finally, we introduce some results about the equations of greater degree.

Several areas, such as physical and health sciences, require the use of matrices as fundamental tools for solving various problems. Matrices are used in real-life contexts, such as control, automation, and optimization, wherein results are expected to improve with increase of computational precision. However, special attention should be paid to ill-conditioned matrices, which can produce unstable systems; an inadequate handling of precision might worsen results since the solution found for data with errors might be too far from the one for data without errors besides increasing other costs in hardware resources and critical paths. In this paper, we make a wake-up call, using 2 × 2 matrices to show how ill-conditioning and precision can affect system design (resources, cost, etc.). We first demonstrate some examples of real-life problems where ill-conditioning is present in matrices obtained from the discretization of the operational equations (ill-posed in the sense of Hadamard) that model these problems. If these matrices are not handled appropriately (i.e., if ill-conditioning is not considered), large errors can result in the computed solutions to the systems of equations in the presence of errors. Furthermore, we illustrate the generated effect in the calculation of the inverse of an ill-conditioned matrix when its elements are approximated by truncation. We present two case studies to illustrate the effects on calculation errors caused by increasing or reducing precision to s digits. To illustrate the costs, we implemented the adjoint matrix inversion algorithm on different field-programmable gate arrays (FPGAs), namely, Spartan-7, Artix-7, Kintex-7, and Virtex-7, using the full-unrolling hardware technique. The implemented architecture is useful for analyzing trade-offs when precision is increased; this also helps analyze performance, efficiency, and energy consumption. By means of a detailed description of the trade-offs among these metrics, concerning precision and ill-conditioning, we conclude that the need for resources seems to grow not linearly when precision is increased. We also conclude that, if error is to be reduced below a certain threshold, it is necessary to determine an optimal precision point. Otherwise, the system becomes more sensitive to measurement errors and a better alternative would be to choose precision carefully, and/or to apply regularization or preconditioning methods, which would also reduce the resources required.

We consider a problem of the construction of feedback regulator which synthesizes the assigned stochastic sensitivity of the equilibrium in stochastically forced nonlinear dynamic system. In the case of complete information, it is shown that this problem can be reduced to the solution of the matrix algebraic equation. A presence of noise in measurements deforms the stochastic sensitivity. We find conditions when such deformation is extremely large, and the considered problem is ill-posed. For this ill-posed problem, a regularization method is suggested. We propose an analytical approach which allows us to take into account a presence of noise in measurements when we construct an optimal feedback regulator. General theoretical results are illustrated by examples.

Trabajo presentado al 20th IFAC (International Federation of Automatic Control) World Congress, celebrado en Toulouse (Francia) del 9 al 14 de julio de 2017., This paper addresses a robust periodic economic model predictive control (EMPC) based on interval arithmetic with unknown-but-bounded additive disturbances for the management of water distribution networks (WDNs). The system constraints in presence of system disturbances are tightened along the prediction horizon by means of the proposed interval arithmetic and considering that system variables in the WDN model are also subject to some algebraic equations. These algebraic equations should be satisfied when the disturbances have effects on those system variables. The EMPC controller is designed with adding the terminal state constraint. The periodically optimal steady states are obtained by employing a periodic EMPC planner with the nominal model. This periodic steady states are subsequently used to be terminal states. The on-line robust constraint satisfactions are implemented into the robust periodic EMPC controller. Finally, the proposed control strategy is verified using a case study., This work has been partially funded by the Spanish Government and FEDER through the projects CICYT ECOCIS (ref. DPI2013-48243-C2-1-R), CICYT DEOCS (ref. DPI2016-76493-C3-3-R) and CICYT HARCRICS (ref. DPI2014-58104-R).

In this note we study multivariate perturbations of algebraic equations. In general, it is not possible to represent the perturbed solution as a Puiseux-type power series in a connected neighborhood. For the case of two perturbation parameters we provide a sufficient condition that guarantees such a representation. Then, we extend this result to the case of more than two perturbation parameters. We motivate our study by the perturbation analysis of a weighted random walk on the Web Graph. In an instance of the latter the stationary distribution of the weighted random walk, the so-called Weighted PageRank, may depend on two (or more) perturbation parameters in a manner that illustrates our theoretical development.

The concept of generalized imaginary numbers and the theory of special polynomials as well are pivoting elements in pure and applied Mathematics. Special polynomials can be viewed as a realization of the algebraic abstract notion of generalized imaginary unit. We unify different concepts, emerged in the past or in more recent times, like Cardan polynomials, Chebyshev exponents and Ultra-Radicals, within a common framework yielding new tools to deal with the theory of algebraic equations. © 2014, Springer Basel.

D'Alembert's proof of the fundamental theorem of algebra (FTA), the first published, is still widely misunderstood. Typical of d'Alembert, his work is bold and imaginative but in need of significant repair. The proof is examined in detail, in both the 1746 and 1754 versions, along with commentary over 250 years and recent efforts to revive d'Alembert's reputation. A particular challenge is to work with algebraic equations while avoiding dependence on the FTA itself. A repaired version is offered.

We consider in this paper the conditions under which the iterative methods of Euler-Chebyshev's type fail and a numerical typical example is presented.

This paper deals with an index-2 notion for linear implicit difference equations (LIDEs) and with the solvability of initial value problems (IVPs) for index-2 LIDEs. Besides, the cocycle property as well as the multiplicative ergodic theorem of Oseledets type are also proved. (C) 2010 Elsevier Inc. All rights reserved.

In this article, the collocated parameter models are used to estimate the effective thermal conductivity of the two-phase materials. The algebraic equations are derived using a unit-cell-based isotherm approach for a two- and three-dimensional spatially periodic medium. The geometry of the medium is considered as matrix of touching and nontouching in-line square and circular cylinders as well as touching and nontouching in-line solid and hollow cubes. The models are used to predict the thermal conductivity of numerous two-phase materials (maximum conductivity ratio of 1000 and concentration ranging between 0 and 1). The estimated thermal conductivity data is in good agreement with experimental data within ± 16.67% deviation for various two-phase systems. The results are compared with the standard models and experimental data for different types of two-phase systems; it is shown that the touching in-line solid cube model predicts better as compared to other geometrical configurations.

In this paper, a lumped parameter model of electromechanical systems is developed and its solution is simplified by using look-up tables (LT). The model is developed in two steps. First, a Simulink model of the electromechanical system is built using algebraic equations of the magnetic circuit which contains nonlinear effects. In the second step, the model is simplified using LT which replace the nonlinear effects. LT are created by the program developed in MATLAB environment. The simulation run-times of the simplified model is about 50% less than those of the first model while the accuracy of the solutions remains same. Experimental studies are evaluated on two different electromechanical systems to show the accuracy of the new model. Simulation results were compared with the experimental results and a good agreement is obtained between them.

It may happen that the equations governing the response of dynamical systems have some parameters whose values may not be known a priori and have to be obtained using parameter estimation schemes. In this article, we present a parameter estimation scheme for a class of sequential hybrid systems. By hybrid systems, we refer to those systems whose response is described by different governing equations corresponding to various regimes/modes of operation along with some criteria to switch between the same. In a sequential hybrid system, the different modes are arranged in a specific sequence and the system can switch from a given mode to either the previous mode or the following mode in this sequence. Here, we consider those systems whose governing equations consist of ordinary differential equations and algebraic equations. The conditions for switching between the various modes (referred to as transition conditions) are in the form of linear inequalities involving the system output. We shall first consider the case where the transition conditions are known completely. We present a parameter update scheme along with sufficient conditions that will guarantee bounded parameter estimation errors. Then, we shall consider the case where the transition conditions are not known in the sense that some parameters in these conditions are not known. We present a parameter estimation scheme for this case. We illustrate the performance of the parameter estimation scheme in both cases with some examples.

In this paper, we first present a componentwise existence and nonexistence theorem for solutions of nonlinear equations in R n or C n, which improves those of Alefeld (1986) and Yamamoto (1978,1979,1980,1984). Next, the theorem is applied to algebraic equations. Finally, the results are illustrated with examples.

The method of identifying first order plus time delay transfer function model proposed for unstable systems by Ananth and Chidambaram [Closed loop identification to transfer function model for unstable systems, J. Franklin Inst. 336 (1999) 1055-1061] is modified to avoid the stability problems [Cheres, Parameter estimation of an unstable system with a PID controller in a closed loop configuration, J. Franklin Inst., 2005, accepted for publication] of the method. Two modifications are proposed. In the first modification of the method, the under-determined algebraic equations problem is converted into an optimization problem for calculation of the three parameters of the first order plus time delay (FOPTD) model. A simple method is given for the initial guess values of the model parameters. In the second approach, from the definition of Laplace transform of the output response, a third equation is formulated. The resulted three equations, in terms of the three parameters of the transfer function model, are then numerically solved. Simulation results are given for the second order plus time delay transfer function considered by Cheres 2005 [Parameter estimation of an unstable system with a PID controller in a closed loop configuration, J. Franklin Inst., 2005, accepted for publication]. The responses of the identified models with the same PID controllers are compared with that of the actual system. PID controllers are designed based on the identified models. The closed loop responses of the controllers on the original system are evaluated and compared. The present methods give better control performances. � 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Published under the name Marie-Paule Gascuel; International audience; 3D models of the lips have been developed in the framework of an audiovisual articulatory speech synthetizer. Unlike most of the regions of the human face, the lips are essentially characterized by their border contours. The internal and external contours of the vermilion zone can be fitted by means of algebraic equations. The coefficients of these equations must be controlled so that the lip shape can be adapted to various speakers conformations and to any speech gesture. To reach this goal, a 3D model of the lips has been worked out from geometrical analysis of the natural lips of a French speaker. Our lip model was developed to adjust a set of continuous functions best fitting the contours of 22 reference lip shapes. Only five parameters are necessary to predict all the equations of the contours of the lip model. From this model, a volumic model based on implicit surfaces was also developped to take in account lip contact.

We present two applications of jacobian nullwerte, both related with the resolution of algebraic equations of any degree. We give a very simple expression of the roots of a polynomial of arbitrary degree in terms of derivatives of hyperelliptic theta functions. This expression can be understood as an explicit proof of Torelli's theorem in the hyperelliptic case. We also give geometrical expressions of the discriminant of a polynomial. Both applications are based on a jacobian version of Thomae's formula.

A numerical method based on orthogonal triangular functions (TFs) is proposed to approximate the solution of Fredholm integral equations systems. The powerful properties of orthogonal triangular functions are utilized in a direct method to reduce a system of Fredholm integral equations to a system of mere algebraic equations. The proposed method does not need any integration for obtaining the constant coefficients hence, it can be applied in a simple and fast technique. Convergence analysis and associated theorems are considered. Some numerical examples illustrate the accuracy and computational efficiency of the proposed method.