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In this article, we prove that the double inequalities α1[7C(a,b)16+9H(a,b)16]+(1−α1)[3A(a,b)4+G(a,b)4]0\\(a, b>0\\) with a≠b\\(a b\\) if and only if α1≤3/16=0.1875\\(_1 3/16=0.1875\\), β1≥64/π2−6=0.484555…\\(_164/^2-6= 0.484555\\), α2≤3/16=0.1875\\(_23/16=0.1875\\) and β2≥(5log2−log3−2logπ)/(log7−log6)=0.503817…\\(_2(52-3-2 )/(7-6)= 0.503817\\), where E(a,b)=(2π∫0π/2acos2θ+bsin2θdθ)2\\(E(a,b)= (2ınt^/2_0a^2 +b^2\\,d )^2\\), H(a,b)=2ab/(a+b)\\(H(a,b)=2ab/(a+b)\\), G(a,b)=ab\\(G(a,b)=ab\\), A(a,b)=(a+b)/2\\(A(a,b)=(a+b)/2\\) and C(a,b)=(a2+b2)/(a+b)\\(C(a,b)=(a^2+b^2)/(a+b)\\) are the quasi-arithmetic, harmonic, geometric, arithmetic and contra-harmonic means of a and b, respectively.

In the article, we prove that λ1=1/2+[(2+log(1+2))/2]1/ν−1/2\\( _1=1/2+ [ (2+ (1+2) )/2 ]^1/ -1/2\\), μ1=1/2+6ν/(12ν)\\( _1=1/2+6 /(12 )\\), λ2=1/2+[(π+2)/4]1/ν−1/2\\( _2=1/2+ [( +2)/4 ] ^1/ -1/2\\) and μ2=1/2+3ν/(6ν)\\( _2=1/2+3 /(6 )\\) are the best possible parameters on the interval [1/2,1]\\([1/2, 1]\\) such that the double inequalities Cν[λ1x+(1−λ1)y,λ1y+(1−λ1)x]A1−ν(x,y)0\\(x, y>0\\) with x≠y\\(x y\\) and ν∈[1/2,∞)\\( ın [1/2, ınfty )\\), where A(x,y)\\(A(x, y)\\) is the arithmetic mean, C(x,y)\\(C(x, y)\\) is the contraharmonic mean, and RQA(x,y)\\(R_QA(x, y)\\) and RAQ(x,y)\\(R_AQ(x, y)\\) are two Neuman means.

In the article, we present the best possible parameters λ = λ ( p ) and μ = μ ( p ) on the interval [ 0 , 1 / 2 ] such that the double inequality G p [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] A 1 − p ( a , b ) < E ( a , b ) < G p [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] A 1 − p ( a , b ) holds for any p ∈ [ 1 , ∞ ) and all a , b > 0 with a ≠ b , where A ( a , b ) = ( a + b ) / 2 , G ( a , b ) = a b and E ( a , b ) = [ 2 ∫ 0 π / 2 a cos 2 θ + b sin 2 θ d θ / π ] 2 are the arithmetic, geometric and special quasi-arithmetic means of a and b , respectively.

This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of n positive real numbers. Together with this interpretation, we prove some relevant properties that will allow us to define numerical approximation methods in several dimensions adapted to discontinuities.

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

This study aimed to investigate the arithmetic mean of surgically induced astigmatism (M-SIA) and the centroid of surgically induced astigmatism (C-SIA) after standard trabeculectomy. We comprised 185 eyes of 143 consecutive patients (mean age ± standard deviation, 67.7 ± 11.6 years) who underwent trabeculectomy and completed at least a 3-month routine follow-up. In all cases, the scleral flap was made at the nasal-superior location. Corneal astigmatism was measured with an automated keratometer. We calculated the M-SIA and the C-SIA using vector analysis and applied the astigmatism double angle plot. The magnitude of corneal astigmatism increased significantly, from 1.17 ± 0.92 D preoperatively to 1.77 ± 1.05 D postoperatively (paired t-test, p < 0.001). The M-SIA was 1.12 ± 0.55 D, and the C-SIA was 0.73 D @64° ± 1.02 D in the right eye group, and the M-SIA was 1.08 ± 0.48 D and the C-SIA was 0.60 D @117° ± 1.03 D in the left eye group. The C-SIA showed an astigmatic shift toward the nasal-superior location of the scleral flap creation. Our results revealed that trabeculectomy induced the SIA in the direction of the scleral flap location and that the C-SIA was much lower than the M-SIA in eyes undergoing trabeculectomy.

In the article, we present the best possible parameters α1, β1, α2, β2 ∈ ℝ and α3, β3 ∈ [1/2, 1] such that the double inequalitiesα1C(a,b)+(1−α1)A(a,b) 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, Q(a,b)=a2+b2/2 $\\begin{array}{}\\displaystyleQ(a, b)=\\sqrt{\\left(a^{2}+b^{2}\\right)/2}\\end{array}$is the quadratic mean, C(a, b) = (a2 + b2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and T3(a,b)=(2π∫0π/2a3cos2⁡θ+b3sin2⁡θdθ)2/3 $\\begin{array}{}T_{3}(a,b)=\\Big(\\frac{2}{\\pi}\\int\\limits_{0}^{\\pi/2}\\sqrt{a^{3}\\cos^{2}\\theta+b^{3}\\sin^{2}\\theta}\\text{d}\\theta\\Big)^{2/3}\\end{array}$is the Toader mean of order 3.

We present in this paper a necessary and sufficient condition to establish the inequality between generalized weighted means which share the same sequence of numbers but differ in the weights. We first present a sufficient condition, and then obtain the more general, necessary and sufficient, condition. Our results were motivated by an inequality, involving harmonic means, found in the study of multiple importance sampling Monte Carlo technique. We present new proofs of Chebyshev’s sum inequality, Cauchy-Schwartz, and the rearrangement inequality, and derive several interesting inequalities, some of them related to the Shannon entropy, the Tsallis, and the Rényi entropy with different entropic indices, and to logsumexp mean. Those inequalities are obtained as particular cases of our general inequality, and show the potential and practical interest of our approach. We show too the relationship of our inequality with sequence majorization.

The distribution of diversity can vary considerably from clade to clade. Attempts to understand these patterns often employ state-dependent speciation and extinction models to determine whether the evolution of a particular novel trait has increased speciation rates and/or decreased extinction rates. It is still unclear, however, whether these models are uncovering important drivers of diversification, or whether they are simply pointing to more complex patterns involving many unmeasured and co-distributed factors. Here we describe an extension to the popular state-dependent speciation and extinction models that specifically accounts for the presence of unmeasured factors that could impact diversification rates estimated for the states of any observed trait, addressing at least one major criticism of BiSSE (Binary State Speciation and Extinction) methods. Specifically, our model, which we refer to as HiSSE (Hidden State Speciation and Extinction), assumes that related to each observed state in the model are \"hidden\" states that exhibit potentially distinct diversification dynamics and transition rates than the observed states in isolation. We also demonstrate how our model can be used as characterindependent diversification models that allow for a complex diversification process that is independent of the evolution of a character. Under rigorous simulation tests and when applied to empirical data, we find that HiSSE performs reasonably well, and can at least detect net diversification rate differences between observed and hidden states and detect when diversification rate differences do not correlate with the observed states. We discuss the remaining issues with state-dependent speciation and extinction models in general, and the important ways in which HiSSE provides a more nuanced understanding of trait-dependent diversification.

The present study was conducted to analyze the factors influencing the hydrological behavior of the Riozinho do Rôla hydrographic basin, and was based on descriptive analysis tools. Fourteen pluviometers were set up in order to conduct a representative analysis of the rainfall in the basin. Residents of the area voluntarily participated in collection of the rainfall data in the years 2007 and 2008; the residents were trained to collect the data before the pluviometers were installed. ArcGis 9.2 software was used to outline and calculate the area of the Thiessen polygons, and both arithmetic and Thiessen precipitation means were calculated using Excel software. The average precipitation values were 1,428 mm and 1,450 mm, as calculated using the arithmetic mean and Thiessen method, respectively. Using data from the Agência Nacional de Águas (ANA), the rainfall and flow rate in the basin were calculated for the period from 1998 to 2005. The seasonality of precipitation is reflected in the temporary flow rate activity, which reached a peak of 1,276.9 m³/s during the flood period and 4.1 m³/s during periods of the dry season. This behavior is fundamentally related to natural and social aspects of the basin, such as the occurrence of a hydrographic network with characteristics of a headwater associated with a hydrological regime marked by high seasonality; the low permeability of the soils in the basin; and the intensification of deforestation in the region in order to develop livestock as a privileged form of land use.