Arithmetic Geometric Mean Research Articles

Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales

This paper is concerned with the oscillatory properties of even order advanced type dynamic equation with mixed nonlinearities of the form $$\bigl[r(t)\varPhi_\alpha\bigl(x^{\Delta^{n-1}}(t) \bigr) \bigr]^\Delta+ p(t)\varPhi_\alpha\bigl(x\bigl(\delta(t)\bigr) \bigr) +\sum_{i=1}^kp_i(t) \varPhi_{\alpha_i} \bigl(x\bigl(\delta(t)\bigr) \bigr)=0 $$ on an arbitrary time scale \(\mathbb{T}\), where Φ∗(u)=|u|∗−1u. We present some new oscillation criteria for the equation by introducing parameter functions, establishing a new lemma, using a Hardy-Littlewood-Polya inequality and an arithmetic-geometric mean inequality and developing a generalized Riccati technique. Our results extend and supplement some known results in the literature. Several examples are given to illustrate our main results.

Contribuição para o estabelecimento de níveis de referência para a concentração de mercúrio no sangue de crianças na cidade do Rio de Janeiro

INTRODUÇÃO: O mercúrio é um metal de elevada toxicidade e as crianças representam um dos subgrupos mais susceptíveis da população, tornando relevante estudos sobre valores de referência deste metal em populações não expostas. OBJETIVO: Estabelecimento de níveis de referência de mercúrio no sangue de crianças em uma área urbana. MATERIAL E MÉTODOS: Foi realizado um estudo transversal da concentração de mercúrio total no sangue capilar em 220 crianças escolares de 8 a 10 anos de ambos os sexos de 2 escolas da rede municipal de ensino do Município do Rio de Janeiro. Foram aplicados dois questionários e coletadas amostras de sangue. A análise do teor de mercúrio no sangue foi realizado por espectrometria de massa acoplado a plasma indutivo. O teste não paramétrico de Mann-Whitney foi utilizado na análise estatística dos dados. RESULTADOS: As médias aritmética, geométrica e a mediana foram respectivamente de 0,89, 0,51 e 0,71 µg/L de mercúrio total das crianças participantes. Estes resultados, que podem contribuir para comparação com outras pesquisas em áreas impactadas ambientalmente pelo mercúrio, são compatíveis com inquéritos internacionais e abaixo dos limites para populações não expostas indicados pela Organização Mundial da Saúde.

Adaptive Synchronization of One-Dimensional Discrete Chaotic Systems on Complex Networks

A method of adaptive synchronization of one-dimensional discrete chaotic systems on complex networks is proposed. The nodes of complex networks are constructed by one-dimensional discrete chaotic systems, we consider a general drive-response synchronization model of one-dimensional discrete chaotic systems on complex dynamical networks. Based on the adaptive control technique, the parameter adaptive laws and property conversion laws are given to achieve synchronization and parameters identification simultaneously. Simulation results show that the arithmetic average and geometric mean of all the nodes states are equal, furthermore, the unknown node parameters can be successfully identified, all nodes are transformed to drive nodes. This indicates that chaos synchronization is reached in the whole networks.

Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging

The traditional economic production quantity (EPQ) model assumes that the production products are all perfect. It is not always true in the real production system, due to imperfect production process or other factors, imperfect quality items may be produced. Furthermore, it is well-known that the total production-inventory costs can be reduced by reworking the imperfect quality items produced with a relatively smaller additional reworking and holding costs. In addition, the permissible delay in payments offered by the supplier is widely adopted in the practical business market. In this study, we explore the effects of the reworking imperfect quality items and trade credit on the EPQ model with imperfect production processes and complete backlogging. A mathematical model which includes the reworking and shortage costs, interest earned and interest charged is presented. Besides, an arithmetic-geometric mean inequality approach is employed and an algorithm is developed to find the optimal production policy. Furthermore, some numerical examples and sensitivity analysis are provided to demonstrate the proposed model.

Seven Means, Generalized Triangular Discrimination, and Generating Divergence Measures

Jensen-Shannon, J-divergence and Arithmetic-Geometric mean divergences are three classical divergence measures known in the information theory and statistics literature. These three divergence measures bear interesting inequality among the three non-logarithmic measures known as triangular discrimination, Hellingar’s divergence and symmetric chi-square divergence. However, in 2003, Eve studied seven means from a geometrical point of view, which are Harmonic, Geometric, Arithmetic, Heronian, Contra-harmonic, Root-mean square and Centroidal. In this paper, we have obtained new inequalities among non-negative differences arising from these seven means. Correlations with generalized triangular discrimination and some new generating measures with their exponential representations are also presented.

The complex AGM, periods of elliptic curves over C and complex elliptic logarithms

We give an account of the complex Arithmetic–Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over C, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over C. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we have implemented in both MAGMA and Sage, is illustrated with several examples using elliptic curves defined over number fields with real and complex embeddings.

Refinements and reverses of means inequalities for Hilbert space operators

In this paper we derive some improvements of means inequalities for Hilbert space operators. More precisely, we obtain refinements and reverses of the arithmetic-geometric operator mean inequality. As an application, we also deduce an improved variant for the refined arithmetic--Heinz mean inequality. We also present some eigenvalue inequalities for differences of certain operator means.

Nested Inequalities Among Divergence Measures

In this paper we have considered an inequality having 11 divergence measures. Out of them three are logarithmic such as Jeffryes-Kullback-Leiber (4) (5) J-divergence. Burbea-Rao (1) Jensen-Shannon divergenceand Taneja (7) arithmetic-geometric mean divergence. The other three are non-logarithmic such as Hellinger discrimination, symmetric ´ 2 idivergence, and triangular discrimination. Three more are considered are due to mean divergences. Pranesh and Johnson (6) and Jain and Srivastava (3) studied different kind of divergence measures. We have considered measures arising due to differences of single inequality having 11 divergence measures in terms of a sequence. Based on these differences we have obtained many inequalities. These inequalities are kept as nested or sequential forms. Some reverse inequalities and equivalent versions are also studied.

More Results on Singular Value Inequalities for Compact Operators

The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and are compact operators on a complex separable Hilbert space, then Hirzallah has proved that if are compact operators, then We give inequality which is equivalent to and more general than the above inequalities, which states that if are compact operators, then

The improved arithmetic-geometric mean inequalities for matrix norms

In this paper, we prove a scalar inequality such that this inequality is an improvement of the classical arithmetic-geometric mean inequality. We obtain its matrix version and investigate Hilbert-Schmidt and trace norm of this matrix version

MM algorithms for geometric and signomial programming

This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the MM algorithm. In this setting, one can apply the geometric-arithmetic mean inequality and a supporting hyperplane inequality to create a surrogate function with parameters separated. Thus, unconstrained signomial programming reduces to a sequence of one-dimensional minimization problems. Simple examples demonstrate that the MM algorithm derived can converge to a boundary point or to one point of a continuum of minimum points. Conditions under which the minimum point is unique or occurs in the interior of parameter space are proved for geometric programming. Convergence to an interior point occurs at a linear rate. Finally, the MM framework easily accommodates equality and inequality constraints of signomial type. For the most important special case, constrained quadratic programming, the MM algorithm involves very simple updates.

An Eloquent Formula for the Perimeter of an Ellipse

T he values of complete elliptic integrals of the first and the second kind are expressible via power series representations of the hypergeometric function (with corresponding arguments). The complete elliptic integral of the first kind is also known to be eloquently expressible via an arithmetic-geometric mean, whereas (before now) the complete elliptic integral of the second kind has been deprived such an expression (of supreme power and simplicity). With this paper, the quest for a concise formula giving rise to an exact iterative swiftly convergent method permitting the calculation of the perimeter of an ellipse is over!

An easy method to derive the integrated vendor–buyer production–inventory model with backordering using cost-difference rate comparison approach

In growing competitive markets, close cooperative strategies among the vendors and the buyers are essential to cut the joint inventory cost and to have less response time to the supply chain players. Recently, Yang et al. (2007) [2] have presented an integrated two-stage production–inventory model and used the classical differential calculus method to derive the optimal lot size for the vendor and the buyer, and the number of deliveries in supply chain. Moreover, numerous researchers adopted other algebraic methods to derive the optimal solution for inventory models. However, in these papers, the cost comparisons method and arithmetic–geometric mean inequality do not focus on explicitly developing the mathematical expressions for two-stage inventory system with the backorders. In this study, we extend the inventory model proposed by Yang et al. (2007) [2], and consider a three-variable inventory problem and present an alternative approach to determine the optimal replenishment policy for the integrated vendor–buyer inventory system with backordering utilizing the cost-difference rate comparison approach.

On the validity of the arithmetic–geometric mean method to locate the optimal solution in a supply chain system

Cardenas-Barron [Cardenas-Barron, L.E. (2010) ‘A Simple Method to Compute Economic order Quantities: Some Observations’, Applied Mathematical Modelling, 34, 1684–1688] indicates that there are several functions in which the arithmetic–geometric mean method (AGM) does not give the minimum. This article presents another situation to reveal that the AGM inequality to locate the optimal solution may be invalid for Teng, Chen, and Goyal [Teng, J.T., Chen, J., and Goyal S.K. (2009), ‘A Comprehensive Note on: An Inventory Model under Two Levels of Trade Credit and Limited Storage Space Derived without Derivatives’, Applied Mathematical Modelling, 33, 4388–4396], Teng and Goyal [Teng, J.T., and Goyal S.K. (2009), ‘Comment on ‘Optimal Inventory Replenishment Policy for the EPQ Model under Trade Credit Derived without Derivatives’, International Journal of Systems Science, 40, 1095–1098] and Hsieh, Chang, Weng, and Dye [Hsieh, T.P., Chang, H.J., Weng, M.W., and Dye, C.Y. (2008), ‘A Simple Approach to an Integrated Single-vendor Single-buyer Inventory System with Shortage’, Production Planning and Control, 19, 601–604]. So, the main purpose of this article is to adopt the calculus approach not only to overcome shortcomings of the arithmetic–geometric mean method of Teng et al. (2009), Teng and Goyal (2009) and Hsieh et al. (2008), but also to develop the complete solution procedures for them.

Some New Information Inequalities Involving f-Divergences

Abstract New information inequalities involving f-divergences have been established using the convexity arguments and some well known inequalities such as the Jensen inequality and the Arithmetic-Geometric Mean (AGM) inequality. Some particular cases have also been discussed.

Jessen's functional, its properties and applications

AbstractIn this paper we consider Jessen's functional, defined by means of a positive isotonic linear functional, and investigate its properties. Derived results are then applied to weighted generalized power means, which yields extensions of some recent results, known from the literature. In particular, we obtain the whole series of refinements and converses of numerous classical inequalities such as the arithmetic-geometric mean inequality, Young's inequality and Hölder's inequality

HYPERGEOMETRIC SERIES AND ARITHMETIC–GEOMETRIC MEAN OVER 2-ADIC FIELDS

Dwork proved that the Gaussian hypergeometric function on p-adic numbers can be extended to a function which takes values of the unit roots of ordinary elliptic curves over a finite field of characteristic p ≥ 3. We present an analogous theory in the case p = 2. As an application, we give a relation between the canonical lift and the unit root of an elliptic curve over a finite field of characteristic 2 by using the 2-adic arithmetic–geometric mean.

Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators

The main objective of this paper is an improvement of the original weighted operator arithmetic-geometric mean inequality in Hilbert space. We define the difference operator between the arithmetic and geometric means and investigate its properties. Due to the derived properties, we obtain a refinement and a converse of the observed operator mean inequality. As an application, we establish one significant operator mean, which interpolates the arithmetic and geometric means, that is, the Heinz operator mean. We also obtain an improvement of this interpolation.

Binning algorithm for high-resolution IRS-P4 OCM chlorophyll image

Daily chlorophyll-a concentration from the Ocean Colour Monitor (OCM) sensor onboard the Indian Remote Sensing satellite (IRS-P4) is used to make weekly and monthly chlorophyll-a concentration maps. The pathwise swath data at 12 noon for every alternate day over the north Indian Ocean (NIO) during February 2004 and February 2005 were used to compare the existing algorithms for binning the data. Atmospherically corrected and geocorrected OCM data were used in the comparative study of three averaging algorithms – arithmetic mean (AVG), geometric mean (GEO) and maximum likelihood estimator (MLE). The analysis shows that the AVG algorithm is best suited when compared with the two other algorithms. However, for case 1 water, MLE gives nearly the same value as AVG. Based on this result, AVG was selected for operational weekly and monthly averaging of OCM data over the NIO. These high-resolution-derived chlorophyll-a weekly and monthly products will be useful to resolve inter-annual-to-decadal changes in chlorophyll-a concentration over the NIO.

Hyperelliptic integrals and generalized arithmetic–geometric mean

We show how certain determinants of hyperelliptic periods can be computed using a generalized arithmetic-geometric mean iteration, whose initialisation parameters depend only on the position of the ramification points. Special attention is paid to the explicit form of this dependence and the signs occurring in the real domain.