Univariate Equation Research Articles

The choice of a foreign price measure in a Bayesian estimated new-Keynesian model for Israel

We estimate a small DSGE model by full information Bayesian techniques on the basis of Israeli data from 1995 to 2006. The model was first developed and estimated by means of classical GMM in Argov and Elkayam (2010), and since then it has been used at the Bank of Israel for monetary policy analysis. It is widely believed that in 2007 (out of sample year) as elsewhere worldwide, inflation rose in Israel due to high commodity prices in global markets. However, our baseline model attributes most of the high inflation in 2007 to supply shocks. One conjecture is that this model's result derives from the inappropriate original use of the unit value of imported consumer goods (which do not include unprocessed food and energy) as the main foreign price measure. We test this conjecture by re-estimating the model with various other foreign price measures that typically do reflect the global rise in commodity prices and compare the log-marginal likelihoods. We find that no other price measure outperforms the original choice in the sample period. Only the foreign trade-weighted CPI equals the performance of the original choice while improving the 2007 interpretation of inflation, and should therefore be considered the main foreign price measure. The proposed methodology for comparing the suitability of alternative measures for observable variables can be applied to any model with exogenous variables that are characterized by univariate equations.

A note on Newton's method for system of stochastic differential equations

Kawabata and Yamada (1991) proposed an implicit formulation for Newton's method for an univariate stochastic differential equation (SDEs). Amano (2009) used the linearized equation technique and proposed explicit formulation for the Newton scheme. In this note, we extend the Newton method for univariate SDEs to the multivariate cases. The error analysis is given and some examples are proposed. Results show that the method works well.

Root isolation of zero-dimensional polynomial systems with linear univariate representation

In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the complex roots of the polynomial system are represented as linear combinations of the roots of several univariate polynomial equations. An algorithm is proposed to compute such a representation for a given zero-dimensional polynomial equation system based on Gröbner basis computation. The main advantage of this representation is that the precision of the roots of the system can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for isolating the roots of the univariate equations in order to obtain roots of the polynomial system with a given precision. As a consequence, a root isolating algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation.

Forward Displacement Analysis of a Parallel Manipulator

A new algebraic method for the solution of the forward displacement analysis of a parallel manipulator is presented in this paper. Based on the algebraic method, the problem of the forward displacement problem is reduced to a polynomial equation in a single unknown from a constructed matrix which is relative small in the size. From the univariate equation, all closed-form solutions of the different locations of the mechanism can be derived.

Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = ( p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.

Univariate Equations Over Sets of Natural Numbers

It is shown that equations of the form p(X) = ψ(X), in which the unknown X is a set of natural numbers and p, ψ use the operations of union, intersection and addition of sets S + T = {m + n |, m ∈ S, n ∈ T}, can simulate systems of equations pj(X1, …, Xn) = pj(X1, …, Xn) with 1 ≤ j ≤ e, in the sense that solutions of any such system are encoded in the solutions of the corresponding equation. This implies computational universality of least and greatest solutions of equations p(X) = ψ(X), as well as undecidability of their basic decision problems. It is sufficient to use only singleton constants in the construction. All results equally apply to language equations over a one-letter alphabet Σ = {a}.

Forward Kinematics Analysis of the General 6-6 Platform Parallel Mechanism Based on Algebraic Elimination

A new algebraic elimination method for the forward kinematics analysis of the general 6-6 platform parallel mechanism is presented.Six in nine constraint quadratic equations are transformed into linear equations by introducing substitution variables and six in nine variables are eliminated by using Cramer algorithm.The reduced Groebner basis under degree lexicographic ordering for the substitution equations and the remainder closed-form equations are obtained.A univariate equation of higher degree is derived from the determinant of the Sylvester's matrix,constructed by the 4th degree subset of the Groebner basis,the size of which is 15×15. Based on computer symbolic manipulating,it can be concluded that the degree of the univariate polynomial equation is at most 20 and the number of closed-form solutions is at most 40.It is proved theoretically that there are many completely different resultants which can derive all closed-form solutions in terms of different term orderings through changing the degree lexicographic order.The direct kinematics of the general 6-6 Stewart platform can be solved directly by the 15 derived equations.And the mathematical mechanized solution of the problem can be realized.The result is verified by a numerical example,whose solutions agree with the original equations without extraneous roots.

Five precision point-path synthesis of planar four-bar linkage using algebraic method

The problem of synthesizing a planar four-bar linkage with two given fixed pivots such that the coupler curve passes through five given points is considered with the Groebner-Sylvester hybrid approach. First, closed-form equations of a single point are constructed. The reduced Groebner basis in degree lexicographic ordering for the closed-form equations is then obtained using computer algebra. A 23 × 23 Sylvester’s matrix can be constructed by selecting 23 out of 89 Groebner bases. A 36th degree univariate equation is obtained directly from the determinate of the matrix. The same result can be obtained with a continuation method. A numerical example is given and verifies that the problem has at most 36 solutions in the complex field.

Optimal Choice of the Shaft Angle for Involute Gear Hobbing

With reference to the machining of an involute spur or helical gear by the hobbing process, the paper suggests a new criterion for selecting the position of the hob axis relative to the gear axis. By adhering to the proposed criterion, the hob axis is set at the minimum distance from the gear axis, thus maximizing the depth of the tooth spaces of the gear. The new criterion is operatively implemented by solving a univariate equation, which stems from a new, synthetic analysis of the meshing of crossed-axis involute gears. A numerical example shows application of the suggested procedure to a case study and compares the optimal hob setting to the customary one.

Predictive financial models of the euro area: A new evaluation test

This paper investigates the predictive ability of financial variables for euro area growth. Our forecasts are built from univariate autoregressive and single equation models. Euro area aggregate forecasts are constructed both by employing aggregate variables and by aggregating country-specific forecasts. The forecast evaluation is based on a recently developed test for equal predictive ability between nested models. Employing a monthly dataset from the period between January 1988 and May 2005 and setting the out-of-sample period to be from 2001 onwards, we find that the single most powerful predictor on a country basis is the stock market returns, followed by money supply growth. However, for the euro area aggregate, the set of most powerful predictors includes interest rate variables as well. The forecasts from pooling individual country models outperform those from the aggregate itself for short run forecasts, while for longer horizons this pattern is reversed. Additional benefits are obtained when combining information from a range of variables or combining model forecasts.

Relationships between board structure and takeover defenses

PurposeAs corporations continue to face substantial information asymmetries between managers and shareholders, they must decide how to mitigate this agency problem using various mechanisms of corporate governance. Two of these mechanisms include the board of directors and takeover defenses. The purpose of this paper is to show that rather than having an additive effect, the marginal benefit of using one of these mechanisms declines as the use of the other increases.Design/methodology/approachUsing a governance index that measures the firm's takeover defenses and a unique board of directors index, the study employs both univariate analyses and multivariate simultaneous equation modeling in order to test the hypotheses.FindingsThe results of these tests show that these two measures of corporate governance should be viewed as a set rather than as individual components. In other words, a strong board is inversely related to strong shareholder protection in the form of takeover defenses. The study further analyzes the extent to which growth opportunities influence this relationship using a logistic regression approach. It appears that a firm's opportunity for growth is related to the strength of its board of directors, yet not to its governance provisions.Originality/valueBy analyzing a specific combination of current corporate governance practices, these results can assist firms at a practical level by providing information on optimal solutions to the agency problem.

Continuous Collision Detection for Two Moving Elliptic Disks

Collision detection and avoidance are important in robotics. Compared with commonly used circular disks, elliptic disks provide a more compact shape representation for robots or other vehicles confined to move in the plane. Furthermore, elliptic disks allow a simpler analytic representation than rectangular boxes, which makes it easier to perform continuous collision detection (CCD). We shall present a fast and accurate method for CCD between two moving elliptic disks, which avoids any need to sample the time domain of the motion, thus avoiding the possibility of missing collisions between time samples. Based on some new algebraic conditions on the separation of two ellipses, we reduce collision detection for two moving ellipses to the problem of detecting real roots of a univariate equation, which is the discriminant of the characteristic polynomial of the two ellipses. Several techniques are investigated for robust and accurate processing of this univariate equation for two classes, of commonly used motions: planar cycloidal motions and planar rational motions. Experimental results demonstrate the efficiency, accuracy, and robustness of our method

The geometry of rational parameterized representations

We study the projective space \(\mathbb{S}_n^d \) of univariate rational parameterized equations of degree d or less in real projective space \(\mathbb{P}^n. \) The parameterized equations of degree less than d form a special algebraic variety \(\mathcal{K}_1. \) We investigate the subspaces on \(\mathcal{K}_1 \) and their relation to rational curves in \(\mathbb{P}^n, \) give a geometric characterization of the automorphism group of \(\mathcal{K}_1 \) and outline applications of the theory to projective kinematics.

Improved dialytic elimination algorithm for the forward kinematics of the general Stewart–Gough platform

This paper presents the forward kinematics of the general Stewart–Gough platform using dialytic elimination. The proposed algorithm, when implemented in a computer language, provides all the solutions in fairly less computation time compared to the existing ones, mainly due to the fact that it directly leads to a 40th-degree univariate equation from a constructed 15×15 Sylvester’s matrix which is relatively small in size. In addition, the ways to avoid ill-conditioned linear systems in the solution procedure are dealt with for the purpose of enhancing the accuracy of the computed poses. Numerical examples are given to demonstrate the developed algorithm.

Generation and Forward Displacement Analysis of RP_R-PR-RP_R Analytic Planar Parallel Manipulators

Analytic manipulators are manipulators for which a characteristic polynomial of fourth degree or lower can be obtained symbolically. Six types of RP_R-PR-RP_R analytic planar parallel manipulators (APPMs) are first generated using the component approach and the method based on the structure of the univariate equation. Of the six types, four are composed of Assur II kinematic chains while the other two are composed of Assur III kinematic chains. The forward displacement analysis (FDA) of two types of RP_R-PR-RP_R APPMs composed of Assur III kinematic chains is then performed. The FDA of each of the two types of APPMs composed of Assur III kinematic chains is reduced to the solution of a univariate cubic equation and a quadratic equation in sequence. It is also proven that the maximum number of real solutions to the FDA is 4 for the RP_R-PR-RP_R planar parallel manipulator with one aligned platform and one orthogonal platform. Examples with 4 real solutions for the RP_R-PR-RP_R planar parallel manipulator with one aligned platform and one orthogonal platform or 6 real solutions for the RP_R-PR-RP_R planar parallel manipulator with two aligned platforms are given at the end of this paper.

Solving systems of algebraic equations

Let f 1 ,…, f k be k multivariate polynomials which have a finite number of common zeros in the algebraic closure of the ground field, counting the common zeros at infinity. An algorithm is given and proved which reduces the computations of these zeros to the resolution of a single univariate equation whose degree is the number of common zeros. This algorithm gives the whole algebraic and geometric structure of the set of zeros (multiplicities, conjugate zeros,…). When all the polynomials have the same degree, the complexity of this algorithm is polynomial relative to the generic number of solutions.

Forward kinematics of the general 6–6 Stewart platform using algebraic elimination

It is known that the number of solutions of the forward kinematics of the general 6–6 Stewart platform is up to 40 in the complex domain. Finding all the solutions is necessary when all the possible postures are of interest or a unique actual posture is to be determined by choosing the correct one out of the solutions. In such cases, algebraic elimination method can be a powerful tool. In this article, an elimination procedure for the forward kinematics of the general 6–6 Stewart platform is presented, which leads to a 40th-degree univariate equation without factoring out or deriving the greatest common divisor. In addition, the proposed method, when implemented in a computer language, requires less computation time than the existing algorithms do mainly because the size of the final Sylvester's matrix is relatively small. The algorithm is demonstrated by two numerical examples.

A kinematic structure-based classification and compact kinematic equations for six-dof industrial robotic manipulators

A kinematic structure-based classification of six degree-of-freedom industrial robotic manipulators is introduced, and a complete set of compact kinematic equations is given according to this classification. For the classification, 100 industrial robots are surveyed. These robots are first classified into main groups and then into subgroups under each main group, and nine kinematic main groups, each having 1–10 subgroups are obtained. The main groups are based on the end-effector rotation matrices and characterized by the twist angles. On the other hand, the subgroups are based on the wrist point positions and characterized by the link lengths and offsets. Compact kinematic equations are derived for all main groups and subgroups by utilizing the properties of exponential rotation matrices, and simplification tools derived by using these properties. Most of these simplified compact equations can be solved analytically and the remaining few of them can be solved semi-analytically through a single univariate equation. In these solutions, the singularities and the multiple configurations of the manipulators can be determined easily. Using these solutions, the inverse kinematics can also be computerized by means of short and fast algorithms.

Real-Time Forward Kinematics of the 6-6 Stewart Platform with One Extra Linear Sensor

This paper deals with the forward kinematics of the 6-6 Stewart platform of planar base and moving platform using one extra linear sensor. Based on algebraic elimination method, it first derives an 8th-degree univariate equation and then finds tentative solution sets out of which the actual solution is to be selected. In order to provide more exact solution despite the error between measured sensor value and the theoretic alone, a correction method is also used in this paper. The overall procedure requires so little computation time that it can be efficiently used for real-time applications. In addition, unlike the iterative scheme e.g. Newton-Raphson, the algorithm does not require initial estimates of solution and is free of the problems that it does not converge to actual solution within limited time. The presented method has been implemented in C language and a numerical example is given to confirm the effectiveness and accuracy of the developed algorithm.

Concentration fluctuations and nonlinear nonequilibrium thermodynamics

The multivariate master equation for a one-step reaction is transformed into a univariate master equation for the density of fluctuation in extent of the reaction. The density of variance of the fluctuation is incorporated into ordinary nonequilibrium thermodynamics of chemical reactions. This makes it possible to formulate a new type of Markov fluctuation–dissipation relations of nonlinear nonequilibrium thermodynamics.