Arithmetic Geometric Mean Research Articles

Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete Elliptic Integral

We present the best possible parametersα1,β1,α2,β2∈Randα3,β3∈(1/2,1)such that the double inequalitiesQα1(a,b)A1-α1(a,b)<AG[A(a,b),Q(a,b)]<Qβ1(a,b)A1-β1(a,b),α2Q(a,b)+(1-α2)A(a,b)<AG[A(a,b),Q(a,b)]<β2Q(a,b)+(1-β2)A(a,b),Q[α3a+(1-α3)b,α3b+(1-α3)a]<AG[A(a,b),Q(a,b)]<Q[β3a+(1-β3)b,β3b+(1-β3)a]hold for alla,b>0witha≠b, whereA(a,b),Q(a,b), andAG(a,b)are the arithmetic, quadratic, and Gauss arithmetic-geometric means ofaandb, respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.

A note on interpolation between the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities

A note on interpolation between the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities

A numerical radius version of the arithmetic-geometric mean of operators

In this paper, we obtain some numerical radius inequalities for operators, in particular for positive definite operators A; B a numerical radius and some operator norm versions for arithmeticgeometric mean inequality are obtained, respectively as ?2(A#B)? ? (A2+B2/2)- 1/2inf ||x||=1 ?(x), where ?(x) = ?(A - B)x,x?2, and ||A||||B|| ? 1/2 (||A2||+||B2||)-1/2 inf ||x||=||y||=1 ?(x,y), where, ?(x,y) = (?Ay,y? - ?Bx,x?)2.

Reversible versus irreversible thermalization of two finite blocks

We address both irreversible and reversible thermalization of two finite blocks with different thermal capacities and different initial temperatures. An instructive comparison between the two cases is developed and the increase of total entropy for the irreversible case is shown to be a direct consequence of the arithmetic–geometric mean inequality.

A Product of Nested Radicals for the AGM

The arithmetic-geometric mean of two positive numbers a and b (AGM (a, b)) is the common limit of two sequences generated by an iterative process. This has proven to be an important device for calculating numbers and function in recent years. In this paper, we derive an infinite product representation for the AGM. The factors of this product are nested radicals resembling Vieta's famous product for pi.

An arithmetic–geometric mean inequality for products of three matrices

Consider the following noncommutative arithmetic–geometric mean inequality: Given positive-semidefinite matrices A1,…,An, the following holds for each integer m≤n:1nm∑j1,j2,…,jm=1n⦀Aj1Aj2…Ajm⦀≥(n−m)!n!∑j1,j2,…,jm=1all distinctn⦀Aj1Aj2…Ajm⦀, where ⦀⋅⦀ denotes a unitarily invariant norm, including the operator norm and Schatten p-norms as special cases. While this inequality in full generality remains a conjecture, we prove that the inequality holds for products of up to three matrices, m≤3. The proofs for m=1,2 are straightforward; to derive the proof for m=3, we appeal to a variant of the classic Araki–Lieb–Thirring inequality for permutations of matrix products.

General closed-form basket option pricing bounds

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms; hence, they do not suffer from the curse of dimensionality and can be applied also to high-dimensional problems where most existing methods fail. In particular, we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate.

Capacity Enhancement with Joint Subchannel and Power Allocation Scheme for OFDMA Femtocell Networks

Efficient power and subchannel allocation methods are required for orthogonal frequency division multiple access (OFDMA) based femtocell networks to improve the capacity of the system. This paper considers a joint subchannel and power allocation algorithm with capacity maximization for downlink of an OFDMA based femtocell networks. In the proposed algorithm subchannel allocation is first performed based on signal to interference plus noise ratio (SINR) of the channel with equal power distribution. Then for enhancing capacity with optimal power allocation, successive convex approximation (SCA) based power optimization is adopted. The effect of Arithmetic geometric mean (AGM) approximation with SCA on power optimization is also investigated. The optimal power is subsequently distributed by water-filling algorithm.

Singular values of convex functions of operators and the arithmetic–geometric mean inequality

We prove singular value inequalities for convex functions of products and sums of operators that generalize the arithmetic–geometric mean inequality for operators. Among other results, we prove that if Ai, Bi, Xi, Yi, i=1,…,n are operators on a complex separable Hilbert space such that |Xi|2+|Yi|22n≤I, i=1,…,n and if f is a nonnegative increasing convex function on [0,∞) satisfying f(0)=0, thensj(f(|∑i=1nAiXiYi⁎Bi⁎|))≤12sj(⨁k=1n(Xk⁎(∑i=1nf(|Ai⁎Ak|))Xk+Yk⁎(∑i=1nf(|Bi⁎Bk|))Yk)) for j=1,2,… .

Reverse arithmetic-harmonic mean and mixed mean operator inequalities

This note aims to present some scalar inequalities and operator inequalities on a Hilbert space. Firstly, the direct reverse weighted arithmetic-harmonic mean inequalities for scalars are obtained. Secondly, based on these scalar inequalities, the corresponding operator inequalities are established. Finally, we present the mixed arithmetic-geometric and geometric-harmonic means inequalities for two positive operators.

Distributed resource allocation in cognitive and cooperative ad hoc networks through joint routing, relay selection and spectrum allocation

Cooperative relaying and dynamic-spectrum-access/cognitive techniques are promising solutions to increase the capacity and reliability of wireless links by exploiting the spatial and frequency diversity of the wireless channel. Yet, the combined use of cooperative relaying and dynamic spectrum access in multi-hop networks with decentralized control is far from being well understood.We study the problem of network throughput maximization in cognitive and cooperative ad hoc networks through joint optimization of routing, relay assignment and spectrum allocation. We derive a decentralized algorithm that solves the power and spectrum allocation problem for two common cooperative transmission schemes, decode-and-forward (DF) and amplify-and-forward (AF), based on convex optimization and arithmetic–geometric mean approximation techniques. We then propose and design a practical medium access control protocol in which the probability of accessing the channel for a given node depends on a local utility function determined as the solution of the joint routing, relay selection, and dynamic spectrum allocation problem. Therefore, the algorithm aims at maximizing the network throughput through local control actions and with localized information only.Through discrete-event network simulations, we finally demonstrate that the protocol provides significant throughput gains with respect to baseline solutions.

NEW VERSIONS OF REVERSE YOUNG AND HEINZ MEAN INEQUALITIES WITH THE KANTOROVICH CONSTANT

We show new versions of reverse Young inequalities by virtue of the Kantorovich constant, and utilizing the new reverse Young inequalities we give the reverses of the weighted arithmetic-geometric and geometric-harmonic mean inequalities for two positive operators. Also, new versions of reverse Young and Heinz mean inequalities for unitarily invariant norms are established.

New properties of the lemniscate function and its transformation

In this paper, we show several formulas for the lemniscate function which include an infinite product formula for the lemniscate sine. Furthermore, we show the relation between the product formula and Carlson's algorithm which is known as the variant of the arithmetic geometric mean of Gauss.

Asymptotic expansion of the arithmetic-geometric mean and related inequalities

Asymptotic expansion of the arithmetic-geometric mean is obtained and it is used to analyze inequalities and relations between arithmetic-geometric mean and other classical means.

Interpolating between the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities

We prove an inequality for unitarily invariant norms that interpolates between the Arithmetic-Geometric Mean inequality and the Cauchy-Schwarz inequality.

Risk and Return on Investment in Auto and Banking Sectors of Indian Economy

The objective of the paper is to find out the historical risk and return on investment in Auto and Banking sectors of Indian Economy and help the investor to take investment decisions in these sectors. Two measures of risk are used. Standard Deviation of the monthly returns which is the measure of total risk and Beta which is the measure of systematic risk. Monthly Returns are measured with the help of Arithmetic Mean and Geometric Mean over the period of five years. Sectoral indices CNXAUTO, CNXBANK and CNXNIFTY are used to measure the risk and return on investment in these sectors. The study found that risk is higher in Banking Sector and returns are higher in Auto Sector, which is contrary to theory of higher the risk higher should be the returns on Investment. This study guides the investor to take investment decisions in Auto and Banking sector of the Indian Economy. Monthly data from January 2009 to December 2015 of CNXAUTO, CNXBANK and CNXNIFTY from National Stock Exchange's website is used in this study.

Matrix inequalities for the difference between arithmetic mean and harmonic mean

Motivated by the refinements and reverses of arithmetic-geometric mean and arithmetic-harmonic mean inequalities for scalars and matrices, in this article, we generalize the scalar and matrix inequalities for the difference between arithmetic mean and harmonic mean. In addition, relevant inequalities for the Hilbert-Schmidt norm and determinant are established.

Improved arithmetic-geometric mean inequality and its application

Improved arithmetic-geometric mean inequality and its application

On a discrete weighted mixed arithmetic-geometric mean inequality

On a discrete weighted mixed arithmetic-geometric mean inequality

Improved reverse arithmetic-geometric means inequalities for positive operators on Hilbert space

In this paper, employing induction on the given reverse Young inequalities, we obtain more reverse arithmetic-geometric means inequalities for two positive operators. Concretely, following the main result from [13] we obtain reverse ratio type inequalities and reverse difference type inequalities of the refined arithmetic-geometric means inequality for two positive operators on a Hilbert space. Mathematics subject classification (2010): 47A30, 47A63.