Arithmetic Geometric Mean Research Articles

Inequalities for weighted sums of powers and their applications

Two inequalities for weighted sums of powers are established. Applications to Jacobian elliptic functions and Legendre’s elliptic integrals of the first kind are presented. Some known and new inequalities for circular and hyperbolic functions are obtained. Applications to certain iterative means including Gauss’ arithmetic-geometric mean and Schwab-Borchardt mean are included. Mathematics subject classification (2010): Primary: 26D15, 33E05, 26D05, 26D07.

Singular value and‎ ‎arithmetic-geometric mean inequalities for operators

‎A singular value inequality for sums and products of Hilbert space operators‎ ‎is given‎. ‎This inequality generalizes several recent singular value‎ ‎inequalities‎, ‎and includes that if $A$‎, ‎$B$‎, ‎and $X$ are positive operators‎ ‎on a complex Hilbert space $H$‎, ‎then ‎\begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XB^{^{1/2}}\right) \leq \frac{1}{2}\left\Vert‎ ‎X\right\Vert \text{ }s_{j}\left( A+B\right) \text{, ‎\‎ ‎}j=1,2,\cdots\text{,}‎ ‎\end{equation*} ‎which is equivalent to‎ ‎ \begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XA^{^{1/2}}-B^{^{1/2}}XB^{^{1/2}}\right) \leq‎ ‎\left\Vert X\right\Vert s_{j}\left( A\oplus B\right) \text{, ‎\ }j=1,2,\cdots ‎\text{.}‎ ‎\end{equation*}‎ ‎ Other singular value inequalities for sums and products of operators are‎ ‎presented‎. ‎Related arithmetic-geometric mean inequalities are also‎ ‎discussed‎.

On improved arithmetic-geometric mean and Heinz inequalities for matrices

In this paper, we first generalize an inequality and improve another one for unitarily invariant norms, which are established by Kittaneh and Manasrah in [Improved Young and Heinz inequalities for matrices. J. Math. Anal. Appl.361(2010)262-269]. Then we present a new inequality for unitarily invariant norms, which is equivalent to an inequality presented by Kittaneh and Manasrah in the case of the Hilbert-Schmidt norm.

An elementary proof of arithmetic–geometric mean inequality of the weighted Riemannian mean of positive definite matrices

The weighted Riemannian mean of positive definite matrices is a kind of weighted geometric mean of n matrices. Some properties of the weighted Riemannian mean are easily obtained. But some of them are not easy. In this short paper, a simplified proof of arithmetic–geometric–harmonic mean inequalities will be obtained.

Maximizing Sum Rate and Minimizing MSE on Multiuser Downlink: Optimality, Fast Algorithms and Equivalence via Max-min SINR

Maximizing the minimum weighted signal-to-interference-and-noise ratio (SINR), minimizing the weighted sum mean-square error (MSE) and maximizing the weighted sum rate in a multiuser downlink system are three important performance objectives in nonconvex joint transceiver and power optimization, where all the users have a total power constraint. We show that, through connections with the nonlinear Perron-Frobenius theory, jointly optimizing power and beamformers in the max-min weighted SINR problem can be solved optimally in a distributed fashion. Then, connecting these three performance objectives through the arithmetic-geometric mean inequality and nonnegative matrix theory, we solve the weighted sum MSE minimization and the weighted sum rate maximization in the weak interference regimes using fast algorithms. In the general case, we first establish optimality conditions to the weighted sum MSE minimization and the weighted sum rate maximization problems and provide their further connection to the max-min weighted SINR problem. We then propose a distributed weighted proportional SINR algorithm that leverages our fast max-min weighted SINR algorithm to solve for local optimal solution of the two nonconvex problems, and give conditions under which global optimality is achieved. Numerical results are provided to complement the analysis.

Generalized Pólya–Szegö type inequalities for some non-commutative geometric means

In this paper, we shall show generalized Pólya–Szegö type inequalities of n positive invertible operators on a Hilbert space for any integer n⩾3 in terms of the following two typical non-commutative geometric means, that is, one is the higher order weighted geometric mean due to Lawson–Lim which is an extension of the Ando–Li–Mathias geometric mean, and the other is the weighted chaotic geometric mean. Among others, the Specht ratio plays an important role in our discussion, which is the upper bound of a ratio type reverse of the weighted arithmetic–geometric mean inequality.

Using simple methods to derive EOQ and EPQ models with shortage and imperfect quality

Huang [ Journal of Statistics and Management Systems, Vol. 6 (2003), No. 2, pp. 171– 180 ] studied the EOQ (Economic Order Quantity) and EPQ (Economic Production Quantity) models with backlogging and defective items using the algebraic approach. He assumed 100% inspection policy and the known proportion of defective items was removed prior to storage or use after the screening process. In this paper, we will off er both basic inequalities, arithmetic-geometric mean inequality (AGM) and Cauchy-Bunyakovsky-Schwarz inequality (CBS), to derive both the optimal lot size and backorder level under the total relevant cost per unit time minimized. Without the method of completing perfect square, the proposed methods derive the global minimum of the total relevant cost per unit time more easily than the algebraic approach used by Huang (2003)

Estimation of concentration and exposure doses due to radon by using CR-39 plastic track detectors in the residences of Sudhnuti, Azad Kashmir, Pakistan

This paper presents the results of indoor radon concentration measurements in 120 dwellings of district Sudhnuti of Azad Kashmir. Measurements were taken with CR-39 passive alpha track detector. CR-39 based box type radon detectors were installed in a bedroom and living rooms of each house. The detectors were retrieved after exposing to indoor radon for period of 6 months and then etched in 6 M NaOH at 80°C for 16 h, the observed track densities were converted in to the indoor radon concentration. Indoor radon concentration varied from 20 ± 12 to 170 ± 4 Bq m−3 for the houses of the district Sudhnuti. Arithmetic mean (AM), geometric mean (GM) and geometric standard deviations (GSD) were found to be 82 ± 6, 77 ± 6 and 1.51, respectively. The minimum value of weighted average radon concentration was recorded in one of the house of Mang town, whereas the maximum value was found in the Pattan Sher Khan region. Doses due to indoor radon exposure vary from 0.50 ± 0.31 to 4.28 ± 0.11 mSv year−1 AM, GM and GSD. of mean effective doses were found to be 2.06 ± 0.13, 1.95 ± 0.18 and 1.51, respectively. According to the recommendations made by the Health Protection Agency, UK (200 Bq m−3) all the houses surveyed are within the safe limits.

The numerical assessment of path planning for an autonomous robotic vacuum cleaner using an ant colony optimization method

Wu and Ouyang (Production Planning & Control, Vol. 14 (2003), No. 6, pp.555–561) derived the integrated vendor-buyer system with shortages using algebraic method to determine the optimal replenishment policy. In this note, we will off er both basic inequalities, arithmetic-geometric mean inequality (AGM) and Cauchy-Bunyakovsky-Schwarz inequality (CBS), to derive both the optimal buyer's lot size and backorder level under the integrated total cost per unit time of the vendor and buyer minimized. Without the method of completing perfect square, the proposed methods derive the global minimum of the integrated total cost per unit time of the vendor and buyer more easily than the algebraic approach used by Wu and Ouyang (2003).

Using easy methods to derive an integrated vendor-buyer inventory system under shortage allowed

Wu and Ouyang (Production Planning & Control, Vol. 14 (2003), No. 6, pp.555–561) derived the integrated vendor-buyer system with shortages using algebraic method to determine the optimal replenishment policy. In this note, we will off er both basic inequalities, arithmetic-geometric mean inequality (AGM) and Cauchy-Bunyakovsky-Schwarz inequality (CBS), to derive both the optimal buyer's lot size and backorder level under the integrated total cost per unit time of the vendor and buyer minimized. Without the method of completing perfect square, the proposed methods derive the global minimum of the integrated total cost per unit time of the vendor and buyer more easily than the algebraic approach used by Wu and Ouyang (2003).

A Sequence of Inequalities among Difference of Symmetric Divergence Measures

In this paper we have considered two one parametric generalizations. These two generalizations have in particular the well known measures such as: J-divergence , Jensen-Shannon divergence and arithmetic-geometric mean divergence . These three measures are with logarithmic expressions. Also, we have particular cases the measures such as: Hellinger discrimination , symmetric $\chi ^2$- divergence , and triangular discrimination . These three measures are also well-known in the literature of statistics, and are without logarithmic expressions. Still, we have one more non logarithmic measure as particular case calling it d-divergence . These seven measures bear an interesting inequality. Based on this inequality, we have considered different difference of divergence measures and established a sequence of inequalities among themselves.

Fast evaluation of modular functions using Newton iterations and the AGM

We present an asymptotically fast algorithm for the numerical evaluation of modular functions such as the elliptic modular function $j$. Our algorithm makes use of the natural connection between the arithmetic-geometric mean (AGM) of complex numbers and modular functions. Through a detailed complexity analysis, we prove that for a given $\tau$, evaluating $N$ significative bits of $j(\tau )$ can be done in time $O(\mathcal {M}(N)\log N)$, where $\mathcal {M}(N)$ is the time complexity for the multiplication of two $N$-bit integers. However, this is only true for a fixed $\tau$ and the time complexity of this first algorithm greatly increases as $\mathrm {Im}(\tau )$ does. We then describe a second algorithm that achieves the same time complexity independently of the value of $\tau$ in the classical fundamental domain $\mathcal {F}$. We also show how our method can be used to evaluate other modular forms, such as the Dedekind $\eta$ function, with the same time complexity.

GALOIS THEORY OF THE CANONICAL THETA STRUCTURE

In this article, we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain p-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application, we prove some 2-adic theta identities which describe the set of canonical theta null points of the canonical lifts of ordinary abelian varieties in characteristic 2. The latter theta relations are suitable for explicit canonical lifting. Using the theory of canonical theta null points, we are able to give a theoretical foundation to Mestre's point counting algorithm which is based on the computation of the generalized arithmetic geometric mean sequence.

Refined Young inequality with Kantorovich constant

The Specht ratio S(h) is the optimal constant in the reverse of the arithmetic-geometric mean inequality, i.e., if 0 0, μ ∈ (0,1) ,w hereh = b and r = min{μ,1 − μ}. In this paper, we improve it by virtue of the Kantorovich constant, utilizing the refined scalar Young inequality we establish a weighted arithmetic-geometric-harmonic mean inequality for two positive operators. In the remainder of this work we focus on extending the refined weighted arithmetic-harmonic mean inequality to an operator version for another type of improvement.

Biomonitoring of urinary mercury in Korean school children

Exposure to mercury was assessed in schoolchildren 7-13 years of age (n=274) using urine mercury analysis. The arithmetic mean (AM) and geometric mean (GM) of total urinary Hg were measured as 2.66±1.12 μg/g-ct and 2.22±1.68 μg/g-ct, respectively, for 274 school children. GM of urinary Hg level was 2.11±2.00 μg/g-ct for male and 2.33±2.70 μg/g-ct for female. The influence of independent factors on the total urinary mercury (Hg) level was evaluated utilizing a multiple regression analysis to find a possible correlation with urinary Hg concentration. Independent factors are such as: (1) gender; (2) age; (3) location (inland and coastal); (4) school; (5) nearness to traffic road and industrial complex; (6) residential house types; (7) preference of fish (8) frequency of fish intake (8) amalgam; (9) medication and (10) smoking status as independent variables. The most statistically important cofactors turned out were school location, age and daily fish consumption. Amalgam fillings did not affect any relevance to mercury concentrations in urine, while daily fish consumption was associated with the increasing amount of fish consumed, in females alone. The statistical significance of the effect of each variable was calculated through Ttest, which has shown specific statistical significance among co-founding variables. As a result, the natural logarithm of urinary Hg showed a weak correlation with age (r 2=0.085).

The evaluations among the three solution procedures to an integrated single-vendor single-buyer inventory systems with shortage

In this journal [Production Planning and Control], there are three solution procedures: (1)The algebraic approach (2)Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality approach, and (3)The calculus approach, to solve the integrated single-vendor single-buyer inventory systems with shortage. Two criteria are used to evaluate the three approaches. As argued in this note, we conclude that adopting the calculus approach to solve the inventory model is more recommended.

The EOQ with defective items and partially permissible delay in payments linked to order quantity derived algebraically

Most researchers established their inventory lot-size models under trade credit financing by assuming that the supplier offers the retailer fully permissible delay in payments and the products received are all non-defective. However, in the real business environment, it often can be observed that the supplier offers the retailer a fully permissible delay in payments only when the order quantity is greater than or equal to the predetermined quantity Q d . In addition, an arriving order lot usually contains some defective items due to imperfect production processes or other factors. To capture this reality, the paper extends Huang (2007) economic order quantity (EOQ) model with partially permissible delay in payments to consider defective items. We formulate the proposed problem as a profit maximization EOQ model in which the replenishment cycle time is the decision variable. Then we use the arithmetic-geometric mean inequality approach to determine the optimal solution under various situations. An algorithm to obtain the optimal solution is also provided. Finally, the numerical examples and sensitivity analysis are given to illustrate the results.

ARITHMETIC-GEOMETRIC MEANS FOR HYPERELLIPTIC CURVES AND CALABI–YAU VARIETIES

In this paper, we define a generalized arithmetic-geometric mean μg among 2g terms motivated by 2τ-formulas of theta constants. By using Thomae's formula, we give two expressions of μg when initial terms satisfy some conditions. One is given in terms of period integrals of a hyperelliptic curve C of genus g. The other is by a period integral of a certain Calabi–Yau g-fold given as a double cover of the g-dimensional projective space Pg.

Weighted Geometric Means of Positive Operators

A weighted version of the geometric mean of k () positive invertible operators is given. For operators and for nonnegative numbers such that , we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to if commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

A simple method to investigate the effect of service constraint on EPQ model

Huang [Mathematical Problems in Engineering, Vol. 2006, ID 79028, pp. 1–5] offered the algebraic approach to investigate the effect of service level constraint on the economic production quantity (EPQ) model with random defective rate. In this paper, we propose an arithmetic-geometric mean inequality method to simplify the algebraic method of completing perfect square established by Huang (2006) to find the optimal solution under the expected annual cost minimized. The proposed method is easy-to-understand and simple2 to-use.